Political Institutions and Investment in PublicInfrastructure1

Marco Battaglini2 Salvatore Nunnari3

Thomas R. Palfrey4

February 2009 (Current Version June 1, 2009) VeryPreliminary and Incomplete.

1We thank Dustin Beckett for excellent assistance. We gratefully acknowledgefinancial support from NSF (SES-0547748 and SES-0617820) and The Gordon andBetty Moore Foundation.

2Department of Economics, Princeton University, Princeton, NJ 08544. mbat-tagl@princeton.edu

3Division of the Humanities and Social Sciences, California Institute of Tech-nology, Pasadena, CA 91125.snunnari@hss.caltech.edu

4Division of the Humanities and Social Sciences, California Institute of Tech-nology, Pasadena, CA 91125. trp@hss.caltech.edu

Abstract

We present a theoretical model of the provision of a durable public good overan infinite horizion. In each period, there is a societal endowment of whicheach of n districts owns a share. This endowment can either be invested inthe public good or consumed. We characterize the planner’s optimal solutionand time path of investment and consumption. We then consider alterna-tive political mechanisms for deciding on the time path, and analyze theMarkov perfect equilibrium of these mechanisms. One class of these mech-anisms involves a legislature where representatives of each district bargainwith each other to decide how to divide the distribute current period’s so-cietal endowment between investment in the public good and transfers toeach district. The second class of mechanisms involves the districts makingindependent decisions for how to divide their own share of the endowmentbetween consumption and investment. We conduct an experiment to assessthe performance these mechanisms, and compare the observed allocations tothe Markov perfect equilibrium.Keywords: Durable public goods, legislative bargaining, dynamics, voting,experiments.JEL Classification: D71, D72, C78, C92, H41, H54

1 Introduction

Most public goods that are provided by governments have a dynamic nature:it takes time to accumulate them; and they depreciate slowly, projectingtheir benefits for many years. Prominent examples are national defense, en-vironmental protection and public infrastructure. Considering the dynamicnature of public goods is particularly important when public policies are nottaken by a benevolent planner, but are the result of the dynamic politicalstruggle of many constituencies that can not be assumed to be benevolent,and therefore depend on the details of the institutional setting. The insti-tutional environment does not only determine the extent to which the policywill reflect the welfare of the citizens when the policy is chosen: but it alsodetermines the extent to which it will internalize the benefits that will accruein future periods; in a word, how ”shortsighted” the policy is. The dynamicnature of public goods and the institutional setting, therefore can not bestudied separately: on the one hand, the institutional setting will determinethe shortsightedness of the policy and the nature of the free rider problem;on the other hand, the dynamic free rider problem should be an importantfactor in evaluating the institutional setting.In this paper we present a comparative study of two alternative institu-

tional frameworks in the provision of dynamic public goods. We first proposea theoretical model and then use it to design a laboratory experiment to testthe theory and collect empirical data on how the free rider problem impactpublic good provision. Experimental analysis is particularly important whenstudying dynamic problem, and can not be easily replaced by field data: thisbecause strategic behavior can be observed only if there is a precise measure-ment of the ’state variable” etc. (to be continued).The economy that we study is comprised by a continuum of infinitely

lived citizens who live in n districts. The policy making problem consists inusing taxation, targeted transfers to the districts and public good provision toallocate resources between a general public good that benefits all districts,and private consumption in the districts. The public good depreciates arate d < 1. We consider two institutional mechanisms in which the policycan be taken. In the first, the policy is taken buy a centralized body, theLegislature, composed by representatives from all districts. The legislatureis endowed with the power to tax and allocate revenues. Representativesbargain in the legislature over the allocation of resources. The second is apurely decentralized mechanism, which we call Autarky, whereby each district

1

retains full property rights over its endowment and in each period chooses onits own how to allocate its endowment between investment in the public good(which is shared by all districts) and private consumption, taking as given thestrategies of the other districts. The total economy-wide investment in thepublic good in any period is then given by the sum of the district investments.For both these mechanisms, we characterize the public policies that will resultfrom a symmetric Markov perfect equilibrium and we compare them withthe optimal policy of an utilitarian planner.The predictions of these models are used to evaluate the experimental

data, etc. (to be continued).This work is related to two strand of the literature that previously studied

public good provision in dynamic settings. First there is the literature thatlooked at investment in public goods in decentralized environment in whicheach citizen individually and voluntarily Second of all there is the work oflegislative bargaining.There is a large literature on the inefficiencies caused by the free rider

problem associated with public goods provision in market economies. Theseinefficiencies have been used to justified an increasingly important role ofgovernments in providing a wide range of public goods: pure public goodslike national defense and environmental protection, but also other partiallyexcludable goods like bridges and other types of infrastructure, and othertypes of expenditures that induce important externalities and can not strictlyspeaking be considered public goods (like social security) The inefficienciesthat are associated with public prevision of these goods, however, have beenstudied less extensively. Two main features characterizes governments’ ac-tivity in public good provision. First, most (if not all) public goods areinvestment goods that are accumulated over time and that, once accumu-lated, will last for many periods. Second public good provision depends onthe institutional setting and cannot be assumed to be benevolent. These twofeatures are interrelated and must be studied in connection with eacy other.The institutional environment does not only determine the incentive of thepolicy make in internalizing the utility of the public good of the citizens, butalso determines the incentives to internalize the benefits that will accrue infuture periods, that is how ”shortsighted” the government is.[intro under construction]

2

2 The model

Consider an economy in which a continuum of infinitely lived citizens livein n districts and each district contains a mass one of citizens. Thereare two goods: private good x and a public good g. An allocation is aninfinite nonnegative sequence public policies, z = (x∞, g∞) where x∞ =(x11, ..., x

n1 , ..., x

1t , ..., x

nt , ...) and g∞ = (g1, ..., gt, ...). We refer to zt = (xt, gt)

as the public policy in period t. The utility U i of a representative citizenin district i is a function of zi = (xi∞, g∞), where x

i∞ = (xi1, ..., x

it, ...) We

assume that U i can be written as:

U i(zI) =∞Xt=1

δt−1£xit + u(gt)

¤,

where u(·) is continuously twice differentiable, strictly increasing, and strictlyconcave on [0,∞), with limg→0+ u

0(g) = ∞ and limg→∞+ u0(g) = 0. Thefuture is discounted at a rate δ.There is a linear technology by which the private good can be used to

produce public good, with a marginal rate of transformation p. While theprivate consumption good is nondurable, the public good is durable, and thestock of the public good depreciates at a rate d ∈ [0, 1] between periods.Thus, if the level of pubic good at time t − 1 is gt−1 and the investment ofprivate goods in public goods is It, then the level of public good at time twill be

gt = (1− d)gt−1 + It.

Because all citizens in district i are identical, we refer collectively to the“behavior of a district” as described by the behavior of a representative citi-zen i. Henceforth we will simply refer to district i. In period t, each districti is endowed with wi

t units of private good, and we denote W t=Pn

i=1wit.

Except where noted, we will restrict attention in this paper to symmetriceconomies, where wi

t =Wt

n ∀i, and W t=W∀t. The initial stock of public goodis g0 ≥ 0, a exogenous constant.The public policy in period t is required to satifsy three feasibility condi-

tions:

xit ≥ 0 ∀i

It ≥ −gt−1 ∀t

It +nXi=1

xit ≤ W t

3

Conditions 1 and 2 guarantee that allocations are nonnegative. Condition3 requires that the current economy-wide budget be balanced. These condi-tions can be rewritten slightly. If, therefore, we denote y ≡ gt = (1−d)gt−1+Itas the new level of public good after an investment It when the last period’slevel of the public good is gt−1, then the public policy in period t can berepresented a vector (y, x1t , ..., x

nt ).

