Dynamic Legislative Bargaining with Veto Power:
Theory and Experiments∗
Salvatore Nunnari
Bocconi University and IGIER
First Version: September 14, 2011
This Version: May 19, 2017
Abstract
In many domains, committees bargain over a sequence of policies and a policy remains
in effect until a new agreement is reached. In this paper, I argue that, in order to assess
the consequences of veto power, it is important to take into account this dynamic
aspect. I analyze an infinitely repeated divide-the-dollar game with an endogenous
status quo policy. I show that, irrespective of legislators’ patience and the initial
division of the dollar, policy eventually gets arbitrarily close to full appropriation by
the veto player. However, convergence to this outcome takes an infinite number of
periods: along the path, the veto player shares resources with a coalition partner and,
when resources are dispersed, is willing to accept a reduction to his allocation. The
main predictions of the theory find support in controlled laboratory experiments.
Keywords: Dynamic Legislative Bargaining; Standing Committees; Endogenous Status Quo; Veto
Power; Markov Perfect Equilibrium; Laboratory Experiments
∗
Salvatore Nunnari is Assistant Professor, Bocconi University, Department of Economics, Via Rontgen
1, Milan, Italy, 20136, [email protected]. I would like to thank Andrea Mattozzi, Massimo
Morelli, Thomas Palfrey, Erik Snowberg, Ewout Verriest, and Jan Zapal for feedback and comments. The
paper has also benefited from discussions with seminar participants at the California Institute of Technology,
the 2011 APSA Conference in Seattle, UC Merced, UC San Diego, the 2012 MPSA Conference in Chicago, the
2012 Petralia Applied Economics Workshop, Boston University, New York University, Columbia University,
the Institute for Advanced Studies, and the University of Pennsylvania.
1
Introduction
A large number of important voting bodies grant one or several of their members the right
to block decisions even when a proposal has secured the necessary majority—a veto right.
One prominent example is the United Nations Security Council (UNSC), where a motion is
approved only with the support of nine members, including the concurring vote of the five
permanent members. Another example is the U.S. President’s ability to veto congressional
decisions or the European Parliament’s power to block legislation proposed by the European
Commission. Additionally, in assemblies with asymmetric voting weights and complex voting
procedures, veto power may arise implicitly: this is the case of the U.S. in the International
Monetary Fund and the World Bank governance bodies (Leech and Leech 2004).1
The existence of veto power raises two frequent concerns among practitioners and in the
public opinion. First, the ability of an agent to veto policies increases the possibility of
legislative stalemate, or “gridlock”. Second, although the formal veto right only grants the
power to block undesirable decisions, it could de facto allow veto members to impose their
ideal decision on the rest of the committee. These concerns were expressed by the delegates
of the smaller countries when the founders of the UN met in San Francisco in June 1945
(Russell 1958), and have been a crucial point of contention in the ongoing discussion over
UNSC reform (Blum 2005). A similar debate has arisen regarding the IMF and WB voting
weights (Woods 2000).
Formal models concur with this conventional wisdom: Tsebelis (2002) asserts that veto
players make significant policy change difficult or impossible; O’Neill (1996) applies the
Shapley-Shubik index to the UNSC and finds that each non-permanent member has less than
1
Many other institutions grant veto power to some of their decision makers. For example, some corporate
boards of directors grant minority shareholders a “golden share”, which confers the privilege to veto any
decision. This share is often held by members of the founding family or governments in order to maintain
some control over privatized companies and was widely used in the European privatization wave of the late
1990s and early 2000s. For instance, the British government had a golden share in BAA, the UK airports
authority; the Spanish government had a golden share in Telefónica; and the German government had a
golden share in Volkswagen.
1
0.2 percent of the power; Winter (1996) uses non-cooperative game theory and concludes
that “the bargaining power of the non-veto members is effectively null” (p. 820). At the
same time, recent empirical studies suggest that non-veto members do have substantial
leverage on the deliberations and decisions of the committees they belong to, and that veto
players are not stalwarts of the status quo. Kuziemko and Werker (2006) and Vreeland and
Dreher (2014) find that UNSC permanent members spend significant resources on aid to nonpermanent members in an attempt to buy their votes; Stone (2011) finds that countries with
veto power cede disproportionate influence in the International Monetary Fund to members
with small voting weights; König and Junge (2009) show that veto players in the EU do not
use their power and support policy changes even when they prefer the status quo; Gehlbach
and Malesky (2010) find that veto players encouraged economic reforms in Eastern Europe
and the former Soviet Union. Existing formal models fail to predict these robust empirical
patterns.
In this paper, I argue that, in order to assess veto members’ incentive to uphold the
status quo and the balance of power between veto and non-veto members in a committee,
it is important to take into account the dynamic aspect of policy-making. Veto power has
mostly been investigated from the perspective of an ad hoc committee bargaining over a single
policy: players have no option to walk out and receive an exogenous status quo until the
committee reaches an agreement, when negotiations end. This is not a realistic description of
many bargaining environments. Committees are often dynamic: their members bargain over
a sequence of policies—that is, the committee is standing—and a policy remains in effect until
a new agreement is reached—that is, the status quo is endogenous. These are key features
of Congressional legislation on mandatory spending (e.g., Social Security and Medicare),
tax exemptions or trade protection and European legislation on environmental standards
or competition policy. Similarly, Supreme Court opinions remain in force unless revisited
and some UNSC resolutions—for example, on disputed borders or economic sanctions—and
IMF decisions—for example, on its basket of global reserve currencies—hold until explicitly
2
addressed by a new vote.
This is the first paper to investigate, theoretically and experimentally, the consequences
of veto power in a dynamic bargaining setting where the location of the current status quo
policy is determined by the policy implemented in the previous period. In each of an infinite
number of periods, one of three legislators—one of whom is a veto player—is recognized to
propose the allocation of a fixed endowment. The proposal is implemented if it receives at
least two affirmative votes, including the vote of the veto player. Otherwise, the status quo
policy prevails and the endowment is allocated as it was in the previous period. In this sense,
the status quo policy evolves endogenously.
In this setting, I answer three basic questions: To what extent is the veto player able to
leverage his veto power into favorable outcomes, both in the short and in the long run? How
does this depend on the legislators’ patience and the initial agreement? What are the effects
of institutional measures meant to reduce the power of the veto player?
In particular, I fully characterize a Markov Perfect Equilibrium (MPE) and prove it exists
for any discount factor and any initial divisions of the resources.2 In this MPE, the veto player
is able to move the status quo policy closer to his ideal point whenever he sets the agenda,
irrespective of the legislators? discount factor and the status quo policy. At the same time, it
takes him an infinite number of proposals to converge to full appropriation of the resources.
This happens because, in contrast with non-veto members of ad hoc committees, non-veto
members of standing committees who receive a proposal increasing the veto player’s share
take into account the associated reduction in their future bargaining power and demand more
than their current status quo share to support it. This premium is always positive and, thus,
some benefits accrue to non-veto players in all periods of the game. The speed of convergence
2
The only general existence result for dynamic bargaining games applies to settings with stochastic
shocks to preferences and the status quo (Duggan and Kalandrakis 2012). As these features are not present
in my model, proving existence is a necessary step of the analysis. Moreover, if I were to consider a model
with stochastic shocks to preferences and the status quo, the results in Duggan and Kalandrakis (2012)
would guarantee existence of an equilibrium but would not provide a characterization of its dynamics or
comparative statics with respect to patience, the initial division of the dollar or recognition probabilities.
