Speculative Partnership Dissolutionwith Auctions∗

Ludwig Ensthaler† Thomas Giebe‡ Jianpei Li§

March 2, 2012

Abstract

The literature on partnership dissolution generally takes the dissolution decision

as given and examines whether the outcome is efficient. A well-known result is that

k + 1-price auctions dissolve a partnership efficiently when the share structure is

sufficiently close to equal. We endogenize the dissolution decision by adding a stage

where partners decide whether to call for dissolution. This introduces the possibility

of inefficient breakups and makes continuation possible. We show that, if k+1-price

auctions are used, an equal-share partnership cannot be dissolved ex post efficiently.

While allowing for veto or requiring consent does not help, adding a proper reserve

restores efficiency.

JEL: C78, D23, D44, J12, L24

Keywords: Partnership Dissolution, k + 1-price Auctions, Efficiency.

∗We thank R. Preston McAfee, Jimmy Chan, Uwe Dulleck, Roland Strausz, Cédric Wasser, ElmarWolfstetter for helpful discussions. Financial support from the German Science Foundation throughSFB/TR 15 and the Chinese National Natural Science Foundation through Project No. 71002001 aregratefully acknowledged.

†University College London, UK. l.ensthaler@ucl.ac.uk.

‡TU Berlin, Germany. thomas.giebe@tu-berlin.de.

§University of International Business and Economics, China. lijianpei@uibe.edu.cn.

1 Introduction

The economics literature on partnership dissolution under incomplete information was

started by Cramton, Gibbons, and Klemperer (1987) (henceforth CGK). The now stan-

dard model introduced in CGK and the many works following it assume that a partner’s

valuation of the jointly owned asset is unchanged by dissolution. Thus, a partner who

owns half of the firm values sole ownership of the firm exactly twice as much as he values

the status quo (with the partners present). Hence, the presence of the partners does not

affect the valuation of the asset and it is economically efficient to break up the part-

nership rather than to continue if the partners have different values for sole ownership.

This is of course a valid model for non-producing partnerships where partners are merely

co-owners of a good, e.g., siblings who jointly inherit a house. But for most business

partnerships, the presence of partners matters for the value of the partnership and its

impact on dissolution should be taken into account.

Partnerships that jointly produce goods or services typically form because partners

benefit from complementary skills (see, e.g., Farrell and Scotchmer, 1988).1 If such part-

nerships are dissolved and one of the partners is to take sole ownership, the other partners

usually leave the company and the complementarities are lost, making dissolution poten-

tially inefficient.2 Thus, when judging the efficiency of a partnership dissolution mecha-

nism it seems natural to ask whether it encourages break-up only in those cases where

continuation is not desirable. In other words, an efficient dissolution mechanism should

not only ensure that the sole ownership of the partnership asset goes to the partner that

values it the most, but moreover prevent break-up if economically undesirable.

Furthermore, the literature generally takes the dissolution decision as given. It seems

natural to enrich the model by assuming that dissolution does not happen automatically,

but must be triggered by a decision of the partners. If partners hold private information

about the value of sole ownership, proposing dissolution may be a signal of that private

information.

The k + 1-price auctions are a well-studied and widely used class of dissolution mech-

1Examples abound. Lawyers with complementary fields of expertise jointly run a legal practice. Achef and a service person jointly operate a restaurant. Many firms use business alliances in order to entera new market, to obtain new skills, or to share risk and resources.

2For example, a senior partner in a law firm might take important clients with him, a specializedengineer with years of experience might not easily be replaced or banks might lose confidence in thebusiness.

2

anisms. They are appealing because of their simplicity and ease of implementation.3

Applied to a two-player partnership, the two partners simultaneously submit sealed bids

and the partner who submitted the larger bid buys the other partner’s share of the part-

nership. The transaction price for the whole asset is a convex combination of the low and

high bid: kbL + (1 − k)bH with k ∈ [0, 1]. For k = 0 the auction is known as a winner’s

bid auction (WBA), while for k = 1 it is a loser’s bid auction (LBA). A very influential

result in CGK and the subsequent papers is that if the ownership structure is sufficiently

symmetric, k+1-price auctions implement efficient allocation of ownership in a symmetric

private values framework.

We extend the model of CGK by adding a proposal stage where partners decide

whether to enter a k + 1 price auction in order to dissolve the partnership. This adds the

possibility that the partnership is continued and it makes inefficient breakups feasible.

Moreover we introduce complementarities by assuming a commonly known continuation

value different from the private values for sole ownership. We show that, in this extended

model, ex post efficiency is not achievable by standard k + 1-price auctions.

Adding a proposal stage induces a signaling game: A partner’s decision to call for

dissolution may reveal private information. This affects the other partner’s beliefs and,

thus, bidding behavior at the auction stage. Naturally, a partner who has a high value

for sole ownership would call for dissolution and bid aggressively in the auction in order

to gain sole ownership. Anticipating this, a low-value partner might have an incentive to

signal a high valuation by calling for dissolution, speculating that the other partner has

a high valuation of sole ownership and will, thus, bid aggressively and pay a large price.

Of course, in equilibrium this incentive is taken into account and results in inefficient

dissolution.

We then examine the performance of different variants of the k + 1-price auctions.

While allowing a partner to veto or requiring consent for a dissolution to occur does not

restore efficiency, a WBA with a reserve equal to the continuation value achieves ex post

3The k+1-price auction is referred to as k-double auction in Kittsteiner (2003). In the business world,the auction is also referred to as the ‘Mexican Shoot-out Clause”, “Dutch auction”, or as a variant of themost popular “Texas Shoot-out” (see, for example, http://www.businessninja.co.za/mexican-shoot-out-clause-business-partners.php, and http://www.mediate.com/articles/spoelstra6.cfm.). Real life examplesthat used auctions include the breakup of the 3Com and Huawei joint venture in 2007 (see Ming, Xu, andChen, 2007) and the termination of Chagoyansk Joint Venture between Peter Hambro Mining Plc, RioTinto Mining and Exploration Limited (see the Peter Hambro Mining announcement at FE Investigateon 31 January, 2006, http://investegate.co.uk/Article.aspx?id=200601310700476640X). Jade trading inHetian (Xinjiang, China) usually takes place via k + 1-price auctions. Traders who are interested in anitem of jade buy it jointly from the jade panner. Afterwards, they hold an ascending price auction amongthemselves to decide the sole owner of the item. See Saimaiti (2011).