Dropping the t subscripts and substituting y, the budget balance con-

straint It +nXi=1

xit ≤W t can be rewritten as:

nXi=1

xi + p [y − (1− d)g] ≤W,

recalling that we use y to denote the post-investment level of public goodattained in period t, and (1 − d)g the pre-investment level of public goodinherited from period t−1. The one-shot utility to district i from this publicpolicy, (y, x1, ..., xn), is U i = xi + u(y).Our interest in this paper is to compare the performance of different mech-

anisms for building public infrastructure, i.e., generating a feasible sequenceof public policies, z. While more general formulations are possible, we willconsider mechanisms that are time independent and have no commitment.That is, the mechanism is played in every period, the rules of the mechanismare the same in every period, and the outcome of the mechanism is a publicpolicy for only the current period.1 In such mechanisms we will study theoutcomes associated with symmetric Markov perfect equilibria. Before pro-ceding to the analysis of equilibria in mechanisms, we first characterize theoptimal planner’s solution to this dynamic allocation problem, which pro-vides a natural benchmark against which the different mechanisms can becompared.

3 The planner’s problem

As a benchmark with which to compare the equilibria in mechanisms, we firstanalyze the sequence of public policies that would be chosen by a benevolentplanner who maximizes the sum of utilities of the districts. This is the

1We discuss more general mechanisms at the end of the paper.

4

welfare optimum in this case because the private good enters linearly in eachdistrict’s utility function. To solve this problem, we look a the dynamicprogram for the planner to solve, where the state space is the level of thepublic good, g. We hypothesize the existence of a concave and differentiablevalue function for the planner, vp(g), which is the discounted present valueof the optimal planner’s solution if the initial state is g.2 The Planner’soptimization problem at state g can be written as:

maxy,x

½X + nu(y) + δvp(y)

s.t X + y − (1− d)g ≤W, x ≥ 0 , y ≥ 0

¾(1)

where X =Pn

i=1 xi. is the sum of private transfers to the districts. Sincethe we can assume without loss of generality that the budget constraint,X + y − (1 − d)g ≤ W , is binding and the y ≥ 0 is never binding,3 we canrewrite the problem as:

maxy

½W + (1− d)g − y + nu(y) + δvp(y)

X =W + (1− d)g − y ≥ 0.

¾(2)

Next we solve for the optimal policy, byp(g). Depending on the state g,there are two possible cases depending on whether the constraint is binding.In the first case, where it is binding, the planner would like to invest anamount I (which equals y− (1 − d)g, by definition) greater than W butcannot because of the constraint: i.e., the planner would like to choose y >W + (1 − d)g. Thus, the solution is this case is byp(g) = W + (1 − d)g,

so bXp(g) = 0. In the second case, the constraint is not binding and the

unconstrained optimization yields byp(g) ≤ W + (1− d)g > y bXp(g) > 0. Inthis case a necessary condition for byp(g) is characterized by the first orderequation:

nu0(byp(g)) + δv0p(byp(g)) = 1However, by the concavity of u and vp, the second order condition is satisfied,and furthermore the first order condition has a unique solution for byp(g),independent of g, which we denote y∗p, and call the steady state solution forthe planner’s problem.

2After characterizing the solution, we verify that indeed the value function is con-cave, and differentiable. [[[from old version: ??As it can be verified, (2) is a contractionand defines a unique equilibrium value function that is continuous, strictly concave anddifferentiable in g.??]]]

3The constraint y ≥ 0 is never binding because limg→0+ u0(g) =∞.

5

This implies the following very simple rule-of-thumb optimal policy forinvesting in the public good as a function of its current level g. Thus, forany values of g such that y∗p− (1−d)g ≤W , invest Ip(g) = y∗p− (1−d)g and

private good consumption is bXp(g) =W +(1− d)g− y∗p. For any values of gsuch that y∗p−(1−d)g > W , invest Ip(g) =W and private good consumption

is bXp(g) = 0. This second case is possible only if y∗p − (1− d)g ≥ W , i.e., if

g is lower or equal to a threshold gp:

gp = max

½y∗p −W

1− d, 0

¾The value function is easily derived from this characterization of the

optimal policy. A particularly simple case arises if y∗p ≤ W . Then thebudget constraint is never binding and byp(g) = y∗p for all g. thus vp(g) =W + (1 − d)g − y∗p+ u(y∗p) + δvp(y

∗p) and therefore vp is actually linear in g

and hence continuously differentiable and concave. Note that for g = y∗p, we

have vp(y∗p) =W − dg+ u(y∗p) + δvp(y

∗p), or vp(y

∗p) =

W−dg+u(y∗p)1−δ .

Derivation of vp is only slightly more involved if y∗p > W . In this case,

the budget constraint is initially binding, until we pass gp. Above gp, thevalue function is linear and hence concave. Below gp we have v

00p(g) = u00(g+

W ) + δv00p(g +W ) < 0. Thus, for all g, v00p(g) < 0 as long as v00p(g +W ) ≤ 0.

It follows immediately that v00p ≤ 0 for all g ≥ gp −W , and therefore v00p ≤ 0for all g ≥ gp − 2W and so forth. We summarize the above argument asProposition 1, below:

Proposition 1. The optimal solution to the planner’s problem is uniquelycharacterized by a steady state y∗p solving nu0(byp(g))+ δv0p(byp(g)) = 1 and aninvestment policy function ip(g) = min

©W,y∗p − (1− d)g

ª.

This implies that the optimal public policy at time t if the public goodlevel at t− 1 was g is given by ezt(g) = ((ext(g), eyt(g)) ,where:

eyt(g) = min©W + (1− d)g, y∗p

ª(3)

and (4)

exit(g) =W−min

©W,y∗p − (1− d)g

ªn

for all i (5)

This has a clear intuition. When the state g is sufficiently low, the plan-ner invests all currently available resources (W ) in the public good: this is

6

y*p

gp

W1-d

gp/(1-d)

y*p

gp

W

gp/(1-d)

Figure 1: The Planner’s problem

because the marginal value of an additional unit of the public good is so highthat each dollar invested in the public good yields more than a dollar in value.

For g >y∗p−W1−d , however, the marginal value of the public good investment is

lower than 1, so it is more valuable to leave some resources to the districtsfor private consumption. In this region of g, he planner expends resourcesjust enough to cover depreciation and maintain the level of the public goodat y∗p, where the marginal value of investment is exactly 1. Note that if the

stock of public good at period t − 1 is, for some reason,4 equal to g >y∗p1−d

then optimal investment is negative in period t. For future reference, whenwe will compare this solution with the equilibria in the political systems, it isinteresting to note that this investment function ip(g) is (weakly) monotonedecreasing in g.

3.1 Example

To illustrate the planner’s solution, we derive the analytical solution for thecase when the utility function is given by u(y) = 1

αyα. From Proposition 1,

one can see that there can actually be two types of equilibria, depending onhow high the optimal steady state,y∗p, is relative to the parameters of themodel {n,W, d, α, δ}. This is illustrated in Figure 1.

4For example, it could be that g0 >y∗p−W1−d .