3
to the veto player’s ideal outcome is decreasing in the discount factor of the committee, as
the premium demanded by non-veto legislators increases in their patience. Moreover, when
the status quo policy prescribes a positive allocation to all legislators, the veto player of a
standing committee is willing to accept a reduction to his allocation, something that the
member of an ad hoc committee would never do. This occurs because the current decision
becomes the future status quo policy and, in turn, affects the future leverage the veto player
has when he is the proposer.
This dynamic model suggests that giving a committee member the power to oppose does
not lead to legislative stalemate nor deprives completely other members of their bargaining power but it guarantees a strong leverage on long run outcomes. Therefore, I analyze
institutional mechanisms to weaken veto power and promote more equitable outcomes for
longer. First, I investigate the effect of reducing the agenda setting power of the veto player
in committees with three legislators. Second, I assume that only veto players have the power
to propose, and characterize an MPE for committees with an arbitrary number of veto and
non-veto players. This allows me to study the effect of competing veto powers, committee size and majority requirements on veto players’ ability to appropriate resources in the
short and in the long run. I show that these institutional measures do not prevent complete
expropriation of non-veto players in the long run but do affect short run outcomes.
While the model I study is the simplest one to account for veto power in committees, the
incentives it lays bare are representative of tensions we observe in real world institutions.
For example, the logic of gradual appropriation by the veto player might help to explain the
struggle between the U.S. Congress and the U.S. President in allocating federal resources and
the historical trends in mandatory spending. The U.S. Constitution establishes Congress as
the body with controlling power over the federal budget and, while it gives the President the
power to veto legislative decisions, it does not assign any budgetary authority to the executive branch. This changed with the Budget and Accounting Act of 1921 (BAA). The BAA
required the President to submit a budget to Congress on an annual basis and introduced
4
the Office of Management and Budget, which granted the executive branch a monopoly on
budgetary information and a strong advantage in setting the agenda (Gailmard and Patty
2012). Between the 1920s and the 1970s, the scale and scope of the federal budget expanded and—especially between the late 1950s and the early 1970s—the share of the pie
absorbed by entitlement spending grew disproportionately, with discretionary spending being progressively scaled down (see Figure 1).3 Eventually, the increased policymaking power
of the President and the reduced resources available for discretionary spending culminated
in a fiscal conflict with Congress. The result was the 1974 Congressional Budget and Impoundment Act (CBA), designed to help Congress regain power over the budget process
(Wildavsky 1986). The CBA not only limited the ability of the President to impound funds
already appropriated by Congress—that is, it reduced the power to veto of the President—
but also created the Congressional Budget Office to give Congress independent economic
analysis—that is, it decreased the proposal power of the President.
In spite of this suggestive anecdotal evidence, it would be challenging to evaluate the
ability of this complex theoretical model to predict empirical behavior using observational
data. Instead, I test the predictions from the theoretical analysis with laboratory experiments, which allow a tight control of the decision environment and the evolution of the
status quo policy. I consider an experimental design that varies legislators’ long run incentives, comparing legislatures with different degrees of patience. The theory is consistent
with many features of the data: the allocation to the veto player gradually increases over
time, more rapidly in impatient committees; veto proposers expropriate the richer non-veto
legislator and give a premium to the poorer non-veto legislator, higher in more patient committees; voting behavior is forward-looking, especially in patient committees, with non-veto
legislators less likely to support a proposal with a higher premium to the veto player and
3
The literature on U.S. budgetary politics typically sees the President as counterweight to the distributive
politics of the legislature (Carter and Schap 1987, Lohmann and O’Halloran 1994). Even allowing the
President to have preferences over the distribution of benefits across districts, veto power is predicted to
lower local transfers (McCarty 2000a).
5
0.7 0.6 0.5 0.4 0.3 0.2 0.1 19
62
19 65
19 68
19 71
19 74
19 77
19 80
19 83
19 86
19 89
19 92
19 95
19 98
20 01
20 04
20 07
20 10
20 13
0 Discre3onary Mandatory Figure 1: U.S. mandatory versus discretionary spending as a percentage of total spending, 19622014. Source: Historical Tables, Budget of the United States Government, Fiscal Year 2016.
veto legislators more likely to support a proposal reducing the future bargaining power of a
non-veto legislator.
2
Related Literature
This paper contributes primarily to the theoretical literature on the consequences of veto
power in legislatures. A large number of studies build on models of legislative bargaining à
la Baron and Ferejohn (1989) to examine the role of veto power in policy making. Most of
these papers model specific environments and focus on the case of the U.S. Presidential veto
(Diermeier and Myerson 1999, Cameron 2000, McCarty 2000a,b, Groseclose and McCarty
2001). More closely related to this paper, Winter (1996) shows that the share of resources
to veto players is decreasing in the cost of delaying an agreement, so that the share of
resources to non-veto players declines to zero as the cost of delay becomes negligible, that is,
6
as legislators become infinitely patient. Banks and Duggan (2000) derive a similar result in
a more general model of collective decision making. A common limitation of this literature,
and the main point of departure with my paper, is the focus on static settings: the legislative
interaction ceases once the legislature has reached a decision, and policy cannot be modified
after its initial introduction. In this paper, the status quo policy is not exogenously specified
but is rather the product of policy makers’ past decisions.
In this sense, this study belongs to a recent literature on legislative policy making with
an endogenous status quo and farsighted legislators (Baron 1996, Kalandrakis 2004, Penn
2009).4 This literature does not explore the consequences of veto power, with three exceptions: Duggan, Kalandrakis and Manjunath (2008) model the specific institutional details
of the American presidential veto and limit their analysis to numerical computations. In
subsequent work, Diermeier, Egorov and Sonin (2016) focus on the case where sufficiently
patient agents allocate indivisible objects and veto players have the monopoly of agenda
setting power. They show that increasing the number of veto players may increase the expropriation of agents without veto power. I show that, in the setup of this paper, changing
the number of veto players does not change the ability of non-veto players to retain a share
of the resources in the long run. Anesi and Duggan (Forthcoming) study a model where
the set of alternatives is finite, players have strict preferences and are sufficiently patient.
They show that, if there is a veto player with positive recognition probability, then starting
from any given alternative, there is a unique absorbing point which the equilibrium process
transitions to.
Finally, this paper contributes to the literature on laboratory experiments testing models
of legislative bargaining (Frechette, Kagel and Lehrer 2003, Frechette, Kagel and Morelli
2005, Diermeier and Gailmard 2006). In particular, Kagel, Sung and Winter (2010) provide
experimental evidence on the consequences of veto power in ad hoc committees à la Winter
4
See also Seidmann and Winter (1998), Bowen and Zahran (2012), Dziuda and Loeper (2016), Diermeier
and Fong (2011), Duggan and Kalandrakis (2012), Anesi and Seidmann (2015), Baron and Bowen (2015).
