The Pennsylvania State University

The Graduate School

College of the Liberal Arts

BARGAINING AND MECHANISM DESIGN

IN NETWORKS

A Dissertation in

Economics

by

Joosung Lee

c© 2014 Joosung Lee

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Doctor of Philosophy

December 2014

The dissertation of Joosung Lee was reviewed and approved1 by the following:

Kalyan ChatterjeeDistinguished Professor of Economics and Management ScienceDissertation Advisor, Chair of Committee

Sona N. GolderAssociate Professor of Political Science

Edward GreenProfessor of Economics

Vijay KrishnaDistinguished Professor of EconomicsDiretor of Graduate Program

Neil WallaceProfessor of Economics

1Signatures on file in the Graduate School.

ii

Abstract

This dissertation consists of three chapters. In the first two chapters, I study a new nonco-

operative coalitional bargaining model in which each player can buy out other players with

upfront transfers. Chapter 1 considers general transferable utility environments and studies

players’ strategic alliance behavior. In Chapter 2, I extend the model to network-restricted

environments in which players can communicate only with their neighbors and they can

trade their communication links. Chapter 3 analyzes mechanisms with referrals where each

agent’s social network is private information.

Chapter 1: I introduce a noncooperative coalitional bargaining model with buyout options.

The model enables us to investigate players’ strategic alliance behavior and gradual agree-

ment phenomena. The main theorem characterizes conditions for a strategic delay: If there

is both a strict subcoalition with a strictly positive worth and a player with a strictly positive

marginal contribution to the grand-coalition, then forming an efficient coalition is delayed

with positive probability for a sufficiently high discount factor. Two special results are es-

tablished. First, in a general transferable utility environment, a grand-coalition equilibrium

is impossible when players are sufficiently patient, unless it is a unanimity game. Second, in

legislative bargaining, allowing vote buying causes a non-minimal winning coalition, if both

a veto player and a non-veto player coexist. As an application, I characterize an equilibrium

payoff vector in three-player simple games.

Chapter 2: I introduce a noncooperative multilateral bargaining model for a network-

restricted environment, in which players can communicate only with their neighbors. Each

iii

player can strategically choose the bargaining partners among the neighbors and buy their

communication links with upfront transfers. I characterize a condition on network struc-

tures for efficient equilibria: An efficient stationary subgame perfect equilibrium exists for

all discount factors if and only if the underlying network is either complete or circular. I also

provide an example of a Braess-like paradox, in which the more links are available, the less

links are actually used. Thus, network improvements may decrease social welfare.

Chapter 3: I study mechanisms for environments with asymmetric contact information.

Each agent has a connection type, which determines his neighbors, in addition to a payoff

type, which represents a preference over allocations. The participants and the feasible allo-

cations are endogenously determined by agents’ strategic referrals. First, I generalize VCG

mechanisms, in which the mechanism designer compensates each agent for his marginal con-

tribution. Alternatively, I introduce multi-level mechanisms, in which each agent is compen-

sated by the agents who would not be able to participate without his referrals. I characterize

conditions for these mechanisms to be individually rational, budget surplus, and incentive

compatible in both fully referring their neighbors and truthfully telling their preferences.

iv

Contents

List of Figures vii

List of Tables viii

Acknowledgments ix

1 Bargaining and Buyout 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 A Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Formal Descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4 Strategic Delay and Inefficiency . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.5 Vote Buying in Simple Games . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.6 Non-transferability of Recognition Probabilities . . . . . . . . . . . . . . . . 42

2 Multilateral Bargaining in Networks 49

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.2 A Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.3 Efficient Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.4 Non-transferability of Recognition Probabilities . . . . . . . . . . . . . . . . 73

2.5 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3 Mechanisms with Referrals 81

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.2 A model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

v

3.3 A VCG mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.4 A Multilevel mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Bibliography 105

vi

List of Figures

1.1 An Equilibrium in a Three-Party Weighted Majority Game (Example 1.1) . 20

1.2 Workers’ Equilibrium Payoff in an Employer-Employee Game: p = (13, 1

3, 1

3)

and δ → 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.1 A Coalition Formation in a Network . . . . . . . . . . . . . . . . . . . . . . 50

2.2 Examples of Pre-complete Networks . . . . . . . . . . . . . . . . . . . . . . . 61

2.3 A Braess-Like Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.1 A Connection Profile and the Induced Contribution Tree. . . . . . . . . . . . 87

vii

List of Tables

1.1 The Models of Noncooperative Coalitional Bargaining . . . . . . . . . . . . . 6

1.2 The Role of Vote Buying in a Three-Party Weighted Majority Game [w∗; 4, 3, 2] 42

viii

Acknowledgments

I am most grateful to my advisor Professor Kalyan Chatterjee. This dissertation would

not have been possible without his continuous guidance, support, and encouragement. I

also thank Professor Edward Green, Professor James Jordan, Professor Vijay Krishna, and

Professor Neil Wallace – they kept on providing me with thoughtful comments and helpful

suggestions over all the chapters. Help from from Mehmet Ekmekci, Shih En Lu, Andrew

McLennan, Bruce Bueno de Mesquita, Maria Montero, Akira Okada, Alastair Smith, and

Rajiv Vohra is greatly acknowledge.

ix

Chapter 1

Bargaining and Buyout

1.1 Introduction

When three or more players bargain over their joint surplus, an intermediate subcoalition

occasionally forms though it is inefficient. Rather than immediately forming an efficient

coalition, players may increase their bargaining power by forming an inefficient coalition

as a transitional state. In wage bargaining, for instance, workers form a labor union even

though the union itself produces nothing. Similarly, in legislative bargaining, minor parties

form a coalition though the coalition is still minor. In most of coalitional bargaining models,

however, an efficient coalition tends to form immediately.

To investigate players’ strategic alliance behavior and gradual agreement phenomena,

we introduce a new noncooperative bargaining model in which a player can buy out other

players’ resources and rights. In practice, such a transaction among players appears in various

forms in many societies, economies, and organizations. For simple game1 environments, vote

buying in legislatures, unifying candidates in elections, logrolling among political parties,

and proxy voting for corporate control and takeover battles are some examples. For general

transferable utility environments, as in Gul (1989), we interpret a transaction among players

as a coalition formation. That is, a player who buys out some other players forms a coalition

with them and controls the coalition thereafter.

1In a simple game, each coalition is either all-powerful, called a winning coalition, or completely ineffectual.Simple games are introduced by von Neumann and Morgenstern (1944). See Shapley (1962) for mathematicalproperties of simple games.

1

Given a transferable utility environment, which is represented by a characteristic func-

tion defined on the set of possible coalitions, a noncooperative coalitional bargaining game

proceeds as follows in discrete time. In each period, a proposer is randomly selected. The

proposer makes an offer specifying a coalition to bargain with and monetary transfers to

each member in the proposed coalition. If all the members in the coalition accept the offer,

then the proposer inherits other respondents’ resources and rights and controls the coalition

thereafter. If any of the respondents rejects the offer, then it dissolves without changing

the coalitional state. At the end of each period, each remaining player derives a per-period

payoff from her coalition according to the given characteristic function. We assume that all

the players have a common discount factor.

Vote Buying in Legislatures – A Motivating Example

As an example, consider a voting game with three parties: One big party has two votes, and

two small parties have one vote apiece. They are supposed to divide a dollar by supermajority

rule, that is, at least three votes are needed to pass a certain division. In the existing

noncooperative legislative bargaining model, for instance, Baron and Ferejohn (1989) and

Winter (1996), a randomly selected proposer proposes a division of the dollar, and then all

the parties vote for the proposal. Since the big party’s agreement is essential to implement

any division, the small parties must compete with each other to attract the big party, but

they have no reason to cooperate. Due to the lack of strategic alliances, it turns out the big

party takes all the surplus as the discount factor converges to 1 in any stationary subgame

perfect equilibrium.

In order to model strategic alliance behaviors, we allow vote buying. Suppose a party

who obtains more than three votes wins a dollar. Now small parties may want to buy out

each other’s vote and form an intermediate coalition, rather than directly bargaining with

the big party. Once the small parties form a coalition, the coalition with two votes and

the big party play a standard two-player bargaining game, in which the small parties get a

substantial amount of payoff. Thus, if the discount factor is greater than a certain level, each

2

small party plays a mixed strategy in any stationary subgame perfect equilibrium: with some

positive probability, a small player buys out the big party and the game ends immediately;

but with some other positive probability small parties form a coalition. That is, they not

only compete but also cooperate with each other. In this particular example, if each party

is recognized as a proposer proportionally to its votes, then the payoff in any stationary

subgame perfect equilibrium coincides to the Shapley value, (23, 1

6, 1

6), as the discount factor

converges to 1.

Through this paper, we extend the idea of vote buying to general transferable utility

environments. Then we characterize conditions for a delay and investigate the effect of al-

lowing strategic alliances both on efficiency and equality. The equilibrium payoff vector is

also compared to some well-known cooperative solution concepts.

Delay and Inefficiency – Main Results

When players have buyout options, they not only consider the surplus from the current

coalition, but also take into account their future bargaining power in the subsequent game.

This additional strategic consideration may hinder efficiency. Our main theorem shows

that an efficient equilibrium is generically impossible when players are sufficiently patient.

Suppose a characteristic function form game has the following two properties: 1) it is not a

unanimity game, that is, there is a strict subcoalition that generates a strictly positive worth,

and 2) it has an essential player, that is, at least one player has a strictly positive marginal

contribution to the grand-coalition. Then the corresponding noncooperative bargaining game

has no efficient stationary subgame perfect equilibrium if the discount factor is sufficiently

high.

The source of inefficiency is strategic delay in forming an efficient coalition. In any

equilibrium, at least one player forms an inefficient coalition with positive probability as

an intermediate bargaining step. Strategic delay has been a central issue in bargaining

literature. Under incomplete information, delay in equilibrium is common even in a two-

3

player bargaining game2, but we assume complete information. Under complete information,

Chatterjee et al. (1993) and Cai (2000) provided examples of a delayed equilibrium, rely on

a specific characteristic function or a restrictive bargaining protocol, while our model shows

that delay generically occurs. Gomes (2005) investigated delay in a partition function form

game with externalities, but our result does not rely on explicit externalities.

In our model, inefficiency emerges when the discount factor is sufficiently high but strictly

less than 1. If the discount factor is very low so that each player only considers the current

payoff and does not take into account the future bargaining power, then a proposer immedi-

ately forms an efficient coalition and each respondent accepts any offer. On the other hand,

as the discount factor converges to 1, delay is more likely to occur, but it becomes less and

less costly. Since an efficient coalition eventually forms within finite periods, an equilibrium

is asymptotically efficient so that inefficiency disappears as the discount factor converges to

1.

As a first special case, the main theorem establishes an impossibility result of a grand-

coalition equilibrium. That is, for any transferable utility game other than a unanimity

game, a strict subcoalition forms with positive probability if the discount factor is sufficiently

high. This shows the pervasiveness of gradualism in coalition formation. Recall that in

standard coalitional bargaining models including Chatterjee et al. (1993) and Okada (1996),

the grand-coalition always forms immediately for a sufficiently high discount factor, if the

grand-coalition has the largest per-capita worth among all coalitions.3 Our main result

implies, however, a grand-coalition equilibrium is impossible when each player has a buyout

option, no matter how large the worth of the grand-coalition is, as long as there is a strict

subcoalition whose worth is strictly positive.

One of the novel aspects of our model is allowing multiple efficient coalitions. If a grand-

coalition is the unique efficient coalition, then an efficient equilibrium must be a grand-

coalition equilibrium. However, when there is a strict subcoalition that generates a surplus

2For instance, Myerson (1997), Abreu and Gul (2000), and Compte and Jehiel (2002) among many.3Also see Proposition 12 in Ray and Vohra (2013).

4

as much as the grand-coalition does, the grand-coalition may not form even in an efficient

equilibrium. In such cases, the existence of an essential player is a necessary condition

for delay in an equilibrium. For any characteristic function form game with an essential

player and multiple efficient coalitions, if the discount factor is sufficiently high, then at

least one player forms an inefficient coalition with positive probability and hence an efficient

equilibrium is impossible.

The most well-known and practically important class of characteristic function form

games with multiple efficient coalitions is a set of simple games. As a second special result,

we investigate players’ strategic alliance behavior in simple games. For a simple game with

a veto player, a non-winning coalition may form as a transitional state and the final winning

coalition is not necessarily minimal, unless all the players are veto. However, in most of

noncooperative legislative bargaining models, including Baron and Ferejohn (1989), Win-

ter (1996), Montero (2002), Morelli and Montero (2003), Montero (2006) and Montero and

Vidal-Puga (2011), only a minimal winning coalition occurs in stationary subgame perfect

equilibria.4

Bargaining with Buyout – Literature

The notion of buyout in multi-player bargaining is introduced by Gul (1989).5 In his model,

however, a player cannot choose the partners to bargain with and her strategic decision is

limited on splitting the joint surplus in a randomly selected bilateral meeting. As Hart and

Mas-Colell (1996) pointed out, a random-meeting-model does not entirely capture players’

strategic behavior. We allow players to strategically choose their bargaining partners instead

of imposing random meetings.

Vote buying has been studied in formal theory of political economics to explain the preva-

lence of non-minimal winning coalitions in practice, for instance, Groseclose and Snyder Jr

4In our model, a winning coalition may always form immediately if there is no veto player. For instance,in an apex game or a simple majority game, players may not exercise buyout options in equilibrium.

5Krishna and Serrano (1996) also use the notion of buyout to refine a unique subgame perfect equilibriumin unanimity games.

5

GeneralTU

Environments

MultipleEfficient

Coalitions

GradualCoalitionFormation

StrategicCoalitionFormation

Baron-Ferejohna) X © X ©Seltenb) © X X ©

Gulc) © X © XSeidmann-Winterd) © X © ©

Mine © © © ©a) Baron and Ferejohn (1989), Winter (1996), Montero (2002), Morelli and Montero (2003), Montero

(2006), Montero and Vidal-Puga (2011), and so on.

b) Selten (1981), Chatterjee et al. (1993), Okada (1996, 2011), Compte and Jehiel (2010), and so on.

c) Gul (1989), Hart and Mas-Colell (1996), Hart and Levy (1999), Gul (1999), and so on.

d) Seidmann and Winter (1998), Okada (2000), Gomes (2005), and so on.

Table 1.1: The Models of Noncooperative Coalitional Bargaining

(1996); Banks (2000); Dal Bo (2007); Dekel et al. (2008, 2009) among many. In those models,

however, only two lobbyists or parties compete with each other to buy non-strategic voters in

a specific political environment. To the best of our knowledge, our model is the first attempt

to allow vote buying in an abstract legislative bargaining model like Baron and Ferejohn

(1989). Our model also provides a general occurrence of non-minimal winning coalitions as

a special result for simple games.

Strategic decision on coalition formation has been widely studied in the noncooperative

coalitional bargaining literature. In most of the models, each player has at most one chance to

form a coalition. For instance, Selten (1981) and Compte and Jehiel (2010) assume one-stage

property, that is, the game is terminated right after any coalition formation. Other models,

including Chatterjee et al. (1993) and Okada (1996, 2011), assume exclusion property ; once

players form a coalition, all the players in the coalition must exit the game and they are

excluded in further bargaining. In such environments, when a player chooses bargaining

partners, she only concerns the surplus from the current coalition, but not the coalitional

structure induced by the coalition formation.

Our model is close to Seidmann and Winter (1998), Okada (2000), and Gomes (2005),

which allow renegotiations in coalitional bargaining. In Seidmann and Winter (1998) a rejec-

6

tor becomes a proposer in the subsequent period; while in our model a proposer is randomly

selected in each period. Okada (2000) assumes that, once a subcoalition forms, any other

disjoint subcoalition cannot form and the subsequent bargaining is limited to a renegotia-

tion with the previously formed coalition. Gomes (2005) developed a coalitional bargaining

model for environments with externalities. Compared to them, our contribution is to allow

multiple efficient coalitions.6 Allowing multiple efficient coalitions is technically involved, but

practically important particularly for legislative bargaining and political party formation.7

The paper is organized as follows. Section 2 describes a noncooperative coalitional bar-

gaining model with buyout options. Section 3 provides formal descriptions of the noncoop-

erative coalitional bargaining model and fundamental results on stationary subgame perfect

equilibria. Section 4 characterizes a condition on coalitional structure for efficient equilibria.

In Section 5, we apply the result to simple games and characterize the equilibrium payoff

vector in three-player simple games. Section 6 studies an alternative environment, in which

players cannot trade their chances of being a proposer.

1.2 A Model

1.2.1 An Environment

Let N be a set of players and v : 2N \ ∅ → RN+ be a characteristic function. A tuple (N, v)

is an underlying characteristic function form game, or shortly an underlying game.8 We

assume that (N, v) is zero-normalized, essential, and superadditive that is, v(i) = 0 for all

i ∈ N ; v(N) > 0; and v(S∪S ′) ≥ v(S) +v(S ′) for all S, S ′ ⊆ N such that S∩S ′ = ∅. Define

6Seidmann and Winter (1998) assumed semi-strict superadditivity, which is the weakest assumption toimply that a grand-coalition is the unique efficient coalition. Okada (2000) assumed that one and only oneprofitable coalition can form. Gomes (2005) also limited his analysis on a grand-coalition equilibrium in aparticipation function form game.

7Hyndman and Ray (2007) allow both multiple efficient coalitions and ongoing renegotiation in real-timeframework. To the best of our knowledge, this paper is the first attempt in discrete-time framework.

8We follows Gul (1989)’s interpretation. Each player initially has a specific resource. Each coalitionrepresents a combination of resources that initially belong to the players in the coalition which generates aflow of surplus according to the characteristic function.

7

v = maxS⊆N v(S), then, due to superadditivity, v = v(N). An underlying game (N, v) is a

unanimity game if v(S) = 0 for all S ( N and v(N) > 0.

Given (N, v), a coalition S ⊆ N is efficient if v(S) = v. Let E be a set of efficient

coalitions.9 A player i ∈ N is essential if v(S) < v for any S ⊆ N \ i. Let K be a set of

essential players. It is easy to see K = ∩E = k ∈ N | v(N \ k) < v, that is a player

is essential if and only if her marginal contribution to the grandcoalition is strictly positive.

Note also that N = K ⇐⇒ E = N.

A (coalitional) state π specifies a set of active players Nπ and a partition Sii∈Nπ of N

such that Si ∩Nπ = i, representing a distribution of resources. Abusing notation, π also

refers the partition in the state. Let [i]π be a player i’s partition block in π. Denote π by

the initial state, that is, Nπ = N and [i]π = i for all i ∈ N . A state π is efficient if∑i∈Nπ v([i]π) = v(N). That is, in any efficient state, there is no unrealized surplus. Let Π

be a set of all states.

For each π ∈ Π and i ∈ Nπ, denote Nπi = S ⊆ Nπ | i ∈ S. For each π ∈ Π, i ∈ Nπ,

and S ⊆ Nπi , player i’s S-formation, or (i, S) formation, yields a subsequent state π(i, S),

where Nπ(i,S) = (Nπ \ S) ∪ i, [i]π(i,S) = ∪k∈S[k]π, and [j]π(i,S) = [j]π for all j ∈ Nπ \ i.

For each (π′, π) ∈ Π×Π, π′ succeeds π, if there exists a sequence of formations (i`, S`)L`=1

such that π′ = π(i1, S1) · · · (iL, SL); i1 ∈ Nπ and S1 ⊆ Nπi1

; and i` ∈ Nπ(i1,S1)···(i`−1,S`−1) and

S` ⊆ Nπ(i1,S1)···(i`−1,S`−1)i`

for all ` = 2, · · · , L. Given π ∈ Π, let Π|π be a set of succeeding

states of π.

1.2.2 A Noncooperative Game

A noncooperative coalitional bargaining game, or shortly, a bargaining game is a tuple Γ =

(N, v, p, δ), where p ∈ R|N |++ is the initial recognition probability with∑

i∈N pi = 1, and

0 < δ < 1 is the common discount factor. For each π ∈ Π, we define the induced recognition

9We omit (N, v) when there is no danger of confusion.

8

probability pπ in the following way:

pπi =

∑j∈[i]π

pj if i ∈ Nπ

0 otherwise.(1.1)

That is, if a player forms a coalition, then the player takes other players’ recognition proba-

bilities as well.10

A bargaining game proceeds as follows. In each period t = 1, 2, · · · , they begins with the

previous state πt−1. If πt−1 is an efficient state, then only a production stage occurs without

bargaining stages and hence πt = πt−1. Otherwise, the period consists of three bargaining

stages and one production stage. Each stage is defined as follows:

i) Recognition: Nature selects a player i ∈ Nπt−1as a proposer with probability pπ

t−1

i .

ii) Proposal: The proposer i chooses a pair (S, y) of a coalition S ⊆ Nπt−1

i and monetary

transfers yjj∈S with∑

j∈S yj = 0.

iii) Response: By a given order, each respondent j ∈ S \ i sequentially either accepts the

offer or rejects it. If any j ∈ S \ i rejects then the current state does not change and

hence πt = πt−1. If all j ∈ S \ i accept the offer, then the current state transitions

to πt = πt−1(i, S), that is, each j ∈ S \ i leaves the game with receiving yj from the

proposer i.

iv) Production: Each partition block generates a surplus to the owner. That is, each active

player i ∈ Nπt derives (1− δ)v([i]πt).11

Given a sequence of states π = πt∞t=0 and a sequence of transfers among players y =

yt∞t=0, player i’s discounted sum of payoffs is

Ui(π, y) =∞∑t=1

δt−1((1− δ)v([i]πt) + yti

)1(i ∈ Nπt).

10This is a generalization of the partnership game in Gul (1989).11The coefficient 1− δ normalizes the discounted sum of streams of surplus. Thus, a coalition S generates

(1 − δ)v(S) for each period so that the sum of streams of surplus is v(S) = (1 − δ)v(S) + δ(1 − δ)v(S) +δ2(1− δ)v(S) + · · · .

9

When there is no danger of confusion, we omit π in notations, for instance, Nπ(i,S) =

N (i,S), pπ(i,S) = p(i,S), and so on. For notational simplicity, for any z ∈ Rn and S ⊆ N ,

denote zS =∑

j∈S zj. For a characteristic function v, denote vi = v(i) and vS =∑

i∈S vi.

See Appendix 1.3 for formal descriptions of the bargaining game.

1.2.3 Stationary Subgame Perfect Equilibria

A player’s strategy is stationary if it does not depend on the histories of past periods. Note

that even in the class of stationary strategies, players’ decision may depend on the current

state and within-period histories, which involve the identity of the proposer, the proposed

coalition, preceding respondents’ reactions, and so on.

Our equilibrium concept is a stationary subgame perfect equilibrium (SSPE). An SSPE

consists of each player’s stationary strategy, which is the player’s best response to other

players’ strategies in every subgames. It is worth mentioning that in an SSPE each player

has no incentive to play any other strategies including even non-stationary strategies, as long

as other players are supposed to play stationary strategies. The existence of SSPE is proved

by Eraslan (2002), Gomes (2005), and Eraslan and McLennan (2013). Formal definitions of

a stationary strategy and an SSPE appear in Appendix 1.3.

The notion of SSPE is widely adopted as a focal equilibrium in the coalitional bargaining

literature. Unlike in two-player noncooperative bargaining models, it is well-known that any

feasible allocation can be achieved by a subgame perfect equilibrium for high discount factors

in most of multi-player bargaining models.12 Without the notion of stationarity, therefore, a

noncooperative multi-player bargaining model usually fails to provide a sharp prediction.13

To analyze SSPE, we introduce a special form of stationary strategies, so-called a cutoff

strategy. Before defining a cutoff strategy and a cutoff strategy equilibrium, we state Propo-

sition 1.1, which provides a payoff equivalence result between a cutoff strategy equilibrium

and a general SSPE. Thus we may focus on cutoff strategy equilibria instead of all SSPE

12Shaked (reported by Sutton (1986)) for a multi-lateral unanimity game, Baron and Ferejohn (1989) fora legislative bargaining, and Chatterjee et al. (1993) for a general transferable utility game

13A notable exception for a unanimity game is Krishna and Serrano (1996).

10

without loss of generality, when we are interested in players’ equilibrium payoffs. The proof

of Proposition 1.1 is described in Appendix 1.3.

Proposition 1.1. For any SSPE, there exists a cutoff strategy equilibrium which yields the

same payoff vector.

A cutoff strategy profile (x,q) consists of a cutoff value profile x = xπi i∈Nππ∈Π and a

coalition formation strategy profile q = qπi i∈Nππ∈Π, where xπi ∈ R and qπi ∈ ∆(2Nπi ) for

each π ∈ Π and it specifies the behaviors of an active player i ∈ Nπ in any coalitional state

π in the following way:

• player i proposes (S, y) with probability qπi (S) such that

yk =

xπk if k ∈ S \ i−xπS\i if k = i

0 otherwise;

• player i accepts any proposal (S, y) if and only if i ∈ S and yi ≥ xπi .

Given x, define an active player i’s excess surplus of forming S in π:

eπi (S,x) = xπ(i,S)i − xπS.

LetDπi (x) = argmaxS⊆Nπieπi (S,x) be a demand set of player i in π andmπ

i (x) = maxS⊆Nπieπi (S,x)

be a net proposal gain of player i in π. Given a cutoff strategy profile (x,q), player i’s con-

tinuation payoff in π is:

uπi (x,q) = pπi∑S⊆Nπ

qπi (S)eπi (S,x) +∑j∈Nπ

pπj∑S⊆Nπ

qπj (S)[1(i ∈ S)xπi + 1(i 6∈ S)x

π(j,S)i

]. (1.2)

If the cutoff strategy profile (x,q) under consideration is fixed, then it is omitted when there

is no danger of confusion. If xπ(i,S)i = x

π(j,S)j for all i, j ∈ S, then the subscript is omitted

and the excess surplus is simply written as eπ(S) for all i ∈ S.

The main virtue of a cutoff strategy equilibrium is its tractability. Proposition 1.2 char-

acterizes a cutoff strategy equilibrium in terms of a value profile and a coalition formation

strategy profile. The proof of Proposition 1.2 appear in Appendix 1.3.

11

Proposition 1.2. A cutoff strategy profile (x,q) is an SSPE if and only if for all π ∈ Π

and i ∈ Nπ,

i) Optimality: qπi ∈ ∆(Dπi (x)); and

ii) Consistency: xπi = (1− δ)v([i]π) + δuπi (x,q).

1.3 Formal Descriptions

In this section, we formally describe the noncooperative bargaining game. Then, we show

the payoff equivalence between a cutoff strategy equilibrium and an SSPE, and characterize

a cutoff strategy equilibrium with two conditions, optimality and consistency. Note that this

section is developed largely based on Eraslan and McLennan (2013) and Yan (2003).

Fix Γ = (N, v, p, δ). Let n be a cardinality of N and define S = S ⊆ N | S 6= ∅,

Si = S ⊆ N | i ∈ S, and X = Rn+. First, we consider the initial state π. Since a proposer

is selected at the beginning of each period, each within-period history always specifies a

current proposer. Let H0 = N × S × X be a set of within-period histories right after the

proposer makes an offer; and for all i ∈ N , H i = H i−1 × 0, 1 be a set of within-period

histories right after player i responses. The set of all possible within-period histories is

H = N ∪H0 ∪H1 ∪ · · · ∪Hn.