7

The steady state is at the intersection point between the 45o line and theinvestment curve (3). The first case, shown in the left panel of the figure iswhen the steady state is gp = y∗p = y(gp) and, therefore, X(gp) > 0. Thesecond case is when the steady state gp satisfies gp = y(gp) < y∗p. In thiscase W + (1 − d)g = g, so gp = W/d and X(gp) = 0. In the Appendix weprove:

Proposition 2. Let u(y) = 1αyα. If 1 − δ(1 − d) > n

¡dW

¢1−α, the long

run steady state in the planner’s solution is gp =³

n1−δ(1−d)

´ 11−α

= y∗p. If

1− δ(1− d) ≤ n¡dW

¢1−αthe steady state is gp =

Wd< y∗p.

This result gives us a complete characterization of the equilibrium dy-namics and long term behavior. When the economy has relatively large

resources as measured by W (i.e. 1 − δ(1 − d) > n¡dW

¢1−α) eventually the

level of the public good will reach a saturation point at which its marginalvalue is equal to the marginal utility of consumption (one), and the steady

state is y∗p. When 1−δ(1−d) ≤ n¡dW

¢1−αthis optimal saturation point, y∗p,

will never be reached if one starts at g0 = 0. Depreciation is too high com-pared toW for the saturation point to ever be reached (W

d< y∗p). The actual

time path depends on initial conditions. If the economy starts at g0 >Wd, de-

preciation in the stock is greater thanW and gt declines over time to gp =Wd.

If g0 <Wdthen gp =

Wdis only asymptotically approached. For example, if

g0 = 0, then g1 =W , g2 =W + (1− d)W , g3 =W + (1− d)W + (1− d)2W ,and so forth. So gt =

Ptτ=0(1−d)τW < W

dfor all t. In both cases investment

in public goods is a non decreasing function of g: It is constant (= W ) for

low values of g ≤ g1 = maxny∗p−W1−d , 0

o, and strictly decreasing above g1.

An immediate corollary of Proposition 2 that will be useful in the exper-imental application is that if d = 0, then we are always in the first case, and

the steady state is¡

n1−δ¢ 11−α .

4 Political Mechanisms for Building Public

Infrastructure

The set of possible mechanisms to implement sequences of public policies isobviously huge. We limit ourselves to two different classes of mechanisms.The second class of mechanisms we consider are bargining mechanisms

8

for a centralized economy-wide representative legislature, which we call theLegislative mechanism. In these mechnsims, each district cedes its propertyrights over its share of the economy wide endowment in exchange for 1/nrepresentation in the legislature. In each period, the legislature decides ona uniform lump sum tax on all districts, which cannot exceed a district’sendowment, W/n, and a level of investment in the public good. The legisla-tive policy also includes an allocation of the budgetary surplus (tax revenueminus investment) to the districts, which is non-negative for all districts,but not necessarily uniform. Investment can be negative, but the amount ofnegative investment cannot exceed the current stock of public good. Thus,as before, we can represent a policy by the legislature at time t, by an publicpolicy (x1t , ..., x

nt , yt) that satisfies the same feasibility constraints as in the

planner’s problem.The second is a purely decentralized mechanism, which we call Autarky,

whereby each district retains full property rights over its endowment (Wn)

and in each period chooses on its own how to allocate its endowment betweeninvestment in the public good (which is shared by all districts) and privateconsumption, taking as given the strategies of the other districts. The totaleconomy-wide investment in the public good in any period is then given bythe sum of the district investments. We characterize the public policies thatwill result from the unique symmetric Markov perfect equilibrium of thismechanism.5

4.1 Centralized Provision: The Legislative (L) Mech-anism

To characterize behavior when policies are chosen by a legislature, we lookfor a symmetric Markov perfect equilibrium. In this type of equilibrium anyrepresentative selected to propose at some time t uses the same strategyand this depends only on the current stock of public good (g). Similarly,at the voting stage of a round τ , the probability a legislator votes for aproposal depends only on the proposal itself and the state g. As is standardin the theory of legislative voting, we focus on weakly stage undominatedstrategies, which implies that legislators vote for a proposal if they prefer it

5Fershtman and Nitzan (1991) study a similar mechanism in a different environment,with continuous time and quadratic payoffs. The symmetric Markov equilibrium of ourdiscrete time game corresponds to their “feedback” Nash equilibrium.

9

(weakly) to the status quo. The status quo is taken to be a “null” decisionby the legislature. That is, if the proposal fails, taxes and investment areboth equal to zero. We assume without loss of generality that proposals arealways accepted with probability one.As it is easy to verify, in a symmetric Markov equilibrium, a proposer

would either make no monetary transfer to the other districts, or wouldmake a transfer only to q−1 legislators, selected randomly each with the sameprobability of being selected. An equilibrium can therefore be described by acollection of proposal functions {y(g), s(g)} which specify the proposal madeby the proposer in a period in which the state is g. Here y(g) is the proposednew level of public good and s(g) is a transfer offered to the districts of theq− 1 randomly selected representatives. The proposer’s district receives thesurplus revenues x(g) =W − y(g) + (1− d)g − (q − 1)s(g). Associated withany symmetric Markov perfect equilibrium in the L game is a value functionvL(g) which specify the expected future payoff of a legislator when the stateis g. In the reminder we will study equilibria in which v is non decreasingin g.To study this equilibrium, it is useful to start from the problem of a

proposer who is selected to choose a policy. The proposer maximizes theutility of his own district under a series of feasibility constraints:

maxx,y,s

⎧⎪⎪⎨⎪⎪⎩x+ u(y) + δv(y)

s.t s+ u(y) + δv(y) ≥ Wn+ u [g(1− d)] + δv(g(1− d))

(q − 1)s+ x+ y − (1− d)g ≤Wx ≥ 0, s ≥ 0

⎫⎪⎪⎬⎪⎪⎭ (6)

where x is the transfer to the proposer. This problem is similar to the plan-ner’s problem (2): the second inequality is the budget balance constraint, andthe last two inequalities are the feasibility constraints.6 The first inequalityis however new: it is the incentive compatibility constraint that needs to besatisfied if a proposal is to be accepted by the floor.We will consider two cases. First the case when the incentive compatibil-

ity constraint is not binding, so the proposer can effectively ignore the otherlegislators. Second, when it binds and so the proposer has either to mod-ify the level of public good, or provide pork transfers to a minimal winningcoalition or both.

6It can be verified that the constraint y ≥ 0 is never binding and therefore it can beignored without loss of generality.

10

Non binding IC. Assume first that we can ignore the incentive compati-bility constraint and set s = 0. The problem becomes:

maxy

½W − [y − (1− d)g] + u(y) + δv(y)

s.t. W − y + (1− d)g ≥ 0

¾(7)

The solution of this problem can take one of two forms: either W − y(g) +(1− d)g = 0 (so x(g) = 0) or W − y(g)+ (1− d)g > 0 (and so x(g) > 0). Inthe second case, an optimal choice is y∗1 such that:

7

y∗1 = argmaxy{u(y)− y + δv(y)} (8)

Given this choice, it is easy to see that at y(g) = y∗1 we can have x(g) > 0if and only if W +(1− d)g ≥ y∗1, which is satisfied if and only if g is lower orequal than the threshold g1(y

∗1):

g1(y∗1) = max

½y∗1 −W

1− d, 0

¾(9)

When g < g1(y∗1), the proposer is constrained to choose y(g) =W +(1−d)g.