7
Allocation to Non-Veto 1
Allocation to Non-Veto 1
x1
xv
x2
Allocation to Non-Veto 2
Allocation to Non-Veto 2
(a)
(b)
Figure 2: The set of possible legislative outcomes in each period, ∆
(1996). All this work focuses on static environments where a given amount of resources is
allocated only once. The sole exception is Battaglini and Palfrey (2012) and Baron, Bowen
and Nunnari (2016) who investigate a dynamic model of bargaining with endogenous status
quo in the absence of veto power.
3
Model and Equilibrium Notion
Three agents repeatedly bargain over a legislative outcome xt for each period of an infinite
horizon, t = 1, 2, . . . .5 One of the three agents is endowed with the power to veto any
proposed outcome in every period. I denote the veto player with the subscript v and the
two non-veto players with the subscript j = {1, 2}. The possible outcomes in each period
are all possible divisions of a fixed resource among the three players. Figure 2 represents the
set of possible legislative outcomes, x ∈ ∆. The vertical dimension represents the share to
(non-veto) player 1, while the horizontal dimension represents the share to (non-veto) player
2. The remainder is the share that goes to the veto player.
5
An extension with committees composed of an arbitrary number of agents is presented in Section 5.2.
8
The Bargaining Protocol. At the beginning of each period, one agent is randomly selected to propose a new policy, z ∈ ∆. Each agent has the same probability of being
recognized as policy proposer, that is 13 .6 This proposal is voted up or down by the committee. A proposal passes if it gets the support of the veto player and at least one other
committee member. If a proposal passes, xt = z is the implemented policy at t. If a proposal
is rejected, the policy implemented is the same as it was in the previous period, xt = xt−1 .
Thus, the previous period’s decision, xt−1 , serves as the status quo policy in period t. The
initial status quo x0 is exogenously specified.
Stage Utilities. Agent i derives stage utility ui = xi from the implemented policy xt .
Players discount the future with a common factor δ ∈ [0, 1), and their payoff in the game is
given by the discounted sum of stage payoffs.
Strategies and Equilibrium Notion. In what follows, I look for a Markov perfect equilibrium. In this type of equilibrium, strategies depend only on payoff-relevant effects of past
behavior (Maskin and Tirole 2001). I define the state in period t as the status quo policy, or the previous period’s decision, st = xt−1 . In a Markov perfect equilibrium, agents
behave identically in different periods with the same state s, even if that state arises from
different histories. In this dynamic game, the expected utility of agent i from the allocation
implemented in period t does not only depend on his stage utility, but also on the discounted
expected flow of future stage utilities, given a set of strategies. The continuation value, vi (s),
is the expected payoff of legislator i when the state is s before the proposer is selected. We
can write the expected utility of legislator i from the allocation implemented in period t, xt ,
as:
Ui (xt ) = xti + δvi (xt )
Given that non-veto legislators are otherwise identical, I focus on Markov proposal and
6
An extension in which recognition probabilities can differ among agents is presented in Section 5.1.
9
voting strategies that are symmetric with respect to the two non-veto legislators. Finally, as
is standard in the theory of legislative voting, I require that agents use stage-undominated
voting strategies—that is, they vote yes if and only if their expected utility from the status
quo is not greater than their expected utility from the proposal.
4
Equilibrium Analysis
In this Section, I propose natural conditions on strategies, and show that these conditions
define an equilibrium. First, equilibrium proposals involve minimal winning coalitions, such
that at most one non-veto player receives a positive amount in each period. Second, the
proposer proposes the acceptable allocation—that is, an allocation that defeats the status
quo—that maximizes his current share of the dollar. The set of allocations each agent prefers
to the status quo policy changes with the discount factor, as legislators take more or less
into account the impact of the current allocation on future periods. Not surprisingly, this
has important consequences for the dynamics of the game.
I first discuss the case when the proposer is a non-veto player, and then the case when
the proposer is a veto player. To help with the exposition, I partition the space of possible
divisions of the dollar into two subsets, ∆ and ∆\∆. Define ∆ ⊂ ∆ as the set of states
x ∈ ∆ in which at least one non-veto legislator gets zero. Define the demand of legislator i,
di , as the minimum amount he requires to accept a proposal x ∈ ∆.
4.1
Non-Veto Proposer
When a non-veto player is proposing, he needs to secure the vote of the veto player in order
to change the current status quo. If the non-veto proposer wants to maximize his current
share of the dollar, he will propose the veto player’s demand to the veto player, and assign
the remainder to himself. Therefore, to characterize the equilibrium proposal strategies of a
non-veto player, we need to identify the acceptance set of the veto player.
10
A perfectly impatient veto player values only his current allocation and, thus, only supports proposals that give him as much as the status quo or more. On the other hand, a
patient veto player is not indifferent between all states in which he receives the same allocation, and might be better off with allocations that reduce his current share when these
decrease his future coalition building costs. In particular, he is willing to move from an
interior allocation where he gets a higher share, to an allocation where both he and one nonveto player have a smaller share. This occurs because the future status quo policy affects the
future leverage the veto player has when he is the proposer. In this event, he needs to secure
the vote of just one non-veto player, and he will, thus, build a coalition with the non-veto
player who demands the least. As shown below, the demand of each non-veto player is an
increasing function of what he gets in the status quo and, therefore, a veto player’s coalition
building costs with status quo s are a positive function of min{s1 , s2 }. Thus, a veto player
prefers an allocation s0 where he gets s0v and min{s01 , s02 } = s0nv to an alternative allocation
s00 with s00v = s0v but min{s001 , s002 } = s00nv > s0nv .
Figure 3 depicts the acceptance set of a patient veto player for two different values of
δ > 0. In the Appendix, I characterize the amount the veto player demands to accept a
proposal that brings the status quo into ∆—where one non-veto player gets nothing—as:
dv = max{sv −
δ
s , 0}
3−2δ nv
(1)
where snv is the allocation of the poorer non-veto player in the status quo. The reduction
accepted by the veto player increases with his discount factor δ and the share to the poorer
non-veto player snv . A veto player does not accept any division of the dollar that gives him
less than the status quo when s ∈ ∆. Note also that the reduction a veto player is willing
to accept could be more than what he has in the status quo, in which case his demand is
bounded below by 0.
The non-veto proposer proposes the acceptable policy that maximizes his current alloca-
11
s1
Acceptance set of veto player
Acceptance set of veto player
Acceptance set of veto player
Allocation to Non-Veto 1
s1=s2
Allocation to Non-Veto 2
s0
s1=s2
Allocation to Non-Veto 2
(a)
s1=s2
Allocation to Non-Veto 1
s0
Allocation to Non-Veto 1
s0
s1
Allocation to Non-Veto 2
(b)
(c)
Figure 3: Veto’s acceptance set and non-veto’s proposal strategies for state s0 : (a) Av (s0 ) when
δ = δ 1 > 0; (b) Av (s0 ) δ = δ 2 > δ 1 , (c) equilibrium proposal of non-veto 1 (blue arrow) and
non-veto 2 (green arrow).
tion. These are depicted in the right-most panel of Figure 3. A non-veto proposer completely
expropriates the other non-veto player, gives the veto player his demand, and allocates the
remainder to himself. When the state is in ∆, the non-veto proposer can only get 1 − sv ,
but when the state is in ∆\∆ he can extract an higher amount, namely 1 − dv .