For all h ∈ H, typically denoted by h = (φ, S, y, r1, r2, · · · , ri), the first element of the

history specifies the current proposer, denoted by φ(h) = φ ∈ N . For all h ∈ ∪n`=0H`,

the second element and the third element specify the proposed coalition and the proposed

allocation, denoted by S(h) = S ∈ S and y(h) = y ∈ X. For all i ∈ N and all h ∈ ∪n`=iH`,

the (i+ 3)th element specifies the player i’s response, denoted by ri(h) = ri ∈ 0, 1, where

0 represents i’s rejection and 1 represents i’s acceptance.

For a measurable space (Ω,A), let ∆(Ω) be the set of probability measures on Ω. For

a pair of measurable spaces (Ω1,A1) and (Ω2,A2), let ∆(Ω1,Ω2) be the set of transition

probabilities from Ω1 to Ω2. Player i’s (stationary) proposal strategy in the initial state is

12

αi ∈ ∆(Si×X). Define a proposal transition probability α ∈ ∆(N,S×X) so that α(i)(S, y) =

αi(S, y). Player i’s (stationary) response strategy in the initial state is βi ∈ ∆(H i−1, 0, 1)

such that βi(h)(1) = 1 if either φ(h) = i or i 6∈ S(h).

The three stochastic components, the recognition probability p, the proposal transition

probability α, and the response strategy profile β ≡ βii∈N , induce a unique probability

measure on Hn.14 The induced probability measure is denoted by p⊗ α⊗ β1 ⊗ · · · ⊗ βn.

Let O = (S × X) ∪ π be the outcome space in π. Define an outcome function

o : Hn → O, such that for all h ∈ Hn,

o(h) =

(S(h), y(h)) if ×j∈N rj(h) = 1

π otherwise.

Let (p⊗ α⊗ β1 ⊗ · · · ⊗ βn) o−1 be the induced measure on O by (p, α, β). We also define

the induced measure κ on O by any history h ∈ H and the initial-state stationary strategy

profile (α, β):

κ(h, α, β) =

(δh ⊗ α⊗ β1 ⊗ · · · ⊗ βn) o−1 if h ∈ N(δh ⊗ β`+1 ⊗ · · · ⊗ βn) o−1 if h ∈ H` ` = 0, 1, · · · , n,

where δh is the Dirac probability measure on H.

Let x be a value profile, that is, x = xπj j∈Nππ∈Π. Given an initial-state stationary

strategy profile (α, β) and a value profile x, player i’s expected payoffs at h ∈ H is:

wi(h, α, β,x) = κ(h, α, β)(π)xi

+∑S∈Si

∫y∈X

(yi +

(x

(i,S)i −

∑j∈S

yj

)1(i = φ(h))

)κ(h, α, β)(S, dy)

+∑

S∈S\Si

x(φ(h),S)i

∫y∈X

κ(h, α, β)(S, dy).

Then an initial-state stationary strategy profile (α, β) is an initial-state stationary subgame

perfect x-equilibrium if, for all h ∈ H, all i ∈ N , and i’s all possible initial-state stationary

strategies αi and βi,

wi(h, α, β,x) ≥ wi(h, (αi, α−i), (βi, β−i),x). (1.3)

14This is known as a generalized Fubini Theorem. See Eraslan and McLennan (2013).

13

A stationary strategy profile specifies all the active players’ initial-state stationary strate-

gies for each subgame. Let (α,β) be a stationary strategy profile, where α ≡ αππ∈Π and

β ≡ βππ∈Π. Given x, for each π ∈ Π, denote x|π = xπ′π′∈Π|π and xπ = x|π ∪ xπ. The

studies about stationary discounted dynamic programming characterize an SSPE as a col-

lection of initial-state stationary subgame perfect equilibria with respect to a specific value

profile. The proof is omitted.

Proposition 1.3.

i) If (α,β) is an SSPE, then it induces a value profile x, and for all π ∈ Π a partial

strategy profile (απ, βπ) is an initial-state stationary subgame perfect xπ-equilibrium of

the subgame starting with π.

ii) If there exist a stationary strategy profile (α,β) and a value profile x such that for

all π ∈ Π, (απ, βπ) is an initial-state stationary subgame perfect xπ-equilibrium of the

subgame starting with π, then (α,β) is an SSPE.

In order to prove Proposition 1.1 and Proposition 1.2, some lemmas are presented.

Lemma 1.1. Let (α,β) be an SSPE and x be the induced value profile. For all π ∈ Π,∑j∈Nπ xπj < v(N).

Proof. Fix π ∈ Π. (απ, βπ) is an initial-state stationary subgame perfect xπ-equilibrium of

the subgame starting with π. Thus, for all j ∈ Nπ, it must be wj(h, απ, βπ,xπ) ≥ xπj and

hence we have ∑j∈Nπ

wj(h, απ, βπ,xπ) ≥

∑j∈Nπ

xπj .

The left-hand side, the sum of expected payoffs of all the active players must be less

than or equal to v(N). Suppose, for contradiction,∑

j∈Nπ xπj ≥ v(N). This yields that∑j∈Nπ wj(h, α

π, βπ,xπ) = 1, and hence we have wj(h, απ, βπ,xπ) = xπj for all j ∈ Nπ, since

wj(h, απ, βπ,xπ) ≥ xπj for all j ∈ Nπ. However, this contradicts that π is inefficient.

Lemma 1.2. For all π ∈ Π, i ∈ N , and h ∈ H i(π) such that

14

i) X1≤`≤i−1

r`(h) = 1; and

ii) (∀` ∈ S(h) \ φ(h)) ` ≥ i =⇒ y`(h) > xπ` ,

the current proposal (S(h), y(h)) will be implemented for sure in any SSPE.

Proof. Fix π ∈ Π and we divide the proof into two cases.

Case 1: i = n.

subcase 1-1: Suppose that n 6∈ S(h)\φ(h). It must be βπn(h) = 1 no matter what yn(h).

Thus, the outcome at the history of h′ = (h, rn) must be o(h′) = (S(h), y(h)), that is,

(S(h), y(h)) will be implemented for sure.

subcase 1-2: Suppose that n ∈ S(h) \ φ(h). If βπn(h) = 1, then the outcome at the

history of h′ = (h, rn) is o(h′) = (S(h), y(h)), and hence the player n’s payoff at the

state is yn(h). If βπn(h) < 1, then the outcome at the history of h′ = (h, rn) will be

o(h′) = π with positive probability, and hence they face the same state π in which the

player n’s value is xπn. Thus, for the player n, βπn(h) = 1 is optimal at the history h

and hence (S(h), y(h)) is implemented.

Case 2: i ≤ n− 1.

As induction hypothesis, suppose that, for all j > i and all h ∈ Hj(π) if

[X

1≤`≤j−1r`(h) = 1

]and

[(∀` ∈ S(h) \ φ(h)) ` ≥ j =⇒ y`(h) > xπ`

], then (S(h), y(h)) is implemented for sure.

subcase 2-1: Suppose that i 6∈ S(h) \ φ(h). It must be βπi (h) = 1 no matter what yi(h).

Thus, the outcome at the history of h′ = (h, ri, · · · , rn) must be o(h′) = (S(h), y(h)),

that is, (S(h), y(h)) will be implemented for sure.

subcase 2-2: Suppose that i ∈ S(h)\φ(h). If βπi (h) = 1, then the outcome at the history

of h′ = (h, ri, · · · , rn) is o(h′) = (S(h), y(h)), and hence the player i’s payoff at the

state is yn(h). If βπi (h) < 1, then the outcome at the history of h′ = (h, ri, · · · , rn) will

be o(h′) = π with positive probability, and hence they face the same state π in which

15

the player i’s value is xπi . Thus, for the player i, βπi (h) = 1 is optimal for the player i

at the history h and hence (S(h), y(h)) is implemented.

Lemma 1.3. For all π ∈ Π, i ∈ N , and h ∈ H i(π) such that

i) X1≤`≤i−1

r`(h) = 1; and

ii) (∃` ∈ S(h) \ φ(h)) ` ≥ i and y`(h) < xπ` ,

the current proposal (S(h), y(h)) will never be implemented in any SSPE.

Proof. Fix π ∈ Π and we divide the proof into two cases.

Case 1: i = n ∈ S(h) \ φ(h) and yn(h) < xπn.

If βπn(h) < 1, then the outcome at the history of h′ = (h, rn) will be o(h′) = (S(h), y(h))

with positive probability, and hence the player n’s payoff at the state is yn(h), which less

than the stationary value xπn. Thus, βπn(h) = 0 is optimal for the player n and the current

proposal (S(h), y(h)) is not implemented.

Case 2: i ≤ n − 1. As induction hypothesis, suppose that, for any h ∈ H i+1(π), if

X1≤`≤i

r`(h) = 1 and there exists j ∈ S(h) \ φ(h) ∩ j ≥ i + 1 such that yj(h) < xπj ,

then the proposal (S(h), y(h)) will not be implemented.

subcase 2-1: If there exists j ∈ S(h) \ φ(h) ∩ j ≥ i+ 1 such that yj(h) < xπj , then by

the induction hypothesis, the proposal (S(h), y(h)) will not be implemented no matter

what βπi (h) is.

subcase 2-2: Suppose that yj(h) ≥ xπj for all j ∈ S(h) \ φ(h) ∩ j ≥ i + 1. It must

be i ∈ S(h) \ φ(h) and yi(h) < xπi . For all continuation histories of (h, ri = 1),

(h, ri = 1, ri+1 = 1), · · · , (h, ri = 1, · · · , rn−1 = 1), if βπ` (h, ri = 1, · · · , r`−1 = 1) > 0

for all ` = i + 1, i + 2, · · · , n, then βπi (h) = 0 is optimal for the player i and the

proposal (S(h), y(h)) will not be implemented. If there exists ` = i + 1, i + 2, · · · , n,

such that βπ` (h, ri = 1, · · · , r`−1 = 1) = 0, again the proposal (S(h), y(h)) will not be

implemented no matter what βπi (h) is.

16

For any S ∈ 2N , define an allocation yS ∈ X as ySj = xπj for all j ∈ S and ySj = 0

otherwise.

Lemma 1.4. Let (α,β) be an SSPE and x be the induced value profile. For all π ∈ Π and

h ∈ H(π), if απφ(h)(S, y) > 0 then S ∈ Dπφ(h)(x) and y = yS. Furthermore, every proposal is

implemented for sure and the proposal gain of φ(h) is mπφ(h)(x).

Proof. Fix π ∈ Π and h ∈ H(π). First, by Lemma 1.3, the proposal gain of φ(h) in an SSPE

(α,β) is less than or equals to mπφ(h)(x). Suppose, for contraction, that the proposal gain of

φ(h) in an SSPE (α,β) is strictly less than mπφ(h)(x) and let απφ(h) be the proposal strategy.

There must exist (S, y) such that απφ(h)(S, y) > 0 and yj ≥ xπj for all j ∈ S and yj′ > xπj′ for

some j′ ∈ S. By Lemma 1.2, φ(h) can be strictly better off by slightly decreasing j′ share

in the proposal, which is a contradiction. Thus, for all player i ∈ Nπ, the proposal gain of

the player i in an SSPE (α,β) equals to mπi (x). For the proposal gain mπ

i (x) in order to be

obtained, the player i must make a proposal (S, yS) for any S ∈ Pi(Nπ), that is, the player

i chooses S in Dπi (x).

Now we are ready to prove Proposition 1.1 and Proposition 1.2.

Proof of Proposition 1.1: We will show that for any SSPE (α,β) that induces a value

profile x, there exists a cutoff strategy equilibrium (x,q). Let x be the value profile induced

by an arbitrary SSPE (α,β).

Case 1: For π ∈ Π with n(π) = 2, Γπ = (Nπ, v, pπ, δ) has a unique subgame perfect Nash

equilibrium, in which the active players play cutoff strategies.

Case 2: Consider π ∈ Π with n(π) ≥ 3. As induction hypothesis, suppose for all the suc-

ceeding states π′ ∈ Π|π, there exists a cutoff strategy SSPE (xπ′,qπ

′) of Γπ

′= (Nπ′ , v, pπ

′, δ).

Now we show that (xπ,qπ) is a cutoff strategy SSPE of Γπ. By Lemma 1.4, since the pro-

posed allocation is determined by x, a player i’s proposal strategy in π can be represented by

qπi ∈ ∆(Dπi (x)), which is a proposal cutoff strategy. Since this proposal must be implemented

for sure by Lemma 1.4, an active player i’s expected payoff when i is not selected as a pro-

poser must equal to the value of current state, that is, xπi = (1−δ)v([i]π)+δuπi (xπ,qπ). Since

17

all the active proposer plays a cutoff proposal strategy, for all i ∈ Nπ and all h ∈ H i−1(π)

such that i ∈ S(h) \ φ(h), player i’s optimal response strategy is βπi (h) = 1 if yi(h) ≥ xπi

and otherwise βπi (h) = 0. Thus all the respondents follows the cutoff strategies.

Proof of Proposition 1.2:

(only-if part) Suppose (x,q) is an SSPE. Consider π ∈ Π and i ∈ Nπ. By Lemma 1.4, the

player i’s equilibrium proposal strategy must maximizes the proposal gain xπ(i,S)i −

∑j∈S

xπj ,

and hence it must be qπi ∈ ∆(Dπi (x)). When i is supposed to response, i can get at most

(1−δ)v([i]π)+δuπi (x,q) by rejecting any proposal. Thus in equilibrium, each respondent must

indifferent between accepting and rejecting, which requires that xπi = (1−δ)v([i]π)+δuπi (x,q).

(if part) Suppose all the players except i follow the given cutoff strategies (x−i,q−i). For

any π such that i ∈ Nπ, if xπi = (1 − δ)v([i]π) + δuπi (x,q), then it is impossible for i to

deviate profitably from the given response strategy. When i is supposed to propose, forming

a subcoalition which is not in Dπi (x) is not optimal for i. On the other hand, a proposer i

can propose a grand coalition, in which i’s proposal gain is

xπ(i,Nπ)i −

∑j∈Nπ

xπj = v(N)−∑j∈Nπ

xπj > v(N)− v(N) = 0,

where the first equality is from the fact π(i, Nπ) is efficient; and the inequality is from Lemma

1.1. Thus, given x, we have mπi (x) > 0 and the proposer i always has a strictly positive

proposal gain as long as the current state is inefficient. That is, making an acceptable

proposal is strictly better than a proposal which will be rejected. Therefore, in an SSPE,

a proposal i makes a proposal (S, yS) with S ∈ ∆(Dπi (x)), which is the proposal cutoff

strategy.

1.4 Strategic Delay and Inefficiency

Given Γ = (N, v, p, δ), a cutoff strategy profile (x,q) is efficient if∑

i∈N ui(x,q) = v. Since

any proposal is always accepted under a cutoff strategy profile, (x,q) is efficient if and only

18

if, for all i ∈ N and S ⊆ N , qi(S) > 0 implies v(S) = v. The main theorem shows that an

efficient equilibrium is generically impossible.

Theorem 1.1.

i) Let (N, v) be a unanimity game. For any p and δ, any stationary subgame perfect

equilibrium of (N, v, p, δ) is efficient.

ii) Suppose (N, v) has an essential player. For any p there exists δ < 1 such that for all

δ > δ, (N, v, p, δ) has no efficient stationary subgame perfect equilibrium, unless (N, v)

is a unanimity game.

Remark. As long as the current state is inefficient, the remaining players continue to bargain

over the remaining surplus and hence any inefficient state transitions into an efficient state in

a finite period. Thus, as the discount factor increases, two different effects on efficiency are

intertwined: a strategic delay occurs more and more frequently, but it becomes less and less

costly. Inefficiency occurs for a sufficiently high discount factor; however it eventually disap-

pears as a discount factor converges to 1. That is, any equilibrium must be asymptotically

efficient.

Theorem 1.1 provides three necessary conditions for strategic delay. That is, a strategic

delay occurs with positive probability and an efficient equilibrium is impossible, if 1) the

discount factor is high enough, 2) at least one strict subcoalition generates a strict positive

worth, and 3) at least one player is essential. Before proving this, we illustrate the role of

each condition with examples.

If the discount factor is sufficiently low, then players do not care about future payoffs.

Thus, any respondent immediately accepts any positive offer, and hence any proposer chooses

an efficient coalition to extract most of the surplus from the efficient coalition. In other

words, players have no incentive to form an intermediate coalition if the discount factor is

lower than a certain level. If the discount factor is greater than the threshold, players may

form an inefficient coalition as an intermediate step, expecting more bargaining power in

19

Oδ

u

δ 1

1

12

14

16

23

u2 = u3

u1

Welfare: u1 + u2 + u3

Efficiency Loss

(a) Equilibrium Payoffs

Oδ

q

δ 1

1

12

q1(1, 2) = q1(1, 3)

q2(2, 3) = q3(2, 3)

(b) Coalition Formation Strategies

Figure 1.1: An Equilibrium in a Three-Party Weighted Majority Game (Example 1.1)

the subsequent bargaining, and hence a strategic delay may occurs. The following example

illustrates the role of a discount factor on efficiency.

Example 1.1 (A Three-Party Weighted Majority Game). Suppose there are three parties:

one big party has two votes and other two small party has one vote each. They split a

dollar by a supermajority rule, that is, at least 3 votes are required to pass a division of

the dollar. Their recognition probabilities are proportional to their voting weights. Then

the corresponding noncooperative bargaining game consists of N = 1, 2, 3, v(S) = 1 if

S ∈ 1, 2, 1, 3, N and v(S) = 0 otherwise, and p =(

12, 1

4, 1

4

). In any subgame with

two active players, the equilibrium strategy is uniquely determined in a standard way, and

hence x(2,2,3)2 = x

(3,2,3)3 = (p2 + p3)δ and x

(2,2,3)1 = x

(3,2,3)1 = p1δ. Thus we concentrate

on the strategies in the initial state. Figure 1.1 depicts the equilibrium payoffs and coalition

formation strategies in the game. There are two types of equilibria which depend on δ. Let

δ = 6−2√

33≈ 0.845.

i. (Efficient Equilibrium) If δ ≤ δ, either 1, 2 or 1, 3 always immediately forms. There

exists an equilibrium which consists of

20

• x =(

(2−δ)δ4−3δ

, (1−δ)δ4−3δ

, (1−δ)δ4−3δ

);

• q1(1, 2) = q1(1, 3) = 12; and

• q2(1, 2) = q3(1, 3) = 1.

To see the proposed strategies construct an equilibrium, we verify two conditions in

Proposition 1.2, optimality and consistency. To verify optimality, first we observe:

e(1, 2) ≥ e(2, 3) ⇐⇒ 1− (3− 2δ)δ

4− 3δ≥ 1

2δ − 2(1− δ)δ

4− 3δ⇐⇒ δ ≤ δ.

In addition, e(N) = 1 − xN = 1 − δ is strictly less than than e(1, 2) = e(1, 3) =

1− x1− x2 = 1− (3−2δ)δ4−3δ

. Thus, we have argmaxS e(S) = 1, 2, 1, 3 and optimality

holds. To verify consistency, compute each player’s expected payoff:

u1 = p1 [q1(1, 2)e(1, 2) + q1(1, 3)e(1, 3)] + (p1 + p2 + p3)x1

= 12

(1− (3−2δ)δ

4−3δ

)+ 1 · (2−δ)δ

4−3δ= 2−δ

4−3δ,

u2 = p2e(1, 2) + (p2 + p1q1(1, 2))x2

= 14

(1− (3−2δ)δ

4−3δ

)+ 1

4· (1−δ)δ

4−3δ= 1−δ

4−3δ.

Thus, xi = δui for all i ∈ N , and hence consistency holds.

ii. (Inefficient Equilibrium) If δ > δ, then the small parties form an intermediate coalition

2, 3 with positive probability. There exists an equilibrium with

• x =(1− δ+2e

4, δ−2e

4, δ−2e

4

);

• q1(1, 2) = q1(1, 3) = 12;

• q2(2, 3) = 1− q2(1, 2) = q3(2, 3) = 1− q3(1, 3) = 2(2δ−δ2−4e)δ(δ−2e)

> 0,

where e is the solution to 2δ−δ2−4eδ−2e

= 5δ−δ2−4+2e4−3δ−2e

, that is,

e = 1− 23δ − 1

6

√−12δ3 + 61δ2 − 84δ + 36.

21

Let r = 2(2δ−δ2−4e)δ(δ−2e)

= 2(5δ−δ2−4+2e)δ(4−3δ−2e)

. Given x, it is easy to see e(1, 2) = e(1, 3) =

e(2, 3) = e > e(N) = 1− δ. In addition, observe that

0 < r < 1 ⇐⇒ 2δ − δ2

4< e <

4δ − 3δ2

4− 2δ⇐⇒ δ < δ < 1,

which implies optimality. To verify consistency, compute each player’s expected payoff:

u1 =p1e+ [p1 + (p2 + p3)(1− r)]x1 + [(p2 + p3)r]δp1

=12e+ x1 − 1

2r(x1 − δ

2

)=1− δ

4− 1

2

(2(5δ−δ2−4+2e)δ(4−3δ−2e)

) (4−3δ−2e

4

)= 1

δ

(1− δ+2e

4

)u2 =p2e+ [p1q1(1, 2) + p2 + p3r]x2 + [p1q1(1, 3) + p3(1− r)] · 0

=14e+ 1

2x2 + 1

4rx2

= δ8

+ 14

(2(2δ−δ2−4e)δ(δ−2e)

) (δ−2e

4

)= 1

δ

(δ−2e

4

),

which implies consistency. Note that limδ→1 e = 16, limδ→1 r = 1, and limδ→1 x =(

13, 1

6, 1

6

), which coincides to the Shapley value of the underlying game. As in Figure

1.1, inefficiency occurs due to strategic delay if δ > δ. As δ increases over δ, q2(2, 3)

and q3(2, 3) also increase and hence delay occurs more frequently. At δ = 1, however,

inefficiency eventually disappears.

In a bargaining model without buyout options, either 1, 2 or 1, 3 always forms imme-

diately and any equilibrium must be efficient no matter what the discount factor is. The

equilibrium payoffs and strategies without buyout option are depicted as the dotted lines in

Figure 1.1. For lower discount factors, the buyout options do not affect to the equilibrium.

As the discount factor converges to 1, however, the big party takes all the surplus without

buyout options.

The following example illustrates the role of a strict subcoalition that generates a positive

worth. Such a strict coalition works as a fulcrum for a player to form an intermediate coalition

instead of immediately forming an efficient one.

22

Example 1.2 (An Employer-Employee Game). Consider a three-player game with N =

1, 2, 3. Suppose v(N) = 1, v(1, 2) = w2, and v(1, 3) = w3 with 0 ≤ w2, w3 < 1;

and all other coalition generate zero. Suppose (x,q) be an efficient equilibrium so that

the grand-coalition always forms immediately. Hence each player i’s expected payoff is

ui(x,q) = pi(1− xN) + xi. Since it is efficient, it must be uN(x,q) = 1. Due to consistency,

multiplying by δ yields xi = δpi(1 − δ) + δxi, and hence xi = δpi. Take j ∈ 2, 3 and let

k = 2, 3 \ j. The excess surplus from forming 1, j is:

e(1, j) = x(1,1,j)1 − (x1 + xj) = (wj + δ(p1 + pj)(1− wj))− δ(p1 + pj)

= ((1− δ) + δpk)wj.

Player 1’s optimality implies that e(N) ≥ e(1, j), or equivalently, δ ≤ 1−wj1−wj+pkwj

. If

pkwj > 0, then1−wj

1−wj+pkwj< 1, and hence the optimality is violated for any δ >

1−wj1−wj+pkwj

.

Therefore, in order for an efficient strategy profile (x,q) to be an equilibrium, it must be

p2w3 = p3w2 = 0. As long as p2 > 0 and p3 > 0, an efficient equilibrium is impossible for

a sufficiently high discount factor, unless w2 = w3 = 0. If w2 = w3 = 0, then (N, v) is a

unanimity game and the efficient strategy profile constructs an equilibrium for any discount

factor. An equilibrium in this game is discussed in Example 1.6.

Remark. In Example 1.2 the grand-coalition is the unique efficient coalition. Thus the excess

surplus of forming an efficient coalition converges to zero as the discount factor closes to 1,

and showing inefficiency is straightforward. When there are multiple efficient coalitions,

however, the excess surplus of forming an efficient coalition can be strictly positive even in a

limit, and hence showing inefficiency result is not trivial as we will prove in Subsection 1.4.2.

Lastly, we investigate the role of essential players through a simple majority game.

Example 1.3 (A Simple Majority Game). Let N = 1, 2, 3; and v(S) = 1 if |S| ≥ 2,

otherwise v(S) = 0. E = 1, 2, 1, 3, 2, 3, N and K = ∅. Let p = (13, 1

3, 1

3). For any

δ, there exists an efficient equilibrium (x, q) such that, for each i ∈ N and j ∈ N \ i,

xi = 13δ and qi(i, j) = 1

2. In fact, the equilibrium is exactly the same as in Baron and

23

Ferejohn (1989). Right after forming any two-player coalition, an efficient state is reached

and no further bargaining will occur, and hence there is no room for an intermediate coalition

formation. In any equilibrium, therefore, a two-player coalition always immediately forms

independently of p and δ.

We will prove Theorem 1.1 in the following two subsections. Subsection 1.4.1 investigate

grand-coalition equilibria as a benchmark case. Proposition 1.4 constructs a grand-coalition

equilibrium for a unanimity game, which implies the first part of Theorem 1.1. If the

underlying game is not a unanimity game, Proposition 1.5 shows that a grand-coalition

equilibrium is impossible for a sufficiently high discount factor, which is a special case of the

second part of Theorem 1.1.

To complete the proof of the second part, in Subsection 1.4.2, we extend the impossibility

result of grand-coalition equilibrium into the general inefficiency result. This extension is

a main part of the proof of Theorem 1.1. To do so, we first generalize Winter (1996)’s

result about a simple game with a veto player to a general characteristic function form game

with an essential player: essential players take all the surplus in any efficient equilibrium,

as a discount factor converges to 1. When players have buyout options, however, we will

show that even a non-essential player gets a strictly positive payoff, which contradicts to the

generalized Winter’s result and hence it turns out an efficient equilibrium is impossible.

1.4.1 Grand-Coalition Equilibria

First, in the following lemma, we characterizes the payoff vector in grand-coalition equilib-

ria. Though we assume zero-normalization on initial characteristic functions, but in non-

initial states, the induced characteristic functions may not be zero-normalized. The following

lemma does not rely on zero-normalization.

Lemma 1.5. Suppose (x,q) is a grand-coalition equilibrium of (N, v, p, δ). For each i ∈ N ,

i) ui(x,q) = vi + pi(v − vN); and

ii) xi = vi + δpi(v − vN).

24

Proof. Since (x,q) is efficient, it must be uN(x,q) = v and xN = δv. Take i ∈ N . i always

forms a grand-coalition and i is always included by other players’ proposal as well. Thus, i’s

expected equilibrium payoff is

ui(x,q) = pi(v − xN) + xi = pi(1− δ)v + xi. (1.4)

Consistency condition in Proposition 1.2 requires

xi = (1− δ)vi + δui(x,q) (1.5)

and hence (1.4) yields the first result. Plugging the first result into (1.5), the second result

follows.

Next, we construct a grand-coalition equilibrium in a unanimity game, in which each

player’s expected payoff is proportional to her recognition probability. More precisely, for

each state π ∈ Π and each active player i ∈ Nπ, her value in the state is xπi = δvpπi and her

expected payoff in the initial state is piv. Let δvp be a value profile with xπi = δvpπi for each

π ∈ Π and i ∈ Nπ. Let q be an efficient coalition formation strategy profile with qπi (Nπ) = 1

for each π ∈ Π and i ∈ Nπ.

Proposition 1.4. Let (N, v) be a unanimity game. For any p and δ, the bargaining game

(N, v, p, δ) has an efficient equilibrium (δvp, q). Furthermore, any stationary equilibrium of

(N, v, p, δ) is efficient and the equilibrium payoff vector is vp independently of δ.