When the proposer chooses y(g) = W + (1 − d)g, then his choice mustbe better than the outside option for all the legislators (otherwise the latterwould be preferred by the proposer as well). We can therefore ignore theincentive compatibility constraint if and only if:

u(y∗1) + δv(y∗1) ≥W

n+ u [g(1− d)] + δv(g(1− d)) (10)

Condition (10) requires the legislators to vote for the proposal even if theyreceive no consumption good and only y(g) = y∗1. Monotonicity of v, togetherwith (10) implies that there is a threshold g2(y

∗1) ≥ g1(y

∗1) such that this

condition is satisfied if and only if g ≤ g2(y∗1). So we can conclude that there

are thresholds g1(y∗1), g2(y

∗1) such that y(g) = W + (1 − d)g for g ≤ g1(y

∗1)

and y(g) = y∗1 for g ∈ (g1(y∗1), g2(y∗1)].7In general, argmaxy

©yαα − y + δv(y)

ªmay contain more than one solution because

v may be non concave. If this is the case, the agent may choose a selection from this setwhich may depend on g. This case does not occur in the parametrizations that we considerin the experiment, so we ignore it to keep the discussion cleaner. It would however bestraightforward to extend the characterization.

11

Binding IC constraint. When g > g2(y∗1) the incentive compatibility

constraint can not be ignored. In this case, the problem solved by theproposer is:8

maxy,s

⎧⎨⎩ [W − [y − (1− d)g]− (q − 1)s] + u(y) + δv(y)s.t. s+ u(y) + δv(y) = W

n+ u [g(1− d)] + δv(g(1− d))

s ≥ 0

⎫⎬⎭ (11)

There are two possibilities. First, the proposer continues to provide no con-sumption to the districts of other legislators, but he increases the provision ofthe public good y(g) in order to satisfy the incentive compatibility constraint(no transfer case). Second, he provides consumption to the districts of q− 1other legislators and to his own district (transfers case).Consider the second case first, assuming s > 0. We can write (11) as:

maxy

½W − [y − (1− d)g]

−(q − 1)£Wn+Ψ((1− d)g)−Ψ(y)

¤+Ψ(y)

¾(12)

where Ψ(x) = u(x) + δv(x). Choosing an optimum in problem (12) isequivalent to choosing an optimum in problem: maxy {qΨ(y)− y}. So anoptimal choice for the proposer is to propose y(g) = y∗2 such that:

9

y∗2 = argmaxy{qΨ(y)− y} (13)

This is feasible only if s = Wn+Ψ (g(1− d))−Ψ (y∗2) ≥ 0, that is if g ≥ g3(y

∗2)

where g3(y∗2) is defined as the solution of:

W

n+Ψ (g(1− d)) = Ψ (y∗2) . (14)

In the case in which g ∈ [g2, g3] then the public good is determined bythe incentive compatibility constraint, which implies:

y(g) = ey(g) = Ψ−1µW

n+Ψ (g(1− d))

¶(15)

It is immediate to verify that ey(g) is increasing in g. Moreover, ey(g3 (y∗2)) =y∗2 and ey(g2 (y∗1)) = y∗1 (except at most in a corner solution in which g2 (y

∗1) =

8Note that when g > g2(y∗1), it must be x > 0: if x = 0, then the incentive compatibility

constraint would not be binding, and we would have g ≤ g2(y∗1).

9See footnote 7.

12

g1 (y∗1) = 0). Before we discuss how to use this characterization to compute

an equilibrium, it is useful to note a few properties of the proposer’s choice.From the discussion we can easily derive the investment function:

i(g) =

⎧⎪⎪⎨⎪⎪⎩W g ≤ g1

y∗1 − (1− d)g g ∈ (g1, g2]ey(g)− (1− d)g g ∈ (g2, g3]y∗2 − (1− d)g else

(16)

It is interesting to note that while in the planner’s solution investment is amonotonically decreasing function, in the political equilibrium is not mono-tonic (see panel 3 of Figure 2). The non monotonicity of the investmentfunction is a consequence of the fact that the incentive compatibility con-straint is not always binding and that the value of the status quo is en-dogenous. When g is small the marginal value of the public good is high.The cost if the bargaining proposal fails is therefore high. In this case theproposer can implement his preferred policy ignoring the incentive compat-ibility constraint. When this happens (in g ∈ (g1, g2]), the proposer willnot accumulate more than y∗1 (except, of course, if forced by the incentivecompatibility constraint). When g ≥ g2, however, the proposer is forced tointernalize the utility of at least a minimal winning coalition of other leg-islators: and so it will have to invest until the marginal utility of g is atleast 1/q. The final range in which investment is constant corresponds tothe region in which accumulating more than y∗2 is not profitable even whenthe q− 1 utilities of the other members of the minimal winning coalition areinternalized (which, not surprisingly, occurs for a much higher level of g).

Computing an equilibrium. In general there is a double relationshipbetween the value function v and the solutions y∗1, y

∗2. Given the value

function and two values y∗1 ≤ y∗2, we can construct the thresholds g1, g2, g3using, respectively (9), (10) and (14) and therefore reconstruct the policychoice y(g). This implies that the value function is fully described by y∗1, y

∗2

as:10

10To write the value function for g ≥ g3 note that in this range the value function ofa proposer is: W − [y∗2 − (1− d)g]− (q − 1)

£Wn +Ψ((1− d)g)−Ψ(y∗2)

¤+Ψ(y∗A) and the

probability of being a proposer is 1/n. The value of a legislator who receive pork transfers,on the other hand, is

£Wn +Ψ((1− d)g)−Ψ(y∗A)

¤+Ψ(y∗A), and the probability of receiving

a transfer s(g) conditional on not being a proposer is (q−1)/(n−1). Finally, the value ofa legislator excluded from transfers is simply Ψ(y∗A), and the probability of being in this

13

vL(g) =

⎧⎪⎪⎨⎪⎪⎩u (W + (1− d)g) + δv(W + (1− d)g) g ≤ g1

1n[W − [y∗1 − (1− d)g]] + u(y∗1) + δv(y∗1)) g ∈ (g1, g2]

1n[W − [ey(g)− (1− d)g]] + u(y∗1) + δv(ey(g))) g ∈ (g2, g3]1n[W − [y∗2 − (1− d)g]] + u(y∗2) + δv(y∗2) else

(17)Second, given a value function v, y∗1, y

∗2 are characterized by (8) and (13).

We can indeed conclude:

Proposition 5. If there are two values y∗1, y∗2 such that: a. given y∗1, y

∗2,

vL(g) satisfies (17); b. given vL(g), y∗1, y

∗2 satisfy:

y∗1 = argmaxy{u(y)− y + δv(y)} (18)

y∗2 = argmaxy{qu(y)− y + δqv(y)}

then vL(g) is an equilibrium of the L game and the equilibrium investment isdescribed by (16).

The specific values of y∗1, y∗2 and the shape of vL(g) of course depend on

the parameters of the game. Proposition 5, however, suggests a simplealgorithm to find the equilibrium:11

Step 1. Discretize the state space G = [0, g]. Let the number of states bem.

Step 2. For an initial value function v0, find y∗1, y

∗2 from (18); g1 (y

∗1), g2 (y

∗1),

g3 (y∗2) from (9), (10) and (14); and eyj (gj) for any gj ∈ (g2 (y∗1) , g3 (y∗2)) from

(15). Let y be the vector of these parameters. These parameters fullydescribe the equilibrium investment strategy and will be our initial guess.