4.2
Veto Proposer
When the veto player desires to pass a proposal with a minimal winning coalition, he is
not bound to include any specific legislator. Thus, he selects the legislator who accepts the
highest increase to the veto player’s share—that is, the legislator with the lowest demand—
as his coalition partner. When legislators are perfectly impatient, the veto player builds a
coalition with the poorer non-veto player—the non-veto player who receives the least in the
status quo—giving him as much as he is granted by the status quo. A perfectly impatient
non-veto player accepts this proposal. A patient non-veto player does not.
In fact, the bargaining power of a patient non-veto player decreases with the share held
by the veto player in the status quo. For this reason, a patient non-veto player prefers an
allocation s0 ∈ ∆ where he gets s0j = 0 and the veto player gets s0v to an alternative allocation
s00 ∈ ∆ with s00j = s0j but s00v > s0v . The difference between these allocations arises when he
is recognized in t + 1, as he will gain the support of the veto player only for proposals that
12
s0
s0
Acceptance set of
Non-Veto 2
Allocation to Non-Veto 1
Allocation to Non-Veto 1
Acceptance set of
Non-Veto 2
Acceptance set of
Non-Veto 2
Allocation to Non-Veto 1
s0
Allocation to Non-Veto 2
Allocation to Non-Veto 2
(a)
(b)
Allocation to Non-Veto 2
(c)
Figure 4: Non-veto 2’s acceptance sets for state s0 where s1 > s2 : (a) A2 (s0 ) when δ = 0, (b)
A2 (s0 ) when δ = δ 1 > 0; (c) A2 (s0 ) δ = δ 2 > δ 1 .
give him no more than 1 − sv . Figure 4 depicts the acceptance set of the poorer non-veto
player for a state s0 ∈ ∆ and three increasing values of the discount factor.
With δ > 0, the veto player’s coalition partner demands a premium to vote in favor
of an allocation that increases the veto player’s share: the veto player has to compensate
his coalition partner with a short term gain in stage utility for the long term loss in future
bargaining power. The Appendix shows that the demand of the poorer non-veto player for
states s ∈ ∆ is:
dnv =
δ
s
3−2δ nv
(2)
where snv is the allocation to the richer non-veto player in the status quo.
Some properties of dnv are worth noting. First, dnv is smaller than snv for any δ ∈ [0, 1).
This means that, as long as δ < 1, the veto proposer can increase his share, as he can assign
himself 1 − dnv > sv = 1 − snv . Since the veto player does not accept any reduction to his
allocation once s ∈ ∆, the allocation to the veto player displays a ratchet effect: it can only
stay constant or increase.
Second, the premium paid by the veto player to his coalition partner is monotonically
increasing in δ and linearly increasing in snv : dnv converges to snv as δ converges to 1, and
to 0 as δ converges to 0. This implies that the ratchet effect is slower with more patient
13
s0
s0
Allocation to Non-Veto 1
Allocation to Non-Veto 1
Acceptance set of
Non-Veto 2
Premium
s1
s2
Allocation to Non-Veto 2
(a)
s1
Allocation to Non-Veto 2
(b)
Figure 5: Veto’s equilibrium proposal strategy for state s0 and δ > 0.
legislators. With δ = 0, the premium is 0 and the veto player is able to steer the status quo
policy to his ideal point in at most two proposals, as he can pass any x ∈ ∆ when the poorer
non-veto player has zero. With δ ∈ (0, 1), the premium is always positive and convergence to
the veto player’s ideal point happens only asymptotically. Figure 5(b) shows how the state
would evolve when the veto player always proposes.
When the allocations to the two non-veto players are close, the veto player mixes between
coalition partners. This is necessary to guarantee that the proposer’s choice of a partner is
a best response to what they demand: if the veto player always picked the poorer non-veto
player as coalition partner, this player would become the most expensive coalition partner.7
4.3
Theoretical Results
Proposition 1 provides a summary of the discussion above:
Proposition 1. For any δ ∈ [0, 1) and any s0 ∈ ∆ there exists an MPE such that:
• All proposals give a positive allocation at most to a minimal winning coalition.
• For some s ∈ ∆\∆, the veto proposer mixes between possible coalition partners.For the
7
Mixed proposal strategies are a common feature of stationary equilibria in bargaining models.
14
Allocation to Non-Veto 2
(a)
C
x1=x2
D
Allocation to Non-Veto 2
Allocation to Non-Veto 1
x1=x2
Allocation to Non-Veto 1
Allocation to Non-Veto 1
A
B
A
B
A
x1=x2
C
D
Allocation to Non-Veto 2
(b)
(c)
Figure 6: Partition of ∆ into regions with different equilibrium strategies for allocations where
s1 ≥ s2 : (a) δ=0; (b) δ = δ 1 > 0; (c) δ = δ 2 > δ 1 . In A and B, veto proposer builds a coalition with
poorer non-veto player; in C and D, veto proposer mixes between coalition partners; in B, and C
veto player is willing to accept nothing when he is not proposing.
remaining s ∈ ∆, the veto proposer proposes dnv to the poorer non-veto player.
• For all s ∈ ∆, the non-veto proposers propose dv to the veto player.
• For all s ∈ ∆, dv = sv and dnv ≥ snv , that is, the veto player demands his status quo
allocation, non-veto players demand weakly more.
In the Appendix, I give the exact statement of the equilibrium proposal and voting
strategies for each region of ∆, and show that these strategies and the associated value
functions constitute part of an MPE.
Proposition 2 discusses the long run implications of the equilibrium from Proposition 1:
Proposition 2. There exists a symmetric MPE in which, irrespective of the discount
factor and the initial division of the dollar, the status quo policy eventually gets arbitrarily
close to the veto player’s ideal point, that is ∀ ε > 0 there exists T such that ∀ t ≥ T the
veto player’s allocation in the status quo is greater than or equal to 1 − ε.
Proposition 3 addresses the speed of convergence to this long run outcome:
Proposition 3. In the symmetric MPE characterized in the proof of Proposition 1, if
legislators are impatient, δ = 0, it takes at most two rounds of proposals by the veto player to
15
converge to the irreducible absorbing state where sv = 1. If legislators are patient, δ ∈ (0, 1),
convergence to this absorbing state does not happen in a finite number of bargaining periods,
and the higher the discount factor the slower the convergence.
In the Appendix, I characterize sufficient conditions for full appropriation by the veto
player to be the unique equilibrium outcome8 and show that the equilibrium from Proposition
1 is robust to changes in the legislators’ patience.
5
Institutional Limits to Veto Power
In this section, I explore institutional measures that could, in principle, reduce the leverage of
the veto player and promote more equitable outcomes: reducing the recognition probability
of the veto player, expanding the committee, and changing the voting rule—for example,
requiring a qualified majority of the committee to approve a policy change.
5.1
Heterogeneous Recognition Probabilities
In some settings, the veto player is an outsider with reduced ability to set the agenda—
for example, the U.S. President—while in others the veto player has a privileged position
to set the agenda—for example, committee chairs in the U.S. Congress. In this section, I
relax the assumption of symmetric recognition probabilities and find sufficient conditions on
the discount factor and the recognition probabilities under which the veto player is able to
eventually appropriate all resources for any initial status quo. In these cases, the convergence
to this outcome is slower the lower is the probability the veto player sets the agenda.