Proof: For any two-player game, the statement is clearly true. As an induction hypothesis,

suppose the statement is true for any less-than-n-player game. Now consider an n-player

game (N, v, p, δ).

Step 1: (δvp, q) is an equilibrium.

For any non-initial state π, the strategy profile (δvp, q) constitutes an equilibrium in

the subgame starting with π, due to the induction hypothesis and Lemma 1.5. Now

we verify (δvp, q) constitutes an equilibrium in the initial state. First, consistency

25

condition is hold by Lemma 1.5. Second, for each i ∈ N and S ( Ni,

ei(S, δvp) = x(i,S)i − xS = δvp(i,S) − δvpS = 0.

However, ei(N, δvp) = v − δvpN = (1− δ)v > 0, and hence forming a grand-coalition

satisfies optimality condition.

Step 2: An equilibrium payoff vector is always vp.

Let (x,q) be an equilibrium. If q = q, then Lemma 1.5 implies x = δvp, which yields

the equilibrium payoff vector is vp. Suppose that there exists i ∈ N and S ( N such

that qi(S) > 0. Since (x,q) is inefficient, it must be xN < δv. Player i’s optimality

condition requires that x(i,S)i −xS ≥ v−xN , and hence the induction hypothesis implies

that δvpS − xS ≥ v − xN . Putting xN < δv, we have

xS < δvpS − (1− δ)v. (1.6)

On the other hand, for each j ∈ S, let Qj =∑

k∈N pk∑

S⊆N qk(S)1(j ∈ S). Player

j’s continuation payoff is no less than her payoff from making an offer with the grand-

coalition, and hence we have

uj(x,q) ≥ pj(v − xN) +Qjxj + (1−Qj)δpj v

> pj v +Qj(xj − δpj v). (1.7)

Since δuj(x,q) = xj, rearranging (1.7), we have xj > δpj v. Since this inequality holds

for any j ∈ S, by summing this over S, it follows xS > δvpS, which contradicts to

(1.6). Therefore, for all i ∈ N , it must be qi(N) = 1 in any equilibrium. However, this

contracts to qi(S) > 0 with S ( N .

Now we prove impossibility of grand-coalition equilibria for a non-unanimity game. If a

grand-coalition equilibrium is assumed, then at least one player can be strictly better off by

forming a strict subcoalition for a sufficiently high discount factor.

Proposition 1.5. Suppose (N, v) is not a unanimity game. For any p, there exists δ < 1

such that, for all δ > δ, a bargaining game (N, v, p, δ) has no grand-coalition equilibrium.

26

Proof. Since (N, v) is not a unanimity game, there exists S ( N such that v(S) > 0.

By superadditivity, for all k ∈ N \ S, we have v(N \ k) > 0. Suppose (x,q) is an

efficient equilibrium. Take any i ∈ S. Player i’s optimality condition requires ei(N,x) ≥

ei(N \ k,x), or equivalently,

x(i,N)i − xN ≥ x

(i,N\k)i − xN + xk. (1.8)

First, Lemma 1.5 implies that x(i,N)i = v and xk = δpkv. Secondly, i’s (N \ k)-formation

induces a two-player game, and hence Lemma 1.5 yields x(i,N\k)i = v(N \ k) + δ(1 −

pk) (v − v(N \ k)). Plugging xk, x(i,N)i , and x

(i,N\k)i into (1.8), we have

v ≥ v(N \ k) + δ(1− pk) (v − v(N \ k)) + δpkv.

Rearranging the terms, it follows

(1− δ)v ≥ (1− δ(1− pk))v(N \ k). (1.9)

As δ → 1, the left-hand side of (1.9) converges to zero; while the right-hand side is

strictly positive uniformly on δ, which yields a contradiction. More precisely, let δ =

v(N\k)(1−pk)v(N\k)+pk v

, which is strictly less than 1. Then for all δ > δ, (1.9) does not hold,

and hence forming a grand-coalition is not optimal for i. Therefore, (x,q) cannot be an

equilibrium for any δ > δ.

The following result is a direct consequence of Proposition 1.4 and Proposition 1.5.

Corollary 1.1. Consider an underlying game (N, v) and an initial recognition probability

vector p ∈ R|N |++. A bargaining game (N, v, p, δ) has a grand-coalition equilibrium for all δ if

and only if (N, v) is a unanimity game.

1.4.2 Efficient Equilibria

In this subsection, we consider an efficient equilibrium and complete the proof of the second

part of Theorem 1.1. First, we generalize Winter (1996), which showed that each non-veto

player gets nothing in equilibria for simple games with a veto player as the discount factor

27

converges to 1. Lemma 1.6 extends his result to a general characteristic function form game

with an essential player and it also allows a general recognition probability. We will use

this lemma to show a contradiction when players have buyout options. That is, even a

non-essential player gets strictly positive payoff uniformly on discount factors, and hence an

efficient equilibrium is impossible.

Lemma 1.6. Suppose (x,q) is an efficient equilibrium of (N, v, p, δ). If K 6= ∅, then xK

converges to v as δ approaches 1.

Proof. Suppose (x,q) is an efficient equilibrium. Let x = minS∈E xS and E∗ = argminS∈E xS.

For any i ∈ N , since ui ≥ pie(N) + pixi, it follows xi ≥ pi1−δpi (1− δ) > 0 and hence E∗ ⊆ Em.

Since (x,q) is efficient, for all i ∈ N and S ⊆ N , qi(S) > 0 implies K ⊆ S; and for all k ∈ K

and S ⊆ N , qk(S) > 0 implies S ∈ E∗. Therefore, for any k ∈ K, uk(x,q) = pk(v − x) + xk.

Summing this over K, we have uK(x,q) = pK(v − x) + xK , or equivalently,

(1− δ)xK = δpK(v − x) = pK(xN − δx) > pK(xN − x). (1.10)

Take any S ∈ E∗ so that xS = x. First, suppose K ∪ (N \ S) 6∈ E. For any A ⊆ N \ S,

monotonicity implies thatK∪A 6∈ E. Since S ∈ Em, for any j ∈ S, it follows that S\j 6∈ E.

Therefore, S = K and hence x = xK as desired. Next, suppose K ∪ (N \ S) ∈ E. Since

S ∈ E∗, xK∪(N\S) ≥ x. Note that K ⊆ S and hence K ∩ (N \ S) = ∅. Thus it follows that

xK + xN\S ≥ x, or equivalently, xN − x ≥ x− xK . Furthermore, K ⊆ S implies x− xK ≥ 0.

Thus (1.10) yields

(1− δ)xK = δpK(v − x) > pK(x− xK) ≥ 0. (1.11)

Thus, as δ → 1, xK → x and x→ v, as desired.

Now we are ready to prove the second part of Theorem 1.1. The proof consists of three

cases. In each case, we observe different strategic reasons that cause delay. In the first case,

if K 6∈ E, that is, the set of essential players is not an efficient coalition, then non-essential

players form a coalition among themselves, instead of directly forming an efficient coalition

with the essential players. In the second case, if K ∈ E and there exists an essential player

28

k′ ∈ K such that v(N \ k′) > 0, then there exists another essential player k ∈ K who can

be better off by forming an inefficient coalition excluding k′. In the last case, if K ∈ E and

v(N \ k′) = 0 for all for k′ ∈ K, then essential players compete with each other to buy out

non-essential players rather than immediately forming an efficient coalition.

If K 6∈ E, then the set of non-essential players can be a collective essential player in the

subsequent period by forming a coalition, after which there is a unique efficient coalition.

Lemma 1.7 shows that if there is a unique efficient coalition, then each player gets at least

some positive portion of her marginal contribution to the grand-coalition in addition to her

stand-alone value.

Lemma 1.7. Suppose v(S) < v for all S ( N . If (x,q) is an equilibrium of (N, v, p, δ), for

any i ∈ N

ui(x,q) ≥ vi + δ|N |−2pi (v − v(N \ i)) .

Proof. If |N | = 2, then (N, v) is a unanimity game and Proposition 1.4 proves this case.

As induction hypothesis, suppose the result holds for any less-than-n-player game. Consider

(N, v) with |N | = n and ∀S ( N v(S) < v. Take any i ∈ N .

ui(x,q) ≥ pi · 0 +∑j∈N

pj∑S⊆N

qj(S)(1(i ∈ S)xi + 1(i 6∈ S)x

(j,S)i

)≥ Qi ((1− δ)vi + δui(x,q))

+(1−Qi)((1− δ)vi + δ

(vi + δn−3pπi (v − v(N \ i))

))= δQiui(x,q) + (1− δQi)vi + (1−Qi)δ

n−2(v − v(N \ i)),

where Qi =∑

j∈N pj∑

S⊆N qj(S)1(i ∈ S) and the second inequality comes from the induc-

tion hypothesis. Rearranging the terms, the inequality yields

ui(x,q) ≥ vi +1−Qi

1− δQi

δn−2(v − v(N \ i))

≥ vi + δn−2(v − v(N \ i)),

as desired.

29

Based on Lemma 1.7, under assuming an efficient equilibrium, we show that a non-

essential player can be better off by forming a coalition which consists of non-essential play-

ers, rather than immediately forming an efficient coalition. However, this is a contradiction

and proves impossibility of an efficient equilibrium.

Proof of ii) in Theorem 1.1 (Case 1: K 6∈ E)

If K 6∈ E, then v(K) < v. Suppose that (x,q) is an efficient equilibrium. Take any i ∈ N \K.

Since (x,q) is efficient, there exists A 6= ∅ such that K ∪ A ∈ E and qi(K ∪ A) > 0. Player

i’s optimality condition requires that ei(K ∪ A,x) ≥ ei(N \K,x), which is

v − xK − xA ≥ x(i,N\K)i − xN + xK .

After i’s (N \K)-formation, since there is only one efficient coalition, Lemma 1.7 yields that

v − xK − xA ≥ v(N \K) + δ|K|−2(1− pK)(v − v(K))− xN + xK .

Since xN = δv and v(N \K) ≥ 0, rearranging the terms, we have

(1 + δ)v − 2xK − xA ≥ δ|K|−2(1− pK)(v − v(K)). (1.12)

Due to Lemma 1.6, as δ → 1, the left-hand side of (1.12) converges to 0, while the right-hand

side converges to (1− pK)(v − v(K)) > 0, which yields a contradiction.

Lemma 1.8. If K ∈ E, then Em = K, K∪D = N , K∩D = ∅, and |K| ≥ 2. In addition,

if D = ∅, then K = N and Em = E.

Proof. Suppose K ∈ E. Take any S ∈ E. Since K = ∩S′∈ES ′, we have S ∩ K = K, and

hence K ⊆ S, which implies Em = K. It follows that D ≡ N \ (∪Em) = N \K, and hence

K ∪D = N and K ∩D = ∅. If |K| = 0, then K = ∅ ∈ E. By monotonicity, for any i ∈ N ,

i ∈ E, which violates zero-normalization. If |K| = 1, say K = k, then k ∈ E, which

violates zero-normalization. Therefore, we conclude |K| ≥ 2. The second part is trivial.

30

Lemma 1.9 provides a lower bound of a cutoff value for each essential player under

assuming an efficient equilibrium.

Lemma 1.9. Suppose (x,q) is an equilibrium of (N, v, p, δ). If K ∈ E, then, for any k ∈ K,

xk =δ

1− δpk(v − xK) =

pkpK

xK ≥ δpkv.

Proof. Since (x,q) is efficient, for each i ∈ N , qi(S) > 0 implies K ⊂ S. Thus, for any

k ∈ K, we have uk(x,q) = pk(v − xK) + xk, and hence

(1− δ)xk = δpk(v − xK), (1.13)

which implies the first equality. Summing (1.13) over K, we have (1− δ)xK = δpK(v− xK).

Plugging this into (1.13), we have the second equality. Since uN(x,q) ≤ v, we have xK ≤

xN ≤ δv and hence (1− δ)xk ≥ δpk(1− δ)v, which implies the inequality part.

We will show that at least one of essential players can be better off by excluding the

other essential player if all the players are supposed to play efficient strategies.

Proof of ii) in Theorem 1.1 (Case 2: K ∈ E and ∃k′ ∈ K v(N \ k′) > 0)

By Lemma 1.8, |K| ≥ 2, and hence we can take k ∈ K such that k 6= k′. Let (x,q) be an

efficient equilibrium. Player k’s optimality condition implies that ek(K,x) ≥ ek(N \k′,x),

that is,

v − xK ≥ x(k,N\k′)k − xN\k′. (1.14)

Since k’s (N \ k′)-formation yields a two-player game, k’s value in the subsequent state is

x(k,N\k′)k = v(N \ k′) + δ(1− pk′)(v − v(N \ k′). (1.15)

Plugging (1.15) into (1.14), we have

(1− δ(1− pk′))v − xK ≥ (1− δ(1− pk′))v(N \ k′)− xN + xk′

(1 + δpk′)v − xK ≥ (1− δ(1− pk′))v(N \ k′) + xk′

v − xK ≥ (1− δ(1− pk′))v(N \ k′), (1.16)

31

where the second line comes from efficiency xN = δv; and the third line is due to Lemma

1.9. However, by Lemma 1.6, the left-hand side of (1.16) converges to 0 as δ → 1; while the

right-hand side converges to pk′v(N \ k′) > 0, which yields a contradiction.

If v(N \ k′) = 0 for all k′ ∈ K, then it may not be profitable for an essential player k

to exclude the other essential player k′ as in Case 2. We show that it is profitable for k ∈ K

to form a coalition with non-essential players rather than forming an efficient coalition.

Proof of ii) in Theorem 1.1 (Case 3: K ∈ E and ∀k′ ∈ K v(N \ k′) = 0)

If D = ∅, then Lemma 1.8 implies that there exists a unique efficient coalition, and hence

Proposition 1.5 completes the proof. Now we assume that D 6= ∅. Again Lemma 1.8 yields

K ∪D = N and pK < 1. Take any k ∈ K. Due to Lemma 1.8, |K| ≥ 2 and hence pK > pk.

Thus (1− pK)(pK − pk) > 0. Rearranging terms and using pK + pD = 1, it follows that

1− pD − pk < 1− pkpK

. (1.17)

Now suppose (x,q) is an efficient equilibrium. Player k’s optimality condition implies that

ek(K,x) ≥ ek(D ∪ k,x), that is,

v − xK ≥ x(k,D∪k)k − xD − xk. (1.18)

After k’s (D∪k)-formation, there is only one efficient coalition, and hence, due to Lemma

1.7, (1.18) yields

v − xK ≥ v(D ∪ k) + δ|K|−1(pD + pk)(v − v(K \ k)− xD − xk. (1.19)

Since v(K \ k) = 0 and xk = pkpKxK due to Lemma 1.9, (1.19) yields

(1− δ|K|−1(pD + pk)

)v + xD ≥

(1− pk

pK

)xK . (1.20)

Due to Lemma 1.6, as δ → 1, (1.20) requires that 1− pD − pk ≥ 1− pkpK

, which contradicts

to (1.17).

32

1.5 Vote Buying in Simple Games

In the class of simple games, the model can be interpret as legislative bargaining with vote

buying. In this section, we rephrase our results in simple games and compare them to the

Baron-Ferejohn model to highlight the role of vote buying.

First, we define a simple game in a formal way. Given a set of players N , a class of

subsets W ⊂ 2N is a set of (proper) winning coalitions if (∀i ∈ N) i 6∈ W; N ∈ W;

S ∈ W =⇒ (∀S ′ ⊃ S) S ′ ∈ W; and S ∈ W =⇒ (N \ S) 6∈ W. A characteristic

function form game (N, v) is a (proper) simple game if v(S) = 1 for all S ∈W and v(S) = 0

otherwise. Let Wm = S ∈ W | (∀i ∈ S) S \ i 6∈ W be a set of minimal winning

coalitions, and V = ∩W a set of veto players.

For a simple game with a veto player, a non-winning coalition form with positive proba-

bility as a transitional state for a sufficiently high discount factor, unless all the players are

veto. In this case, the bargaining game ends up with a non-minimal winning coalition and

the equilibrium payoff vector is not necessarily in the core. This is contrast to Baron and

Ferejohn (1989) and Winter (1996), in which vote buying is not allowed.

Corollary 1.2. Let (N, v) be a simple game with a veto player. A bargaining game (N, v, p, δ)

has an efficient equilibrium for all discount factors if and only if all the players are veto.

Corollary 1.3. Let (N, v) be a simple game with a veto player but not a unanimity game.

In any equilibrium of (N, v, p, δ), there exists δ < 1 such that, for all δ > δ, a non-winning

coalition forms with positive probability as a transitional state; and the final winning coalition

is not necessarily minimal.

Remark. For simple games with veto players, a core allocation may not have a desirable

property in terms of equality.15 Most of well-known cooperative power indices, including

Shapley and Shubik (1954) power index and Banzhaf (1964) power index, do not necessarily

15If there is no veto player, then the core is empty unless it is unanimity game. In such games, the limitingequilibrium payoff vector depends on their initial recognition probabilities. Instead of the core-constrainedNash bargaining solution, the relation between the equilibrium payoff vector and the nucleolus (or the kernel)has been studied by Serrano (1993), Serrano (1995), Montero (2002), and Montero (2006).

33

lie in the core and assign even a non-veto player a positive value. However, as Winter (1996)

showed in the noncooperative bargaining model without buyout options, an equilibrium

payoff vector lies in the core and hence the veto players take all the surplus if the discount

factor is sufficiently high. Thus, one can view allowing buyout options as a noncooperative

foundation of the cooperative power indices.

To investigate the role of vote buying in legislative bargaining, we consider a three-player

simple game. Let N = 1, 2, 3 be a set of players. For any π ∈ Π such that |Nπ| = 2, there

exists a unique cutoff strategy equilibrium with xπi = δpπi and qπi (Nπ) = 1 for all i ∈ Nπ.

Thus, specifying strategies (x, q) in the initial state is enough for stationary subgame perfect

equilibria of three-player games. We characterize an equilibrium payoff vector for the cases

depend on the number of veto players. It is remarkable that the equilibrium payoff vector is

uniquely determined in the limit that the discount factor converges to 1, without imposing

any symmetric assumption on players’ strategies.

If there is no veto player, then any two-player coalition is a winning coalition. Fur-

thermore, in an equilibrium (x, q), no matter who proposes first, a winning coalition forms

immediately, and hence uN(x, q) = 1, or equivalently, xN = δ.

Proposition 1.6. Let (x, q) be an equilibrium of a three-player simple game with no veto

player. For any p with p1 ≥ p2 ≥ p3, as δ → 1:

i) If p1 ≥ 12, then x =

(p1

2−p1 ,1−p12−p1 ,

1−p12−p1

);

ii) If p1 <12, then x = (1

3, 1

3, 1

3).

Lemma 1.10. Let (x, q) be an equilibrium of a three-player simple game with no veto player.

For any p with p1 ≥ p2 ≥ p3, there exists δ < 1 such that for all δ > δ, x1 ≥ x2 = x3.

Proof. Note that any equilibrium (x, q) is efficient so that xN = δ. It is also clear that x1 ≥ δ3

and hence

x1 = δ − x2 − x3 ≥δ

3. (1.21)

34

Suppose by way of contraction that x2 > x3. Since∑

S:3∈S∑

i∈N piqi(S) = 1, we have

x3 = δ(p3(1− x2 − x3) + x3) or

x3 =δp3(1− x2)

1− δ(1− p3). (1.22)

By (1.21) and (1.22), we have

δ − x2 −δp3(1− x2)

1− δ(1− p3)>δ

3. (1.23)

Since the left-hand side of (1.23) converges to 0 as δ → 1, there exists δ < 1 such that for

all δ > δ, (1.23) yields a contraction.

Proof of Proposition 1.6. Due to Lemma 1.10, there exists δ such that for all δ > δ,

x1 ≥ x2 = x3, which implies that e(2, 3, x) ≥ e(1, 2, x) = e(1, 3, x).

Case 1: e(2, 3, x) > e(1, 2, x) = e(1, 3, x).

Since∑

S:1∈S∑

i∈N piqi(S) = p1, we have

x1 = δ(p1(1− x1 − x2) + p1x1) = δp1(1− x2). (1.24)

Since any equilibrium is efficient, xN = δ and hence x2 = x3 = δ−x12

. Thus, (1.24)

implies

x1 =δ(2− δ)p1

2− δp1

and x2 = x3 =δ(1− p1)

2− δp1

. (1.25)

The condition e(2, 3, x) > e(1, 2, x) requires x1 > x2, or δ < 3− 1p1

due to (1.25).

If p1 ≥ 12, then 3− 1

p1≤ 1 and (1.25) consists of the equilibrium payoffs for sufficiently

high δ, which completes the first part. If p1 <12, then it contradicts to the condition

e(2, 3, x) > e(1, 2, x) for sufficiently high δ.

Case 2: e(2, 3, x) = e(1, 2, x) = e(1, 3, x).

It must be x1 = x2 = x3 = δ3

for sufficiently high δ. Thus, we have

x1 = δ(p1(1− x1 − x2) + (p1 + p2q2(1, 2) + p3q3(1, 3))x1),

which implies

x1 =δp1(3− 2δ)

3(1− δ(p1 + p2q2(1, 2) + p3q3(1, 3))).

35

Then, x1 = δ3

yields p1(3− 2δ) = 1− δ(p1 + p2q2(1, 2) + p3q3(1, 3)), or

p1 =1− δ(p2q2(1, 2) + p3q3(1, 3))

3− δ. (1.26)

If p1 <12, then it must be q2(1, 2) + q3(1, 3) > 0 as δ → 1. If p1 >

12, (1.26) implies

p2q2(1, 2) + p3q3(1, 3) < 0 for sufficiently high δ, which yields a contradiction.

If there is only one veto player, then the other non-veto players’ payoffs are always the

same, no matter what their recognition probabilities are. In addition, if the veto player’s

recognition probability is greater than 12, then the other non-veto players are always form a

coalition with each other.

Proposition 1.7. Let (x, q) be an equilibrium of a three-player simple game with Wm =

1, 2, 1, 3. For any p, as δ → 1:

i) If p1 ≥ 12, then q2(2, 3) = q3(2, 3) = 1 and

x1 =p1(3− 2p1)

2− p1

, and x2 = x3 =(1− p1)2

2− p1

. (1.27)

ii) If p1 <12, then 0 < q2(2, 3) < 1 and 0 < q3(2, 3) < 1, and

x1 =1 + 2p1

3, and x2 = x3 =

1− p1

3. (1.28)

Remark. This limiting equilibrium payoff vector depends only on the veto player’s recog-

nition probability and the two non-veto players payoffs are the same no matter what their

recognition probabilities are. Note that the Shapley-Shubik power index for this game is

(23, 1

6, 1

6). Thus, the equilibrium payoff vector Lorenz-dominates the Shapley-Shubik power

index if and only if p1 <12.

Lemma 1.11. Let (x, q) be an equilibrium of a three-player simple game with Wm =

1, 2, 1, 3. For any p, there exists δ < 1 such that for all δ > δ, x1 ≥ x2 = x3.

Proof. Suppose x2 > x3. We show a contradiction in each possible case.

36

Case 1: e(23, x) > e(13, x) > e(12, x).

The expected payoffs of player 2 and player 3 are:

x2 = δ(p2(δ(p2 + p3)− x2 − x3) + (p2 + p3)x2);

x3 = δ(p3(δ(p2 + p3)− x2 − x3) + x3),

which implies that

x2 =δ2(1− δ)p2(p2 + p3)

1− δ + p1p3δ2and x3 =

δ2p3(p2 + p3)(1− δ(p2 + p3))

1− δ + p1p3δ2.

As δ → 1, x2 converges to 0 and x3 converges to p2 + p3. Thus, for sufficiently high δ,

it contradicts to x2 > x3.

Case 2: e(13, x) > e(23, x) > e(12, x).

The expected payoffs of player 1 and player 3 are:

x1 = δ(p1(1− x1 − x3) + (p1 + p3)x1);

x3 = δ(p3(1− x1 − x3) + x3),

which implies that

x1 =δ(1− δ)p1

1− δ + p2p3δ2and x3 =

δp3(1− δ(p1 + p3))

1− δ + p2p3δ2.

As δ → 1, x1 converges to 0 and x3 converges to 1. Thus, for sufficiently high δ, it

contradicts to x1 > x3.

Case 3: e(13, x) > e(12, x) > e(23, x).

It must be q1(1, 3) = q3(1, 3) = q2(1, 2) = 1, which implies that a winning

coalition must form immediately, which contradicts to Theorem 1.1.

Proof of Proposition 1.7.

i) Suppose q23 = q32 = 1. It must be e(2, 3, x) ≥ e(1, 2, x), or x1 − x2 ≥ p1. Since

x2 = x3 by Lemma 1.11, the veto player’s expected payoff is:

x1 = p1(1− x2) + p2x(2,2,3)1 + p3x

(3,2,3)1

= p1

(1−

(1

2− x1

2

))+ (1− p1)p1,

37

which yields (1.27). The condition x1 − x2 ≥ p1 requires that p1(3−2p1)2−p1 − 1−p1

3≥ p1.

Solving this inequality, p1 must satisfy −2p21 + 3p1 − 1 ≥ 0, or 1

2≤ p1 ≤ 1. This

completes the proof of the first part.

ii) Suppose 0 < q23 < 1 and 0 < q32 < 1. It must be e(2, 3, x) = e(1, 2, x) =

e(1, 3, x), or x1 − x2 = p1 = x1 − x3. Solving these equations with xN = 1, we have

(1.28). In this case, the veto player’s expected payoff is:

x1 = p1(1− x2) + p2

(q21x1 + q23x

(2,2,3)1

)+ p3

(q31x1 + q32x

(3,2,3)1

)= p1(1− x2) + rx1 + (1− r)p1 − p2

1, (1.29)

where r = p2q21 + p3q31 > 0 is the probability that the veto player is included in the

proposed coalition. Plugging (1.28) into (1.29), it follows that

1 + 2p1

3= p1

(1− 1− p1

3

)+ r

1 + 2p1

3+ (1− r)p1 − p2

1,

which yields r = 1 − 2p1. Since r > 0, it must be r = 1 − 2p1 > 0, or p1 <12. This

completes the proof of the second part.

Now suppose there are two veto players. Then each veto player buys out the non-veto

player with positive probability and hence the non-veto player gets a positive payoff.

Proposition 1.8. Let (x, q) be an equilibrium of a three-player simple game with Wm =

1, 2. For any p, as δ → 1, we have q1(1, 3) > 0 and q2(2, 3) > 0, and

x1 = p1 +p3

3, x2 = p2 +

p3

3, and x3 =

p3

3.

Lemma 1.12. Let (x, q) be an equilibrium of a three-player simple game with Wm =

1, 2. For any p, there exists δ < 1 such that for all δ > δ,

argmaxS⊆N

e(S, x) = 1, 2, 1, 3, 2, 3.

Proof.

38

Step 1: We show qi(N) = 0 for all i ∈ N .