3

efficiency.

The paper proceeds as follows. First, we give a brief overview of the related literature.

In Section 2 we recall the CGK model, explain how we extend it, and introduce the game.

In Section 3 we derive the main result. A discussion on alternative mechanisms is in

Section 4. Section 5 provides a conclusion. All proofs are in the Appendix.

Related Literature

Our paper contributes to the partnership dissolution literature and the literature that

analyzes k + 1-price auctions.

The formal analysis of the partnership dissolution problem in an incomplete infor-

mation setting started with CGK. They show that there exist ex-post efficient, incentive

compatible, individually rational and budget balanced dissolution mechanisms under the

symmetric independent private values framework if and only if the ownership rights are

sufficiently symmetric. Fieseler, Kittsteiner, and Moldovanu (2003) extend the analysis to

interdependent valuations. They find that in contrast to a private values setting, it might

be impossible to find a distribution of ownership rights such that the partnership can

be dissolved efficiently. Jehiel and Pauzner (2006) derive efficient, incentive-compatible,

individually rational, and budget-balanced mechanisms for the case that valuations only

depend on private information of one of the partners.4 Chien (2007) characterizes the

second-best mechanisms when it is impossible to achieve ex post efficiency. Ornelas and

Turner (2007) study the case where control rights matter for the value of the partner-

ship. Galavotti, Muto, and Oyama (2010) study ex post individually rational, efficient

partnership dissolution in a setting with interdependent valuations.

A number of papers have studied the performance of k + 1-price auctions in the part-

nership dissolution environment. CGK first analyzed the k +1-price auctions, and arrived

at the conclusion that these auctions implement ex post efficiency in a symmetric private

values framework if the partners hold equal shares of the partnership. McAfee (1992) ana-

lyzes the performance of the WBA and the LBA assuming that bidders are risk averse and

there is an option to sell the asset to a third party in a two-player equal-share partnership.

He finds that the WBA is efficient, whereas the LBA not. Engelbrecht-Wiggans (1994)

solves for the equilibrium of the WBA and LBA and compares their expected equilibrium

prices in an interdependent environment. Bulow, Huang, and Klemperer (1999) analyze

4Moldovanu (2001) surveys some of these contributions.

4

the WBA and the LBA in a common values model with uniform distribution of types

in a take-over game. They show that a toehold in the target firm helps a buyer win an

auction, sometimes very cheaply. Minehart and Neeman (1999) examine the performance

of the WBA when coordination is important for the success of a project. de Frutos (2000)

provides results on the existence and uniqueness of pure strategy equilibria in the WBA

and the LBA for the case in which partners’ valuations are asymmetrically distributed.

Kittsteiner (2003) analyses the general k + 1-price auctions under interdependent valu-

ations and proves the uniqueness of the equilibrium for these auctions. He also shows

that k + 1-price auctions remain efficient if the partners are forced to participate in the

dissolution. Lengwiler and Wolfstetter (2005) establish revenue equivalence between the

WBA and LBA in the symmetric independent private values framework, assuming the

existence of an auctioneer who collects a fixed fraction of the auction revenue. Athanas-

soglou, Brams, and Sethuraman (2010) analyze auctions with per-unit price that is a

function of the two highest bids when agents have unequal endowments. Li, Xue, and

Wu (forthcoming) analyzes the WBA and the LBA in a common-values framework with

proprietary information.

The literature has largely followed the tradition of CGK by assuming that dissolution

is exogenously given and the partners have the same value for the partnership and for sole

ownership. Thus the impact of a loss of complementarity on the potential efficiency of a

breakup has been neglected.5 We take this missing factor in the literature into account

by adding a proposal stage to the standard dissolution game. Putting k +1-price auctions

in perspective, we find that efficiency is not attainable. This suggests that taking the

dissolution decision as exogenously given may have important limitations.

2 Model

In this section we give a quick review of the CGK model of partnership dissolution and

discuss how we extend it in order to account for the loss of complementarities due to

breakup and in order to endogenize the dissolution decision. There are n partners indexed

i ∈ N = {1, . . . , n}. Partner i owns a share ri of the partnership and has a value of sole

partnership of vi. A partner’s valuation is private information but it is common knowledge

5An exception is Li and Wolfstetter (2010). They examine the interaction of investment incentivesand dissolution incentives. In contrast to our paper, Li and Wolfstetter (2010) focus on the cake-cuttingmechanism.

5

that it is drawn from a distribution F with support [vl, vh] and positive continuous density

f . A partner with valuation vi and share ri values his share with rivi. A partner’s ex

post utility from sole ownership is ui(vi, t) = vi − t where t is the amount of money paid

to the other partner. In their model, a partner needs to get at least rivi in expectation

to voluntarily participate in the mechanism. Thus, a partner who holds a share of ri and

has a value of vi for the sole ownership of the partnership, assigns a value of rivi to the

status quo. Therefore, in the CGK model, it is natural that a dissolution mechanism is

ex post efficient if

si =

1 if vi = max{vj}

0 else,(1)

where si is partner i’s ex post share of ownership. CGK find, among other results,

Proposition 1 (CGK Proposition 6). The set of partnerships that can be dissolved ef-

ficiently using a k + 1-price auction is nonempty and centered around the equal-shares

partnership.

As we have discussed in the introduction, business partnerships typically form to enjoy

the benefits of complementarities. These get lost in a dissolution. In order to account for

this aspect, we extend CGK’s notion of efficiency as follows. In addition to each partner’s

standalone value for the partnership, vi, there is a commonly known continuation value

C for the whole partnership in its present state. Thus vi is partner i’s valuation of the

business without partners and C the valuation inside the partnership. Then a dissolution

mechanism is efficient, if

si =

0 if maxj 6=i{vj} > max {C, vi}

ri if C > max {vj}

1 if vi = max {C, max {vj}} .

(2)

In the next section, we analyze k + 1-price auctions, taking into account the option of

continuation, and evaluate its performance using the extended efficiency benchmark, (2).

In order to illustrate our main point, it suffices to consider a two-partner equal-shares

partnership. As in CGK, we assume that the two partners, i ∈ {1, 2}, are risk neutral,

and each partner has a private valuation vi for sole ownership. It is common knowledge

that private valuations are drawn independently from distribution F with support [0, 1]

and positive continuous density f . We further assume C ∈ (0, 1) because for C ≤ 0, resp.