Denote by pv the probability the veto player is recognized as the proposer in each period,
with pnv =
1−pv
2
being the probability a non-veto player is recognized. Proposition 5 shows
that, when pv ∈ [0, 1/2] or pv = 1, there exists an MPE of this dynamic game that has the
8
I show that a veto proposer can always increase his allocation in any continuous and weakly monotonic
MPE. This is also true in any SPE as long as δ < 3/7. An MPE satisfies weak monotonicity if a non-veto
player is always worse off when transferring some of his share to the other non-veto player.
16
same features as the one characterized in Section 4: all proposals entail positive distribution
to, at most, a minimal winning coalition and the status quo allocation converges to the
ideal point of the veto player irrespective of the discount factor and the initial allocation of
resources. When pv ∈ (0.5, 1), an MPE with these features exists as long as the discount
factor is below a threshold, δ(pv ).
Proposition 4. With different recognition probabilities for veto and non-veto players,
there exists a symmetric MPE in which, irrespective of the initial division of the dollar, the
status quo policy eventually gets arbitrarily close to the veto player’s ideal point, as long as
h
i
1) pv ∈ 0, 12 and δ ∈ [0, 1), or 2) pv = 1 and δ ∈ [0, 1), or 3) pv ∈
1
,1
2
and δ ≤ δ(pv ).9
As in the case with even recognition probabilities, this result hinges on the fact that
the veto player is able to move the status quo to ∆—the set of allocations where at least
one non-veto player gets zero—as soon as he proposes and that, once an allocation is in
this absorbing set, the veto player is able to increase his share whenever he proposes. The
proposing veto player can move the status quo to ∆ only when he is able and willing to
offer a non-veto player his demand. With homogeneous recognition probabilities, this is true
for any discount factor and initial allocation of resources. With heterogeneous recognition
probabilities, this is not always the case.
The proposal power of the veto player influences the speed of convergence to his ideal
outcome both directly and indirectly. The direct effect is given by the change in the frequency
at which the veto player can increase his allocation—which happens only when he proposes.
The indirect effect is given by the change in the amount the veto player can extract from the
non-veto players when he proposes. The probability of recognition of the veto player affects
the continuation value of the status quo policy for all legislators, and, thus, it affects how
much they demand to support a policy change.
9
The threshold δ is characterized in the proof of Proposition 4. The MPE exists for all but a small
fraction of parameters: the lowest value of δ is 0.875, reached when pv ≈ 0.857. This means that, for
δ ∈ [0, 0.875), the MPE exists for any pv ∈ (0, 1] and any s0 ∈ ∆.
17
Consider status quo allocations where one non-veto player has nothing. As pv increases,
the poorer non-veto player is less likely to be recognized and, thus, he is less concerned about
the endowment of the richer non-veto player, which represents the resources he can appropriate when he has the power to set the agenda. This reduces the premium he demands to
support an allocation that increases the share to the veto player. In the proof of Proposition
4, I show that this premium is monotonically decreasing in pv . When pv = 1, the poorer
non-veto player does not demand a premium and supports any allocation. Thus, with a
higher pv , the veto player is more likely to increase his share in each period, and he can also
extract more from the non-veto players when he is the proposer.
Proposition 5 summarizes this discussion:
Proposition 5. In the symmetric MPE from Proposition 4, if δ = 0 or pv = 1, it takes
at most two rounds of proposals by the veto player to converge to the absorbing state where
sv = 1. If δ ∈ (0, 1) and pv ∈ (0, 1), convergence to the absorbing state does not happen in a
finite number of periods. Convergence is slower the higher is δ and the lower is pv .
5.2
Competing Veto Powers, Committee Size and Majority Requirement
In this section, I study committees with n legislators, k ≤ n veto players and m = n − k nonveto players (with m even). A proposal defeats the status quo if it receives the concurring
h
i
support of the k veto players and q ∈ 0, 32 m non-veto players. This includes a wide array of
voting rules, from oligarchies where the coalition of all veto players can change the status quo
without the approval of any non-veto player (q = 0), to qualified majorities where the status
quo is defeated only with the approval of more than 50% of legislators (k + q > n/2). This
more general setup allows me to investigate whether expanding the committee or changing
the majority requirement can reduce the leverage of the veto player(s) and promote more
equitable outcomes.
In order to preserve the analytical tractability of the model, I introduce two assump18
tions.10 Fist, having explored the impact of recognition probabilities in smaller committees,
I assume that only veto players are able to make proposals and that each veto player proposes with the same probability. Second, I restrict the set of feasible allocations to those
with, at most, two types of non-veto players: a subset who receives zero and a subset who
receives the same, non-negative amount. In particular, a feasible allocation is summarized by
s = {sv1 , sv2 , . . . , svk , m, sm }, where svi , i = {1, 2, . . . , k}, denotes the share to veto player i,
m denotes the number of poor non-veto players (whose share is sm = 0), and sm ≥ 0 denotes
the share to each of the (m − m) rich non-veto players.11 I denote with snv = (m − m)sm
the total share of resources allocated to non-veto players in allocation s.
The presence of multiple veto players or qualified majorities do not prevent the complete
expropriation of the resources initially allocated to non-veto players. Proposition 6 shows
that this dynamic game has a symmetric MPE—where symmetry applies to legislators of
the same type—in which the k veto players extract all the surplus in finite time.
Proposition 6. In the game with n legislators, k ≤ n veto players, and q non-veto
players needed for passage, there exists a symmetric MPE in which, irrespective of δ, k, n,
q and s0 , the committee transitions to an absorbing state where the k veto players get the
whole pie in at most three periods.12
Complete appropriation by veto players is possible because a veto proposer can always
pass an allocation that increases his allocation. The committee converges to this outcome
10
The existence proofs for the equilibria proposed in this paper rely on constructing the equilibrium
strategies, and the associated continuation values, for any allocation of the dollar, s ∈ ∆. With more than
three legislators, the dimensionality of the state space increases and tractability is quickly lost. An extension
to four legislators—two veto players and two non-veto players—which does not employ these additional
assumptions is available from the author.
11
This assumption does not restrict the number of legislators who receive a positive allocation and it
allows proposers to give different amounts to different legislators. The only role of this assumption is to
simplify the identification of the cheapest coalition: the proposer randomizes among coalition partners with
the same status quo allocation but does not need to employ a different mixing probability for each feasible
allocation (as it was the case in regions C and D for the model with 3 legislators).
12
For the case where veto players are decisive (q = 0), Proposition 8 in the Appendix proves an analogous
result for the more general setup where pv ∈ [0, 1] and all vectors of non-negative transfers which sum to 1
are feasible agreements.
19
in finite time because poor non-veto players do not demand a premium and support any
allocation of resources. This means that the veto player can expropriate non-veto players
completely as soon as the status quo gives zero to at least q non-veto legislators.
At the same time, non-veto players might enjoy positive resources in the initial periods
and larger majority requirements reduce the speed of convergence to the absorbing state
where non-veto players hold nothing. This is because rich non-veto players do demand a
premium to support an allocation which increases the allocation to veto players, and this
premium is increasing in the discount factor and in the fraction of the resources to non-veto
players. The cumulative demand of the (q − m) rich non-veto players needed for a minimal
winning coalition is less than the cumulative amount to non-veto players in the status quo.
Thus, the proposing veto player can increase his allocation and increase the number of poor
non-veto players in the future status quo.13 However, with a larger q or a lower m, it takes
more periods to have at least q poor non-veto players and for veto players to appropriate all
resources.