Since x3 > 0 for all δ ∈ (0, 1), it must be e(1, 2, x) > e(N, x) and hence q1(1, 2) =

q2(1, 2) = 1. If q3(N) = 1, then the equilibrium is efficient which contradicts to

Theorem 1.1. Thus it must be q3(N) < 1, which implies either e(1, 3, x) ≥ e(N, x) or

e(2, 3, x) ≥ e(N, x). Without loss of generality, suppose e(1, 3, x) ≥ e(N, x). If the

inequality is strict, then q3(N) = 0 and the proof is completed. Now suppose it holds

with equality. If e(2, 3, x) > e(1, 3, x) = e(N, x), then again we have q3(N) = 0,

which completes the proof. Thus it must be e(1, 3, x) = e(N, x) ≥ e(2, 3, x), which

implies δ(p1 +p3)−x1−x3 = 1−xN and 1−xN ≥ δ(p2 +p3)−x2−x3, or equivalently,

x2 = δp2 + (1 − δ) and x1 ≤ δp1 + (1 − δ). Note that x3 ≤ δ(p3(1 − xN) + p3x3) and

hence x3 = δp31−δp3 (1−xN). Putting altogether, we have xN ≤ δp1 + (1− δ) + δp2 + (1−

δ) + δp31−δp3 (1− xN), or

xN ≤ (1− δp3) [δ(1− p3) + 2(1− δ)] + δp3. (1.30)

Note the right-hand side of (1.30) converges to 1− p3(1− p3) < 1 as δ → 1. However,

(1.30) is a contradiction for sufficiently high δ, because xN → 1 as δ → 1.

Step 2: We show that | argmaxS e(S, x)| ≥ 2.

Suppose by way of a contradiction, | argmaxS e(S, x)| = 1. There must be i, j such

that

e(i, j, x) > e(i, k, x) and e(i, j, x) > e(j, k, x). (1.31)

Since 1− xS ≥ e(S, x) ≥ δpS − xS for any S ⊂ N , (1.31) implies

xi < 1− δ(1− pi) + xk, and (1.32)

xj < 1− δ(1− pj) + xk. (1.33)

On the other hand, since∑

S:k∈S(qi(S) + qj(S)) = 0, (1.31) implies

xk < δ(pke(i, j, x) + pkxk) ≤ δpj(1− xN + 2xk) ≤ δpj(1− δ2 + 2xk)

39

and hence

xk <δpk

1− 2δpk(1− δ2). (1.34)

From (1.32), (1.33), and (1.34), we have

xN < pi + pj + 2(1− δ) + 3xk < pi + pj + 2(1− δ) + 3δpk

1− 2δpk(1− δ2). (1.35)

Since the right-hand side of (1.35) converges to pi+pj < 1 as δ → 1, it is a contradiction

again for sufficiently high δ.

Step 3: We show that argmaxS e(S, x) 6= 1, 3, 2, 3.

Suppose argmaxS e(S, x) = 1, 3, 2, 3, Then xN = δ2 and we have

x1 = δ(p1e(1, 3, x) + (p1 + p3q3(1, 3))x1)

= δ(p1 [δ(p2 + p3)− xN + x1]︸ ︷︷ ︸e(1,3,x)=e(2,3,x)

+(p1 + p3q3(1, 3))x1)

= δ(p1

[δ(1− p1)− δ2

]+ (2p1 + p3q3(1, 3))x1),

and hence

[1− δ(2p1 + p3q3(1, 3))]x1 = δ2p1(1− δ − p1). (1.36)

For sufficiently high δ, the right-hand side of (1.36) is strictly negative and hence it

must be

δ(2p1 + p3q3(1, 3)) > 1. (1.37)

Similarly, by solving x2, we have

δ(2p2 + p3q3(2, 3)) > 1. (1.38)

Recall that q3(1, 3) + q3(2, 3) = 1 due to Step 1. By (1.37) and (1.38), we have

δ(p1 + p2) > 1, which yields a contradiction.

Step 4: We show that argmaxS e(S, x) = 1, 2, 1, 3, 2, 3.

A cutoff strategy equilibrium must exists, it suffices to show argmaxS e(S, x) 6= 1, 2, 1, 3

40

and argmaxS e(S, x) 6= 1, 2, 2, 3. Without loss of generality, suppose argmaxS e(S, x) =

1, 2, 1, 3. Then

xN = (p1q1(1, 2) + p2)δ + (p1q1(1, 3) + p3)δ2 (1.39)

and we have

x2 = δ(p2e(1, 2, x) + (p2 + p1q1(1, 2))x2)

= δ(p2 [δ(p1 + p3)− xN + x2] + (p2 + p1q1(1, 2))x2)

= δ(p2 [δ(1− p2)− xN ] + (2p2 + p1q1(1, 2))x2).

Rearranging the terms, we have

[1− δ(2p2 + p1q1(1, 2))]x2 = δp2(δ − xN − δp2). (1.40)

From (1.39), note that xN → 1 as δ → 1, and hence the right-hand side of (1.40) is

strictly negative for sufficiently high δ. Thus, we have

δ(2p2 + p1q1(1, 2)) > 1. (1.41)

On the other hand, we have

x3 = δ(p3e(1, 3, x) + (p3 + p1q1(1, 3))x3)

= δ(p3(1− xN) + (2p2 + p1q1(1, 3))x3)

and hence [1− δ(2p3 + p1q1(1, 3))]x3 = δp3(1 − xN). Since the right-hand side

converges to 0 as δ → 1, it must be either 2p3 + p1q1(1, 3) → 1 or x3 → 0.

If 2p3 + p1q1(1, 3) → 1 then it yields a contradiction to (1.41) for sufficiently

high δ. Now assume x3 → 0. Note that e(1, 2, x) = e(1, 3, x) > e(2, 3, x)

implies x2 = 1 − δ(1 − p2) + x3 and x1 < 1 − δ(1 − p1) + x3. Thus, we have

xN = 2(1−δ)+(p1 +p2)+3x3, which converges to p1 +p2 < 1, and hence it contradicts

to (1.39) for sufficiently high δ.

41

w∗ VPower Index (%)with Vote Buying

Power Index (%)without Vote Buying

5 ∅ (33,33,33) (33,33,33)

6 1 (63,19,19) (100,0,0)

7 1, 2 (52,41,7) (57,43,0)

8 1, 2, 3 (44,33,22) (44,33,22)

Table 1.2: The Role of Vote Buying in a Three-Party Weighted Majority Game [w∗; 4, 3, 2]

Proof of Proposition 1.8. By Lemma 1.12, for sufficiently high δ, we have e(1, 2, x) =

e(1, 3, x) = e(2, 3, x), which implies

x1 − δp1 = x2 − δp2 = x3 + (1− δ), (1.42)

and hence xN = 2(1 − δ) + δ(1 − p3) + 3x3. Since δ2 ≤ xN ≤ δ, we have limδ→1 xN = 1,

which implies x3 converges to p33

. Plugging x3 into (1.42), we have the desired result.

If all players are veto, that is, Wm = N, then it is a unanimity game. Due to Propo-

sition 1.4, the equilibrium payoff vector is equivalent to the initial recognition probability

no matter what the discount factor is. The following example summarizes the role of vote

buying in legislative bargaining.

Example 1.4 (Three-Party Weighted Majority Games). Consider a three-party weighted

majority game, in which each party has 4, 3, and 2 votes; and w∗ votes are required to win

for a unit surplus. Suppose that the initial recognition probabilities are proportional to their

votes, and hence p =(

49, 3

9, 2

9

). Table 1.2 shows the role of vote buying, taking a limiting

equilibrium payoff vector as a power index.

1.6 Non-transferability of Recognition Probabilities

We have assumed that recognition probabilities are transferable: when players trade their

resources and rights, they also trade their chances of being a proposer as well. With transfer-

able recognition probabilities, our result verifies that an efficient equilibrium does not exist

42

for a sufficiently high discount factor, if the underlying game has an essential player and it

is not a unanimity game.

The inefficiency comes from players’ strategic delay on coalition formation. To be con-

crete, when each player chooses a coalition to form, the player considers three different

factors: 1) a surplus directly from combining resources, 2) changes in bargaining power by

changing coalitional state, and 3) increasing the chance of being a proposer by taking other

players’ chances. In this section, in order to investigate the net effect of varying coalitional

structure caused by a transitional coalition formation, we eliminate the last factor. Then

we explore the role of transferability of recognition probabilities, comparing the result from

that in the previous section.

Assume that players’ recognition probabilities are their innate right which cannot be

traded. That is, each proposer inherits other respondents’ resources but not their chances of

being a proposer. With non-transferable recognition probabilities, instead of (1.1), we define

the recognition probabilities in any state π in the following way:

pπi =

pi∑

k∈Nπ pkif i ∈ Nπ

0 otherwise.(1.43)

Remark. In Gul (1989), each possible bilateral meeting occurs with equal probability and one

of two players in the meeting becomes a proposer. That is, the initial recognition probability

is uniform and players cannot trade their recognition probabilities. The uniformity and non-

transferability of recognition probabilities are crucial to implement the Shapley value. He

also considered an alternative model, namely a partnership game in which players can trade

their recognition probabilities, to observe the implementation of the Shapley value is not

robust.

In an environment with non-transferable recognition probabilities, since players cannot

accumulate their chances of being a proposer in the subsequent bargaining game, they have

less incentive to form a transitional coalition. Thus, imposing non-transferability of recog-

nition probabilities may relax the inefficiency result. We characterize a sufficient condition

43

for efficient equilibria in this alternative environment. In this section, we concentrate on a

grand-coalition equilibrium. For any π ∈ Π and S ⊆ Nπ, let [S]π =∑

j∈S[j]π. Also define

vπ(S) = v([S]π) −∑

j∈S v([j]π), which represents a marginal surplus of S-formation in π.

Note that vπ(Nπ) = v −∑

j∈Nπ v([j]π). Define a value profile x such that for all π ∈ Π and

i ∈ Nπ,

xπi = v([i]π) + δpπi vπ(Nπ). (1.44)

Lemma 1.13. For all π ∈ Π, i ∈ Nπ, and S ( Ni such that |S| ≥ 2,

eπi (Nπ, x) ≥ eπi (S, x) ⇐⇒ vπ(Nπ) ≥ ρπi (S; δ)vπ(S),

where ρπi (S; δ) =1− pπS + pπi − δpπi

(1− δ + δpπS)(1− pπS + pπi )− δpπi.

Proof. Fix π ∈ Π, i ∈ Nπ, and S ( Ni with |S| ≥ 2. Using (1.44), calculate eπi (Nπ, x) and

eπi (S, x):

eπi (Nπ, x) = v − xπNπ

= v −∑j∈Nπ

v([j]π)− δvπ(Nπ)

= (1− δ)vπ(Nπ), (1.45)

and

eπi (S, x) = xπ(i,S)i − xπS

= v([S]π) + δpπi

1− pπS + pπi

(v −

∑j∈Nπ

v([j]π) +∑j∈S

v([j]π)− v([S]π)

)

−

(∑j∈S

v([j]π) + δpπSvπ(Nπ)

)

= δ

(pi

1− pS + pi− pS

)vπ(Nπ) +

(1− δ pi

1− pS + pi

)vπ(S). (1.46)

From (1.45) and (1.46), we have the desired result.

Proposition 1.9. Suppose that players cannot transfer their recognition probabilities. There

exists an equilibrium in which a grand-coalition always forms immediately in all states for

44

all discount factors, if and only if, for all π ∈ Π, i ∈ Nπ, and S ( Ni such that |S| ≥ 2,

vπ(Nπ) ≥(

1

pπS − pπi

)vπ(S). (1.47)

Proof. Suppose q constructs an equilibrium. Due to Lemma 1.5, the corresponding value

profile is uniquely determined as in (1.44). By Lemma 1.13, optimality condition implies for

all δ,

vπ(Nπ) ≥ ρπi (S; δ)vπ(S).

Observe that 1pπS−p

πi≥ ρπi (S; δ) for all δ ≤ 1, and ρπi (S; δ)→ 1

pπS−pπi

as δ → 1. This completes

the necessary condition. Now suppose that for all π ∈ Π, i ∈ Nπ, and S ( Ni such that

|S| ≥ 2, the inequality (1.47) holds. Then it can be easily shown that a strategy profile

(x, q) satisfies optimality condition and consistency condition for all δ.

Proposition 1.9 provides a sufficient condition for a grand-coalition equilibrium. A grand-

coalition equilibrium requires that all the players form a grand-coalition only in the initial

state. As long as players are supposed to form a grand-coalition immediately, their strategies

in proper subgames do not effect on efficiency. In Proposition 1.9, we investigate a stronger

notion of efficiency; it requires efficiency even in off-equilibrium paths.16 Thus the condition

in Proposition 1.9 is a sufficient condition for a grand-coalition equilibrium to exist. The

condition may not be a necessary condition. However, at least for three-player games, the

condition in Proposition 1.9 is also a necessary condition for a grand-coalition equilibrium.

For any non-initial inefficient state in three-player games, there are only two active players

and hence they form a grand-coalition as long as it is the unique efficient coalition.

Remark. Non-transferability of recognition probabilities discourage players from forming a

transitional coalition. Thus, compared to Proposition 1.5, Proposition 1.9 implies that ban-

ning players from trading recognition probabilities improves efficiency. If all the players can

immediately form an efficient coalition, then non-transferability of recognition probabilities

discourage some players from forming a transitional coalition and press them to form an

16This notion is called subgame efficiency by Okada (1996).

45

efficient coalition. However, its effect on efficiency can be negative when cooperation re-

strictions are imposed. If some players cannot directly form an efficient coalition due to

cooperation restriction, then they may decline to be a proposer to avoid forming transitional

coalitions with non-transferable recognition probabilities. That is, discouraging players from

forming a transitional coalition causes additional inefficiency. See Lee (2014) for details in

network-restricted games.

The following result is for a special case in which each player is selected as a proposer

with equal probability.

Corollary 1.4. Suppose pπi = 1|Nπ | for all π ∈ Π and i ∈ Nπ. There exists an equilibrium

in which a grand-coalition always forms immediately in all states for all discount factors, if

and only if,

vπ(Nπ)

|Nπ|≥ vπ(S)

|S| − 1for all π ∈ Π and S ( N such that |S| ≥ 2. (1.48)

Remark. This result is reminiscent of the result of Chatterjee et al. (1993) and Okada (1996),

which characterizes a condition for a grand-coalition equilibrium without buyout options:

there exists a grand-coalition equilibrium for a sufficiently high discount factor if and only if

v(N)

|N |≥ v(S)

|S|for all S ⊆ N. (1.49)

Noting that (1.48) is stronger that (1.49), buyout options hinder an efficient equilibrium

even though players cannot trade their recognition probabilities.

The following example illustrates the condition on the discount factor under which an

efficient equilibrium exists.

Example 1.5. N = 1, 2, 3; and v(S) = |S| if |S| ≥ 2, and v(i) = 0 for all i ∈ N .

• Without buyout options, players always immediately form N for all discount factor.

Note that each player’s cutoff value is δ in any efficient equilibrium. Given this cutoff

value, since v(N)− 3δ ≥ v(1, 2)− 2δ ⇐⇒ δ ≤ 1, an efficient equilibrium exists for

any δ.

46

Oa

u2

13

12

23

1

13

16

29

Eqm. Payoff

C-Nash

Shapley

Nucleolus

(a) Transferable Recognition Probabilities

Oa

u2

13

12

23

1

13

16

29 Eqm. payoff

C-Nash

Shapley

Nucleolus

(b) Non-transferable Recognition Probabilities

Figure 1.2: Workers’ Equilibrium Payoff in an Employer-Employee Game: p = (13, 1

3, 1

3) and

δ → 1

• With buyout options and non-transferable recognition probabilities, an efficient equi-

librium exists, if v(N)− 3δ ≥ v(1, 2)− 2δ + δ 12(v(N)− v(1, 2)), or equivalently if

δ ≤ 23.

• With buyout options and transferable recognition probabilities, an efficient equilibrium

exists, if v(N)− 3δ ≥ v(1, 2)− 2δ + δ 23(v(N)− v(1, 2)), or equivalently if δ ≤ 3

5.

To investigate the role of transferability of recognition probabilities, we consider an

employer-employee game as in Example 1.2, but for tractability assume two workers are

symmetry, that is a = w2 = w3. First, we consider the limiting equilibrium payoff vector

with transferable recognition probabilities. As we discussed in Example 1.2, strict sub-

coalitions form with positive probability in any equilibrium as long as a > 0. Thus, the

equilibrium payoff vector is generically different from the core-constrained Nash bargaining

solution, which can be implemented when a grand-coalition always forms immediately. See

(a) in Figure 1.2.

Example 1.6 (An Employer-Employee Game with Transferable Recognition Probabilities).

The limiting equilibrium payoff is 3+2a9

for the firm and 3−a9

for each worker. Thus, the

47

equilibrium wage (each worker’s payoff) is strictly higher than the Shapley value for any

a > 0. If a < 34, then the core is relatively large and hence the core-constrained Nash

bargaining solution assigns a higher value to each worker than the equilibrium wage in this

model. However, if a > 34, that is the core is relatively small, then the equilibrium payoff

vector Lorenz-dominates the core-constrained Nash bargaining solution.

However, when players cannot trade their recognition probabilities, the limiting equilib-

rium payoff may coincide with the core-constrained Nash bargaining solution, nucleolus, or

the Shapley value, depend on a. See (b) in Figure 1.2.

Example 1.7 (Employer-Employee Game with Non-transferable Recognition Probabilities).

i) If 0 ≤ a ≤ 13, then all the players get 1

3, which coincides with the (core-constrained)

Nash bargaining solution, Note that the core is relatively large and all the players always

immediately form a grand-coalition. Thus, players do not exercise their buyout options.

ii) If 13≤ a ≤ 1

2, then the firm’s payoff is a and each worker’s payoff is 1−a

2, and this

coincide to nucleolus. In this case, the firm is indifferent between immediately forming

a grand-coalition and sequentially hiring one by one, while each worker still prefers to

forming a grand-coalition.

iii) If 12≤ a ≤ 1, then the firm expects 1+a

3and each worker expects 2−a

6. In this case,

the excess surpluses of all two-player coalitions are the same and greater than that of a

grand-coalition. Hence, the equilibrium payoff coincides with the Shapley value.

48

Chapter 2

Multilateral Bargaining in Networks

2.1 Introduction

Bargaining sometimes involves an agreement among three or more players. Examples are

abundant; in order to construct a railroad, a number of landowners must agree on it; to

undertake a profitable project, an unanimous agreement should be reached among several

individuals who have different but indispensable skills or resources. Sometimes each player

can cooperate with all the other players so that the unanimous agreement is immediately

reached; while other times, some players can cooperate only with a part of other players for

informational or physical reasons, and hence an gradual agreement is inevitable.1

To analyze multilateral bargaining under cooperation restrictions, we introduce a non-

cooperative bargaining model in which each player can communicate only with the directly

connected players in a given network. In each period, a proposer is randomly selected and

the proposer makes an offer specifying a coalition among the neighbors and monetary trans-

fers to each member in the proposed coalition. If all the members in the coalition accept

the offer, then the coalition forms and the proposer controls the coalition thereafter, inher-

iting other members’ network connections (See Figure 2.1). Otherwise, the offer dissolves.

The game repeats until the grand-coalition forms, after which the player who controls the

grand-coalition obtains the unit surplus. All the players have a common discount factor.2

1Since Aumann and Dreze (1974), cooperation restrictions have been studied mainly in cooperative games.Myerson (1977) uses a network to described the structure of cooperation restrictions.

2This model can be extended to a more general class of network-restricted games. In this paper, however,in order to concentrate on network structures and control network-irrelevant factors, we confine our attention

49

1

2

3

4

5

6

7

(a) An initial network

1

2

3

4

5

6

7

(b) Player 1 forms 1, 2, 3

1

2

3

4

5

6

7

(c) The induced network

Figure 2.1: A Coalition Formation in a Network

The model has two important features which distinguish it from the existing noncoop-

erative bargaining models in networks. First, we allow strategic coalition formation so that

each player can choose the partners to bargain with. In the literature, however, players’

strategic interaction is limited in a randomly selected meeting. A bilateral meeting (Manea,

2011a,b; Abreu and Manea, 2012a,b) or a multilateral meeting (Nguyen, 2012) randomly oc-

curs, then the players in the random meeting bargain over their joint surplus.3 As Hart and

Mas-Colell (1996) pointed out, however, a random-meeting-model does not entirely capture

players’ strategic behaviors, and hence strategic decision on coalition formation should be

considered.

Next, we allow players to buy out other players and it enables them to gradually form a

coalition.4 In the Manea/Abreu-Manea/Nguyen model, given a coalition, players’ strategic

decision is limited on how to split the coalitional surplus: All the players in the coalition, once

they reach an agreement, must exit the game and they are excluded in further bargaining.

Therefore, those models are not applicable to an environment in which gradual coalition

formation is inevitable to generate a surplus.5 On the other hand, when players can buy

out other players as an intermediate step, they not only consider the surplus of the current

to a unanimity game but allow any possible network. Chapter 1 considers general transferable utilityenvironments but not network restrictions.

3Abreu and Manea (2012a) also consider an alternative model in which a proposer chooses a bargainingpartner. However, their model is still limited to a bilateral bargaining.

4The notion of buyout option is based on Gul (1989). In this sense, we succeed Gul (1989). On theother hand, since his model assumes random meeting, we develop the model by allowing strategic coalitionformation.

5For instance, in a four-player circle, since no one can immediately form the grand-coalition, the surpluscannot be realized in their models.

50

coalition itself, but also take into account the subsequent bargaining game. Thus players

may even form a zero-surplus coalition strategically.

The main result characterizes a condition on network structures for efficient equilibria.

If the underlying network is either complete or circular, then for any discount factor there

exists an efficient stationary subgame perfect equilibrium. In such an efficient equilibrium,

each player buys out all the neighbors whenever she is selected as a proposer. However, if the

underlying network is neither complete nor circular, then some of the players strategically

delay a unanimous agreement when they are sufficiently patient. Thus, an efficient stationary

subgame perfect equilibrium is impossible for a sufficiently high discount factor.

We also provide an interesting example in which adding a new communication link de-

creases social welfare. This observation is reminiscent of the Braess’s paradox which first

appeared in Braess (1968).6 In the original context, the Braess’s paradox refers a situation

that constructing a new route reduces overall performance when players choose their route

selfishly. Analogously in our model, each player strategically chooses communication links

to use for bargaining. Similarly, the more links are available, the less links are actually used.

As the result, network improvements decrease social welfare.7 To our best knowledge, this

is the first observation of an analog of the Braess’s paradox in the bargaining literature.

The paper is organized as follows. In section 2, we introduce a noncooperative multi-

lateral bargaining model for a network-restricted environment. Section 3 provides the main

characterization result with leading examples. Section 4 considers an alternative model in

which players cannot trade their chances of being a proposer. Missing proofs are presented

in Appendix.

6See Braess et al. (2005) for translation from the original German. In recent years, numerous studies onBraess’s paradox have been made particularly in computer science and related disciplines. See Roughgarden(2005) for more details.

7In the Braess’s paradox with the traffic network context, all the players are worse off; while in thisbargaining in communication networks, some players may be better off even though overall performancedeteriorates.

51

2.2 A Model

2.2.1 Networks

A network (or a graph) g = (N,E) consists of a finite set N = 1, 2, · · · , n of players (or

nodes) and a set E of links (or edges) of N . When g = (N,E) is not the only network

under consideration, the notations N(g) and E(g) are occasionally used for the player set

and the link set rather than N and E to emphasize the underlying network g. Through this

paper, we assume that g is simple8 and connected. Given g = (N,E) and S ⊆ N , a subgraph

restricted on S is g|S = (S, ij ∈ E | i, j ⊆ S). The (closed) neighborhood of i ∈ N is

given by Ni(g) ≡ j ∈ N | ∃ij ∈ E ∪ i. Let degi(g) ≡ |Ni(g)| − 1 be a degree of i and

d(i, j; g) be a (geodesic) distance between i and j in g.

A set S ⊆ N is dominating in g if, for all i ∈ N , either i ∈ S or there exists j ∈ S

such that ij ∈ E. A player i ∈ N is dominating in g if i is a dominating set. Let D(g)

be a set of dominating players in g. A dominating set S is minimal if no proper subset is

a dominating set. A network is trivial if |N(g)| = 1. For any integer k = 2, · · · , n − 1,

a network is k-regular if degi(g) = k for all i ∈ N(g). A network g is complete if it is

(n − 1)-regular, or equivalently if D(g) = N(g). A connected network g is circular if it is

2-regular.9

A complete cover of g is a collection M of subsets of N(g), such that, ∪M = N(g) and

g|M is a complete network for all M ∈M. A complete covering number of g is the minimum

cardinality of a complete cover of g. A minimal complete cover is a complete cover of which

cardinality is minimum.

2.2.2 A Noncooperative Bargaining Game

A noncooperative bargaining game, or shortly a game, is a triple Γ = (g, p, δ), where g is

a underlying network, p ∈ R|N |++ is an initial recognition probability with∑

i∈N pi = 1, and

8A simple network is an unweighted and undirected network without loops or multiple edges.9A circular network (or a circle) should not be confused with a cycle in a network. A circular network is

a network that consists of a single cycle.

52

0 < δ < 1 is a common discount factor.

A game Γ = (g, p, δ) proceeds as follows. In each period, Nature selects a player i ∈ N as

a proposer with probability pi. Then, the proposer i makes an offer, that is, i strategically

chooses a pair (S, y) of a coalition S ⊆ Ni(g) and monetary transfers y ∈ R|N |+ with∑

j∈N yj =

0. Each respondent j ∈ S \ i sequentially either accepts the offer or rejects it.10 If any

j ∈ S \i rejects the offer, then the offer dissolves and all the players repeat the same game

in the next period. If each j ∈ S \ i accepts the offer, then i buys out S \ i, that is,

each respondent j ∈ S \ i leaves the game with receiving yj from the proposer i and the

remaining players (N \ S) ∪ i play the subsequent game Γ(i,S) in the next period. All the

players have a common discount factor δ.

After i buys out S \ i, or i forms S, the subsequent game Γ(i,S) =(g(i,S), p(i,S), δ

)is

defined in the following way:

i) The induced network g(i,S) =(N (i,S), E(i,S)

), where N (i,S) = (N \ S) ∪ i and

E(i,S) = ij | (∃i′j ∈ E) i′ ∈ S and j ∈ N \ S⋃jk | (∃jk ∈ E) j, k ∈ N \ S.

That is, after i’s S-formation, S \i leaves the network, but i inherits all the network

connections from S.

ii) The induced recognition probability p(i,S):

p(i,S)j =

pS if j = i

pj if j ∈ N \ S0 if j ∈ S \ i.

That is, the proposer i takes the respondents’ chances of being a proposer as well.

The game continues until only one player remains, after which the last player acquires

one unit of surplus. When the game ends in finite period T , the history h specifies a finite

sequence y(h) = yt(h)Tt=0 of monetary transfers and the last player i∗(h) ∈ N . Given

Γ = (g, p, δ) and a history h, player i’s discounted sum of expected payoffs is

Ui(h) =T∑t=0

δtyti(h) + δT1(i = i∗(h)).

10The result does not depend on the order of responses.

53

If the game does not end within finite periods, then the history h induces a sequence y(h)

of monetary transfers without determining the last player, and hence player i’s discounted

sum of expected payoffs is

Ui(h) =∞∑t=0

δtyti(h).

2.2.3 Coalitional States

A (coalitional) state π is a partition of N , specifying a set of active players Nπ ⊆ N . For

each active player i ∈ Nπ, i’s partition block [i]π represents the players i together with

players whom he has previously bought out. Denote π by the initial state, that is, Nπ = N

and [i]π = i for all i ∈ N . A state π is terminal if |Nπ| = 1.