6

C ≥ 1, it is common knowledge that breakup is efficient, resp. inefficient. An implication

of these assumptions is that it is private information whether or not continuation is

efficient.

We assume that there exists a partnership contract specifying that if the partners

decide to break up, a k + 1-price auction will be used to determine both the sole owner

and the transaction price. There, the high bid wins. The winner pays one half of the

transaction price to the loser and becomes the sole owner.6 The transaction price is

kbL + (1 −k)bW where k ∈ [0, 1] and bL (resp. bW ) is the losing (resp. winning) bid. With

k = 1 (resp. k = 0), the auction is the LBA (resp. WBA).

The time structure of the game is as follows:

1. (“proposal stage”.) Partners 1 and 2 learn their private values, v1 and v2, for sole

ownership. They simultaneously choose an action, either D or N , where D means

“calling for dissolution” and N means “not calling for dissolution”. The action profile

(a1, a2) ∈ {D, N} × {D, N} determines how the game continues. Profile (N, N)

implies that the partnership continues and each player’s payoff is C/2. Otherwise,

the game proceeds to the next stage.

2. (“dissolution stage”.) The action profile (a1, a2) becomes common knowledge. The

partnership is dissolved using a k + 1-price auction.

In case of a tie (i.e., equal bids), the proposing partner wins the auction if only that

partner has proposed.7 If both have proposed dissolution, then the winner is determined

randomly.

This paper focuses on whether ex post efficiency can be achieved through k + 1-price

auctions. Therefore, we do not make an attempt to solve for all equilibria of the game.

Instead, it is sufficient to concentrate on a class of equilibria, symmetric “cutoff equilibria”,

that, intuitively, contain all equilibria that possibly implement ex post efficiency, given

that k + 1-price auctions are the stipulated dissolution mechanism. Formally, a cutoff

equilibrium is defined as follows.

Definition 1 (Cutoff Equilibrium). Partner i plays a cutoff strategy if he proposes disso-

lution iff his value, vi, is above a cutoff value, vi, where vi ≥ vi ≥ 0. A cutoff equilibrium

is a Perfect Bayesian Nash equilibrium where the partners play cutoff strategies.

6The transaction price refers to the whole partnership. However, the winner already owns half of thepartnership and, therefore, only pays one half of the transaction price to the loser.

7Alternatively, the non-proposing partner could be made the winner. This would not have a qualitativeeffect on the results.

7

Since the k + 1-price auction automatically dissolves the partnership once the disso-

lution stage is reached, efficiency requires that partners call for dissolution if and only

if they know that dissolution is efficient.8 In a cutoff equilibrium, efficiency as defined

in benchmark (2) is achieved if and only if there is a common cutoff value equal to the

continuation value of the partnership, v = v1 = v2 = C.

3 Analysis

We proceed by preparing the main result in a series of lemmas.

If the game ever proceeds to the dissolution stage, the action profile (a1, a2) of the

proposal stage may reveal private information. Therefore, we have a signaling issue.

For the class of cutoff equilibria with cutoff value v ∈ [0, 1], the only consistent belief

system is as follows. After observing that partner j has played D, partner i’s updated

belief is that j’s value must be distributed according to the c.d.f. F conditional on

vj ≥ v. We denote this conditional c.d.f. by G(x) := F (x|x ≥ v) and the belief of

partner i by µi(x | aj) := Pr{Vj ≤ x | aj}, i, j ∈ {1, 2}, j 6= i. Thus, partner i’s prior

belief is µi(x) = F (x) and the posterior belief—having observed partner j’s action choice

aj ∈ {D, N}—is

µi(x|aj) =

G(x) if aj = D

F (x)F (v)

if aj = N, where G(x) :=

0 if x < 0

F (x)−F (v)1−F (v)

if x ∈ [v, 1]

1 if x > 1.

(3)

Denote g := G′.

We first derive the unique equilibrium of a symmetric k + 1-price auction stage as-

suming that it is common knowledge that both partners have valuations in the range

v1, v2 ∈ [v, 1], which, assuming that cutoff strategies are played, corresponds to the con-

tinuation game after both partners have played D at the proposal stage.

Lemma 1 (Symmetric Auction). Consider a k + 1-price auction where both bidders’

valuations are independently distributed according to the c.d.f. G on [v, 1].

8Note that ex post efficiency rules out mixed strategies. Furthermore, it rules out that a partner pro-poses dissolution if his valuation is below the continuation value, because there, with positive probabilitythe other partner also has a low value and the partnership is broken up inefficiently.

8

The unique equilibrium bid function is symmetric and is given by

β(vi) = vi −∫ vi

G−1(k)

(G(t) − k)2

(G(vi) − k)2dt, (4)

and parter i’s interim expected payoff is

U(vi) =1

2+

1

2

∫ 1

vG2(t)dt −

∫ 1

vi

G(t)dt. (5)

The following lemma gives an equilibrium of the auction for the case that exactly one

partner has played D at the proposal stage, given that both partners play cutoff strate-

gies. For our purposes, it is not necessary to consider the uniqueness of the equilibrium

characterized in Lemma 2. However, we need to rule out that there are other equilibria

that can possibly induce efficiency of the whole game.

Lemma 2 (Asymmetric Auction). Suppose both partners play cutoff strategies and exactly

one partner has proposed dissolution. Then the k + 1-price auctions have an equilibrium

with bid functions

b1 = b2 = v. (6)

The corresponding equilibrium interim expected payoffs are

Ui = vi −v

2, Uj =

v

2, vi ∈ [v, 1], vj ∈ [0, v), i, j ∈ {1, 2}. (7)

There is no other equilibrium that is consistent with ex post efficiency in the whole game.

In the next lemma, we state the cutoff equilibrium of the whole game that corresponds

to the equilibrium stated in Lemmas 1 and 2. Define vm implicitly by F (vm) = 1/2.

Lemma 3 (Cutoff Equilibrium). Assume the continuation equilibrium derived in Lemma

2. If C ∈ [vm + 4∫ 1

vm (1 − F (t))2 dt, 1], the game has a unique cutoff equilibrium, where

the cutoff value is implicitly given by

C = v +∫ 1

v

(1 − F (t))2

F (v)(1 − F (v))dt. (8)

There, the cutoff value v is strictly below C. For C < vm + 4∫ 1

vm (1 − F (t))2 dt, a cutoff

equilibrium does not exist.