Step 3. Given y, we can solve the system (17) of m equations in the munknowns (v (0) , ..., v (g)), and find a value function vL(g;y) for any g ∈ G.

state conditional on not being a proposer is 1 − (q − 1)/(n− 1). The expression in (17)follows from these expressions by taking expectations. The other cases can be computedin a similar way.11Existence of a Markov Nash equilibrium can be proven if the state and actions spaces

are discrete (as in the experimental setting) following the same argument as in Fudenbergand Tirole [1991]. With the numerical algorithm below we find a monotonic pure strategyMarkov equilibrium as discussed above.

14

Step 4. Given vL(g;y), we can reestimate the vector of parameters y0 given

vL(g;y). We now have a correspondence y0 ∈ Y (y). We can therefore find

the equilibrium solving for the fixpoint y ∈ Y (y).12

Figure 2 provides a representation of the equilibrium for one of the two pa-rameter configurations that we use in the experiment with u(y) = B 1

αyα. In

the case if power utility functions, we get a steady state of y∗2 =³

Bnn−(1−d)δ

´ 11−α.

For the experiment, we use n = 3, q = 2, W = 15, α = 0.5, d = 0, δ = .75,B = 1; and n = 5, α = 0.5, d = 0, δ = .75, B = 1. For n = 3 we havey∗2 = 16, and for n = 5 we have y∗2 ≈ 29.75 for n = 5. The figure is forthe n = 3 parameters, but with a discount factor of δ = .83. In this casey∗1 = 7 and y∗2 = 20. The threshold g∗1 is zero, because W is sufficiently highto finance y∗1; g

∗2 = 2 and g∗3 = 10. The first and second panel represent, re-

spectively, the investment function i(g) described in (16) and the cumulatedlevel of the public good y(g):

y(g) =

⎧⎪⎪⎨⎪⎪⎩W + (1− d)g g ≤ g1

y∗1 g ∈ (g1, g2]ey(g) g ∈ (g2, g3]y∗2 else

(19)

This curve fully describes the dynamics of public good provision and thesteady state. The steady state level of public good g∗ corresponds to thepoint where the 45o line intersects the investment curve. We can have differ-ent types of steady states and corresponding equilibria. If the intersectionis before g1 we have an efficient equilibrium in which no player receives porktransfers; if g∗ ∈ (g1, g3], in the steady state the proposer can extract trans-fers from his district without paying any transfer to the other districts; ifg∗ > g3, instead, in the steady state there is always a minimal winning coali-tion of legislators who receive positive transfers for their own districts. InFigure 2, the steady state is indeed in this case, so g∗ = y∗2.The panel representing the value function and y(g) makes clear the com-

plications involved with computing and studying the equilibrium. When gpasses from the region in which the incentive compatibility constraint is notbinding to the region with binding incentive compatibility constraint, the

12Depending on the parametrization, we solved for the fixpoint either by iterating Steps1-4, or when this procedure did not converge by directly minimizing the norm ky− Y (y)kin the space of parameters y.

15

Figure 2:

16

expected marginal value of g increases, because the IC constraint forces theproposer to internalize the utility of more agents. As it can be seen fromFigure 2, investment in g increases reducing the inefficiency, and the valuefunction becomes non concave in correspondence to g2.

4.2 Decentralized Provision: The AutarkyMechanism

To study the properties of the Autarky mechanism we focus on symmetricMarkov-perfect equilibria, where all districts use the same strategy, and thesestrategie are time-independent functions of the state, g. A strategy is apair (x(·), I(·)): where x(g) is the level of consumption in the district andI(g) is the level of contribution to the public good in state g. Given thesestrategies, by symmetry, the public good in state g is y(g) = (1−d)g+nI(g).Associated with any equilibrium is a value function vA(g) which specify theexpected future payoff of a legislator when the state is g. In the reminderwe characterize the equilibrium, hypothesizing that vA is concave, and laterverify that the value function at the derived equilibrium is indeed concave.The optimization problem for district j if the current level of public good

is g, and the district’s value function is vA(y) is:

maxy,x

⎧⎪⎪⎨⎪⎪⎩x+ u(y) + δvA(y)

s.t x+ y − (1− d)g =W − (n− 1)exW − (n− 1)ex+ (1− d)g − y ≥ 0

x ≤ (1− d)g/n+W/n

⎫⎪⎪⎬⎪⎪⎭ (20)

where y is the level of public good tomorrow, and ex = x(g) is the consumptionin state g of each of the other districts (which the proposer takes as given).The first constraint is a rewritten form of i + x = W , substituting out fori. The second constraint is x ≥ 0. The third constraint is i ≥ − (1−d)g

n: it

requires that no legislator can reduce y by more than his share (1− d)g/n.13

Note that since ex is an endogenous variable that depends on vA, (20) is notnecessarily a contraction.To characterize the equilibrium, first suppose the second constraint is

not binding. Depending on the state g the solution of (20) falls in one of

13In equilibrium, legislators will not reduce y. As in the legislative model, however,legislators can reduce it if they want. In a decentralized system as the VC game, y ≥ n−1

n gguarantees that (out of equilibrium) the sum of reductions in g can not be larger than thestock of g.

17

two cases: we may have W + (1 − d)g = y(g) + (n − 1)ex, so x(g) = 0; orW + (1− d)g > y(g) + (n− 1)ex, so x(g) > 0.Using concavity of vA, we can see that if W +(1− d)g > y(g) + (n− 1)ex,

then the solution is characterized by a unique public good level y∗A satisfyingthe first order equation:

u0 (y∗A) + δv0A(y∗A) = 1 (21)

The investment by each district is equal to IA(g) =1n[y∗A − (1− d)g] and

per capita private consumption is x(g) =W+(1−d)g−y∗A

n.

In the second possible case, if x(g) = 0, then y(g) =W+(1−d)g and andinvestment by each district of ip(g) =

Wn. This second condition is possible

only if and only if W = y∗A − (1− d)gA, that is if g is below some thresholdgA defined by:

gA = max

½y∗A −W

1− d, 0

¾To complete the characterization note that if (1 − d)g ≥ y∗A, then we have

y(g) = y∗A, i(g) = y∗A − (1− d)g, and x(g) = W−i(g)n

for each district. So wecan conclude that:

Proposition 3. In the decentralized non-cooperative dynamic public goodprovision game, the solution is that there is a y∗A such that:

y∗A(g) = min {w + (1− d)g, y∗A} .

The investment function y∗A(g) is qualitatively similar to the planner’sinvestment function. The main difference is that gA < gp, so public goodprovision is typically smaller. This is a dynamic version of the usual freerider problem associated with public good provision: each agent invests lessthan optimally because he/she fails to internalize the other agent’s utilities.Part of the free rider problem can be seen from (21): in choosing investment,legislators only count only the benefit to their district, u(y) + δvA(y), ratherthan nu(y) + δnvA(y)), but all the costs (−y). In this dynamic model,however, there is an additional effect that reduces incentives to invest, calleddynamic free riding.14 To see this, consider the value function for g > gA

14The same dynamic free riding problem arises in the Fershtman and Nitzan (1991)model.