A state π is feasible in g, if there exists a sequence of coalition formations (i`, S`)L`=1

such that i1 ∈ N and Si1 ⊆ Ni1 ; and i` ∈ N (i1,S1)···(i`−1,S`−1) and S` ⊆ N(i1,S1)···(i`−1,S`−1)i`

for

all ` = 2, · · · , L; and Nπ = N (i1,S1)···(iL,SL). Let Π(g) be a set of all feasible states in g. For

each π ∈ Π(g), the induced network gπ = (Nπ, Eπ) is uniquely determined by

Eπ ≡⋃i∈Nπ

ij | ∃i′j′ ∈ E (i′ ∈ [i]π and j′ ∈ [j]π)

,

and the induced recognition probability pπ is determined by

pπi =

∑j∈[i]π

pj if i ∈ Nπ

0 otherwise.(2.1)

When there is no danger of confusion, we omit π in notations, for instance, gπ

= g,

gπ(i,S) = g(i,S), and so on. The description of the underlying network g may also be omitted,

when it is clear. For notational simplicity, for any v ∈ R|N | and any S ⊆ N , we denote

vS =∑

j∈S vj.

2.2.4 Stationary Subgame Perfect Equilibria

We focus on stationary subgame perfect equilibria. A stationary strategy depends only on

the current coalitional state and within-period histories, but not the histories of past periods.

The existence of a stationary subgame perfect equilibrium is known in the literature including

54

Eraslan (2002) and Eraslan and McLennan (2013). See Section 1.3 for the formal description

of stationary strategies. In the literature, instead of considering all the possible stationary

strategies, a simple stationary strategy, namely a cutoff strategy, is usually accepted.

A cutoff strategy profile (x,q) consists of a value profile x = xπi i∈Nππ∈Π and a

coalition formation strategy profile q = qπi i∈Nππ∈Π, where xπi ∈ R and qπi ∈ ∆(2Nπi ) for

each π ∈ Π(g).11 A cutoff strategy profile (x,q) specifies the behaviors of an active player

i ∈ Nπ: in the following way:

• player i proposes (S, y) with probability qπi (S) such that

yk =

δxπk if k ∈ S \ i−δxπS\i if k = i

0 otherwise;

• player i accepts any offer (S, y) with i ∈ S if and only if yi ≥ δxπi .

Note that player i can decline to make an offer by choosing S = i. A cutoff strategy profile

(x,q) induces a probability measure µx,q on the set of all possible histories. Given history h,

let π(h) = πt(h)Tt=0 be a sequence of states which is determined by h. Given (x,q), define

the set of inducible states:

Πx,q(g) = π ∈ Π(g) | (∃h ∃t) µx,q(h) > 0 and π = πt(h).

Given x, for each π ∈ Π(g), i ∈ Nπ, and S ⊆ Nπi , define a player i’s excess surplus of

S-formation:

eπi (S,x) =

δx

π(i,S)i − δxπS if S ( Nπ

1− δxπNπ if S = Nπ.

LetDπi (x) = argmaxS⊆Nπieπi (S,x) be a demand set of player i in π andmπ

i (x) = maxS⊆Nπieπi (S,x)

be a (net) proposal gain of player i in π. Given a cutoff strategy profile (x,q), define an

11Through this paper, for a finite set X, ∆(X) is the set of all possible probability measures in X.

55

active player i’s continuation payoff in π:

uπi (x,q) = pπi∑S⊆Nπ

qπi (S)eπi (S,x) +∑j∈Nπ

pπj

( ∑S:i∈S⊆Nπ

qπj (S)δxπi + δ

( ∑S:i 6∈S⊆Nπ

qπj (S)xπ(j,S)i

))= pπi

∑S⊆Nπ

qπi (S)eπi (S,x)

+δ

(∑j∈Nπ

pπj∑S⊆Nπ

qπj (S)(1(i ∈ S)xπi + 1(i 6∈ S)x

π(j,S)i

)). (2.2)

We close this section with two important lemmas which provide fundamental tools for our

analysis. Lemma 2.1 shows that any stationary subgame perfect equilibrium can be uniquely

represented by a cutoff strategy equilibrium in terms of a equilibrium payoff vector. Thus,

when we are interested in players’ equilibrium payoffs or efficiency, without loss of generality,

we may consider only cutoff strategy equilibria. Through this paper, an equilibrium refers a

cutoff strategy equilibrium. Lemma 2.2 characterizes a cutoff strategy equilibrium with two

tractable conditions, optimality and consistency.

Lemma 2.1. For any stationary subgame perfect equilibrium, there exists a cutoff strategy

equilibrium which yields the same equilibrium payoff vector.

Lemma 2.2. A cutoff strategy profile (x,q) is an stationary subgame perfect equilibrium if

and only if, for all π ∈ Π and i ∈ Nπ, the following two conditions hold,

i) Optimality: qπi ∈ ∆(Dπi (x)); and

ii) Consistency: xπi = uπi (x,q).

2.3 Efficient Equilibria

In this section, we characterize a necessary and sufficient condition on network structures

for efficient equilibria. Given g, define a maximum coalition formation strategy profile q =

qπi i∈Nππ∈Π(g) with

qπi (S) =

1 if S = Nπ

i

0 otherwise,

56

that is, for each state π ∈ Π(g), each proposer i ∈ Nπ chooses a maximum coalition Nπi to

bargain with. Given Γ = (g, p, δ), let u(Γ) be a maximum welfare. Note that u(Γ) is obtained

by any cutoff strategy profile involves with a maximum coalition formation strategy profile.

A strategy profile (x,q) is efficient if∑i∈N

ui(x,q) = u(Γ). (2.3)

Example 2.1. Let N = 1, 2, 3, 4. Consider two game Γ = (g, p, δ) and Γ′ = (g′, p, δ),

where g = (N, 12, 23, 34, 41) is a circular network and g′ = (N, 12, 23, 34, 41, 13) is a

chordal network. It is easy to see u(Γ) = δ and u(Γ′) = (p1 + p3) + δ(p2 + p4).

An efficient strategy profile does not necessarily consist of maximum coalition formation

strategies. For each π ∈ Π(g), define a set of i’s coalitions which maximizes the sum of

players’ expected payoffs in the subsequent state:

Eπi ≡ argmaxS⊆Nπ

i

u(Γπ(i,S)).

Lemma 2.3. Given Γ = (g, p, δ), an equilibrium (x,q) is efficient if and only if,

∀π ∈ Πx,q(g) ∀i ∈ Nπ qπi ∈ ∆(Eπi ).

Proof. If |N(g)| = 2, then the statement is obviously true. As an induction hypothesis,

suppose the statement is true for any less-than-n-player game. Consider g with |N(g)| = n.

For any π ∈ Π(g), observe that summing (1.2) over Nπ yields

∑i∈Nπ

uπi (x,q) =∑i∈Nπ

pπi∑S⊆Nπ

qπi (S)

[eπi (S,x) + δ

(∑j∈S

xπj +∑j 6∈S

xπ(i,S)j

)]=

∑i∈Nπ

pπi∑S⊆Nπ

qπi (S)Xπ(i,S), (2.4)

where Xπ(i,Nπ) = 1 and Xπ(i,S) = δ∑

j∈Nπ(i,S) xπ(i,S)j for all S ( Nπ.

Sufficiency: Let (x,q) is an efficient equilibrium. By the consistency condition, for all

S ( Nπ, ∑j∈Nπ(i,S)

xπ(i,S)j =

∑j∈Nπ(i,S)

uπ(i,S)j (x,q).

57

Since (x,q) is efficient, the induction hypothesis and the definition of efficiency yieldXπ(i,S) =

δu(Γπ(i,S)) for all S ( Nπ. Suppose for contradiction that there exists π ∈ Πx,q(g), i ∈ Nπ,

and S, S ′ ⊆ Nπi such that qπi (S) > 0 and u(Γπ(i,S)) < u(Γπ(i,S′)). Then i can strictly improve

the sum of the players’ payoff by putting more weight on S ′ in his coalition formation strategy

and hence qπi cannot be a part of an efficient equilibrium.

Necessity: Given g, π ∈ Π(g), and (x,q), define a partial strategy profile (x|π,q|π) =

(xπ′ , qπ′)π′∈Π(gπ). By induction hypothesis, for all π ∈ Πx,q(g) \ π, (x|π,q|π) is an

efficient equilibrium for a game with gπ. Consider the initial state. By (2.4), in order to

maximize∑

i∈N ui(x,q), each player i must maximize∑

S⊆N qi(S)X(i,S). Since, for all i ∈ N

and all S ∈ Ei, (x|(i,S),q|(i,S)) is an efficient equilibrium for a game with g(i,S), the condition

qi ∈ ∆(Ei) maximizes∑

i∈N ui(x,q) and hence (x,q) is efficient.

We are ready to state our main theorem, which characterizes a condition on network

structures for efficient equilibria.

Theorem 2.1. An efficient stationary subgame perfect equilibrium exists for all discount

factors if and only if the underlying network is either complete or circular.

We prove the theorem through four propositions. For the sufficient condition, in sub-

section 2.3.1, we construct an efficient equilibrium in a complete network (Proposition 2.1)

and in a circular network (Proposition 2.2). Moreover, in a complete network, the equi-

librium payoff vector is unique and hence any stationary subgame perfect equilibrium is

efficient. For the necessary condition, Proposition 2.3 proves the inefficiency result for a

specific class of networks, namely pre-complete networks, in subsection 2.3.2. That is, if the

underlying network is pre-complete and non-circular, then any stationary subgame perfect

equilibrium is inefficient for a sufficiently high discount factor. In subsection 2.3.3, Proposi-

tion 2.4 completes the necessary condition by showing that, for any game with an incomplete

non-circular network, any efficient strategy induces a pre-complete non-circular network with

positive probability.

58

2.3.1 The Sufficient Condition

First, we consider a complete network. Proposition 2.1 shows that a unanimous agreement

is always immediately reached for any p and δ. Furthermore, any equilibrium is efficient and

its payoff vector equals to p. Let p = pπi i∈Nππ∈Π.

Proposition 2.1. Let g be a complete network. For any Γ = (g, p, δ),

i) there exists a cutoff strategy equilibrium (p, q);

ii) for any equilibrium, the equilibrium payoff vector equals to p.

Example 2.2 (A Three-Player Complete Network). Let g be a complete network with

N(g) = i, j, k and p be an initial recognition probability. In the first period, a proposer i

forms a grand-coalition by buying out other two players at the prices of δpj and δpk. Thus

the unit surplus belongs to i and his payoff is 1− (pj + pk)δ = 1− (1− pi)δ. Thus, before a

proposer is selected, player i’s expected payoff is pi · (1− (1− pi)δ) + (pj + pk) · δpi = pi.

Next, in a circular network, we construct an efficient equilibrium in which each player

always forms a maximum coalition and the equilibrium payoff vector is proportional to the

initial recognition probability. Recall that bxc is the largest integer not greater than x.

Proposition 2.2. Let g be a circular network. For any Γ = (g, p, δ), there exists a cutoff

strategy equilibrium (x, q), where for all π ∈ Π(g) and all i ∈ Nπ,

xπi = δ

⌊|Nπ |

2

⌋−1pπi . (2.5)

Example 2.3 (A Four-Player Circular Network). Let g be a circular network with |N(g)| =

4. For all π with 2 ≤ |Nπ| ≤ 3, since gπ is complete, the equilibrium strategies in a non-initial

state π are xπ = pπ and qπ = qπ, which are consistent with (2.5). For the initial state, take

any i ∈ N and let Ni = i, j, k. For any i ( S ⊆ Ni, since S-formation induces a complete

network, the excess surplus from S-formation is ei(S,x) = pSδ − δxS = δ(1 − δ)pS, which

59

implies Di = Ni. For all ` ∈ N , then q`(N`) = 1. Thus, we have∑

`∈N p`∑

S3i q`(S) = pNi

and∑

`∈N p`∑

S 63i q`(S) = 1− pNi . Therefore, i’s expected payoff is:

ui(x, q) = pi · δ(1− δ)pNi + δ [pNi · δpi + (1− pNi) · pi] = δpi,

which satisfies consistency condition.

2.3.2 The Necessary Condition : Pre-complete Networks

To prove the necessary condition, we will show that any efficient strategy profile cannot be

an equilibrium in any incomplete non-circular network if the discount factor is sufficiently

high. First, we need to define a special class of networks, namely pre-complete networks, in

which all the players can induce a complete network. Given g, denote a set of i’s feasible

coalitions which yield a complete network by

Ci(g) = S ⊆ Ni(g) | g(i,S) is complete.

Definition 2.1. A graph g is pre-complete if

∀i ∈ N(g) i /∈ Ci(g) 6= ∅.

See Figure 2.2 for examples of pre-complete networks.12

Proposition 2.3. Let g be a pre-complete non-circular network. For any p, there exists

δ < 1 such that for all δ > δ, any efficient strategy profile (x,q) cannot be an equilibrium in

Γ = (g, p, δ).

In a pre-complete network, there may or may not be exist a dominating player. We divide

the proof into two disjoint cases, D(g) 6= ∅ and D(g) = ∅.

Case 1: Dominating Players

We provide two leading examples to illustrate an occurrence of a strategic delay. Based on

the examples, we discuss a Braess-Like paradox. Then, we prove Proposition 2.3 in a case of

D(g) 6= ∅.12A dark node represents a dominating player.

60

(a) Single Dominating Player: Inefficient

(b) Multiple Dominating Players: Inefficient

(c) No Dominating Player (Non-circular): Inefficient

(d) Circular: Efficient.

Figure 2.2: Examples of Pre-complete Networks

The first example is of a three-player chain, in which there is only one dominating player.

In such a chain, the unique dominating player has a stronger bargaining power than the

other players so that her value is too high for the other players to buy her out. Thus, when

non-dominating players are recognized as a proposer, they decline to make an offer and a

delay occurs.

Example 2.4 (A Chain). Let g = (1, 2, 3, 12, 13). First, we show an impossibility of

an efficient equilibrium. Suppose there exists an efficient equilibrium (x,q). Then player

1 is always included in a proposed coalition, that is, q1(N) = q2(1, 2) = q3(1, 3) = 1.

Thus player 1’s expected payoff is u1(x,q) = p1(1 − δxN) + δx1. Since x1 = u1(x,q) and

xN = p1 + (1− p1)δ, it follows (1− δ)x1 = p1(1− δ(p1 + (1− p1)δ)), or equivalently,

x1 = p1(1 + (1− p1)δ). (2.6)

61

On the other hand, player 2’s expected payoff is

u2(x,q) = p2m2(x) + δ((p1 + p2)x2 + p3p2) ≥ δ(1− p3)x2 + p3p2δ.

By consistency, we have x2 ≥ δp2p31−δ(1−p3)

and similarly x3 ≥ δp2p31−δ(1−p2)

. Together with (2.6), it

requires that

xN ≥ p1(1 + (1− p1)δ) +δp2p3

1− δ(1− p3)+

δp2p3

1− δ(1− p2).

To see a contradiction, as δ converges to 1, observe that the right-hand side converges to

1+p1(1−p1), which is strictly greater than 1 as long as p1 > 0. However, xN never exceeds 1.

Thus, for a sufficiently high δ, the efficient strategy profile (x,q) cannot be an equilibrium.

Next, we construct an inefficient equilibrium. Let δ = max

p2(p1+p2)(1−p1)

, p3(p1+p3)(1−p1)

so that δ < 1. Consider a strategy profile (x,q) such that

• x1 = p11−(1−p1)δ

; x2 = x3 = 0; and

• q1(N) = q2(2) = q3(3) = 1,

and in any two-player subgame the active players follow the strategy according to Proposition

2.1. Since player 2 and player 3 decline to be a proposer in the initial state, the strategy

profile is inefficient. To see that (x,q) constructs an equilibrium for δ > δ, due to Lemma

2.2, it suffices to verify the following two conditions.

i) Optimality: Calculate each player’s excess surpluses. It is easy to see that e1(N,x) >

0 and ei(i,x) = 0 for all i ∈ N . For all i ∈ 1, 2, due to Proposition 2.1,

x(i,1,2)i = p1 + p2, and hence

ei(1, 2,x) = δ(p1 + p2)− δ(x1 + x2) = δ(p1 + p2)− δ(

p11−(1−p1)δ

+ 0)

= δ1−(1−p1)δ

(p2 − (p1 + p2)(1− p1)δ).

Then, δ > δ implies ei(1, 2,x) < 0. Similarly, we have ei(1, 3,x) < 0 for all

i ∈ 1, 3. Given x, therefore, D1 = N, D2 = 2, and D3 = 3.

ii) Consistency: Compute each player’s expected payoff:

62

• u1(x,q) = p1e(N,x) + δx1 = p1(1− δx1) + δx1 = p1 + (1− p1)δ(

p11−(1−p1)δ

)= p1

1−(1−p1)δ

• u2(x,q) = p2e(2,x) + δx2 = p2 · 0 + δ · 0 = 0

• u3(x,q) = p3e(3,x) + δx3 = p3 · 0 + δ · 0 = 0.

Therefore, ui(x,q) = xi for all i ∈ N .

Even if there are multiple dominating players, as see (b) in Figure 2.2, they can generate

an additional advantage by forming a coalition with other dominating players and splitting

non-dominating players into two isolated groups. In the next example, we construct an

equilibrium in a chordal network in which there are two dominating players.

Example 2.5 (A Chordal Network). Let g = (1, 2, 3, 4, 12, 23, 34, 41, 13) and p =

(14, 1

4, 1

4, 1

4). Suppose δ > δ ≈ 0.91.13 We construct an equilibrium (x,q) such that

• x1 = x3 = (6−δ)δ4(4−δ)(2−δ) ; x2 = x4 = (6−6δ+δ2)δ

4(4−δ)(2−δ) ;

• q1(1, 3) = q3(1, 3) = 1; q2(1, 2) = q2(2, 3) = q4(1, 4) = q4(3, 4) = 12.

In any subgame in which the number of active players is less than or equal to three, they fol-

lows the equilibrium strategies according to Proposition 2.1 and Example 2.4. Note that the

equilibrium welfare is xN = δ(3−δ)2(2−δ) . The equilibrium payoff vector converges to

(512, 1

12, 5

12, 1

12

)as δ → 1. Now we verify the equilibrium conditions.

i) Odd Players’ Optimality: Since δ > 34, Example 2.4 implies that x

(1,1,3)1 = p1+p3

1−(1−p1−p3)δ=

12−δ and x

(1,1,3)1 = x

(1,1,3)4 = 0. Given x, calculate player 1’s excess surpluses:

• e1(1, 2,x) = δx(1,1,2)1 − δ(x1 + x2) = δ(1−δ)(4−δ)

4(2−δ)

• e1(1, 3,x) = δx(1,1,3)1 − δ(x1 + x3) = δ(8−8δ+δ2)

2(2−δ)(4−δ)

• e1(1, 2, 4,x) = δx(1,1,2,4)1 − δ(x1 + x2 + x4) = δ(6−6δ+δ2)

2(4−δ)

• e1(N,x) = 1− δxN = (1−δ)(4+2δ−δ2)2(2−δ)

13Note that δ is a solution to δ(8− 8δ + δ2) = (4− δ)(1− δ)(4 + 2δ − δ2).

63

Given e1(S,x) for all S ⊆ N1, it is routine to see that δ > δ implies D1(x) = 1, 3.

Similarly, we also have D3(x) = 1, 3.

ii) Even Players’ Optimality: For any 2 ( S ⊆ N2, player 2’s S-formation induces a

complete network. Thus, given x, one can compute player 2’s excess surpluses:

• e2(1, 2,x) = e2(2, 3,x) = δx(2,1,2)2 − δ(x1 + x2) = δ(1−δ)(4−δ)

4(2−δ)

• e(1, 2, 3,x) = δx(2,1,2,3)2 − δ(x1 + x2 + x3) = δ(24−36δ+11δ2−δ3)

4(2−δ)(4−δ)

Observe that e2(1, 2,x) = e2(2, 3,x) > 0 for all δ; while e(1, 2, 3,x) is strictly

negative if δ > δ. Thus, for any δ > δ, we have D2(x) = 1, 2, 2, 3 and similarly

D4(x) = 1, 4, 3, 4.

iii) Consistency: Given (x,q), compute each players’ expected payoffs:

• u1(x,q) = p1e(1, 3,x) + δ[(p1 + p3 + 1

2(p2 + p4)

)x1 + 1

2(p2 + p4)p1

]= 1

4· δ(8−8δ+δ2)

2(2−δ)(4−δ) + δ[

34· (6−δ)δ

4(4−δ)(2−δ) + 12· 1

8

]= (6−δ)δ

4(4−δ)(2−δ) = x1

,

• u2(x,q) = p2e(1, 2,x) + δ [p2x2 + p4p2 + (p1 + p3) · 0]

= 14· δ(1−δ)(4−δ)

4(2−δ) + δ[

14· (6−6δ+δ2)δ

4(4−δ)(2−δ) + 14· 1

4

]= (6−6δ+δ2)δ

4(4−δ)(2−δ) = x2

,

and similarly u3(x,q) = x3 and u4(x,q) = x4, and hence consistency holds.

Remark (Braess-Like Paradox). Comparing between Example 2.3 and Example 2.5, we ob-

serve a negative welfare effect of adding a new communication link.14 In the four-player circle

with pi = 14

for all i ∈ N , the maximum welfare δ is achieved in an equilibrium. If we add a

link between player 1 and player 3 in the circular network, then it becomes a chordal network

as in Example 2.5. Since odd players can form a grand-coalition immediately, the maximum

welfare is 12(1 + δ), which is strictly greater than that of the circular network. However,

the equilibrium welfare in Example 2.5 is δ(3−δ)2(2−δ) which is strictly less than δ, which is the

14This question has been raised by Vijay Krishna.

64

1

2

3

4

(a) Bargaining in a Circular Network (See Example 2.3): It takes exactly 2 periods for agrand-coalition in any equilibrium. In the first period, any proposer forms a three-playercoalition by buying out two neighbors. Then the induced game is of two players.

1

2

3

4

(b) Bargaining in a Chordal Network (See Example 2.5): The expected periods for a grand-coalition is strictly greater than 2. In the first period, if the even players are selected asa proposer, then they choose one of the odd players as a bargaining partner to induce athree-player circle. In the circle, grand-coalition immediately forms. However, if the oddplayers are initially selected as a proposer, then they induce a three-player chain. In thechain, the leaf players decline to make an offer and hence an additional delay occurs withpositive probability.

Figure 2.3: A Braess-Like Paradox

maximum welfare in the circle. This observation is reminiscent of the Braess’s paradox. In

fact, this result does not depend on the initial recognition probability p, as long as p2 > 0

and p4 > 0.

One can observe the negative welfare effect of adding a new link by computing the

expected periods for a unanimous agreement. See Figure 2.3. In the circle, it takes exactly

2 periods for a grand-coalition in the equilibrium. Note that all the players fully use their

communication links whenever they are recognized as a proposer. In the chordal network,

65

however, the expected periods for a unanimous agreement is 2.5.15 If the even players are

recognized as a proposer in the first period, then they chooses one of the odd players as a

bargaining partner to induce a three-player circle. In the circle, grand-coalition immediately

forms. However, if the odd players are initially recognized as a proposer, they induces a

three-player chain. In the chain, then the leaf players decline to make an offer and hence an

additional delay occurs with positive probability.

Remark. In the random-proposer bargaining model, the equilibrium may not be unique even

in the class of stationary subgame perfect equilibria.16 However, the equilibrium constructed

in Example 2.3, Example 2.4, and Example 2.5 is unique in the class of symmetric cutoff-

strategy equilibria, in which identical players in terms of a position in a network and a

recognition probability play the identical cutoff strategy.

Now, we are ready to prove Proposition 2.3 in a case of D(g) 6= ∅. Since g is a pre-

complete, note that there exists j1 and j2 such that d(j1, j2; g) = 2. Let J1(g) = Nj1(g)\D(g),

J2(g) = Nj2(g) \D(g), and J(g) = J1(g) ∪ J2(g). Lemma 2.4 provides a lower bound of the

unique dominating player’s expected payoff.

Lemma 2.4. Let g be a pre-complete network with D(g) = i. If (x,q) is an equilibrium

of Γ = (g, p, δ), then

xi ≥ pi + pi(1− pi)δ. (2.7)

Proof. Step 1: Consider a three-person chain, that is, J1 = j1 and J2 = j2. Since

15 (p2 + p4)× 2 + (p1 + p3)[(p1 + p3)× 2 + (p2 + p4)

((p1 + p3)× 3 + (p2 + p4)

((p1 + p3)× 4 + · · ·

))]= 1

2 × 2 + 14 × 2 + 1

8 × 3 + 116 × 4 + · · ·

= 1 +

∞∑k=2

k1

2k= 2.5.

16To overcome multiplicity of equilibria, the uniqueness of equilibrium payoffs has been studied in therandom-proposer bargaining model. Eraslan (2002) shows the equilibrium payoff uniqueness for a weightedmajority game and Eraslan and McLennan (2013) generalizes this result to a general simple game using fixedpoint index theorem. Unfortunately, those results cannot be applied to the model in which a player has abuyout option, because a player can expect some partial payoff by forming an intermediate subcoalition andhence the actual characteristic function that the players play is not of a simple game. The uniqueness ofstationary equilibrium payoffs is conjectured in a broader class of characteristic function form games, but itstill remains as an open question. See Eraslan and McLennan (2013) for a discussion.

66

x(j1,J1)i = x

(j2,J2)i = xi and uN(x,q) ≤ u(Γ) = pi + δ(1− pi), player i’s expected payoff is

xi ≥ piei(N,x) +∑k∈N

pk∑S3i

qk(S)δxi + δ∑k∈N

pk∑S 63i

qk(S)xi

≥ pi(1− δ(pi + δ(1− pi))) + δxi.

Rearranging the terms, we have the desired result.

Step 2: As an induction hypothesis, assume that for any pre-complete network g′ with

D(g′) = i, ≤ |J1(g′)| ≤ a, and 1 ≤ |J2(g′)| ≤ b, x′i ≥ p′i + p′i(1 − p′i)δ. Now we consider

a pre-complete network g with D(g) = i, |J1(g)| = a, and |J1(g)| = b + 1. Player i’s

expected payoff is

xi ≥ piei(N,x) +∑k∈N

pk∑S3i

qk(S)δxi + δ∑k∈N

pk∑S 63i

qk(S)x(k,S)i . (2.8)

For any k ∈ N and S ⊆ N such that i 6∈ S, the induction hypothesis implies x(k,S)i ≥

pi + pi(1 − pi)δ. Suppose by way of contradiction that pi + pi(1 − pi)δ > xi. Then, (2.8)

can be written as xi ≥ pi (1− δ(pi + δ(1− pi))) + δxi, or equivalently, xi > pi + pi(1− pi)δ,

which yields a contradiction. Similarly, induction argument completes the proof.

Proof of Proposition 2.3 (Case 1: D(g) 6= ∅)

Take any j ∈ J1. Since (x,q) is efficient, we have (∀j′ ∈ J2)∑

S∈Cj′qj′(S) = 1 and (∀i ∈

D ∪ J1)∑

j∈S⊆N qi(S) = 1. Thus, player j’s payoff is

uj(x,q) = pjmj(x) + δ(pD + pJ1)xj + δ∑j′∈J2

pj′∑S⊆N

qj′(S)x(j′,S)j

≥ δ(pD + pJ1)xj + δpJ2pj,

which implies that xj ≥pjpJ2δ

1−(1−pJ2 )δ. Summing j over J1, we have xJ1 ≥

pJ1pJ2δ

1−(1−pJ2 )δ. Similarly

for J2, we have xJ2 ≥pJ1pJ2δ

1−(1−pJ1 )δ, and hence

xJ = xJ1 + xJ2 ≥ pJ1pJ2δ

(1

1− (1− pJ1)δ+

1

1− (1− pJ2)δ

)(2.9)

Now take any i ∈ D. Player i’s optimality implies ei(N,x) ≥ ei(D,x), or equivalently,

1 − δxN ≥ δx(i,D)i − δxD. Since g(i,D) has a single dominating player, Lemma 2.4 implies

67

x(i,D)i ≥ pD + pD(1− pD)δ and it follows that

1− pDδ(1 + δ − pDδ) ≥ δxJ (2.10)

By (2.9) and (2.10), we have

1− pDδ(1 + δ − pDδ) ≥ pJ1pJ2δ2

(1

1− (1− pJ1)δ+

1

1− (1− pJ2)δ

). (2.11)

As δ → 1, the right hand side of (2.11) converges to pJ ; while the left hand side converges

to p2J . Since pJ < 1, there exists δ < 1 such that the inequality (2.11) yields a contradiction

for δ > δ.