9

Recalling that efficiency can be achieved if and only if there is a cutoff equilibrium

with v = C, we arrive at the following main result.

Proposition 2. Ex post efficiency is unattainable in equilibrium.

The inefficiency result is intuitive. Ex post efficiency requires that both partners play

a cutoff strategy with cutoff value v = C. Suppose this equilibrium candidate is played

by partner 2 and consider partner 1’s incentives to deviate. Suppose partner 1’s value is

“slightly” below C. Partner 1’s payoff from playing the same strategy is C/2 since the

partnership continues. If partner 1 deviates and proposes dissolution instead, then his

payoff depends on partner 2’s value. If v2 ≥ C, then 2, believing that v1 ≥ C as well,

will bid b2 = β(v2) ≥ C and, thus, increase partner 1’s payoff above C/2. If v2 < C and

partner 2 bids b2 = v = C in the auction, partner 1 wins the auction with payoff v2 −C/2

which is only “slightly” below C/2. For v1 sufficiently close to C, partner 1’s expected

gain from a higher price when he sells strictly dominates the expected loss from becoming

the buyer. Therefore it is beneficial for partner 1 to deviate rather than to follow the

cutoff strategy with cutoff value C. In equilibrium, this is taken into account, and both

partners propose dissolution more often than is efficient.

In our setting, as in CGK, each partner has private information about the value of sole

ownership and, given that the auction stage is reached, it allocates sole ownership to the

partner with the higher value. In CGK, it is always efficient to run the auction. However,

in our case, the continuation value is different from the private values which implies that

it is not always efficient to run the auction. The partners, as in CGK, maximize their

expected payoffs, regardless of efficiency. In our game, in addition to making bids, they

can influence their payoffs by strategically calling for dissolution, making the game a

signaling game. As a result, efficiency cannot be achieved.

4 Alternative Mechanisms

Veto Suppose after the simultaneous proposal stage there is a veto stage where, if only

one partner has called for dissolution, the other partner can veto the dissolution.9 Similar

to our base game, we cannot achieve efficiency: Suppose there is an efficient equilibrium.

As in the base game, this requires a cutoff equilibrium where partners call for dissolution

9k + 1-price auctions with veto are analyzed in Kittsteiner (2003). In his game, there is no proposalstage and the “veto” is submitted at the auction before any information is revealed. In this sense, theveto right in Kittsteiner (2003) is more like the “Consent” requirement we discuss below.

10

if and only if their valuation is above the continuation value and the veto right is not

exercised. Given this candidate, foreseeing that the veto right will not be exercised, a

partner with valuation just below C has an incentive to deviate in the way that we have

argued for the base game.

Consent Suppose consent of both partners is required in order for dissolution to occur.10

In terms of our game this implies that the auction stage is reached if and only if both

partners have called for dissolution. It is easy to see that ex post efficiency cannot be

achieved: If at least one partner has a high value (above C) then efficiency requires

dissolution, i.e., that both partners call for dissolution. If neither partner has a high

value, then efficiency requires continuation, i.e., that at least one partner does not call for

dissolution. An efficient equilibrium candidate does not exist because strategies cannot

be conditional on the other partner’s private information.

Reserve Price Efficiency can be restored by a WBA with reserve price equal to the

continuation value.11 There, dissolution only occurs if the price determined in the auction

exceeds the reserve price, and, thus, the continuation value. In terms of our game, as

before, the auction is efficient, given that the dissolution stage is reached. At the proposal

stage, the reserve price makes it a dominant strategy for the partners to always call for

dissolution: By playing N , a partner’s payoff is C/2, while by playing D, the ex-post

payoff is C/2, if the auction price is below the continuation value, or more, if the auction

price exceeds the continuation value. The partners leave it to the auction to first aggregate

the private information before “deciding” whether to dissolve the partnership or not and

the right reserve price makes sure that dissolution only occurs when it is efficient.

Note that in the our game without reserve price, only the higher types found it prof-

itable to enter the auction because calling for dissolution implies the risk of becoming

the buyer. Thus, calling for dissolution reveals information. A reserve price, in contrast,

allows every type, in particular the low types, to avoid becoming the buyer in the auc-

tion, while possibly selling at a profit. This makes it profitable for every type to enter the

auction. As a result, signaling becomes costless and, thus, uninformative.

10If there is no prior agreement on how to dissolve the partnership then dissolution might requiremutual consent.

11See Proposition 3 in the Appendix for the detailed solution. A WBA with outside option is analyzedin McAfee (1992). That outside option corresponds to our continuation value.

11

5 Conclusion

The literature on partnership dissolution generally takes the dissolution decision as given

and examines whether dissolution is efficient, i.e., whether the asset is allocated to the

partner with the highest valuation. A well-known result is that k + 1-price auctions

dissolve a partnership efficiently when the share structure is sufficiently close to equal. In

this paper, we extend the CGK model by adding a stage where the two partners decide

on whether to call for dissolution. Moreover, we add a continuation value different from

the private valuations. These changes introduce the possibility of inefficient breakup and

model the complementarities that are typically present in business partnerships. We find

that, if k+1-price auctions are used, the partnership cannot be dissolved ex post efficiently.

While allowing a partner to veto or requiring consent for a dissolution to occur does not

restore efficiency, a WBA with a reserve equal to the continuation value achieves ex post

efficiency.

6 Appendix

Proof of Lemma 1. The equilibrium bid function, (4), has been derived by Cramton, Gib-

bons, and Klemperer (1987, p.630) and Kittsteiner (2003, p.70). The latter shows that

this equilibrium is symmetric and unique. In the following, we compute the interim

expected payoff, (5).

If both partners bid according to (4), the interim expected payoff of a partner with

valuation v ∈ [v, 1] (when the rival’s valuation is the random variable V ) is given by

U(v) = G(v)

(

v −1

2(kE [β(V )|V < v] + (1 − k)β(v))

)

+ (1 − G(v))1

2(kβ(v) + (1 − k)E [β(V )|V > v])

= G(v)v −k

2

∫

v

v

β(s)g(s)ds −G(v)

2(1 − k)β(v) +

1 − G(v)

2kβ(v) +

1 − k

2

∫

1

v

β(s)g(s)ds

= G(v)v −k

2

∫

v

v

β(s)g(s)ds −1

2(G(v) − k) β(v) +

1 − k

2

∫ 1

v

β(s)g(s)ds

= G(v)v −1

2(G(v) − k) β(v) +

1

2

∫ 1

v

β(s)g(s)ds −k

2

∫ 1

v

β(s)g(s)ds. (9)

Denote parts of (9) as follows:

A : = (G(v) − k) β(v), B :=∫ 1

vβ(s)g(s)ds, C :=

∫ 1

vβ(s)g(s)ds.