18

(where we have an interior solution):

v(g) = W − (n− 1)x(g)− (y∗A − (1− d)g) + u (y∗A) + δvA(y∗A)

=W − (y∗A − (1− d)g)

n+ u (y∗A) + δvA(y

∗A)

where the last equation follows by the fact that in a symmetric equilibrium:x(g) =W − (n− 1)x(g)− (y∗A− (1− d)g). A marginal increase in g has twoeffects. A direct effect, corresponding to the increase in resources availablein the following period: (1 − d)g. But there is also an indirect effect: theincrease in g triggers a reduction in the investment of all the other legislatorsthrough an increase in x(g): for any level of g > gA, y(g) will be kept aty∗A. In a symmetric equilibrium, if legislator i increases the investment by1 dollar, he will trigger a reduction in investment by all future legislators by1/n dollars: the net value of the increase in g for i will be only δ/n.To establish that the value function is concave in g, the argument is

essentially identical to the proof for the planner’s problem. One first notesthat for g > gA, vp is actually linear in g and hence continuously differentiableand concave. Then it easily follows that vp concave for smaller values of g.The exact solution for y∗A and gA, depend on n, d, δ, and the functionalform of u. In the experiment we consider two parametrizations as in theexample, where u(y) = B 1

αyα. In this case, from equation (??) we get

y∗A =³

Bnn−(1−d)δ

´ 11−α. For the experiment, we use n = 3, α = 0.5, d = 0,

δ = .75, B = 1; and n = 5, α = 0.5, d = 0, δ = .75, B = 1. For both these

parameterizations, we obtain that gA = 0, and y∗A =¡43

¢2, for n = 3 and

y∗A =¡2017

¢2for n = 5.

5 Experimental Design

The experiments were all conducted at the Social Science Experimental Lab-oratory (SSEL) using students from the California Institute of Technology.Subjects were recruited using a database of volunteer subjects, maintainedby SSEL. Eight sessions were run, using a total of 102 subjects. No sub-ject participated in more than one session. Half of the sessions used thedynamic bargaining/voting mechanism using simple majority rule, and halfof the sessions used the dynamic voluntary contributions mechanism wherethe committee endowment was split equally between the individuals. Half

19

were conducted using 3 person committees, and half with 5 person commit-tees. In all sessions there was no depreciation (d = 0), the discount factorwas δ = 0.75, and the current-round payoff from the public good was pro-portional to the square root of the stock at the end of that round (α = .5).In the 3 person committees, we used the parameters W = 15, while in the5 person committees W = 20. Payoffs were renormalized so subjects couldtrade in fractional amounts. Table 1 summarizes the theoretical predictionof the mechanisms studied in the previous sections for this parametrization.

Table 1: Experimental parameters and equilibriumMechanism N B W g = (g1, g2, g3) y = (y1, y2) gA y∗A gp y∗pLegislative 3 2 15 (·, ·, ·) (·, 16)Legislative 5

√3 20 (·, ·, ·) (·, 29.83)

Autarky 3 2 15 0 1.77Autarky 5 2 20 0 1.38Planner 3 2 15 129 144Planner 5 2 20 380 400

Discounted payoffs were induced by a random termination rule by rollinga die after each round in front of the room, with the outcome determiningwhether the game continued to another round (with probability δ) or wasterminated (with probability 1−δ). The n = 5 sessions were conducted with15 subjects, divided into 3 committees of 5 members each. The n = 3 sessionswere conducted with 12 subjects, divided into 4 committees of 3 memberseach.15 Committees stayed the same throughout the rounds of a given match,and subjects were randomly rematched into committees between matches. Amatch consisted of one multiround play of the game which continued untilone of the die rolls eventually ended the match. As a result, different matcheslasted for different lengths, but all committees ended a match at the sametime. Table 2 summarizes the design.

Table 2: Experimental DesignMechanism n #Committees #SubjectsLegislative 3 70 21Legislative 5 60 30Autarky 3 70 21Autarky 5 60 30

15Two of the N = 3 sessions used 9 subjects.

20

Before the first match, instructions16 were read aloud, followed by a prac-tice match and a comprehension quiz to verify that subjects understood thedetails of the environment including how to compute payoffs. The currentround’s payoffs from the public good stock (called project size in the experi-ment) was displayed graphically, with stock of public good on the horizontalaxis and the payoff on the vertical axis. The horizontal axis ranged fromthe beginning-of-round stock to W plus the beginning of round stock. Thesample screen in the appendix illustrates this. Subjects could click anywhereon the curve and the payoff for that level of public good appeared on thescreen.For the bargaining/voting mechanism each round had two separate stages,

the proposal stage and the voting stage. At the beginning of each match, eachmember of a committee was randomly assigned a committee member numberwhich stayed the same for all rounds of the match. In the proposal stage, eachmember of the committee submitted a provisional budget for how to dividethe budget between the public good, called project investment, and privateallocations to each member. After everyone had submitted a proposal, onewas randomly selected and became the proposed budget. Members were alsoinformed of the committee member number of the proposer, but not informedabout the other provisional budgets that were not selected. Each memberthen casts a vote either for the proposed budget or for the backup budget withzero public investment and equal private allocations. The proposed budgetpassed if and only if it received at least N+1

2votes. Payoffs for that round

were added to each subject’s earnings and a die was rolled to determinewhether the match continued to the next round. If it did continue, thenthe end-of-round project size became the next round’s beginning-of-roundproject size.At the end of the last match each subject was paid privately in cash the

sum of his or her earnings over all matches plus a showup fee of $10. Earningsranged from approximately $20 to $50, with sessions lasting between one andtwo hours. There was considerable range in the earnings and length acrosssessions because of the random stopping rule.

16Instructions from one of the sessions are in the appendix.

21

6 Experimental Results

6.1 Time path of the stock of public good

As a first cut at the data, we computed the time path of the stock of the pub-lic good across rounds for each treatment. Figure 1 compares the time pathsfor the 3-district and 5-district Autarky mechanism. Figure 2 compares thetime paths for the Legislative mechanism and Autarky mechanisms in thethree member treatments. Figure 3 compares the time paths for the Leg-islative mechanism and Autarky mechanisms in the five member treatments.Superimposed on the graphs are the theoretical time paths, corresponding tothe Markov perfect equilibrium. Figure 4 shows all the observed time pathson one graph.

Figures 1-4 here

These time paths exhibit some systematic properties, which we list anddiscuss below (incomplete):

1. The Legislative mechanism leads to much greater public good produc-tion than the Autarky mechanism.

2. Both mechanism lead to public good levels significantly below the op-timal steady state.

3. In both mechanisms there is overinvestment relative to equilibrium, butit is relatively small in magnitude.

4. In both mechanisms, there is overshooting of the steady state, followedby significant disinvestment approaching the steady state.

5. In both mechanisms, the public good level overshoots the equilibriumsteady state and then declines.

6. The Legislative mechanism leads to similar outcomes for n=3 and n=5.This contrasts with a theoretical prediction of fairly large differences.For n=5, equilibrium is approached in the long run in most committees.For n=3, there is less investment early on, compared with n=3, but lessleveling out.

7. The Autarky mechanism leads to somewhat higher public good levelswith n=5 than n=3, but the difference is small.

22

6.2 Proposed public good investment

The results for proposed public good investment, which includes all proposals(including provisional proposals and failed proposals). The results are verysimilar to the data for the state variable, y. Slight overinvestment early on,declining to equilibrium levels in later rounds. See figures 5, 6, 7.