Moreover, the presence of other legislators with veto power reduces the amount a single
legislator can extract. A non-proposing veto player asks more than what he receives in the
status quo to support the expropriation of non-veto players. He demands this premium
because a higher current allocation to another veto player decreases the amount he is able
to extract when he proposes in the future. In particular, when the non-veto players are
completely expropriated, the policy moves to a gridlock region where no future proposer will
be able to pass a policy he prefers to the status quo. In order to offset this loss and gain
their vote, the proposing veto player has to share part of the amount he expropriates from
the non-veto players with the other k − 1 veto players. Nonetheless, the proposing veto
player gets a greater share of the resources expropriated from the non-veto players for any
δ ∈ [0, 1).
Proposition 7 summarizes this discussion.
13
This is because mt+1 = m − (q − mt ) = mt + (m − q) > mt .
20
Proposition 7. In the symmetric MPE from Proposition 6: (a) the number of periods
with positive resources to non-veto players is weakly increasing in q; (b) the cumulative value
of the game for non-veto players is weakly increasing in δ, q and s0nv , and weakly decreasing in
m; (c) the share to each veto player in the absorbing state is strictly larger than his starting
share, unless s0 is an absorbing state (that is, s0nv = 0) or δ = 0.
6
Experimental Design
The theory provides sharp empirical implications, in particular on the shadow of the future
in standing committees, that is, on how different degrees of patience affects legislators?
bargaining behavior and the allocation of resources. In the remainder of the paper, I assess
the empirical validity of these theoretical predictions with the use of controlled laboratory
experiments, which have some important advantages over field data when studying a highly
structured dynamic environment such as the one in this paper (Falk and Heckman 2009).
The experiments were conducted at the Rady Behavioral Lab and at the Columbia Experimental Laboratory in the Social Sciences, using students from the University of California,
San Diego and from Columbia University. Subjects were recruited from a database of volunteer subjects. Ten sessions were run, using a total of 120 subjects. No subject participated
in more than one session.
The experimental treatment is the discount factor, that is, the degree of patience of
the committee. I conduct two sessions with no patience committees (δ = 0), four sessions
with low patience committees (δ = 0.50), and four sessions with high patience committees
(δ = 0.75). Discount factors were induced by a random termination rule: after each round
of the same game, a fair die was rolled by the experimenter at the front of the room, with
the outcome determining whether the game continued to another round (with probability δ).
This is a standard technique used in the experimental literature to preserve the incentives
21
Treatment
High Patience (Dynamic)
Low Patience (Dynamic)
No Patience (One Shot)
n
3
3
3
δ
0.75
0.50
0
Sessions
4
4
2
Committees
160
320
320
Subjects
48
48
24
Table 1: Experimental design.
of infinite horizon games in the laboratory (Roth and Murnighan 1978).14
All sessions were conducted with 12 subjects, divided into 4 committees of 3 members
each—one veto player and two non-veto players. Veto players were selected randomly at the
beginning of the session, with their role as veto players remaining fixed throughout the session. Committees stayed the same throughout the rounds of a given game, and subjects were
randomly rematched into committees between games. The exogenous amount of resources
in each round was 60 experimental units (corresponding to $2). At the beginning of each
game, an initial status quo was randomly chosen by the computer. After being informed
of the initial status quo, each committee member was prompted to enter a provisional proposal. After all members had entered a provisional proposal, one was selected at random to
become the proposed budget. This proposal was then voted on against the status quo, which
was referred to as the standing budget. Whichever budget passed the voting stage was the
policy that was implemented in that round, and each member received earnings accordingly.
Instructions were read aloud and subjects were required to correctly answer all questions on
a short comprehension quiz before the experiment was conducted. The experiments were
conducted via computers.15
Table 1 lists the number of subjects and committees in each treatment condition.
14
In the sessions with δ = 0, no die was rolled at the end of a round and subjects played a sequence of
one-shot games. In sessions with δ > 0, the length of a game ranged from 1 to 12 rounds. In order to ensure
the same number of expected rounds (40), each of the high patience sessions lasted for 10 games, each of the
low patience sessions lasted for 20 games, and each of the no patience sessions lasted for 40 games.
15
Sample instructions are available in the Appendix. The computer program used in the experiment was
an extension to the open source software Multistage.
22
Region DV
D1
Allocation to Non-Veto 1
Region MVX
Region U+M12
M12
U
Allocation to Non-Veto 2
Figure 7: Allocation regions. Notes: DV groups allocations that give at least 2/3 of the pie to
the veto player; MVX groups the other majoritarian allocations between the veto player and one
non-veto player; U+M12 groups allocations where every player receives at least 1/4 of the pie.
7
Experimental Results
7.1
Transition Between Policies
The evolution of policies over time provides a clear picture of outcomes dynamics, since
it provides a synthetic description of aggregate behavioral data on both proposal making
and voting. One way to represent the data compactly is to cluster policies in three regions,
as shown in Figure 7: the off-the-equilibrium path allocations that give a positive and
substantial share to both non-veto players (U+M12); all the allocations that give most of
the pie to the veto player (DV); and all the other majoritarian allocations between the veto
player and one non-veto player (MVX).16
Before discussing the results, it is useful to recall the theoretical predictions. If the
status quo is in region U+M12 (panel (a) in Figure 8), the equilibrium predicts we move
16
The U+M12 region consists of all allocations where both non-veto players receive at least 1/4 of the
pie, that is 15 units out of 60. The DV region consists of all allocations where the veto player receives at
least 2/3 of the pie, that is 40 units out of 60. The remaining allocations are in region MVX.
23
Region DV
Region DV
D1
Region U+M12
M12
SQ
U
Region DV
D1
Region MVX
Allocation to Non-Veto 1
Allocation to Non-Veto 1
Region MVX
SQ
Region U+M12
M12
U
Region MVX
Allocation to Non-Veto 1
D1
Region U+M12
M12
U
SQ
Allocation to Non-Veto 2
(a)
Allocation to Non-Veto 2
(b)
Allocation to Non-Veto 2
(c)
Figure 8: Transition between policies, summary of theoretical predictions. Notes: DV groups
allocations that give at least 2/3 of the pie to the veto player; MVX groups the other majoritarian
allocations between the veto player and one non-veto player; U+M12 groups allocations where
every player receives at least 1/4 of the pie. Wider arrows mean a larger transition probability.
immediately away from it, regardless of the identity of the proposer, towards a majoritarian
allocation where the pie is shared by one non-veto player and the veto player. If the status
quo is in the region composed of minimal winning coalitions (panel (b) in Figure 8), the
theory predicts that we either keep the status quo or move to the opposite side of ∆. When
moving to the opposite side, if the veto player is proposing and the initial status quo gives
him enough, we will transition to the DV region. Finally, if we are in the DV region (panel
(c) in Figure 8), we will almost always stay there.17 The predicted evolution of policies
between these coarse regions is very similar between the three treatments. The theory does
predict sharp differences between high, low, and no patience committees for finer details of
behavior and I investigate them below.
The transitions and outcomes for the three treatments are summarized in Table 2. For
each panel, the last row gives the overall outcome frequencies, excluding the initial status
quos, which were decided randomly by the computer to start each game. Each cell in the
other three rows gives the probability of moving to a policy in the column region when
starting from a policy in the row region. The numbers in parentheses are the theoretical
17
If δ = 0 and the status quo is in DV, the predicted outcome is always in DV.