12

Inserting (4), we get

A = G(v)v − kv −1

G(v) − k

∫ v

G−1(k)(G(t) − k)2dt,

B =∫ 1

v

s −∫ s

G−1(k)

(

G(t) − k

G(s) − k

)2

dt

g(s)ds

=∫ 1

vsg(s)ds −

∫ 1

v

∫ s

G−1(k)(G(t) − k)2dt

g(s)

(G(s) − k)2ds

=∫ 1

vsg(s)ds +

∫ 1

v

∫ s

G−1(k)(G(t) − k)2dtd

1

(G(s) − k)

= 1 − vG(v) −∫ 1

vG(s)ds +

1

1 − k

∫ 1

G−1(k)(G(t) − k)2dt

−1

G(v) − k

∫ v

G−1(k)(G(t) − k)2dt −

∫ 1

v(G(s) − k)ds

= 1 − vG(v) − 2∫ 1

vG(s)ds + k(1 − v) +

1

1 − k

∫ 1

G−1(k)(G(t) − k)2dt

−1

G(v) − k

∫ v

G−1(k)(G(t) − k)2dt,

C =∫ 1

v

s −∫ s

G−1(k)

(

G(t) − k

G(s) − k

)2

dt

g(s)ds

=∫ 1

vsg(s)ds +

∫ 1

v

∫ s

G−1(k)(G(t) − k)2dtd

1

G(s) − k

= 1 −∫ 1

vG(s)ds +

1

1 − k

∫ 1

G−1(k)(G(t) − k)2dt

−1

k

∫ G−1(k)

v(G(t) − k)2dt −

∫ 1

v(G(s) − k)ds

= 1 − 2∫ 1

vG(s)ds + k(1 − v) +

1

1 − k

∫ 1

G−1(k)(G(t) − k)2dt

−1

k

∫ G−1(k)

v(G(t) − k)2dt.

Inserting A, B and C back into (9) and simplifying, we get (5).

Proof of Lemma 2. W.l.o.g. suppose partner 2 plays the equilibrium candidate. We show

that partner 1 cannot deviate profitably. We distinguish two cases. In the first case,

partner 1 is the partner who has played D, i.e., the action profile is (a1, a2) = (D, N). In

the second case partner 2 has proposed, i.e., the action profile is (a1, a2) = (N, D).

13

1. (a1, a2) = (D, N). Given that partner 2 bids b2 = v and partner 1 is favored by

the tie-breaking rule, partner 1 wins the auction with any bid b1 ∈ [v, 1] and loses with

b1 < v. The payoff of partner 1 from winning is uw = v1 − (kv + (1 − k)b1)/2 and his

payoff from losing is ul = (kb1 + (1 − k)v)/2.

We differentiate k ∈ (0, 1), k = 0, and k = 1. For k ∈ (0, 1), uw decreases in b1. Its

maximum value equals v1 − v/2 at b1 = v. ul increases in b1 and is less than v/2 as losing

means b1 < v. Since v1 ≥ v, uw > ul. Winning dominates losing for partner 1. Therefore,

partner 1’s best response to b2 = v is indeed b1 = v.

If k = 1, uw = v1 − v/2 is independent of b1 given that b1 ≥ v. ul = b1/2 increases

in b1 and is less than v/2 as losing means b1 < v. Since v1 ≥ v, uw > ul and winning

dominates losing. For partner 1, any bid b1 ≥ v is a best response to b2 = v and hence

bidding b1 = v is also a best response for partner 1.

If k = 0, uw = v1 − b1/2 decreases in b1, and its maximum value v1 − v/2 is reached

at b1 = v. ul = v/2 and is independent of b1 given that b1 < v. Since v1 ≥ v, winning

dominates losing and bidding b1 = v is a best response for partner 1 to b2 = v.

Thus, for k ∈ [0, 1], in equilibrium, partner 1 wins by the tie-breaking rule with payoff

v1 − v/2. Partner 2’s payoff is v/2.

2. (a1, a2) = (N, D). Given that partner 2 bids b2 = v and is favored by the tie-

breaking rule, partner 1 wins the auction with bids b1 ∈ (v, 1], with payoff uw = v1 −

(kv + (1 − k)b1)/2, and loses with bids b1 ∈ [0, v], with payoff ul = (kb1 + (1 − k)v)/2.

If k ∈ (0, 1), uw is decreasing in b1 and since b1 > v in case of winning, uw < v1 − v/2.

ul increases in b1 and takes its maximum value v/2 at b1 = v. Since v1 < v, uw < ul

and losing dominates winning for partner 1. Hence, bidding b1 = v is a best response to

b2 = v.

If k = 1, uw = v1 − v/2 is independent of b1. The losing payoff ul = b1/2 takes its

maximum value v/2 at b1 = v. Since v1 < v, uw < ul and losing dominates winning for

partner 1. Hence, bidding b1 = v is a best response to b2 = v.

If k = 0, uw = v1 −b1/2 decreases in b1. Since b1 > v in case of winning, uw < v1 − v/2.

ul = v/2 is independent of b1. Since v1 < v, uw < ul and losing dominates winning for

partner 1. Hence, any b1 ≤ v, including b1 = v, is a best response to b2 = v.

Thus, for k ∈ [0, 1], in equilibrium partner 2 wins by the tie-breaking rule and obtains

a payoff v2 − v/2. Partner 1 obtains a payoff v/2.

It remains to be shown that there is no other equilibrium of the asymmetric auction

game that can possibly induce ex post efficiency for the whole game.

14

Given that both partners play cutoff strategies, it is w.l.o.g. to consider the case v1 ∈

[0, v) and v2 ∈ [v, 1]. Ex post efficiency requires that partner 2 always wins the auction,

i.e., β1(v1) ≤ β2(v2), where partner 2 wins by the tie-breaking rule if β1(v1) = β2(v2). We

differentiate three cases: k ∈ (0, 1), k = 0, and k = 1. In each case, we first show that in

an efficient equilibrium candidate, the partners necessarily submit the same bids. Then

we show that the bid equals v.