Case 2: No Dominating Player

Now we consider a network without a dominating player. See (c) in Figure 2.2 for instance.

Even in a case of that there is no dominating player, we will show that some players can be

a dominating player in the induced network by buying out only a part of their neighbors.

Before proving this, some lemmas are presented. First, whenever there is a dominating player

in an incomplete network, Lemma 2.5, Lemma 2.6, and Lemma 2.7 show dominating players

have some additional bargaining power compared to other non-dominating players. In a

network without a dominating player, Lemma 2.8 shows that each player’s payoff should

be strictly less than her recognition probability under any efficient equilibrium. For any

non-circular pre-complete network without a dominating player, Lemma 2.9 finds a player

who can be a dominating player avoiding a complete network. Combining those lemmas,

therefore, when an efficient equilibrium is assumed, at least one player can be strictly better

off by strategically delaying a unanimous agreement, which is a contradiction in turn.

Lemma 2.5. Let g be a pre-complete network with ∅ ( D(g) ( N(g) and (x,q) be an

equilibrium of (g, p, δ) with δ < 1. If i ∈ D(g) then xi > pi.

Proof. If |N(g)| = 3, due to Lemma 2.4, then xi ≥ pi + pi(1 − pi)δ > pi for any i ∈ D. As

an induction hypothesis, suppose the statement is true for any g′ with |N(g′)| < n. Now

consider g with |N(g)| = n. Take any i ∈ D(g). For any k ∈ N and any S such that

68

i 6∈ S, if g(k,S) is complete then x(k,S)i = pi; and if g(k,S) is incomplete then x

(k,S)i > pi

by the induction hypothesis. Thus, letting Qi =∑

k∈N pk(∑

S3i qi(S) + qk(k)), we have

xi ≥ pi(1− δxN) +Qiδxi + δ(1−Qi)pi, and hence xi ≥ pi + δ(1−δ)1−δQi pi > pi.

Lemma 2.5 says that for any dominating player, her expected payoff is strictly greater

than her recognition probability. However, we need a stronger result: the difference between

the expected payoff and the recognition probability is strictly positive even in the limit of that

the discount factor converges to one. Lemma 2.6 shows that there exists such a dominating

player and Lemma 2.7 proves it for all dominating players. For notational convenience,

denote ∆i = xi − pi and ∆(j,S)i = x

(j,S)i − p

(j,S)i . If g(i,S) is complete and non-trivial, by

Proposition 2.1, note that ei(S,x) = δ(x(i,S)i − xS) = δ(pS − xS) = −δ∆S.

Lemma 2.6. Let g be a pre-complete network with ∅ ( D(g) ( N(g) and (x,q) be an

equilibrium of (g, p, δ). There exists h ∈ D(g) such that

xh − ph ≥ph (pD(1− pD)δ2 − (1− δ))

1 + (|D| − 1)phδ.

Furthermore, limδ→1(xh − ph) ≥ phpD(1−pD)1+(|D|−1)ph

> 0

Proof. Take any h ∈ argmaxi∈N ∆i and let Qh =∑

i∈N∑

S3h piqi(S). For any i ∈ N and

S ⊆ N such that h 6∈ S, since h ∈ D(g(i,S)), Lemma 2.5 implies x(i,S)h ≥ ph, and hence we

have

xh ≥ pheh(D,x) +Qhδxh + δ(1−Qh)ph

≥ phpD(1− pD)δ2 − ph∆Dδ +Qhδ(ph + ∆h) + (1−Qh)δph

≥ phpD(1− pD)δ2 − ph|D|∆hδ + δph + ph∆hδ, (2.12)

where the second inequality is due to Lemma 2.4, which implies

eh(D,x) = δ(x

(h,D)h − xD

)≥ δ(pD + pD(1− pD)δ − xD) = pD(1− pD)δ2 −∆Dδ,

and the last inequality comes from ph ≤ Qh, ph ≥ xh, and ∆D ≤ |D|∆h. Subtracting ph

from both sides of (2.12), we have ∆h ≥ph(pD(1−pD)δ2−(1−δ))

1+(|D|−1)phδ, as desired. Since D ( N , it

must be pD < 1 and hence limδ→1 ∆h ≥ phpD(1−pD)1+(|D|−1)ph

> 0.

69

Lemma 2.7. Let g be a pre-complete network with ∅ ( D(g) ( N(g) and (x,q) be an

equilibrium of (g, p, δ). For any i ∈ D(g), there exists ∆i > 0 such that xi − pi ≥ ∆i as δ

converges to 1.

Proof. We will show that limδ→1 mini∈D ∆i > 0. Let L = argmini∈D ∆i. Since g is a

pre-complete, as before there exists j1 and j2 such that d(j1, j2; g) = 2, and let J1(g) =

Nj1(g) \ D(g), J2(g) = Nj2(g) \ D(g), and J(g) = J1(g) ∪ J2(g). Recall Lemma 2.5, which

implies (∀i ∈ D) ∆i > 0. Thus, for any j ∈ J1 and S ( N , if qj(S) > 0 then either S ⊆ J1

or S ∩D = ` for some ` ∈ L.

Case 1: Suppose |J1| = |J2| = 1. Then, for each j ∈ J , qj(j) +∑

`∈L qj(j, `) = 1,

and hence there exists ` ∈ L such that∑

j∈J pj (qj(j) + qj(j, `)) ≥ pJ|L| . Let Q` =∑

j∈J pj (qj(j) + qj(j, `)) +∑

i∈D∑

S3` piqi(S), then Q` ≥ pJ|L| + p`. Since x` ≥ p`e`(J ∪

`,x) +Q`δx` + (1−Q`)δp`, it follows

∆` ≥ δp`(∆` + ∆J) + δ

(pJ|L|

+ p`

)∆` − (1− δ)p`,

which implies ∆` ≥ −δp`∆J−(1−δ)p`1−δ pJ|L|

. Since xN − pN = ∆N < 0, we have −∆J ≥ ∆D ≥ ∆h.

Thus, by Lemma 2.6, we have the desired result,

limδ→1

∆` ≥ −|L|p`|L| − pJ

∆J ≥|L|p`|L| − pJ

∆h ≥p`phpD(1− pD)|L|

(|L| − pJ)(1 + (|D| − 1)ph)> 0.

Case 2: As an induction hypothesis, for any pre-complete network g′ with ∅ ( D(g′) (

N(g′) and 1 ≤ |J1(g′)| ≤ a and 1 ≤ |J2(g′)| ≤ b and any equilibrium (x′,q′) of (g′, p′, δ),

assume that limδ→1 mini∈D(g′)(x′i − p′i) > 0. Now we consider a pre-complete network g with

∅ ( D(g) ( N(g) and |J1(g)| = a and |J2(g)| = b+1. Due to the induction hypothesis, there

exists ∆′` > 0 such that ∆′` ≥ limδ→1(x(j,J ′)` − p`) for all α ∈ 1, 2, j ∈ Jα, and J ′ ⊆ Jα.

Then, we have

x` ≥ p`e`(J ∪ `,x) +

p` +∑

α∈1,2

∑j∈Jα

pj (qj(j) + qj(Jα ∪ `))

δ(p` + ∆`)

+

∑α∈1,2

∑j∈Jα

∑J ′⊆Jα

pjqj(J′)

δ(p` + ∆′`) + pD\`δp`. (2.13)

70

If limδ→1 ∆` ≥ ∆′`, then there is nothing to prove. Suppose that limδ→1 ∆` ≤ ∆′`. As δ → 1,

then (2.13) yields x` ≥ −p`δ∆J + δp` + (1 − pD)δ∆`, or equivalently, (1 − (1 − pD)δ)∆` ≥

−δp`∆J − (1− δ)p`. Take any h ∈ argmaxi∈D ∆i. Since −∆J > ∆D > ∆h, it follows that

(1− (1− pD)δ)∆` > δp`∆h − (1− δ)p`.

By Lemma 2.5, we have the desired result, limδ→1 ∆` ≥ p`ph(1−pD)1+(|D|−1)ph

> 0.

Lemma 2.8. Let g be a pre-complete network with D(g) = ∅. If (x,q) is an efficient

equilibrium of Γ = (g, p, δ), then for all i ∈ N , xi = δpi.

Proof. Since g is pre-complete and (x,q) is efficient, for all j ∈ N , qj(S) > 0 implies g(j,S)

is complete. Thus, each player i can expect pi in the next period by rejecting any offer.

Suppose player i gets an offer with yi < δ2pi. By rejecting yi, i can be strictly better since

the stationary strategy profile guarantees δpi in the next period. Hence, xi ≥ δpi for all

i ∈ N . If there exists i ∈ N such that xi > δpi, then it must be xN > δpN = δ, which is

infeasible.

Lemma 2.9. Let g be a pre-complete non-circular network with D(g) = ∅. There exist

i, j ∈ N(g) such that i ∈ D(g(i,i,j)) ( N(g(i,i,j)).

Proof. Since g is pre-complete non-circular, its complete covering number is 2. Let M be a

minimal complete cover of g. Since D(g) = ∅, M must be disjoint. Given i ∈ N , then let

Mi ∈ M such that i ∈ Mi. Since D(g) = ∅, for all k ∈ N , there exists at least one k′ ∈ M ck

such that kk′ 6∈ E(g), that is, it must be |M ck \ Nk(g)| ≥ 1. We will show that there exists

i ∈ N and j ∈M ci such that i ∈ D(g(i,i,j)) ( N(g(i,i,j)), by constructing such a pair of i and

j in the following two cases. First, suppose there exists k ∈ N such that |M ck \Nk(g)| ≥ 2.

Take i ∈ M ck \ Nk(g) and j ∈ M c

i with ij ∈ E(g). Take i′ ∈ M ck \ Nk(g) with i′ 6= i.

Since g|Miand g|Mc

iare complete, i ∈ D(g(i,i,j)). Since d(k, i′; g) = d(k, i′; g(i,i,j)) = 2,

k 6∈ N(g(i,i,j)), as desired. Second, suppose, for all k ∈ N , |M ck \ Nk(g)| = 1. Take any

i ∈ N and j ∈ M ci such that ij ∈ E(g). Take k ∈ Mi \ i and k′ ∈ M c

i such that

71

d(k, k′; g) = 2. Again we have i ∈ D(g(i,i,j)) and d(k, k′; g) = d(k, k′; g(i,i,j)) = 2, as

desired.

Proof of Proposition 2.3 (Case 2: D(g) = ∅)

Suppose (x,q) is an efficient equilibrium. Due to Lemma 2.8, for all i ∈ N and all S ∈ Ci,

ei(S,x) = δ(x

(i,S)i − xS

)= δ (pS − δpS) = δ(1− δ)pS,

which converges to 0 as δ → 1. By Lemma 2.9, there exists i, j ∈ N(g) such that i ∈

D(g(i,i,j)) and i, j 6∈ Ci. Due to Lemma 2.7, there exists ∆i such that x(i,i,j)i − p(i,i,j)

i ≥

∆i. By Lemma 2.8, then we have

ei(i, j,x) = δ(x

(i,i,j)i − (xi + xj)

)≥ δ

((p

(i,i,j)i + ∆i − δ(pi + pj)

)= δ∆i + δ(1− δ)(pi + pj).

As δ → 1, note that ei(i, j,x) ≥ ∆i > 0. Thus for a sufficiently high δ, ei(i, j,x) >

ei(S,x) for all S ∈ Ci, which contradicts to optimality of player i.

2.3.3 The Necessary Condition : Incomplete Networks

We have considered pre-complete non-circular networks. To complete the necessary condi-

tion, we have to allow any incomplete non-circular network. Proposition 2.4 implies that for

any game with an incomplete non-circular network, if the players play efficient strategies,

then a pre-complete non-circular network must be induced with positive probability.

Proposition 2.4. Let g be an incomplete network. For any efficient strategy profile (x,q),

there exists π ∈ Πx,q(g) such that gπ is a pre-complete network. In addition, if g is a non-

circular network, then there exists π ∈ Πx,q(g) such that gπ is a pre-complete non-circular

network.

Proof. Suppose that g is neither pre-complete nor complete. Now we construct a sequence of

coalition formations which is consistent with (x,q) and the sequence induces a pre-complete

72

network. Take i∗ ∈ argmaxi∈N(g) degi(g). Let I(g) = i ∈ N(g) | Ci(g) = ∅. Let g1 = g

and take i1 ∈ argmaxi∈I(g1) d(i, i∗; g1). Pick any S1 such that qi1(S1) > 0. Let g2 = g(i1,S1).

Similarly, pick i2 ∈ argmaxi∈I(g2) d(i, i∗; g2). Pick any S2 such that qi2(S2) > 0. Since (x,q)

is efficient, |S1| ≥ 2, |S2| ≥ 2, and so on; and I(g1) ) I(g2) ) · · · . Thus, one can repeat this

process until I(gT ) = ∅, after which gT is a pre-complete network. This proves the first part.

In addition, assume that g is not circular. If g is a tree, then any induced network cannot

be circular and hence gT is not circular. If g has a cycle but not a circular network, then

degi∗(g) = degi∗(gT ) ≥ 3, and hence gT cannot be circular.

2.4 Non-transferability of Recognition Probabilities

We have assumed that the initial recognition probabilities are transferable. That is, when

they trade their communication links, they also trade their chances of being a proposer

as well. In some other environment, however, players cannot trade their recognition prob-

abilities. With non-transferable recognition probabilities, instead of (2.1), we define the

recognition probabilities in any state π in the following way:

pπi =

pi∑

k∈Nπ pkif i ∈ Nπ

0 otherwise.(2.14)

With non-transferable recognition probabilities, obtaining an efficient equilibrium is im-

possible even in circular networks.

Theorem 2.2. With non-transferable recognition probabilities, an efficient equilibrium ex-

ists for all discount factors if and only if the underlying network is complete.

Before proving the theorem, we construct an inefficient equilibrium in a four-player cir-

cular network as an example.

Example 2.6 (A Circular Network with Non-Transferable Recognition Probabilities).

Let g = (1, 2, 3, 4, 12, 23, 34, 14) and p =(

14, 1

4, 1

4, 1

4

). If δ > 2

3, then there exists a cutoff

strategy equilibrium (x,q) such that

73

• x1 = x2 = δ3(2−δ) ; x3 = x4 = δ

6(2−δ) ;

• q1(1) = q2(2) = 1; q3(3, 4) = q4(3, 4) = 1.

That is, player 1 and player 2 decline to make an offer and wait for a three-player complete

network induced by player 3 or player 4.

i) Optimality: Since recognition probabilities are not transferable, note that x(1,1,2)1 =

x(1,1,4)1 = 1

3and x

(1,1,2,4)1 = 1

2. Given x, player 1’s excess surpluses are:

• e1(1, 2,x) = 13δ − δ(x1 + x2) = δ(2−3δ)

3(2−δ)

• e1(1, 4,x) = 13δ − δ(x1 + x4) = δ(1−2δ)

6(2−δ)

• e1(1, 2, 3,x) = 12δ − δ(x1 + x2 + x3) = δ(1−3δ)

6(2−δ) .

Given δ > 23, since e1(S,x) for all 1 ( S ⊆ N1, we have D1(x) = 1, which

implies q1(1) = 1. Similarly we have q2(2) = 1. Now calculate player 3’s excess

surpluses:

• e3(2, 3,x) = 13δ − δ(x2 + x3) = δ(2−3δ)

3(2−δ)

• e3(3, 4,x) = 13δ − δ(x3 + x4) = δ(1−δ)

3(2−δ)

• e3(2, 3, 4,x) = 12δ − δ(x2 + x3 + x4) = δ(2−3δ)

6(2−δ) .

Given δ > 23, since e3(3, 4,x) > 0 and e1(S,x) < 0 for any other 1 ⊆ S ⊆ N1, we

have q3(3, 4) = 1, and similarly q4(3, 4) = 1.

ii) Consistency: Given (x,q), calculate each player’s expected payoff:

• u1(x,q) = p1e1(1,x) + δ((p1 + p2)x1 + (p3 + p4)1

3

)= 1

4· 0 + δ

(12

δ3(2−δ) + 1

213

)= δ

3(2−δ) = x1,

• u3(x,q) = p3e3(3, 4,x) + δx3

= 14

(13δ − 2 · δ2

6(2−δ)

)+ δ2

6(2−δ) = δ6(2−δ) = x3,

and similarly, u2(x,q) = x2 and u4(x,q) = x4.

74

The equilibrium payoff vector converges to (13, 1

3, 1

6, 1

6). Furthermore, the equilibrium payoff

vector is not unique even as δ → 1. Note that there exists another class of equilibrium payoff

vectors which converge to (13, 1

3, 1

3, 0) or its permutations. However, there is no symmetric

equilibrium.

Now we prove Theorem 2.2. Since Proposition 2.1 still holds with non-transferable recog-

nition probabilities, this directly proves the sufficient condition. Proposition 2.5 shows that

an efficient equilibrium is impossible for any pre-complete network. Due to the first part of

Proposition 2.4, then, the necessary condition is proved for any incomplete network.

Proposition 2.5. Suppose that recognition probabilities are not transferable. Let g be a pre-

complete. For any p, there exists δ < 1 such that for all δ > δ, any efficient strategy profile

(x,q) cannot be an equilibrium in Γ = (g, p, δ).

Proof. Since g is incomplete, N(g)\D(g) 6= ∅. After renaming, let N(g)\D(g) = 1, · · · , L.

For each 1 ≤ ` ≤ L, take any S` such that q`(S`) > 0. Since (x,q) is efficient, S` ∈ C`. Due

to Proposition 2.1, we have x(`,S`)` = p

(`,S`)` . Since non-transferable recognition probabilities

are assumed, we have

x(`,S`)` = p

(`,S`)` =

p`∑j∈N\S` pj + p`

=∑j∈S`

pj −

(∑j∈N\S` pj

)(∑j∈S`\` pj

)∑

j∈N\S` pj + p`. (2.15)

In addition, since S`-formation is optimal for each `, it must be e`(S`,x) ≥ e`(`,x) = 0,

or equivalently,∑

j∈S` xj ≤ x(`,S`)` . Summing this over ` and plugging (2.15), we have

L∑`=1

∑j∈S`

xj ≤L∑`=1

∑j∈S`

pj − ρ, (2.16)

where

ρ =L∑`=1

(∑j∈N\S` pj

)(∑j∈S`\` pj

)∑

j∈N\S` pj + p`.

Note that ρ is strictly positive and does not depend on δ.

75

On the other hand, for any k’s S-formation such that qk(S) > 0 and any j 6∈ S, it must

be x(k,S)j = p

(k,S)j =

pj∑`∈N\S pk+pk

> pj. Then, for each j ∈ N , j’s continuation payoff is

uj(x,q) = pjmj(x) + δ∑k∈N

pk

(∑S3j

qk(S)xj +∑S 63j

qk(S)x(k,S)j

)

≥ δ∑k∈N

pk

(∑S3j

qk(S)xj +∑S 63j

qk(S)p(k,S)j

)> Qjxj + (1−Qj)pj.

where Qj =∑

k∈N pk∑

S3j qk(S). By the consistency condition, it follows that

xj >(1−Qj)δpj

1−Qjδ(2.17)

Combining (2.16) and (2.17), we have,

δL∑`=1

∑j∈S`

(1−Qj)pj1−Qjδ

+ ρ <L∑`=1

∑j∈S`

pj. (2.18)

As δ → 1, δ∑L

`=1

∑j∈S`

(1−Qj)pj1−Qjδ converges to

∑L`=1

∑j∈S` pj. However, this contradicts to

the fact that ρ > 0 is fixed. Thus, there exists δ < 1 such that for all δ > δ, the inequality

(2.18) is violated.

Remark. In network-restricted unanimity games, welfare can be improved by allowing for

players to trade their recognition probabilities. If the recognition probabilities are not trans-

ferable, a proposer has less incentive to form a coalition. In general characteristic function

form games, however, the effect of transferability of recognition probabilities on welfare may

be negative. For instance, when there exists a veto player, non-veto players may form a

union, to be a new veto coalition, as a transitional state, rather than immediately forming

an efficient coalition. If the recognition probabilities are transferable, then they have stronger

incentive to form transitional inefficient coalitions. Thus, for a non-unanimity game, ban-

ning players from trading recognition probabilities may improve welfare. See Chapter 1 for

details.

76

2.5 Proofs

Proof of i) in Proposition 2.1.

Case 1: |N(g)| = 2. Let N(g) = i, j and p = (pi, pj) with pi + pj = 1. We show that a

cutoff strategy profile (p1, p2, qi(N) = 1, qj(N) = 1) is an equilibrium by verifying player

i has no profitable deviation strategy given player j’s cutoff strategy. Note that player i’s

expected payoff from following her cutoff strategy is pi(1−δpj)+pj(δpi) = pi. First, consider

player i’s proposal strategy. Either making an offer with yj < δpj or declining to make an

offer yields an expected payoff δpi. Making an offer with yj > δpj is not profitable since the

offer yj = δpj will be accepted. Thus, player i cannot be better off by deviating from the

given proposal strategy. Next, consider player i’s response strategy. By rejecting any offer,

player i expects the payoff pi in the next period. Thus, rejecting any offer with yi < δpi is

optimal. It is clear that accepting any offer with yi ≥ δpi is optimal. Therefore, player i has

no profitable deviation strategy given player j’s cutoff strategy.

Case 2: |N(g)| > 2. Suppose that, for any game (g′, p′, δ) with |N(g′)| < |N(g)|, (p′, q′) is

an equilibrium, where p′ = p′πi i∈Nππ∈Π(g′) and q′ = qπi i∈Nππ∈Π(g′). Note that, in such

an equilibrium, for each i ∈ N(g′), player i’s expected payoff is p′i. We show that a cutoff

strategy profile σ = (p, q) is an equilibrium for (g, p, δ) by verifying player i has no profitable

deviation strategy given other players cutoff strategies. Recall that if player i follows the

cutoff strategy, then her expected payoff is pi(1 − δ) + δpi = pi. Since all the other players

except for i are supposed to play stationary strategies, it is enough to consider the proposal

strategy and the response strategy of player i separately.

• Proposal strategy: Consider player i’s proposal strategy qi such that qi(S) > 0 for some

S ( N instead of qi. By forming S ( N , player i expects p(i,S)i in the subsequent game,

because (g(i,S), p(i,S), δ) is a less-than-n-player game with a complete network. In order

for S to form, it must be yj ≥ δpj for all j ∈ S \ i. Note also that p(i,S)i ≤ pS.17

17With transferable recognition probabilities, it holds with equality. With non-transferable recognitionprobabilities, the inequality is strict.

77

Thus, player i’s proposal gain from S-formation is

δp(i,S)i −

∑j∈S\i

yj ≤ δpS −∑

j∈S\i

δpj = δpi. (2.19)

On the other hand, player i’s proposal gain from following qi is

1−∑

j∈N\i

δpj = (1− δ)pN + δpi = (1− δ) + δpi. (2.20)

Since (2.19) is strictly less than (2.20), any proposal strategy which forms S ( N is

not optimal for i. Among proposal strategies which form N , it is clear that making an

offer with y = δp is optimal.

• Response strategy: Since each j ∈ N \i is supposed to play the given cutoff strategy,

player i is guaranteed at least δpi by rejecting any offer. Thus, it is optimal for i to

accept any offer with yi ≥ δpi and to reject any offer with yi < δpi.

Proof of ii) in Proposition 2.1.

The statement is true for |N(g)| = 2 from the proof of Proposition 2.1. As an induction

hypothesis, suppose that the statement is true for any game with less-than-n-player games

and now consider a game Γ = (g, p, δ) with |N(g)| = n. Due to Lemma 2.1, only cutoff

strategy equilibria are considered. Suppose that there exists a cutoff strategy equilibrium

(x,q).

Case 1: Suppose that q = q. For each i ∈ N , since∑

k∈N pk∑

S⊆N qk(S)1(i ∈ S) = 1, we

have

ui(x, q) = pi (1− δxN) + δxi. (2.21)

Due to consistency, we obtain ui(x, q) = xi and xN = 1. Plugging them into (2.21), we have

xi = pi. Thus, for any cutoff equilibrium involving maximum coalition formation strategies

q yields a payoff vector p.

Case 2: Suppose that there exists i who plays a non-maximum coalition formation strategy

so that qi(S) > 0 with S ( N . This implies that

• xN = uN(x,q) < 1; and

78

• there exists S ( N such that i ∈ S and ei(S,x) ≥ ei(N,x).

Thus for each i ∈ S, we have

δx(i,S)i − δxS ≥ 1− δxN > 1− δ.

By the induction hypothesis, the inequality implies

δxS + 1 < δpS + δ. (2.22)

On the other hand, by letting Qj =∑

k∈N pk∑

S⊆N qk(S)1(j ∈ S), for each j ∈ S, we have

xj = uj(x,q) ≥ pj (1− δxN) + δ(Qjxj + (1−Qj)pj)

> pj (1− δ) + δ(Qjxj + (1−Qj)pj)

= pj + δQj(xj − pj). (2.23)

Rearranging the terms, (2.23) yields xj > pj for all j ∈ S. However, this contradicts to

(2.22) for all δ.

Proof of Proposition 2.2.

Define η(g) = b|N(g)|/2c − 1. If g is circular and η(g) = 0, then g must be a three-player

circle, which is complete. Proposition 2.1 proves this case. As an induction hypothesis,

suppose that, for all circular network g′ such that η(g′) < m, a cutoff strategy profile (x′, q′)

is an equilibrium for (g′, p′, δ), where x′ = δη(g′π)p′πi i∈N(g′π)π∈Π(g′). Now we show that

a cutoff strategy profile (x, q) is an equilibrium for (g, p, δ) with a circular network g and

η(g) = m, where x = δη(gπ)pπi i∈N(gπ)π∈Π(g). Take any i ∈ N and let Ni = i, j, k. We

verify the equilibrium conditions for player i.

i) Optimality: After i’s maximum coalition formation, the active players face a game

with a circular network g′ and η(g′) = m− 1. Due to the induction hypothesis, since

79

x(i,i,j,k)i = δm−1(pi + pj + pk), we have

ei(i, j, k,x) = δm(pi + pj + pk)− δ(xi + xj + xk)

= δm(pi + pj + pk)− δ(δmpi + δmpj + δmpk)

= δm(1− δ)(pi + pj + pk). (2.24)

Suppose i decline to make an offer, that is i forms i. Since ei(i,x) = 0 is strictly

less than (2.24), i’s i-formation is not optimal. Suppose i forms i, j. Note that

x(i,i,j)i =

δm−1(pi + pj) if |N(g)| is even,

δm(pi + pj) if |N(g)| is odd.

Thus, we have

ei(i, j,x) ≤ δm(pi + pj)− δ(xi + xj) = δm(1− δ)(pi + pj),

which is strictly less than (2.24), and hence i’s S-formation with |S| = 2 is not optimal.

ii) Consistency: Since all the players play maximum coalition formation strategies,

player i’s continuation payoff is:

ui(x,q) = piei(i, j, k,x) + δ

(pi + pj + pk)xi +∑

`∈N\i,j,k

p`x(`,N`)i

= piδ

m(1− δ)(pi + pj + pk) + δ(pi + pj + pk)xi + δ(1− (pi + pj + pk))δm−1pi.