Figures 5-7 here

6.3 Variation across committees

6.3.1 L committees

There was remarkble consistency across committees, especially consideringthis was a complicated infinitely repeated game with many non-Markov equi-libria. There were a few committees who invested much more heavily thatpredicted by the Markov perfect equilibrium, but this only happened rarely,and often such cooperation fell apart in later rounds, and both investmentand the stock fell back to approximately equilibrium levels. The next twofigures dislpay the entire set of all time paths of y for all committees us-ing the L mechanism. See figures 8,9 for the n=3 and n=5 L treatments,respectively.

Figures 8-9 here

6.3.2 A committees

In the A mechanism, there was also a lot of uniformity across committees,again with a few exceptions. See figure 10 for the n=3 A treatment.

Figure 10 here

6.4 Coalitions: Types of proposals

We now turn to the analysis of the proposed allocation of pork, as a functionof g and n in the BV mechanism. Figures 11-12 display a categorization ofthe proposal types, and their acceptance rates. For this analysis we focusprimarily on the number of members receiving significant amounts of pork inthe proposed allocation, and whether the proposals had negative investmentin the public good. We break down the proposed allocations into 4 canonicaltypes, and a fifth residual category. These types are: (1) 100% allocation

23

to the public investment; (2) The allocation divided between public invest-ment and private consumption of the proposer; (3) The allocation dividedbetween public investment and a minimum winning coalition that includesthe proposer (2 if n=3; 3 if n=5); (4) Positive allocations to all n member.These are further broken down by whether investment in the public good ispositive, zero, or negative.

Figures 11-12 here

In this table, the observed frequencies of proposal types, broken downby three ranges of g are in bold. In the first range, beginning at g=0, theequilibrium proposal type assigns strictly positive allocation to investmentin the public good and private allocation to the proposer, but zero privateallocation to all other committee members. This corresponds to region of thestate space below g3. The proposer is IC constrained and finds it necessaryto buy off other committee members by allocation more to investment thanhe would if unconstrained. Between g3 and y*, the proposer is constrainedand finds it optimal to “buy off” the other committee members by investingup to y* and also paying off some to another committee member. After g3,the equilibrium involves negative investment of the public good, and becomesa divide the dollar ultimatum game, which requires the proposer to form aminimum winning coalition where the other members in the coalition areallocated xj=w/n (the status quo).

6.5 Voting Behavior

Figures 11-12 also display results about voting behavior, in particular, theprobability that a proposal is accepted, contingent on its type, the round,and the current stock of public good, g.Table below shows the results from a logit regressions of where the de-

pendent variable is vote (0=no; 1=yes). An observation is a single voter’svote on a single proposal. The independent variables are: EU(status quo):the expected value of a ”no” outcome (including the discounted continuationvalue); EU(proposal): the expected payoff of a yes outcome; and pork, theamount of pork offered to the voter under the current proposal. The data is

24

broken down according to the treatment (n=3 or n=5).

n = 3 n = 5

EU(status quo) −0.082∗∗∗ (0.02) −0.22∗∗∗ (0.02)EU(proposal) 0.080∗∗∗ (0.02) 0.22∗∗∗ (0.02)

pork 0.007 (0.007) 0.07∗∗∗ (0.02)constant 1.06 (0.81) 0.1 (0.97)

N 490 576

Table 3 : Logit estimates. Pr{vote=yes}

6.6 Evidence of non-Stationary strategies in the L games:Punishment and Reward

While we observe several small departures from the predicted stationaryequilibrium behavior in the L games, at least two stand out. The first is theoverproduction of public good, especially with n=3. The second is the factthat we rarely see proposals in which the proposer is the only member re-ceiving a private allocation. One possible explanation that may be consistentwith both of these observations is that, rather than playing a stationary equi-librium, many of the committees are supporting more efficient allocations byusing non-stationary strategies. Because this is an infinitely repeated gamewith a random stopping rule we conjecture there are equilibria that can sup-port higher levels of public good provision than the Markov equilibrium wecharacterize in the theoretical section of the paper.In a similar way, there could be punishment strategies imposed on pro-

posers who do not share the residual budget with any other committee meme-bers. Such proposals would be rejected as part of the punishment, or possiblyeven accepted, but then punished by ostracism in the future.We next take a look at the data to see if there is evidence of punishment

strategies. We look at both voting behavior and proposal behavior. Table4 reports the results of a logit regression of voting behavior on the samevariables a table 3, but includes three additional variables that could inprinciple indicate some degree of punishment or reward behavior that couldbe used to affect proposals and support equilibrium outcomes that differ fromthe computed stationary solutions derived earlier in the paper. There are twokinds of such outcomes. The first is what we described above in this section,where outcomes are more efficient that in the stationary solution; that is,

25

the level of g is higher than predicted. In terms of voting behavior, suchequilibrium could be supported by voting against proposals that do not offerenough public good. Thus, we include i and expect the sign to be positive ifthis sort of punishment is occuring. Second, the outcomes tend to be morefair (universalistic) than the theory predicts. While this is not a big effect, itis clearly seen in the data. thus we include a herfindahl index, h, to indicatehow unfair the proposed division of pork is. We expect the sign on this tobe negative, in the sense that unfair proposals are punished by getting morenegative votes. The third variable is ”greed” which measures the amount ofown-private allocation by the previous proposer.17

n = 3 n = 5 n = 3 n

EU(status quo) −0.075∗∗∗ (0.02) −2.33∗∗∗ (0.03) 0.01 (0.02) −0.15∗EU(proposal) 0.069∗∗∗ (0.02) 0.23∗∗∗ (0.03) −0.02 (0.03) 0.14∗∗

pork 0.14∗∗∗ (0.03) 0.17∗∗

i −0.001 (0.003) −0.02∗∗∗ (0.005) 0.06∗∗∗ (0.01) 0.03∗∗

h −2.35∗∗∗ (0.73) −1.19 (1.01) −2.81∗∗∗ (0.80) −1.20greed −2.93∗∗∗(0.93) −2.30∗∗∗ (0.71) −3.02∗∗∗ (0.95) −2.05∗constant 3.16 (0.95) 1.74 (0.96) 1.48 (0.01) 0.62

N 490 576 490 5

Table 4 : Logit estimates. Pr{vote=yes}. Including i, h, and greed

We also look at how current proposals treat the last proposer, dependingon how a proposer was treated by the last proposer. The hypothesis is thatthe amount a proposer gives to the previous proposer is increasing in howmuch the previous proposer gave him. That is, there is a version of tit for

17Note however, that the sign on the herfindahl index is automatically negative if thereare more members in the winning coalition, so this may not be an indication of punish-ment/reward at all, but simply myopic selfish optimizing.