In the following, we denote a constant bid by b and a bid function by β. Suppose

k ∈ (0, 1). For partner 2, β2 > βsup1 := supv1

β1(v1) is strictly dominated by β2 = βsup1

since partner 2’s payoff, v2 − (kβ1 + (1 − k)β2)/2 is strictly decreasing in β2 and he is

favored by the tie-breaking rule. Thus, β2 = βsup1 is the unique best response to partner

1’s strategy β(v1).

Analogously, partner 1’s payoff, (kβ1 + (1 − k)β2)/2, is strictly increasing in β1. Thus,

b1 = βsup1 is the unique optimal bid, independent of β2. Thus, both partners must make

the same bid which we denote as b∗, and partner 2 wins by the tie-breaking rule. The

payoffs are v2 − b∗/2 for the winning partner 2 and b∗/2 for the losing partner 1.

Next, we argue that b∗ = v. First, suppose per absurdum that b∗ = v + s for some

s > 0 and restrict attention to v2 ∈ [v, v + s). If both bid b∗ = v + s, then partner 2’s

payoff is v2 − (v + s)/2 < v2/2. If, however, partner 2 deviates to b′2 = v2 then his payoff

is (kv2 + (1 − k)(v + s)/2 > v2/2, a profitable deviation. Thus, we can rule out b∗ > v.

Second, suppose per absurdum that b∗ = v − s for some s > 0 and restrict attention to

v1 ∈ (v − s, v]. If both bid b∗ = v − s, then partner 1’s payoff is (v − s)/2 < v1/2. If,

however, partner 1 deviates to b′1 = v1 then his payoff is v1−(k(v−s)+(1−k)v1)/2 > v1/2,

a profitable deviation. Thus, we can rule out b∗ < v. Finally, for b∗1 = b∗

2 = v there is no

such profitable deviation.

Now suppose k = 0 (WBA). As above, by efficiency we must have β1(v1) ≤ β2(v2)

which implies that partner 2 always wins. The payoff of partner 2, v2 −β2(v2)/2, is strictly

decreasing in his bid. Thus, given partner 1’s strategy β1(v1), partner 2’s optimal bid is

a constant, equals b∗2 = βsup

1 := supv1β1(v1). Now we show that βsup

1 = v. Suppose b∗2 =

βsup1 = v−s for some s > 0. Then partner 1 prefers winning if v1 −(v−s+ε)/2 > (v−s)/2

for some ε > 0 which, since v1 < v, is equivalent to v1 ∈ (v − s + ε/2, v). In that case,

partner 1 would bid more than b∗2 = βsup

1 , a contradiction. Thus, b∗2 < v can be ruled out.

Similarly, b∗2 = βsup

1 = v + s for some s > 0 can be ruled out since bidding b2 = v2 is a

more profitable bid for partner 2 if v2 ∈ [v, v + s). Finally, for b∗2 = v there is no such

incentive.

15

Now we show that b∗1 = v as well. In order to do that we suppose per absurdum that

partner 1 bids such that there is positive probability mass on bids below v. We show that

partner 2 can then profitably deviate from the bid b∗2 = v.

We consider the type v2 = vt := v + t with t > 0. First note that the status quo profit

of this partner 2 type from bidding b∗2 = v is vt − v/2 = v/2 + t. We want to show that

the deviation profit is above that.

Now suppose this partner 2 type bids a constant bid of b2 = v − s for some s > 0.

Then his expected profit is

pv

v

2+ pW

(

vt −v − s

2

)

+ (1 − pW )(

1

2E[β1(V1)|V1 ∈ [0, v), β1 > v − s]

)

,

where pv ≥ 0 is the probability that partner 1 bids b1 = v and pW is partner 2’s probability

of winning in all other cases (note that pW is a function of s).

Now insert vt = v + t and note that partner 1’s expected bid above satisfies β1 > v − s

as in the relevant events partner 1 wins. Thus, partner 2’s expected profit exceeds

pv

v

2+ pW

(

v + t −v − s

2

)

+ (1 − pW )(

v − s

2

)

This can be written as

(

v

2+ t

)

+(

−t(1 − pW ) + pv

v

2+(

pW −1

2

)

s)

.

The first parenthesis is the status quo profit. We want to know if the second parenthesis

is positive. It is positive if

t <1

1 − pW

(

pv

v

2+(

pW −1

2

)

s)

Obviously, for pW > 12

the RHS is positive (even if pv = 0). Thus, the types v2 ∈ [v, v + t]

can profitably deviate. Now choose s > 0 such that pW > 1/2.

Now suppose k = 1 (LBA), and, as above and w.l.o.g. we only consider the case

(N, D). As above, by efficiency we must have β1(v1) ≤ β2(v2) which implies that partner

1 is always the loser. His payoff, β1(v1)/2, is strictly increasing in his bid. Thus, his

optimal bid is a constant, b∗1 = β inf

2 := infv2β2(v2). Now we show that b∗

1 = β inf2 = v.

Suppose b∗1 = β inf

2 = v − s for some s > 0. Then, if v1 ∈ (v − s, v) bidding b1 = v1 is more

profitable for partner 1 regardless of whether he wins or loses with that bid. Thus, b∗1 < v

16

can be ruled out.

Similarly, b∗1 = β inf

2 = v + s for some s > 0 can be ruled out, since if v2 ∈ [v, v + s)

partner 2 would prefer bidding b2 = v2 regardless of whether he wins or loses with that

bid. Finally, for b∗1 = v there is no such incentive.

Now, similarly to the case k = 0 one can show that there exists t > 0 such that for all

v1 ∈ [v−t, v] there is a s > 0 such that bidding b1 = v+s is more profitable in expectation

than b∗1 = v whenever partner 2 bids more than b2 = v with positive probability.

Summing up, there is no other equilibrium of the asymmetric auction game, that can

possibly induce ex post efficiency.

Proof of Lemma 3. By symmetry of the game, it is w.l.o.g. to restrict attention to part-

ner 1’s incentives to deviate while taking as given that partner 2 plays the proposed

equilibrium strategy of calling for a dissolution if and only if v2 ≥ v.