Since ui(x,q) = xi, rearranging the terms, we have

(1− δ(pi + pj + pk))xi = piδm(1− δ)(pi + pj + pk) + (1− (pi + pj + pk))δ

mpi,

which yields xi = δmpi.

80

Chapter 3

Mechanisms with Referrals

3.1 Introduction

Referral programs are prevalent phenomena in various markets. In labor markets, the im-

portance of referral programs in obtaining job information has been discussed extensively.

For example, Myers and Shultz (1951) showed that 62% of textile workers found their first

job through their friends or relatives; while 23% used a direct application and only 2%

were hired via advertisements. Granovetter (1995) found that 65.4% of workers were hired

through social networks in Newton, MA.

In addition to job referrals, which are classical examples of referral programs, word of

mouth is increasingly considered an important marketing strategy to acquire customers

(Rosen, 2002). From the market of contact lenses and pet supplies to the automobiles,

housing, and financial markets, the practice of rewarded referral is widespread (Schmitt

et al., 2011). Some recent case studies show that the referred customers are more profitable

and more royal. In a study of a German bank, which paid customers 25 euro for brining in a

new customer, Schmitt et al. (2011) estimated that the average value of a referred customer

is 16% to 25% higher than that of nonreferred customer. They said that the return on such

rewards is 60% over a six-year span.

Due to developing internet technologies, customer referral programs have become much

more popular. Referral programs with social media tools or online activities increase the

consumers reach beyond strong and face-to-face networks. For example, Dropbox, a web-

81

based file hosting service, increased customers’ signups by 60% with the referral program,

rewarding more space to both existing and referred customers; users had sent 2.8 million

direct referral invites to their friends. On the other hand, a huge number of customer

referral websites are widely prevalent, in which registered retailers offer customers discount

coupons or money back for referring a new customer.

Referral programs are widely used, but at the same time, many practitioners worry about

incentive problems. First, existing agents may either conceal other potential agents’ contact

information or not refer their friends, especially, in cases that referring increases the level of

competition or it requires a cost. Ryu and Feick (2007) reported that rewards increase referral

likelihood particularly to weak ties and for weak brands. This shows that agents usually do

not fully reveal their knowledge of other potential agents’ contacts without rewards.

Even though the principal provides existing agents some rewards for referring a new

agents, there still remains another kind of incentive problems. Rewards create the potential

for abuse in which existing agents tend to refer low-quality agents just to get referral rewards

(Schmitt et al., 2011). A job referring experiment shows that social networks yields incentives

to refer less qualified workers. Thus many marketing researchers recommended that the

principal must design reward incentives in order to effectively use existing agents to overcome

their screening problem.

In this paper, we adopt a mechanism design approach to find efficient and/or feasible

allocations with asymmetric networks, in which each agent’s social network is private infor-

mation. In such asymmetric networks, agents strategically refer other agents, and hence the

participants are determined endogenously; while most of the mechanism design literature

assumes that the participants are fixed and commonly known.

In order to model asymmetric networks, we introduce a connection type for each agent in

addition to a payoff type. A connection type represents the set of referable (or contactable)

agents and a payoff type determines the agent’s preferences over allocations.1 To highlight

1A connection type is a kind of a knowledge type. In classical mechanism design literature, an agent’s typeusually refers a payoff type. In the recent robust mechanism design theory, a knowledge type is introducedto represent beliefs about others’ payoff types.

82

the role of contact information, we assume that each agent’s preference does not depend on

other agents’ types and his own connection type as well.2

For an efficient mechanism in asymmetric network environments, we generalize a VCG

mechanism, which was introduced by Vickrey (1961), Clarke (1971), and Groves (1973). A

VCG mechanism is efficient, but neither necessarily incentive compatible nor individually

rational. We show that a VCG mechanism is incentive compatible and individually rational

if a social worth increases as the population increases.3 Under this monotonicity condition,

agents truthfully report their payoff types and their connection types as well. However, a

VCG mechanism generically causes a budget problem: it usually requires excessive amount

of subsidies.

In order to overcome the budget problem of a VCG mechanism, we propose an alternative

mechanism, namely a multilevel mechanism. Under this mechanism, each agent’s payment

is based not only on his own marginal contribution but also on that of other agents who are

contacted to the principal only through him; and he is compensated with certain benefit from

them. Under a multilevel mechanism, we show that fully revealing one’s contact information

is a dominant strategy and agents have no incentive to under-report their payoff types if the

social welfare function is monotone and submodular.

We provide two applications, a single-item auction and a public good provision game.

In standard auctions, for any agent, revealing his connection type is a strictly dominated

strategy, since revealing increases competition and there is no compensation on it. A VCG

mechanism, which is different from a second-price auction in these environments, is efficient

but it runs deficit. We show that a multilevel mechanism guarantees a revenue higher than

or equal to any standard auction. In a public good provision game, a principal can achieve

an efficient and budget-balanced allocation by strategically contacting only one agent and

letting the agent contact the remaining agents. This contrasts the existing result that there

2Weakening these independence assumptions, especially in selling environments, are important and chal-lenging issues. There is a growing body of research on collusion and speculation in auction literature; Garrattand Troger (2006), Zheng (2002), Haile (2003), Calzolari and Pavan (2006), Hafalir and Krishna (2008), andZhang and Wang (2011) are some examples.

3This condition has been ignored in the literature, since the population has been assumed to be fixed.

83

is no budget-balanced mechanism which satisfies both incentive compatibility and individual

rationality.

The paper is organized as follows. Section 2 describes a model with asymmetric contact

information and strategic contacting. In Section 3, we show that a VCG mechanism satisfies

both incentive compatibility and individual rationality under a mild condition, but it runs

deficit in general. In Section 4, we propose a multilevel mechanism, which guarantees a

budget surplus, and characterize conditions for incentive compatibility and individual ra-

tionality. In Section 5, we apply our main results to a public good provision game and a

single-item auction.

3.2 A model

Let N = 1, 2, · · · , n be a set of agents. Each agent i ∈ N has two-fold private information:

a payoff type ti ∈ Ti ⊆ R and a connection type ei ∈ Ei ⊆ 2N\i. Let 0 be a mechanism

designer, whose connection type is e0 ∈ E0 ⊆ 2N\i. We denote that N0 = N ∪ 0, T =

×i∈NTi, and E = ×i∈N0Ei. When there is no danger of confusion, we denote t′ = (t′i, t−i)

whereas t = (ti, t−i). A typical element e ∈ E is a connection profile.

Let Γ be a set of allocations. For any S ⊆ N , Γ(S) is a set of allocations restricted on S.

Given ti ∈ Ti, an agent i’s preference over allocations is represented by an utility function,

vi(γ|ti) for all γ ∈ Γ. Note that an agent’s preference is independent on both others’ private

information and his connection type. We assume that, for all i ∈ N , all ti ∈ Ti, and all

γ ∈ Γ, vi(γ|ti) is finite.

Given a reported connection profile e ∈ E, the set of participants N(e) is determined

endogenously,

N(e) = e0 ∪ [∪i∈e0ei] ∪[∪i∈[∪j∈e0ej]ei

]∪ · · · .

A connection profile e is trivial if N(e) = e0 = N .

A mechanism (g, P ) is a pair of an allocation rule g : T × E → Γ and a payment rule

84

P : T × E → Rn satisfying the following three conditions:

i) g(t, e) ∈ Γ(N(e)) for all (t, e) ∈ T × E;

ii) Pi(t, e) = 0 for all (t, e) ∈ T × E and all i ∈ N \N(e); and

iii) tN(e) = t′N(e) implies g(t, e) = g(t′, e) for all e ∈ E.

Agents are asked to report their payoff types and connection types. An agent whose private

information is (ti, ei) may report any t′i ∈ Ti and any e′i ⊆ ei.4 Under a mechanism (g, P ),

if agents report t′ ∈ T and e′ ∈ E, then the agent i whose payoff type is ti gets the payoff

vi(g(t′, e′)|ti)− Pi(t′, e′).

A mechanism (g, P ) is incentive compatible in dominant strategies, if for all (t, e) ∈ T×E,

all i ∈ N(e), all t′i ∈ T−i, and all e′i ⊆ ei,

vi(g(t, e)|ti)− Pi(t, e) ≥ vi(g(t′i, t−i, e′i, e−i)|ti)− Pi(t′i, t−i, e′i, e−i).

An agent’s utility from not participating is normalized to zero. A mechanism (g, P ) is

individually rational, if for all (t, e) ∈ T × E and all i ∈ N(e),

vi(g(t, e)|ti)− Pi(t, e) ≥ 0.

Given a mechanism (g, P ) and a profile of private information (t, e) ∈ T × E, the sum

of agents’ payments∑

i∈N(e) Pi(t, e) is the ex-post surplus to the mechanism designer. A

mechanism (g, P ) runs deficit for (t, e) ∈ T ×E, if the surplus to the mechanism designer is

negative.

For all t ∈ T and all S ⊆ N , a worth function5 is defined by

W (t, S) = maxγ∈Γ(S)

∑i∈S

vi(γ|ti).

4Reporting a connection type e′i ⊆ ei to the mechanism designer can be interpreted as referring a subsete′i among accessible agents ei.

5For simplicity, we assume that no external cost is required for each allocation γ ∈ Γ. One can define aworth function including a cost if it is required.

85

Given t ∈ T and S ⊆ N , a set of efficient allocations is

Γ∗(t, S) ≡

γ ∈ Γ(S)

∣∣∣ ∑i∈S

vi(γ|ti) = W (t, S)

.

An allocation rule g is efficient if, for all (t, e) ∈ T × E, g(t, e) ∈ Γ∗(t, N(e)).

3.3 A VCG mechanism

In this section, we generalize a VCG mechanism in asymmetric contact information. In

complete contact information environments, in which the mechanism designer knows all the

agents directly so that e0 = N(e), any referral does not effect on the allocation and the

payment at all. In this trivial connection profile case, a VCG mechanism consists of an

efficient allocation rule g and a payment rule P such that

Pi(t, e) = W (t, N(e) \ i)−∑

j∈N(e)\i

vj(g(t, e)|tj), (3.1)

which represents an agent i’s marginal effect on the remaining agents, and in fact, it does

not depend on ei. It can be easily shown that revealing agents true types is a dominant

strategy and the payment rule implements the efficient allocation rule as long as e0 = N(e).

Now consider non-trivial cases. An agent affects the economy in two ways: a change in a

allocation due to a payoff type within a fixed participants and a change in participants due

to referrals. However, in cases that referrals increases the participants, that is, e0 ( N(e),

the above payment rule does not guarantee fully referring. Hence, we should take the effect

of referrals into account.

3.3.1 Contribution on New Participants

Given e ∈ E and i ∈ N , the set of i’s outsiders, Oi(e), consists of agents who can be involved

without i, that is Oi(e) ≡ N(e−i)\i. Note that e is trivial if and only if Oi(e) = N(e)\i

for all i ∈ N . The agent i’s group, Gi(e), consists of agents who can not participate without

i, that is Gi(e) ≡ N(e) \Oi(e). Note that e is trivial if and only if Gi(e) = i for all i ∈ N .

86

0

1 2

3 4 5

(a) A connection profile e

0

1 2

3 4 5

(b) The contribution tree induced by e

Figure 3.1: A Connection Profile and the Induced Contribution Tree.

Given e ∈ E, i is a predecessor of j if j ∈ Gi(e). Note that N(e) endowed with this

precedence relation is a tree and we call it a contribution tree. For all i ∈ N , the set of i’s

predecessors is Ri(e) ≡ j ∈ N | i ∈ Gj(e) and the set of i’s referred agents is Fi(e) ≡

ei \ Oi(e). An agent i’s height is the number of predecessors, that is, hi(e) ≡ |Ri(e)|. For

convenience, let R0(e) ≡ ∅ and h0(e) ≡ 0 for all e ∈ E. Let Nd(e) ≡ j ∈ N(e) | hj(e) = d

for each 1 ≤ d ≤ maxi∈N hi(e). Note that Nd(e)1≤d≤maxi∈N hi(e) is a partition of N(e).

A connection profile e ∈ E represented by a directed graph on N , which possibly con-

tains cycles; however, the induced contribution tree Gi(e)i∈N has no cycle. The following

example illustrates the relation between a connection profile and the induced contribution

tree.

Example 3.1. Let N = 1, 2, 3, 4, 5. Let e = eii∈N0 with e0 = 1, 2, e1 = 3, 4,

e2 = 4, e3 = e4 = ∅, and e5 = 2. We have that N(e) = 1, 2, 3, 4 and the induced

contribution tree consists of G0(e) = 0, 1, 2, 3, 4, G1(e) = 1, 3, G2(e) = 2, G3(e) =

3,G4(e) = 4, and G5(e) = ∅. That is, the agent 5 cannot be involved in this connection

profile. Figure 3.1 represents the connection profile e and the induced contribution tree.

Note that 4 6= e0 but 4 ∈ N1(e).

87

3.3.2 Characterizations of a VCG mechanism

We are ready to define a VCG mechanism in asymmetric contact information.

Definition 3.1 (VCG Mechanism). A VCG mechanism consists of an efficient allocation

rule g and a payment rule P V such that

P Vi (t, e) = W (t, Oi(e))−

∑j∈N(e)\i

vj(g(t, e)|tj). (3.2)

The VCG payment can be decomposed in two parts:

P Vi (t, e) =

[W (t, N(e) \ i)−

∑j∈N(e)\i

vj(g(t, e)|tj)]−[W (t, N(e) \ i)−W (t, Oi(e))

].

The first part is the usual VCG payment in the fixed participants N(e), which is exactly

the same with (3.1); and the second part is a reward for effective referrals. If a connection

profile e is trivial, then the second term is zero.

It is well-known that a VCG mechanism satisfies both incentive compatibility and individ-

ual rationality in a fixed population setting. In asymmetric network environments, however,

a VCG mechanism is neither necessarily incentive compatible nor individually rational when

we allow agents to have asymmetric contact information. We show that a VCG mechanism

is incentive compatible and individually rational in dominant strategies if a worth function

W increases as the population increases.

• W is monotone if for all S ′ ⊂ S and all t ∈ T , W (t, S ′) ≤ W (t, S).

Under this monotonicity condition, agents truthfully report their payoff types and their

connection types as well.

Proposition 3.1. If W is monotone, then a VCG Mechanism is incentive compatible and

individually rational. Furthermore, if a VCG Mechanism is incentive compatible, then W is

monotone.

Proof. In Step 1, we show that the monotonicity W is a necessary and sufficient condition

for a VCG mechanism to be incentive compatible. In Step 2, we show that the monotonicity

W is a sufficient condition for a VCG mechanism to be individually rational.

88

Step 1 : First, suppose that W is monotone. Given t−i and e−i, consider an agent i whose

private information is ti ∈ Ti and ei ∈ Ei. Reporting t′i ∈ Ti and e′i ⊆ ei, the agent i’s payoff

is

vi(g(t′, e′)|ti)− P Vi (t′, e′) = vi(g(t′, e′)|ti)−

W (t′, Oi(e′))−

∑j∈N(e′)\i

vj(g(t′, e′)|tj)

=

vi(g(t′, e′)|ti) +∑

j∈N(e′)\i

vj(g(t′, e′)|tj)

−W (t, Oi(e))

≤ W (t, N(e′))−W (t, Oi(e))

≤ W (t, N(e))−W (t, Oi(e)).

The second equality is from independence of private information, that is, W (t′, Oi(e′)) =

W (t, Oi(e)). The third line is from the fact that vi(g(t′, e′)|ti) +∑

j∈N(e′)\i vj(g(t′, e′)|tj) is

maximized at t′i = ti. The last inequality is from the monotonicity of W and the condition

e′i ⊆ ei. Thus reporting ti and ei is a dominant strategy for the agent i.

Conversely, suppose that the VCG mechanism is incentive compatible but W is not

monotone. There exist t ∈ T and S, S ′ ⊂ N such that S∩S ′ = ∅ and W (t, S) > W (t, S∪S ′).

Since either S or S ′ must be nonempty, let S 6= ∅ and choose k ∈ S. Consider a connection

profile e such that e0 = S, ek = S ′, and ei = ∅ for all i 6= 0, k. Let e′k = ∅. Since

N(e) = S ∪ S ′, N(e′) = S, and Ok(e) = Ok(e′), incentive compatibility implies that

W (t, S ∪ S ′) ≥ W (t, S),

which contradicts to monotonicity.

Step 2 : Suppose that W is monotone. Since reporting one’s private information truth-

fully is a dominant strategy from Step 1, it suffices to show, for all (t, e) ∈ T × E and

all i ∈ N , vi(g(t, e)|ti) − P Vi (t, e) ≥ 0 for individual rationality. Definition of W yields

that vi(g(t, e)|ti) − P Vi (t, e) = W (t, N(e)) −W (t, Oi(e)). However, by monotonicity of W ,

W (t, N(e))−W (t, Oi(e)) is nonnegative.

89

Next, we observe a mechanism designer’s budget problem. Unfortunately, a VCG mech-

anism is very hard to provide a positive surplus to the mechanism designer; it runs deficit

as long as there are gains from cooperation. The following propositionosition shows that if

the worth of the economy is greater than the sum of each agent’s per capita worth then a

VCG mechanism runs deficit.

Proposition 3.2. Given (t, e) ∈ T × E, a VCG mechanism (g, P V ) yields nonnegative

surplus if and only if

W (t, N(e)) ≤∑i∈N(e)

W (t, Oi(e))

|N(e)| − 1.

Proof. We have that

∑i∈N(e)

P Vi (t, e) =

∑i∈N(e)

W (t, Oi(e))−∑

j∈N(e)\i

vj(g(t, e)|tj)

=

∑i∈N(e)

W (t, Oi(e))−∑

j∈N(e)\i

vj(g(t, e)|tj)− vi(g(t, e)|ti)

+∑i∈N(e)

vi(g(t, e)|ti)

=∑i∈N(e)

[W (t, Oi(e))−W (t, N(e))] +W (t, N(e)) (3.3)

=∑i∈N(e)

W (t, Oi(e))− (|N(e)| − 1)W (t, N(e)).

Thus, we have∑

i∈N(e) PVi (t, e) ≥ 0 if and only if W (t, N(e)) ≤

∑i∈N(e)

W (t,Oi(e))|N(e)|−1

, as desired.

The following corollary provides a more specific condition for a VCG mechanism to runs

deficit. Roughly speaking, if an agent substantially increases a worth by bring some other

agent, then it runs deficit.

Corollary 3.1. Suppose that W is monotone. Given (t, e) ∈ T×E, if there exist j, k ∈ N(e)

such that

W (t, N(e)) > W (t, Oj(e)) +W (t, Ok(e)), (3.4)

then a VCG mechanism runs deficit.

90

Proof. Plugging the condition (3.4) into that (3.3), we have that

∑i∈N(e)

P Vi (t, e) <

∑i∈N(e)\j,k

[W (t, Oi(e))−W (t, N(e))] ≤ 0,

where the second inequality is due to monotonicity.

Example 3.2. A seller allocates an indivisible good via a VCG mechanism to two existing

agents, say X and Y , and their valuations to the item are x and y, respectively. Suppose

X can refer another potential agent, say Z, whose valuation is z. A VCG mechanism runs

deficit if and only if maxx, y, z > y + maxx, y.

Next corollary investigates the relation between the budget problem of a VCG mechanism

and the existence of core of the underlying environment. Given (t, e) ∈ T × E, define a

network-restricted per-capita game6 (N(e), vt,e), where

vt,e(S) =

W (t,N(e))|N(e)| if S = N(e)

W (t,OT (e))|N(e)|−|T | if S = N \ T0 otherwise.

If the connection profile e is trivial, then the network-restricted per-capita game is equivalent

to the per-capita game.

Corollary 3.2. Given (t, e) ∈ T × E, if a VCG mechanism runs strict surplus, then the

core of (N(e), vt,e) is empty.

Proof. Suppose that the core of (N(e), vt,e) is nonempty. Due to Bondareva (1963) and

Shapley (1967), for all probability measure δ on 2N(e) \ ∅, N(e),

∑S(N(e)

δ(S)vt,e(S) ≤ vt,e(N(e)) =W (t, N(e))

|N(e)|. (3.5)

Now define δ by:

δ(S) =

1

|N(e)| if S = N(e) \ i and i ∈ N(e)

0 otherwise.

6The notion of network-restricted games is proposed by Myerson (1977).

91

With this weight δ, the left hand side of (3.5) becomes

∑S(N(e)

δ(S)vt,e(S) =∑i∈N(e)

1

|N(e)|W (t, Oi(e))

|N(e)| − 1. (3.6)

By (3.5), (3.6), and Proposition 3.2, it contradicts that a VCG mechanism runs strict surplus.

We close this section with stating two impossibility results. First, no mechanism is

incentive compatible, individually rational, and budget surplus at the same time. Second,

any budget surplus mechanism yields an inefficient outcome with positive probability.

3.4 A Multilevel mechanism

This section introduce an alternative mechanism, a multilevel mechanism, which guarantees

a nonnegative surplus to the mechanism designer. As we have seen in the previous sec-

tion, under a VCG mechanism, the mechanism designer provides too much compensation

to agents who contributes to new agents’ participations, and hence it runs deficit for many

environments. Under a multilevel mechanism, an agent is compensated with an appropriate

transfer not from the mechanism designer but from his referred agents, who would not be

able to participate without him.

Definition 3.2 (Multilevel Mechanism). A multilevel mechanism consists of an efficient

allocation rule g and a payment rule PM which satisfies, for all i ∈ N(e),

PMi (t, e) = PG

i (t, e)−∑

j∈Fi(e)

PGj (t, e),

where PGi (t, e) = W (t, Oi(e))−

∑j∈Oi(e)

vj(g(t, e)|tj).

Remark. Note that PGi (t, e) is the marginal effect of the agent i’s group on the economy.

This is different from the VCG payment P Vi (t, e), which is the marginal effect of i alone.

Under a multilevel mechanism, an agent i pays PGi (t, e) to i’s immediate predecessor. Only

the agents in N1(e) pay to the mechanism designer. That is, each agent is responsible for

92

his group when he pays to his predecessor, but he is compensated with some transfer from

his referred agents.

Following lemma shows that the mechanism designer’s net surplus equals to the sum of

the marginal effects of the agents’ group, whose height is 1.

Lemma 3.1. For all (t, e) ∈ T × E, we have∑i∈N(e)

PMi (t, e) =

∑i∈N1(e)

PGi (t, e). (3.7)

Proof. Since PMi (t, e) = PG

i (t, e)−∑

j∈Fi(e) PGj (t, e), we have that∑

i∈N(e)

PMi (t, e) =

∑i∈N(e)

PGi (t, e)−

∑i∈N(e)

∑j∈Fi(e)

PGj (t, e). (3.8)

Let D = maxi∈N hi(e). Since Nd(e)1≤d≤D is a partition of N(e), the last term of (3.8) can

be written by: ∑i∈N(e)

∑j∈Fi(e)

PGj (t, e) =

D∑d=1

∑i∈Nd(e)

∑j∈Fi(e)

PGj (t, e)

=D∑d=2

∑i∈Nd(e)

PGj (t, e), (3.9)

where the second equality follows from the two facts; Fi(e)i∈Nd(e) is a partition of Nd+1(e)

for all 1 ≤ d ≤ D − 1, and Fi(e) = ∅ for all i ∈ ND(e). Plugging (3.9) into (3.8), we have

the desired result.

The following proposition shows that a multilevel mechanism guarantees a nonnegative

revenue.

Proposition 3.3. A multilevel mechanism is budget surplus

Proof. Take any (t, e) ∈ T × E. By Lemma 3.1, it suffices to show∑

i∈N1(e) PGi (t, e) ≥ 0.

By the definition of W , for all i ∈ N(e), we have that

PGi (t, e) = W (t, Oi(e))−

∑j∈Oi(e)

vj(g(t, e)|tj) ≥ 0. (3.10)

Thus,∑

i∈N PMi (t, e) ≥

∑i∈N(e) P

Mi (t, e) ≥ 0, as desired.

93

Our main results related to agents’ incentives about a multilevel mechanism depend on

a submodular and competitive structure.

• W is submodular if for all S, S ′ ⊆ N and all t ∈ T

W (t, S) +W (t, S ′) ≥ W (t, S ∪ S ′) +W (t, S ∩ S ′).

• W is submodular in t if, for all t, t′ ∈ T and all S ⊆ N ,

W (t, S) +W (t′, S) ≥ W (t ∨ t′, S) +W (t ∧ t′, S),

where t ∨ t′ = (maxti, t′i)i∈N and t ∧ t′ = (minti, t′i)i∈N .

• vi is submodular if ti ≤ t′i and vi(γ|ti) ≤ vi(γ′|ti) implies, for all ti, t

′i ∈ Ti and all

γ, γ′ ∈ Γ,

vi(γ′|ti)− vi(γ|ti) ≥ vi(γ

′|t′i)− vi(γ|t′i).

• An environment (v,Γ) is competitive if, for all S, S ′ ∈ N , all t ∈ T , all γ ∈ Γ∗(t, S),

and all γ′ ∈ Γ∗(t, S ′),

S ′ ⊆ S =⇒ vi(γ|ti ) ≤ vi(γ′|ti ) ∀i ∈ S ′, ∀ti ∈ Ti.

The following lemma characterizes submodularity of a general set function and it will be

potentially useful for other applications.

Lemma 3.2. A set function W defined on 2N is submodular if and only if, for all S, S ′ ⊆ N

such that S ′ ⊆ S and all partition P of S \ S ′,

W (S)−W (S ′) ≥∑P∈P

[W (S)−W (S \ P )] . (3.11)

Proof. First, we prove the ‘only-if’ part by mathematical induction. Let S be an arbitrary

subset of N . If S ′ = S, then the claim is obvious. Suppose that S \ S ′ is singleton, say

S \ S ′ = k. Then the claim (3.11) trivially holds. As induction hypothesis, suppose that,

for all S ′ ( S such that |S \ S ′| ≤ l, the statement (3.11) is true. Now consider a subset

94

S ′ ( S such that |S \ S ′| = l + 1 and a partition P of S \ S ′. Pick any k ∈ S \ S ′ and

let P1 ∈ P such that k ∈ P1. Define a new partition P of S \ (S ′ ∪ k) by replacing P1

with P1 ≡ P1 \ k, that is, P = (P \ P1) ∪ P1. Note that P1 is possibly empty. Since

|S \ (S ′ ∪ k)| ≤ l, we have that

W (S)−W (S ′ ∪ k) ≥∑P∈P

[W (S)−W (S \ P )] .

Subtracting W (S ′)−W (S ′ ∪ k) from the both side, we have that

W (S)−W (S ′) ≥∑P∈P

[W (S)−W (S \ P )]− [W (S ′)−W (S ′ ∪ k)]

=∑

P∈P\P1

[W (S)−W (S \ P )]

+W (S)−W (S \ P1)− [W (S ′)−W (S ′ ∪ k)]. (3.12)

On the other hand, by submodularity, we have

W (S ′ ∪ k) +W (S \ P1) ≥ W (S ′) +W (S \ P1). (3.13)

Plugging (3.13) into (3.12), we have

W (S)−W (S ′) ≥∑P∈P

[W (S)−W (S \ P )] ,

which implies that the claim (3.11) holds for S ′ ⊂ S such that |S \ S ′| = l+ 1 and the given

P . Since we have chosen S ′ and P arbitrarily, we get the desired conclusion.