26

tat behavior. the results are presented in Table 5.

n = 3 n = 5 n = 3

Public good 0.086∗∗∗ (0.02) 0.007 (0.006) 0.09∗∗∗ (0.02) 0.0EUratio(prop)(t−1) 122.56∗∗∗ (42.26) 96.48∗∗∗ (47.93) 122.5∗∗∗ (48.23) 89.

i(t−1) −0.02 (0.02) −0h(t−1) −2.11 (5.16) 2

greed(t−1) −4.66 (8.40) −7.constant −36.44 (14.28) −34.74 (17.32) −9

N 384 384 384

Table 5 : OLS estimates. Pork offered to the previous round proposer

7 Discussion

27

8 References

9 Appendix 1: Proofs of Propositions

10 Appendix 2: Details on Computation of

Equilibrium

11 Appendix 3: Sample instructions

28

0

5

10

15

20

25

30

35

40

45

1 2 3 4 5 6 7 8 9 10 11 12 13

mdn(y)A3

mdn(y)A5

y*(g*)A3

y*(g*)A5

0

5

10

15

20

25

30

35

40

45

1 2 3 4 5 6 7 8 9 10 11 12 13

mdn(y)L3

mdn(y)A3

y*(g*)L3

y*(g*)A3

0

5

10

15

20

25

30

35

40

45

50

1 2 3 4 5 6 7 8 9 10 11 12 13

mdn(y)L5

mdn(y)A5

y*(g*)L5

y*(g*)A5

0

5

10

15

20

25

30

35

40

45

1 2 3 4 5 6 7 8 9 10 11 12 13

mdn(y)L3

mdn(y)L5

mdn(y)A3

mdn(y)A5

y*(g*)L3

y*(g*)L5

y*(g*)A3

y*(g*)A5

‐10

‐5

0

5

10

15

20

1 2 3 4 5 6 7 8 9 10 11 12 13

average(prop_i)

median(prop_i)

i(g*)

‐5

0

5

10

15

20

25

1 2 3 4 5 6 7 8 9 10

(avg(prop_inv

(_mdn(prop_inv

(*i*(g

‐4

‐2

0

2

4

6

8

10

1 2 3 4 5 6 7 8 9 10 11 12 13

avg(inv)

mdn(inv)

i*(g*)

round observations% of accepted proposal type observations% of accepted1 70 0.97 inv=w 97 0.952 53 0.94 pork only to proposer 8 0.633 40 0.93 pork to a mwc4 33 0.91 *with positive inv 20 0.905 25 0.88 *with no inv 1 1.006 22 0.77 *with negative in 11 0.557 19 0.89 pork to everyone8 15 0.80 *with positive inv 138 0.979 15 0.93 *with no inv 14 0.93

10 11 0.91 *with negative in 26 0.7311 4 1.0012 4 0.7513 4 1.00

overall 315 0.91

g % of accepted0<=g<=7 (g3) 88 0.977<g<=16 (y*) 23 1.00g>16 204 0.88overall 315

g % of accepted w/ inv<00<=g<=7 (g3) 0 .7<g<=16 (y*) 2 1.00g>16 35 0.66overall 37 0.68

round obs % of accepted proposal type obs % of accepted1 60 0.95 inv=w 88 0.982 48 0.90 pork only to proposer 7 0.713 30 0.60 pork to a mwc4 18 0.83 *with positive inv 16 0.695 15 0.93 *with no inv 13 0.546 12 0.50 *with negative in 9 0.567 6 0.83 pork to a large wc8 6 0.83 *with positive inv 7 1.009 6 0.83 *with no inv 3 1.00

10 3 1.00 *with negative in 4 1.00overall 204 0.84 pork to everyone

*with positive inv 40 0.93g obs % of accepted *with no inv 6 0.170<=g<=18.5 (g3) 93 0.92 *with negative in 9 0.4418.5<g<=29.83 (y* 48 0.85g>29.83 63 0.70overall 204 0.84

g obs % of accepted w/ inv<00<=g<=18.5 (g3) 3 0.3318.5<g<=29.83 (y* 4 0.50g>29.83 15 0.67overall 22 0.59

n = 3 n = 5

EU(status quo) −0.082∗∗∗ (0.02) −0.22∗∗∗ (0.02)EU(proposal) 0.080∗∗∗ (0.02) 0.22∗∗∗ (0.02)

pork 0.007 (0.007) 0.07∗∗∗ (0.02)constant 1.06 (0.81) 0.1 (0.97)

N 490 576

Table X : Logit estimates. Pr{vote=yes}

n = 3 n = 5 n = 3 n = 5

EU(status quo) −0.075∗∗∗ (0.02) −2.33∗∗∗ (0.03) 0.01 (0.02) −0.15∗∗∗ (0.03)EU(proposal) 0.069∗∗∗ (0.02) 0.23∗∗∗ (0.03) −0.02 (0.03) 0.14∗∗∗ (0.03)

pork 0.14∗∗∗ (0.03) 0.17∗∗∗ (0.36)i −0.001 (0.003) −0.02∗∗∗ (0.005) 0.06∗∗∗ (0.01) 0.03∗∗∗ (0.01)h −2.35∗∗∗ (0.73) −1.19 (1.01) −2.81∗∗∗ (0.80) −1.20 (1.07)

greed −2.93∗∗∗(0.93) −2.30∗∗∗ (0.71) −3.02∗∗∗ (0.95) −2.05∗∗∗ (0.71)constant 3.16 (0.95) 1.74 (0.96) 1.48 (0.01) 0.62 (0.99)

N 490 576 490 576

Table X : Logit estimates. Pr{vote=yes}. Including i, h, and greed 1

n = 3 n = 5 n = 3 n = 5

Public good 0.086∗∗∗ (0.02) 0.007 (0.006) 0.09∗∗∗ (0.02) 0.005 (0.007)EUratio(prop)(t−1) 122.56∗∗∗ (42.26) 96.48∗∗∗ (47.93) 122.5∗∗∗ (48.23) 89.55∗ (49.18)

i(t−1) −0.02 (0.02) −0.003 (0.012)h(t−1) −2.11 (5.16) 2.82 (4.65)

greed(t−1) −4.66 (8.40) −7.33∗∗∗ (2.06)constant −36.44 (14.28) −34.74 (17.32) −9.10 (9.74)

N 384 384 384 384

Table X : OLS estimates. Pork offered to the previous round proposer

n = 3 n = 5 n = 3 n = 5

EU(status quo) −0.075∗∗∗ (0.02) −2.33∗∗∗ (0.03) 0.01 (0.02) −0.15∗∗∗ (0.03)EU(proposal) 0.069∗∗∗ (0.02) 0.23∗∗∗ (0.03) −0.02 (0.03) 0.14∗∗∗ (0.03)

pork 0.14∗∗∗ (0.03) 0.17∗∗∗ (0.36)i −0.001 (0.003) −0.02∗∗∗ (0.005) 0.06∗∗∗ (0.01) 0.03∗∗∗ (0.01)h −2.35∗∗∗ (0.73) −1.19 (1.01) −2.81∗∗∗ (0.80) −1.20 (1.07)

greed −2.93∗∗∗(0.93) −2.30∗∗∗ (0.71) −3.02∗∗∗ (0.95) −2.05∗∗∗ (0.71)constant 3.16 (0.95) 1.74 (0.96) 1.48 (0.01) 0.62 (0.99)

N 490 576 490 576

Table X : Logit estimates. Pr{vote=yes}. Including i, h, and greed 1

n = 3 n = 5 n = 3 n = 5

Public good 0.086∗∗∗ (0.02) 0.007 (0.006) 0.09∗∗∗ (0.02) 0.005 (0.007)EUratio(prop)(t−1) 122.56∗∗∗ (42.26) 96.48∗∗∗ (47.93) 122.5∗∗∗ (48.23) 89.55∗ (49.18)

i(t−1) −0.02 (0.02) −0.003 (0.012)h(t−1) −2.11 (5.16) 2.82 (4.65)

greed(t−1) −4.66 (8.40) −7.33∗∗∗ (2.06)constant −36.44 (14.28) −34.74 (17.32) −9.10 (9.74)

N 384 384 384 384

Table X : OLS estimates. Pork offered to the previous round proposer