We distinguish two cases, (L), if v1 ∈ [0, v), and (H), if v1 ∈ [v, 1]. A partner’s expected

payoff at the dissolution stage depends on his private valuation and the proposal stage

action profile. We denote partner 1’s expected profit at the dissolution stage by π(a1,a2)1k ,

where k ∈ {L, H}, (a1, a2) ∈ {D, N} × {D, N}.

(L) Suppose v1 ∈ [0, v). If partner 1 proposes dissolution, D, then his payoff depends

on whether partner 2 also proposes (which happens with probability 1−F (v)) or not (with

probability F (v)). Partner 1’s corresponding payoffs are π(D,D)1L and π

(D,N)1L , respectively.

If, however, partner 1 does not propose, then his corresponding payoffs are either π(N,D)1L

or π(N,N)1L . Since partner 1’s value is low, according to the candidate, partner 1 must prefer

N over D,

F (v)π(N,N)1L + (1 − F (v))π

(N,D)1L ≥ F (v)π

(D,N)1L + (1 − F (v))π

(D,D)1L . (10)

The LHS of (10) is partner 1’s payoff from playing the equilibrium candidate, where

π(N,N)1L = C/2, since the partnership continues, and π

(N,D)1L = v/2 by Lemma 2.

The RHS of (10) is partner 1’s payoff from unilateral deviation. Consider π(D,N)1L . By

Lemma 2, partner 2 bids b2 = v and partner 1 is favored by the tie-breaking rule. Thus,

partner 1 wins with bids b1 ∈ [v, 1], with payoff v1 − (kv +(1−k)b1)/2. This is maximized

at b1 = v with payoff v1 − v/2 which is less than v/2. Partner 1 loses with bids b1 ∈ [0, v),

with payoff (kb1 + (1 − k)v)/2. This is increasing in b1 and, since b1 < v, bounded from

above by v/2. There exist sufficiently small ε > 0 such that the losing payoff from bidding

17

b1 = v − ε is larger than the winning payoff. Since condition (10) needs to hold for every

ε > 0, we use π(D,N)1L = v/2 in (10).

Now consider π(D,D)1L . In the following we first show that β(v) is partner 1’s best bid

among all bids β(x) for x ∈ [v, 1], i.e., bids in the range of the equilibrium bid function,

(4). Then we show that there is no bid that is more profitable than β(v).

Suppose partner 1 bids β(x) with x ∈ [v, 1], where β is given by (4), while partner 2

bids according to (4). Then partner 1’s expected utility and its derivative are

u1(x) = G(x)(

v1 −1

2(kE[β(V2)|V2 ∈ [v, x]] + (1 − k)β(x))

)

+ (1 − G(x))1

2(kβ(x) + (1 − k)E[β(V2)|V2 ∈ [x, 1]])

= G(x)v1 −1 − k

2β(x)G(x) −

k

2

∫ x

vβ(s)g(s)ds

+ (1 − G(x))k

2β(x) +

1 − k

2

∫ 1

xβ(s)g(s)ds.

u′1(x) = g(x)v1 − β(x)g(x) − β ′(x)

G(x) − k

2.

Note that β ′(x) can be written as

β ′(x) =2g(x)

G(x) − k(x − β(x))

This allows us to simplify u′1(x) to

u′1(x) = g(x)(v1 − x).

Since v1 < x ≤ v, β(v) is the most profitable bid for partner 1 of all bids from the range

[β(v), β(1)]. Note that with this bid partner 1 is the sure loser of the auction. For all

k ∈ (0, 1], the payoff of the loser is increasing in his own bid. Thus, there is no lower

bid that is more profitable for partner 1. For the case k = 0, all lower bids are equally

profitable. Thus, β(v) is the most profitable of all bids. Note that bids above β(1) are

dominated by β(1) since these bids also win for sure but cannot have a larger payoff.

Thus, π(D,D)1L is partner 1’s expected profit if he bids β(v) and partner 2 bids β(v2). That

18

profit can be computed as

π(D,D)1L =

1

2(kβ(v) + (1 − k)E [β(V2)|V2 ∈ [v, 1]])

=1

2

(

kv −1

k

∫ v

G−1(k)(G(t) − k)2dt

)

+(1 − k)

2

∫ 1

v

s −∫ s

G−1(k)

(

G(t) − k

G(s) − k

)2

dt

g(s)ds. (11)

Note that∫ 1

vsg(s)ds = 1 −

∫ 1

vG(s)ds. (12)

Also

∫ 1

v

∫ s

G−1(k)

(

G(t) − k

G(s) − k

)2

dtg(s)ds

= −∫ 1

v

∫ s

G−1(k)(G(t) − k)2dtd

1

G(s) − k

= −1

1 − k

∫ 1

G−1(k)(G(t) − k)2dt +

1

k

∫ G−1(k)

v(G(t) − k)2dt +

∫ 1

v(G(t) − k)dt. (13)

Inserting (12) and (13) into (11) and simplifying, we get

π(D,D)1L =

1

2−∫ 1

vG(t)dt +

1

2

∫ 1

vG2(t)dt. (14)

Comparing (14) with (5), note that π(D,D)1L = U(v).

Using the above results, we rewrite condition (10) as

F (v)C

2+ (1 − F (v))

v

2≥ F (v)

v

2+ (1 − F (v))U(v). (15)

(H) Suppose v1 ∈ [v, 1]. Partner 2 plays N with probability F (v) and D with prob-

ability 1 − F (v). The equilibrium requires that partner 1’s payoff from D is at least as

high as the payoff from playing N . Therefore, we have

F (v)π(D,N)1H + (1 − F (v))π

(D,D)1H ≥ F (v)π

(N,N)1H + (1 − F (v))π

(N,D)1H (16)

The LHS of (16) represents partner 1’s expected payoff if he proposes dissolution as

prescribed by the equilibrium candidate. There, by Lemma 2, π(D,N)1H = v1 − v/2 and

π(D,D)1H is given by (5).