24
predictions (given the realized status quo policies and proposers).
There are four results to highlight from these tables. First, consistent with the MPE,
around 79% of all policies give a positive share at most to a minimal winning coalition
(71% in high patience committees, 83% in low patience committees, and 90% in no patience
committees). Second, once we are in these regions (DV and MVX), the policies evolve
according to the predicted transition probabilities. Third, as predicted by the theory, DV is
the unique absorbing region. Fourth, at odds with the theoretical predictions, if committees
start from an off-the-equilibrium-path allocation that gives a positive share to everyone or
to the two non-veto players, they do not move immediately away from there, but there is
higher inertia than predicted. This inertia is higher for more patient committees, even if the
theory is silent in this respect. I discuss this result in Section 8.
7.2
Veto Player’s Allocation
From the transition probabilities in Table 2, we can see that the policies slowly transition
to the DV region, regardless of the initial status quo and degree of patience, and that,
once there, they do not leave this region. Since this is the main implication of the model,
it is interesting to give a closer look to the evolution of the allocation to the veto player.
The theory predicts that the allocation to the veto player increases over time, and that the
convergence to full appropriation is slower in more patient committees.
Figure 9 shows the evolution of the average allocation to the veto player as the number
of rounds played in the same game grows, separately for the three treatments. The first data
point on the left is the average allocation to the veto player in the initial status quo policy
randomly drawn by the computer in all games of the same treatment.18 For the treatments
with δ > 0, the duration of each game is stochastic: the number of observations available
for each round is different and higher rounds have fewer observations.19 For the treatment
18
The initial allocation to the veto player is not statistically different between any pair of treatments.
19
For these two treatments, Figure 9 shows only rounds for which we have at least 12 committees for each
25
Panel A: High Patience (Dynamic)
Status Quo (t)
DV
MVX
U+M12
Frequency
Status Quo (t+1)
DV
MVX
U+M12
0.98 (0.96) 0.01 (0.04) 0.01 (0.00)
0.11 (0.22) 0.79 (0.78) 0.10 (0.00)
0.02 (0.05) 0.18 (0.95) 0.81 (0.00)
0.36 (0.41) 0.35 (0.59) 0.29 (0.00)
Panel B: Low Patience (Dynamic)
Status Quo (t)
DV
MVX
U+M12
Frequency
Status Quo (t+1)
DV
MVX
U+M12
0.97 (0.95) 0.00 (0.05) 0.03 (0.00)
0.21 (0.29) 0.68 (0.71) 0.11 (0.00)
0.03 (0.08) 0.49 (0.92) 0.48 (0.00)
0.43 (0.47) 0.40 (0.53) 0.17 (420)
Panel C: No Patience (One-Shot)
Status Quo (t)
DV
MVX
U+M12
Frequency
Status Quo (t+1)
DV
MVX
U+M12
1.00 (1.00) 0.00 (0.00) 0.00 (0.00)
0.30 (0.37) 0.65 (0.63) 0.05 (0.00)
0.06 (0.18) 0.60 (0.82) 0.34 (0.00)
0.34 (0.42) 0.55 (0.58) 0.10 (0.00)
Table 2: Transition probabilities between policies, observed and predicted (in parentheses).
26
44 45 12 80 40 116 192 35 30 320 44 48 68 88 96 132 320 16 24 148 160 25 20 160 320 320 15 10 5 0 0 1 2 3 HIGH PATIENCE 4 Round LOW PATIENCE 5 6 7 8 NO PATIENCE Figure 9: Average time paths of allocation to veto player. The numbers on top of each bar are
observations (committees).
with δ = 0, each game lasts exactly one round.
Two results are worth noting. First, as predicted by the MPE, we observe a ratchet effect
in the allocation to the veto player: his share gradually increases over time, in all treatments.
Second, as predicted by the MPE, this ratchet effect is slower in patient committees with
the average veto player’s allocation growing faster in lower patience committees. The difference between the High Patience and the Low Patience series is positive for all rounds and
significant at the 1% level for rounds 1, 2, 4, and 5, at the 5% level for rounds 3 and 6, and
the 10% level for round 7.20,21 The difference between the No Patience treatment and the
treatment. This covers 93% of all observations for high patience committees and 96% of all observations for
low patience committees.
20
Significance tests are based on Wilcoxon-Mann-Whitney tests, whose null hypothesis is that the underlying distributions of the two samples are the same. I am treating as unit of observation a single group. The
results are unchanged if I use t-tests for differences in means. I take into account the panel structure of the
data by clustering the standard errors of these tests by groups composed of the same subjects.
21
This difference is not significant for rounds 8 and beyond because, as predicted by the MPE, the
difference between the two treatments is shrinking over time, and because, due to the random termination
rule, the number of observations for high rounds is small in both treatments.
27
Panel A: Veto Proposers
Premium to Proposer
Premium to Non-Veto 1
Premium to Non-Veto 2
% of MWC
% of Universal
Observations
HIGH PATIENCE
8.34 (11.48)
8.81 (12.54)
-17.16 (-24.02)
0.71 (1.00)
0.28 (0.00)
844
LOW PATIENCE
12.65 (19.20)
7.12 (6.49)
-19.77 (-25.69)
0.74 (1.00)
0.26 (0.00)
824
NO PATIENCE
23.89 (33.31)
5.30 (0.00)
-29.19 (-33.31)
0.87 (1.00)
0.11 (0.00)
320
Panel B: Non-Veto Proposers
Premium to Proposer
Premium to Veto
Premium to Other Non-Veto
% of MWC
% of Universal
Observations
HIGH PATIENCE
8.12 (18.66)
-0.83 (-2.99)
-7.29 (-15.67)
0.56 (1.00)
0.38 (0.00)
1688
LOW PATIENCE
10.09 (17.37)
-0.35 (-1.32)
-9.74 (-16.05)
0.68 (1.00)
0.27 (0.00)
1648
NO PATIENCE
6.44 (20.77)
7.39 (0.00)
-13.83 (-20.77)
0.72 (1.00)
0.24 (0.00)
640
Table 3: Proposing behavior, experimental averages and predictions (in parentheses). Non-Veto 1
(Non-Veto 2) is non-veto player receiving the most (least) in a proposal made by a veto player.
first round of the other two treatments is positive and significant at the 1% level.22
7.3
Proposal Making
The experimental data is very rich and allow us to test the finer predictions of the model.
To investigate the origin of the dynamic patterns described above, I decompose the determinants of the transition probabilities and analyze in detail proposal and voting behavior.
Regarding proposing behavior, the model predicts proposals that give a positive allocation
only to a minimal winning coalition. In particular, a veto proposer completely expropriates
the richer non-veto player. With δ > 0, he shares the spoils with the other non-veto player
and the division of resources between coalition partners is more even in more patient com22
The existence of a ratchet effect is confirmed by a Tobit regression. See the Appendix.
28
mittees. With δ = 0, he allocates all the spoils to himself. A non-veto proposer completely
expropriates the other non-veto player, keeps the spoils for himself, and gives the veto player
exactly what he has in the status quo or less (if the status quo gives a positive allocation to
all players).