Next, we prove the ‘if’ part. Pick any U, V ⊆ N . Let S ′ = U ∩ V , S = U ∪ V , and

P = U \ S ′, V \ S ′. The condition (3.11) implies that

W (S)−W (S ′) ≥ [W (S)−W (V )] + [W (S)−W (U)],

or equivalently, W (U) +W (V ) ≥ W (S) +W (S ′), as desired.

For notational simplicity, given (t−i, e−i) ∈ T−i × E−i and ti ∈ Ti, define functions

∆wi : Ti × Ei → R and ∆ui : Ti × Ei → R as follows:

∆wi(t′i, e′i) = [W (t′, N(e′))−W (t′, Oi(e

′))]−∑

j∈Fi(e′)

[W (t′, N(e′))−W (t′, Oj(e′))]

∆ui(t′i, e′i|ti) = vi(g(t′, e′)|ti)− vi(g(t′, e′)|t′i).

95

Note that ∆wi(t′i, e′i) represents the difference between the marginal contribution of i’s refer-

rals and the sum of the marginal contributions of i’s referred agents’ referrals. On the other

hand, ∆ui(t′i, e′i|ti) represents i’s additional gain of misreporting t′i instead of ti from what

he would get if he were a type of t′i. In the following Lemma, we will decompose an agent’s

ex-post payoff into the sum of ∆wi(t′i, e′i) and ∆ui(t

′i, e′i|ti).

Lemma 3.3. Suppose that an agent i’s private information is (ti, ei) ∈ Ti × Ei and others

report (t−i, e−i) ∈ T−i × E−i. Under the multilevel mechanism,

i) the agent i’s ex-post payment from truth telling is

PMi (t, e) = W (t, Oi(e)) +

∑j∈Fi(e)

[W (t, N(e))−W (t, Oj(e))]−∑

j∈N(e)\i

vj(g(t, e)|tj));

ii) the agent i’s ex-post payoff from truth telling is,

vi(g(t, e)|ti)− PMi (t, e) = ∆wi(ti, ei);

iii) the agent i’s ex-post payoff from reporting (t′i, e′i) ∈ Ti × Ei is

vi(g(t′, e′)|ti)− PMi (t′, e′) = ∆wi(t

′i, e′i) + ∆ui(t

′i, e′i|ti);

Proof. Note that

PMi (t, e) = PG

i (t, e)−∑

j∈Fi(e)

PGj (t, e)

=

W (t, Oi(e))−∑

j∈Oi(e)

vj(g(t, e)|tj)

− ∑j∈Fi(e)

W (t, Oj(e))−∑

k∈Oj(e)

vk(g(t, e)|tk)

=

W (t, Oi(e))−∑

j∈Oi(e)

vj(g(t, e)|tj)

−∑

j∈Fi(e)

[W (t, Oj(e))−W (t, N(e))]−∑

j∈Fi(e)

∑k∈Gj(e)

vk(g(t, e)|tk)

= −∑

j∈N(e)\i

vj(g(t, e)|tj)) +W (t, Oi(e)) +∑

j∈Fi(e)

[W (t, N(e))−W (t, Oj(e))] ,

96

where the third equality is due to∑

k∈Oj(e) vk(g(t, e)|tk) = W (t, N(e))−∑

k∈Gj(e) vk(g(t, e)|tk)

and the last equality is from Oi(e) ∪(∪j∈Fi(e)Gj(e)

)= N(e) \ i. This completes the first

part.

Using the first part, we have

vi(g(t, e)|ti)− PMi (t, e) = W (t, N(e))−W (t, Oi(e))−

∑j∈Fi(e)

[W (t, N(e))−W (t, Oj(e))]

= ∆wi(ti, ei),

which completes the second part.

When the agent i reports (t′i, e′i) ∈ Ti × Ei, his payoff is vi(g(t′, e′)|ti)− PM

i (t′, e′). Sub-

tracting and adding vi(g(t′, e′)|t′i), we have

vi(g(t′, e′)|ti)− PMi (t′, e′) = vi(g(t′, e′)|ti)− vi(g(t′, e′)|t′i) + vi(g(t′, e′)|t′i)− PM

i (t′, e′)

= ∆ui(t′i, e′i|ti) + ∆wi(t

′i, e′i),

which completes the last part.

Proposition 3.4. A multilevel mechanism is individually rational if W is monotone and

submodular.

Proof. Due to Lemma 3.3, it suffices to show that ∆wi(ti, ei) ≥ 0 if W is monotone and

submodular. Suppose that W is monotone and submodular. Take any (t, e) ∈ T ×E. Since

Gj(e)j∈Fi(e), i is a partition of N(e) \Oi(e), applying Lemma 3.2, we have that

W (t, N(e))−W (t, Oi(e))

≥∑

j∈Fi(e)

[W (t, N(e))−W (t, N(e) \Gj(e))] + [W (t, N(e))−W (t, N(e) \ i)] . (3.14)

Since W is monotone, W (t, N(e))−W (t, N(e) \ i) ≥ 0, the inequality (3.14) yields that

[W (t, N(e))−W (t, Oi(e))] ≥∑

j∈Fi(e)

[W (t, N(e))−W (t, N(e) \Gj(e))] ,

which is equivalent that ∆wi(ti, ei) ≥ 0.

97

Next, we investigate conditions for incentive compatibility with respect to a connection

type and a payoff type. Under a multilevel mechanism, first we show that agents have no

incentive to under-report their payoff types. Given this result, it turns out that fully referring

is a dominant strategy.

Proposition 3.5. For all (t, e) ∈ T ×E and all i ∈ N(e), under-reporting i’s payoff type is

dominated by reporting the true type ti if W is submodular in t.

Proof. Given (t, e) ∈ T × E and i ∈ N(e), due to Lemma 3.3, the agent i’s payoff from

reporting t′i ∈ Ti, vi(g(t′, e)|ti)− PMi (t′, e), consists of three parts:

vi(g(t′, e)|ti) +∑

j∈N(e)\i

vj(g(t′, e)|tj))︸ ︷︷ ︸(A)

−W (t′, Oi(e))︸ ︷︷ ︸(B)

−∑

j∈Fi(e)

[W (t′, N(e))−W (t′, Oj(e))]︸ ︷︷ ︸(C)

.

(3.15)

The first part (A) is maximized at t′i = ti and the second part (B) does not changed by

t′i. Since W is and submodular in t, for all j ∈ Fi(e), if t′i < ti, then

W (t′, N(e))−W (t′, Oj(e)) ≥ W (t, N(e))−W (t, Oj(e)).

Thus, if t′i < ti, then the last part (C) is less than or equals to

−∑

j∈Fi(e)

[W (t, N(e))−W (t, Oj(e))] ,

which is from truth-telling.

Proposition 3.6. Suppose that W is submodular and (v,Γ) is competitive. Under a multi-

level mechanism, it is a dominant strategy for an agent i to fully reveal his connection type,

if vi is submodular. That is, for all t′i ∈ Ti and all e′i ⊆ ei,

∆wi(t′i, ei) + ∆ui(t

′i, ei|ti) ≥ ∆wi(t

′i, e′i) + ∆ui(t

′i, e′i|ti), (3.16)

Proof. The proof consists of two steps. In Step 1, we will show that submodularity of W

implies that for all t′i ∈ Ti and all e′i ⊆ ei,

∆wi(t′i, ei) ≥ ∆wi(t

′i, e′i). (3.17)

98

In Step 2, we will show that competitiveness of Γ∗ and submodularity of vi imply that for

all t′i ∈ Ti and all e′i ⊂ ei,

∆ui(t′i, ei|ti) ≥ ∆ui(t

′i, e′i|ti). (3.18)

Two conditions, (3.17) and (3.18), implies (3.16), which completes the proof.

Step 1 : Rearranging terms, the inequality (3.17) is equivalent to

W (t′, N(e))−W (t′, N(e′))+∑

j∈Fi(e′)

[W (t′, N(e′)) +W (t′, Oj(e))−W (t′, N(e))−W (t′, Oj(e′))]

≥∑

j∈Fi(e)\Fi(e′)

[W (t′, N(e))−W (t′, Oj(e))] . (3.19)

Since N(e′) ∪ Oj(e) = N(e) and N(e′) ∩ Oj(e) = Oj(e′) for all j ∈ Fi(e′), submodularity of

W implies that

∑j∈Fi(e′)

[W (t′, N(e′)) +W (t′, Oj(e))−W (t′, N(e))−W (t′, Oj(e′))] ≥ 0. (3.20)

Applying Lemma 3.2, submodularity of W implies that

W (t′, N(e))−W (t′, N(e′)) ≥∑

j∈Fi(e)\Fi(e′)

[W (t′, N(e))−W (t′, Oj(e))] . (3.21)

Since (3.20) and (3.21) implies (3.19), we have shown (3.17).

Step 2 : By competitiveness of Γ∗, we have that vi(g(t′, e′)|ti) ≥ vi(g(t′, e)|ti). Due to Propo-

sition 3.5, the agnet i reports t′i ≥ ti in equilibrium, submodularity of vi yields that

vi(g(t′, e′)|t′i)− vi(g(t′, e)|t′i) ≥ vi(g(t′, e′)|ti)− vi(g(t′, e)|ti),

which implies (3.18).

We provide agents’ beliefs in which they truthfully report their types, so that for all the

agents to tell the truth is a maximin strategy.

Proposition 3.7. reporting ti is a dominant strategy if W is monotone and the agent i

believes that, for all t′i ∈ Ti,

W (t′, N(e)) = W (t′, Oi(e) ∪ i). (3.22)

99

Proof. It suffices to show the last part (C) of (3.15) is zero. Since Oi(e)∪i ⊆ Oj(e) ⊆ N(e)

for all j ∈ Fi(e), monotonicity of W implies that, for all t′i ∈ Ti,

W (t′, Oi(e) ∪ i) ≤ W (t′, Oj(e)) ≤ W (t′, N(e)).

However, the condition (3.22) yields that, for all j ∈ Fi(e),

W (t′, Oj(e)) = W (t′, N(e)),

as desired.

Remark. The condition (3.22) is induced by two possibilities. First, the agent i has no

contribution on new agents’ participations, that is, Fi(e) = ∅. Second, the agent i’s group

does not increase the social worth. Therefore, if agents are ambiguity-averse and they cannot

exclude such possibilities, then truthfully reporting their payoff types is a dominant strategy.

3.5 Applications

In this section, we provide two interesting applications; a single-item selling problem and a

public good provision game.

3.5.1 Single-item Auctions

The seller auctions off an indivisible item and each agent i has a private valuation ti ∈ R+

to the item. We assume that resale is prohibited.7 For tractability, we assume that ti 6= tj

for all i, j ∈ N and i 6= j. Let i∗(t, S) ≡ argmaxi∈S ti.

In a second price auction, for all i ∈ e0, it is a dominant strategy to bid his true value

and to conceal his accessible contact information. That is, given (t, e) ∈ T ×E, i∗(t, e0) wins

the auction and the seller’s ex-post revenue is the second highest value among e0,

REV SPA = maxi∈e0\i∗(t,e0)

ti.

7This assumption is crucial for agents’ payoff types and connection types to be independent.

100

If the actual connection profile is not trivial, then a VCG mechanism may be different

from a second price auction. Fix (t, e) ∈ T × E. If i∗(t, N(e)) ∈ e0, then the seller’s ex-

post revenue from a VCG mechanism equals to the one from a second price auction. If

i∗(t, N(e)) ∈ N1(e) \ e0, then a VCG mechanism yields a higher revenue than a second price

auction. If i∗(t, N(e)) 6∈ N1(e), then the seller’s revenue from a VCG mechanism is at most

equals to the one of a second price auction.

Example 3.3. Let N = 1, 2, 3 and t1 = 1, t2 = 2, t3 = 4, and e0 = 1, 2. No matter

what eii=1,2,3 is, the ex-post revenue from a second price auction is 1.

1. Suppose e1 = 3 and e2 = e3 = ∅. Since all the agents are involved, agent 3 wins

the auction and pays 2. However, the seller should pay 2 to agent 1, because agent

1’s marginal contribution to the economy is 2. Hence, the seller’s ex-post revenue is 0,

which is less than the case of a second price auction.

2. Suppose e2 = 3 and e1 = e3 = ∅. Again agent 3 wins and pays 2. However, the seller

should pay 3 to agent 2, and the seller’s ex-post revenue is -1, which runs deficit.

3. Suppose e1 = e2 = 3 and e3 = ∅. Since agent 3 is contacted by two agents, neither

agent 1 nor agent 2 can claim an exclusive contribution on agent 3’s participation and

3 ∈ N1(e). The seller’s ex-post revenue is 2, which is higher than the case of a second

price auction.

While the revenue ranking between a second price auction and a VCG mechanism depends

on a connection profile, a multilevel mechanism guarantees a revenue higher than or equal

to a second price auction.

Lemma 3.4. A multilevel mechanism guarantees a revenue higher than or equal to a second

price auction for single-item cases.

Proof. Fix (t, e) ∈ T × E. Since all agents fully reveal their connection type, N(e) is the

set of participants. Let t be an equilibrium bidding strategy profile. Then i∗(t, N(e)) is the

101

winner. There exists i1 ∈ N1(e) such that i∗(t, N(e)) ∈ Gi1(e), that is, i1 is the existing

buyer who contributes the winner’s participation. Using Lemma 3.1, the seller’s net revenue

is

REV M = maxi∈Oi1 (e)

ti.

Since N1(e) \ i1 ⊆ Oi1(e) and ti∗(t,N1(e)) ≥ ti1 ,

maxi∈Oi1 (e)

ti ≥ maxi∈N1(e)\i1

ti ≥ maxi∈N1(e)\i∗(t,N1(e))

ti.

Since e0 ⊆ N1(e), we have

maxi∈N1(e)\i∗(t,N1(e))

ti ≥ maxi∈e0\i∗(t,e0)

ti.

The remaining proof is divided into two cases:

Case 1: i∗(t, e0) = i∗(t, e0). By Proposition 3.5, since ti ≥ ti for all i ∈ N(e), we have

maxi∈e0\i∗(t,e0)

ti ≥ maxi∈e0\i∗(t,e0)

ti,

which equals to REV SPA.

Case 2: i∗(t, e0) 6= i∗(t, e0). Then, we know

maxi∈e0\i∗(t,e0)

ti ≥ ti∗(t,e0).

By Proposition 3.5 again, we have

ti∗(t,e0) ≥ ti∗(t,e0),

which is greater than REV SPA.

However, this result is not necessarily true for combinatorial auctions. The following

example shows that a multilevel mechanism may yield a lower revenue than a second price

auction.

Example 3.4. The seller auctions two item, X and Y . She knows buyer 1 and buyer 2

directly, but not buyer 3. Buyer 3 is only known to buyer 2. That is e0 = 1, 2, e1 = 3,

and e2 = e3 = ∅. Their values to the items are as follows:

102

Buyer X Y X + Y1 4 4 102 8 0 83 0 8 8

1. Second price auction: Buyer 1 does not refer buyer 3. Buyer 1 and buyer 2 bid their

true values. Buyer 1 wins Y and pays nothing; buyer 2 wins X and pays 6. Hence, the

seller’s revenue is 6.

2. VCG mechanism: Buyer 1 refers buyer 3. All bidders bid their true values. Buyer 2

wins X and pays 4; buyer 3 wins Y and pays 4. Buyer 1 gets a transfer 8 from the

seller, and the seller’s revenue is 0.

3. Multilevel mechanism: Buyer 1 refers buyer 3. Suppose that all bidders bid their true

values. Buyer 2 wins X and pays 4 to the seller; buyer 3 wins Y and pays 4 to buyer

1. Hence, the seller’s revenue is 4, which is less that the case of a second price auction.

Now, we compare a multilevel mechanism and a standard auction in terms of the seller’s

expected revenue. An auction is standard if the highest bidder wins the item, the expected

payment of a bidder with value zero is zero, and there is no compensation on contacting.

The following proposition shows that the expected revenue from a VCG mechanism exceeds

that from any standard auction.

Proposition 3.8. Suppose that values are independently and identically distributed and all

bidders are risk neutral. A multilevel mechanism dominates standard auctions in terms of

seller’s ex-post revenue for single-item cases.

Proof. By Lemma 3.4, for all (t, e) ∈ T × E, a multilevel mechanism guarantees a revenue

higher than or equal to a second price auction. Thus, for any seller’s belief on (t, e), the

seller’s expected payoff from a multilevel mechanism is higher than or equals to the one of

a second price auction. Applying the revenue equivalence principle8, the seller’s expected

8Myerson (1981) shows that any symmetric and increasing equilibrium of any standard auction yields thesame expected revenue to the seller.

103

payoff from a multilevel mechanism is higher than or equals to the one from any standard

auction with the set of bidders e0.

3.5.2 Public Good Provision

Since a public good is both non-excludable and non-rivalrous, it is the ideal example with

which agents’ connection types and payoff types are independent. It is known that there is

no budget-balanced mechanism which satisfies both incentive compatibility and individual

rationality. However, allowing strategic contacting, a multilevel mechanism turns out to

balance the budget. Especially, if each agent has a binary preference, then it satisfies both

incentive compatibility and individual rationality.

Example 3.5 (Public good provision). A principal decides whether or no to provide a public

good at a fixed cost c ∈ (1, 2). There are two agents whose private value is either 0 or 1.

That is, providing the public good is efficient if and only if both agents’ values are 1. Suppose

that both the principal and agents can directly contact with others and that both agents

reveal their true types if they are indifferent between truth-telling and lying.

1. A VCG mechanism with both agents: For both agents, revealing their true values is a

dominant strategy and the allocation is efficient. When both agents’ values are 1, each

pays just c− 1 and hence the budget is 2(c− 1)− c = c− 2 < 0.

2. A VCG mechanism with a randomly chosen agent: For the chosen agent, contacting

with the other agent is a dominant strategy if her value is 1. Again, for both players,

revealing their true value is a dominant strategy. The outcome is exactly the same

with the previous case.

3. A multilevel mechanism with a randomly chosen agent: For the chosen agent, contact-

ing with the other agent is a dominant strategy if his value is 1. For both players,

revealing their true value is a dominant strategy. Suppose that both agents’ values are

1. For the chosen agent, she pays c to the principal and compensated with c− 1 from

the other agent. Thus the allocation is efficient and balances the budget.

104

Bibliography

Abreu, D. and F. Gul (2000): “Bargaining and reputation,” Econometrica, 68, 85–117.

Abreu, D. and M. Manea (2012a): “Bargaining and efficiency in networks,” Journal of

Economic Theory, 147, 43–70.

——— (2012b): “Markov equilibria in a model of bargaining in networks,” Games and

Economic Behavior, 75, 1–16.

Aumann, R. J. and J. H. Dreze (1974): “Cooperative games with coalition structures,”

International Journal of game theory, 3, 217–237.

Banks, J. S. (2000): “Buying supermajorities in finite legislatures,” American Political

Science Review, 677–681.

Banzhaf, J. F. (1964): “Weighted voting doesn’t work: A mathematical analysis,” Rutgers

L. Rev., 19, 317.

Baron, D. and J. Ferejohn (1989): “Bargaining in legislatures,” The American Political

Science Review, 1181–1206.

Bondareva, O. (1963): “Some applications of linear programming methods to the theory

of cooperative games,” Problemy kibernetiki, 10, 119–139.

Braess, D. (1968): “Uber ein Paradoxon aus der Verkehrsplanung,” Un-

ternehmensforschung, 12, 258–268.

105

Braess, D., A. Nagurney, and T. Wakolbinger (2005): “On a paradox of traffic

planning,” Transportation science, 39, 446–450.

Cai, H. (2000): “Delay in multilateral bargaining under complete information,” Journal of

Economic Theory, 93, 260–276.

Calzolari, G. and A. Pavan (2006): “Monopoly with resale,” The RAND Journal of

Economics, 37, 362–375.

Chatterjee, K., B. Dutta, D. Ray, and K. Sengupta (1993): “A noncooperative

theory of coalitional bargaining,” The Review of Economic Studies, 60, 463–477.

Clarke, E. (1971): “Multipart pricing of public goods,” Public choice, 11, 17–33.

Compte, O. and P. Jehiel (2002): “On the role of outside options in bargaining with

obstinate parties,” Econometrica, 70, 1477–1517.

——— (2010): “The coalitional Nash bargaining solution,” Econometrica, 78, 1593–1623.

Dal Bo, E. (2007): “Bribing voters,” American Journal of Political Science, 51, 789–803.

Dekel, E., M. O. Jackson, and A. Wolinsky (2008): “Vote buying: General elec-

tions,” Journal of Political Economy, 116, 351–380.

——— (2009): “Vote buying: legislatures and lobbying,” Quarterly Journal of Political

Science, 4, 103–128.

Eraslan, H. (2002): “Uniqueness of stationary equilibrium payoffs in the Baron–Ferejohn

model,” Journal of Economic Theory, 103, 11–30.

Eraslan, H. and A. McLennan (2013): “Uniqueness of stationary equilibrium payoffs

in coalitional bargaining,” Journal of Economic Theory, 148, 2195–2222.

Garratt, R. and T. Troger (2006): “Speculation in standard auctions with resale,”

Econometrica, 74, 753–769.

106

Gomes, A. (2005): “Multilateral contracting with externalities,” Econometrica, 73, 1329–

1350.

Granovetter, M. (1995): Getting a job: A study of contacts and careers, University of

Chicago Press.

Groseclose, T. and J. M. Snyder Jr (1996): “Buying supermajorities,” American

Political Science Review, 303–315.

Groves, T. (1973): “Incentives in teams,” Econometrica: Journal of the Econometric

Society, 617–631.

Gul, F. (1989): “Bargaining foundations of Shapley value,” Econometrica: Journal of the

Econometric Society, 81–95.

——— (1999): “Efficiency and immediate agreement: a reply to Hart and Levy,” Econo-

metrica, 67, 913–917.

Hafalir, I. and V. Krishna (2008): “Asymmetric Auctions with Resale,” American

Economic Review, 98, 87–112.

Haile, P. (2003): “Auctions with private uncertainty and resale opportunities,” Journal of

Economic Theory, 108, 72–110.

Hart, S. and Z. Levy (1999): “Efficiency does not imply immediate agreement,” Econo-

metrica, 67, 909–912.

Hart, S. and A. Mas-Colell (1996): “Bargaining and value,” Econometrica: Journal

of the Econometric Society, 357–380.

Hyndman, K. and D. Ray (2007): “Coalition formation with binding agreements,” The

Review of Economic Studies, 74, 1125–1147.

Krishna, V. and R. Serrano (1996): “Multilateral bargaining,” The Review of Economic

Studies, 63, 61–80.

107

Lee, J. (2014): “Multilateral Bargaining in Networks,” working paper, available at https:

//sites.google.com/site/joosungecon/Network_posting.pdf.

Manea, M. (2011a): “Bargaining in dynamic markets with multiple populations,” working

paper.

——— (2011b): “Bargaining in stationary networks,” The American Economic Review, 101,

2042–2080.

Montero, M. (2002): “Non-cooperative bargaining in apex games and the kernel,” Games

and Economic Behavior, 41, 309–321.

——— (2006): “Noncooperative foundations of the nucleolus in majority games,” Games

and Economic Behavior, 54, 380–397.

Montero, M. and J. J. Vidal-Puga (2011): “Demand bargaining and proportional

payoffs in majority games,” Games and Economic Behavior, 71, 395–408.

Morelli, M. and M. Montero (2003): “The demand bargaining set: General character-

ization and application to majority games,” Games and Economic Behavior, 42, 137–155.

Myers, C. A. and G. P. Shultz (1951): The dynamics of a labor market; a study of

the impact of employment changes on labor mobility, job satisfactions, and company and

union policies, Prentice-Hall.

Myerson, R. (1981): “Optimal Auction Design,” Mathematics of Operations Research, 6,

58–73.

Myerson, R. B. (1977): “Graphs and cooperation in games,” Mathematics of operations

research, 2, 225–229.

——— (1997): Game theory: analysis of conflict, Harvard university press.

Nguyen, T. (2012): “Coalitional bargaining in networks.” in ACM Conference on Electronic

Commerce, vol. 772.

108

Okada, A. (1996): “A noncooperative coalitional bargaining game with random proposers,”

Games and Economic Behavior, 16, 97–108.

——— (2000): “The Efficiency Principle in Non-Cooperative Coalitional Bargaining,”

Japanese Economic Review, 51, 34–50.

——— (2011): “Coalitional bargaining games with random proposers: Theory and applica-

tion,” Games and Economic Behavior.

Ray, D. and R. Vohra (2013): “Coalition Formation,” mimeo.

Rosen, E. (2002): The anatomy of buzz: How to create word-of-mouth marketing, Crown

Business.

Roughgarden, T. (2005): Selfish routing and the price of anarchy, MIT press.

Ryu, G. and L. Feick (2007): “A penny for your thoughts: Referral reward programs and

referral likelihood,” Journal of Marketing, 71, 84–94.

Schmitt, P., B. Skiera, and C. Van den Bulte (2011): “Referral programs and

customer value,” Journal of Marketing, 75, 46–59.

Seidmann, D. and E. Winter (1998): “A theory of gradual coalition formation,” Review

of Economic Studies, 65, 793–815.

Selten, R. (1981): “A Noncooperative Model of Characteristic-Function Bargaining,” in

Essays in Game Theory and Mathematical Economics in Honor of Oskar Morgenstern,

ed. by V. Bohm and H. Nachtkamp, Mannheim: Bibliographisches Institut.

Serrano, R. (1993): “Non-cooperative implementation of the nucleolus: the 3-player case,”

International Journal of Game Theory, 22, 345–357.

——— (1995): “Strategic bargaining, surplus sharing problems and the nucleolus,” Journal

of Mathematical Economics, 24, 319–329.

109

Shapley, L. (1967): “On balanced sets and cores,” Naval research logistics quarterly, 14,

453–460.

Shapley, L. S. (1962): “Simple games: an outline of the descriptive theory,” Behavioral

Science, 7, 59–66.

Shapley, L. S. and M. Shubik (1954): “A method for evaluating the distribution of

power in a committee system,” American Political Science Review, 48, 787–792.

Sutton, J. (1986): “Non-cooperative bargaining theory: An introduction,” The Review of

Economic Studies, 53, 709–724.

Vickrey, W. (1961): “Counterspeculation, auctions, and competitive sealed tenders,” The

Journal of finance, 16, 8–37.

von Neumann, J. and O. Morgenstern (1944): Theory of Games and Economic Be-

havior, Princeton University Press.

Winter, E. (1996): “Voting and vetoing,” American Political Science Review, 813–823.

Yan, H. (2003): “Noncooperative selection of the core,” International Journal of Game

Theory, 31, 527–540.

Zhang, J. and R. Wang (2011): “Optimal Mechanism Design with Speculation and

Resale,” .

Zheng, C. (2002): “Optimal auction with resale,” Econometrica, 70, 2197–2224.

110

Vita

Joosung LeeEducation

Ph.D. in Economics, Pennsylvania State University, 2014

M.A in Economics, Seoul National University, 2008

B.A in Economics, Korea National Open University, 2005

B.Eng in Nuclear Engineering, Seoul National University, 2001

Publications

“Sequentially Two-Leveled Egalitarianism for TU Games: Characterization and Ap-plication,” (with Theo SH Driessen) European Journal of Operational Research 220.3(2012): 736-743, ISSN: 0377-2217

“Sequential Contributions Rules for Minimum Cost Spanning Tree Problems,” (withYoungsub Chun) Mathematical Social Sciences 64.2 (2012): 136-143, ISSN: 0165-4896

Awards and Honors

Hamilton Best Paper Award, the Seventh Annual Graduate Student Conference in theAlexander Hamilton Center for Political Economy at New York University, 2014