19

The RHS of (16) is partner 1’s expected payoff from unilateral deviation. There,

π(N,N)1H = C/2 since the partnership continues after action profile (N, N). It remains to

determine π(N,D)1H . Partner 2 bids b2 = v and is favored by the tie-breaking rule. Thus,

partner 1 wins with bids b1 > v and loses with b1 ∈ [0, v]. Partner 1’s winning payoff,

v1 − (kv + (1 − k)b1)/2, is nonincreasing in b1, and is bounded from above by v1 − v/2.12

Partner 1’s payoff from losing, (kb1 + (1 − k)v)/2, is nondecreasing in b1 and, therefore,

it is maximized at b1 = v with value equal to v/2. Since v1 ≥ v we find that for v1 = v,

losing is (marginally) more profitable with payoff v/2 which, in that case, can be written

as v1 − v/2. Otherwise, for any v1 > v, there exists sufficiently small ε > 0 such that

winning with a bid of b1 = v + ε is more profitable than losing, with payoff marginally

below v1 − v/2 but above v/2. Since condition (16) must hold for every ε > 0, we conclude

that π(N,D)1H = v1 − v/2 in (16).

Inserting the above results in (16) and simplifying, we get

F (v)(

v1 −v

2

)

+ (1 − F (v))U(v1) ≥ F (v)C

2+ (1 − F (v))

(

v1 −v

2

)

. (17)

Since G(v1) = F (v1)−F (v)1−F (v)

(for v1 ∈ [0, 1]), U(v) is equal to

U(v) =v

2+

1

2(1 − F (v))2

∫ 1

v(1 − F (t))2dt. (18)

Furthermore, U(v) (see (5)) can be written as:

U(v1) =1

2+

1

2

∫ 1

v

(

F (t) − F (v)

1 − F (v)

)2

dt −∫ 1

v1

F (t) − F (v)

1 − F (v)dt

= v1 −v

2+

1

2(1 − F (v))2

∫ 1

v(1 − F (t))2dt −

1

1 − F (v)

∫ vi

v(1 − F (t))dt (19)

Using (18) to simplify and rearrange (15), we get (20) below, and using (19) to simplify

and rearrange (17), we get (21). Naturally, these conditions apply to both partners.

C ≥ v +∫ 1

v

(1 − F (t))2

F (v)(1 − F (v))dt, ∀vi ∈ [0, v), (20)

C ≤ 2vi − v −∫ vi

v2

1 − F (t)

F (v)dt +

∫ 1

v

(1 − F (t))2

F (v)(1 − F (v))dt, ∀vi ∈ [v, 1]. (21)

In a cutoff equilibrium, condition (20) must hold for all vi ∈ [0, v) while condition (21)

12When k = 1, the winning payoff is always v1 − v/2, independent of b1.

20

needs to hold for all vi ∈ [v, 1]. Therefore, (21) also needs to hold for vi = v. Inserting

vi = v into (20) and (21), we see that the two conditions hold simultaneously if and only

if (8) holds.

Second, since (21) also needs to hold for any vi > v, taking the above into account,

we need that the RHS of (21) increases in vi for vi ∈ (v, 1]. The derivative of the RHS

of (21) with respect to vi is 1 − 1−F (vi)F (v)

, which simplifies to F (v)+F (vi)−1F (v)

. This is positive

if and only if v ≥ vm. (Suppose v < vm where F (v) < 12. Then there exists ε > 0 and

vi = v + ε such that F (v) + F (vi) − 1 < 0.) Hence, any v < vm cannot satisfy (21) for all

vi ≥ v. Thus, v ≥ vm is a necessary condition for the existence of a cutoff equilibrium.

Third, taking the derivative of the RHS of (8) with respect to v, we get:

1−f(v)(1 − 2F (v))

F 2(v)(1 − F (v))2

∫ 1

v(1 − F (t))2dt −

1 − F (v)

F (v)

=2F (v) − 1

F (v)+

f(v)(2F (v) − 1)

F 2(v)(1 − F (v))2

∫ 1

v(1 − F (t))2dt. (22)

(22) is negative if v < vm and positive if v > vm and which implies that the RHS of (8)

is decreasing in v for v < vm and increasing in v at v > vm. Therefore, the RHS reaches

its minimum value vm + 4∫ 1

vm (1 − F (t))2 dt at v = vm. For v > vm, the RHS increases,

reaching its maximum value 1 at v = 1. We already know that v ≥ vm is a necessary

condition for the existence of a cutoff equilibrium. Therefore, (8) is satisfied if and only

if v ≥ vm.

Finally, if v ≥ vm, the domain of the RHS of (8) is [vm + 4∫ 1

vm (1 − F (t))2 dt, 1].

Therefore, we conclude that the cutoff equilibrium exists and is unique if and only if

C ∈ [vm + 4∫ 1

vm (1 − F (t))2 dt, 1]. For C < vm + 4∫ 1

vm (1 − F (t))2 dt, there is no positive

v that satisfies both (20) and (21), and therefore a cutoff equilibrium does not exist.

Proof of Proposition 2. Ex post efficiency requires that the partners play cutoff strategies

with common cutoff v = C. From Lemma 3 we know the game has a unique cutoff equi-

librium with v < C assuming the asymmetric continuation equilibrium derived in Lemma

2 is played. But Lemma 2 states that there is no other equilibrium of the asymmetric

auction that can possibly induce ex post efficiency.

Proposition 3 (Reserve Price). Suppose that the dissolution mechanism is a WBA with

reserve price C. Then, the game has PBNE such that:

1. Partners always play D at the proposal stage.

21

2. At the dissolution stage, both partners bid according to

βC(vi) =

C − ǫ if vi < C;

vi −∫ vi

CF (t)2

F (vi)2 dt if vi ≥ C(23)

where C − ǫ is infinitesimally below C.

This equilibrium is ex post efficient.

Proof of Proposition 3. The proof of the bid function, (23), was given in McAfee (1992,

p.272, Lemmas 1 and 2) in a different context and hence is omitted. More precisely,

McAfee (1992) introduces an outside option to sell the item to a third party for a known

market price instead of awarding sole ownership to one of the partners. In our setting the

continuation value corresponds to the outside option.

The remaining part of the proof follows immediately from the fact that proposing can-

not make a partner worse off than not proposing. At the dissolution stage, the partnership

is only dissolved if the high bid is above the reserve C. From (23), βC(C) = C and a

partner only bids above C if he has a value above C. Therefore, the partnership is only

dissolved if at least one of the partners has a value above C, which is ex post efficient.

At the proposal stage, it is a weakly dominant strategy for the partners to propose dis-

solution because the auction has a larger expected payoff than continuation: the auction

payoff is either C/2 or a larger value while continuation has a payoff of C/2. Proposing

dissolution guarantees that the auction takes place while not proposing leaves this open

for the other partner to decide.

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