In the experiments, veto proposers give a significant share of the pie only to a minimal
winning coalition 71%, 74%, and 87% of the time, respectively for high, low, and no patience committees. When the proposer is a non-veto player instead, we see minimal winning
allocations only 56%, 68%, and 72% of the time. These percentages are in line with other
bargaining experiments.
To compare proposals at different status quo policies, I look at the premium proposed to
each committee member, rather than at the absolute amount. The premium to a member is
the difference between the amount proposed to that member and the amount granted to that
same member by the status quo policy. If the premium to a member is positive, this means
the proposer is suggesting an increase in the allocation to that member. Table 3 shows that
both veto and non-veto proposers expropriate resources from one non-veto player and share
the spoils with the other coalition partner. In particular, veto-proposers give a significant
premium to themselves and to the poorer non-veto member. As predicted by the theory, veto
proposers share the spoils more evenly in high patience committees—that is, the premium
to a veto proposer and the premium to the poorer non-veto player are, respectively, smaller
and larger in higher patience committees. The difference between each pair of treatments
are statistically significant at the 1% level.
7.4
Voting Decisions
Consistent with the theory, in all treatments voting is selfish and forward-looking: 79% of the
votes are in favor of the alternative that gives the voter the higher expected utility, taking into
account the continuation value of the allocations. I run Logit regressions for the likelihood of
voting in favor of a proposal. I do this separately for different player’s type—whether they
29
Voter Type
Premium to Me
Premium to Other Non-Veto
Premium to Veto
Constant
Pseudo-R2
Observations
HIGH PATIENCE
Veto
Non-Veto
0.236*** 0.173***
(0.071)
(0.026)
-0.036***
(0.011)
-0.046***
(0.014)
0.010
-0.197**
(0.219)
(0.092)
0.273
0.398
566
1122
LOW PATIENCE
Veto
Non-Veto
0.169*** 0.189***
(0.053)
(0.025)
-0.049***
(0.015)
0.025***
(0.009)
-0.120
-0.162
(0.198)
(0.117)
0.361
0.450
560
1088
NO PATIENCE
Veto
Non-Veto
1.054** 1.285***
(0.463)
(0.231)
0.063
(0.044)
-0.017
(0.016)
-0.167
-0.125
(0.349)
(0.377)
0.577
0.787
208
432
Table 4: Probability of supporting a proposal: Logit estimates. Observations are subject-rounds.
* p < 0.10, ** p < 0.05, *** p < 0.01. Standard errors clustered by subjects in parentheses.
are a veto or non-veto player—and for different treatments and I exclude proposers from
the analysis. Table 4 shows the results. As predicted by the theory, regardless of patience
and role, committee members are more likely to support a proposal when it offers a larger
allocation increase with respect to the status quo policy (that is, a larger premium). More
interestingly, non-veto members are more concerned about the premium to the proposer when
patience is higher: the probability of voting in favor of a proposal is negatively correlated
with the premium to the veto player only for high patience committees. Similarly, veto
members are more concerned about the premium to the non-proposing non-veto player when
the committee is standing.
8
Conclusions
This paper studies the consequences of veto power in a bargaining game with an evolving
status quo policy. As the importance of the right to block a decision crucially depends on the
status quo, a static analysis cannot draw general conclusions about the effect of veto power
on gridlock and policy capture by the veto player. Instead of making ad hoc assumptions
30
on the status quo policy, I study veto power by exploring the inherently dynamic process
via which the location of the current status quo is determined. I prove that there exists an
equilibrium of this dynamic game such that the veto players are eventually able to extract
all resources, irrespective of the discount factor, the initial allocation of resources and the
number of veto and non-veto legislators. This result shows that, in the long run, the right to
veto is extremely powerful, especially if coupled with proposal power. This is true even when
non-veto legislators are patient, and take into account the loss in future bargaining power
implied by making concessions to veto players in the current period. At the same time,
institutional measures to reduce veto power can be effective in reducing the veto players’
ability to extract resources in the short run and can promote more equitable outcomes for
longer.
The main predictions of the theory find support in the behavior of committees bargaining
in controlled laboratory experiments: outcomes evolve according to the predicted transition
probabilities, albeit with stronger persistence; the allocation to the veto player gradually
increases over time; and patient committees exhibit significantly different proposal and voting
behavior than impatient committees. In line with the dynamic bargaining experiments in
Battaglini and Palfrey (2012), we observe a higher frequency and persistence of universal
outcomes than predicted by the MPE with linear preferences. This might be interpreted as
evidence of fairness and prosocial preferences. However, evidence from voting and proposal
behavior does not seem to support such an interpretation: proposers tend to take advantage
of their proposal power and voters consistently support lopsided allocations.
I propose two theoretical explanations of these outcomes. First, I assumed that legislators’ utilities are linear. While the equilibrium with minimal winning coalitions might be
robust to some concavity in legislators’ utilities,23 MPEs without minimal winning coalitions
may arise when utilities are sufficiently concave.24 Second, while I focused on an equilib23
Kalandrakis (2010) shows that this is the case in a dynamic bargaining game without veto players.
24
Battaglini and Palfrey (2012) explored such equilibria in a context without veto players.
31
rium where agents’ strategies depend only on the status quo policy, these bargaining games
usually have other equilibria that can sustain more equitable outcomes through the use of
history-dependent strategies. In the Appendix, I construct an equilibrium such that the initial allocation is an absorbing state and, thus, there is no convergence to full appropriation
by the veto player. This equilibrium exists as long as the committee is sufficiently patient
and the initial allocations to both non-veto players is large enough. While most outcomes
and their evolution are consistent with the MPE, this alternative equilibrium can explain
the occurrence of universal outcomes and their persistence.
This paper is the first to derive theoretical predictions on the consequences of veto power
in a dynamic setting and to test them experimentally. While the results certainly add to our
understanding of the incentives present in real world legislatures, the setup is intentionally
very simple and uses a number of specific assumptions. For example, while I analyzed a
game where legislators’ preferences are purely conflicting, committees decide on many policy
domains where agents’ preferences are partially aligned. Extending the policy space beyond
the pure distributive setting—either considering a unidimensional policy space or allowing
resources to be allocated to a pubic good—is an important direction for future work.
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36
Appendix (For Online Publication)
Proofs of Propositions
Proof of Proposition 1
Before proceeding to the proof, I introduce formal definitions for Markov strategies and
legislators’ demands.
Definition 1 A Markov strategy is a pair of functions, σ i (s) = (µi [·|s], Ai (s)), where µi [z|s]
represents the probability that legislator i makes the proposal z when recognized, conditional
on the state being s; and Ai (s) represents the allocations for which i votes yes when the state
is s.
Definition 2 For a symmetric MPE, non-veto legislator j’s demand when the state is s
is the minimum amount dj (s) ∈ [0, 1] such that for a proposal x ∈ ∆ with xj = dj (s),
xv = 1 − dj (s), we have Uj (x) ≥ Uj (s). Similarly, veto legislator v’s demand when the state
is s is the minimum amount dv (s) ∈ [0, 1] such that for a proposal x ∈ ∆ with xv = dv (s),
xj = 1 − dv (s), for j = 1, 2, we have Uv (x) ≥ Uv (s).
The results of Proposition 1 follow from the existence of a symmetric MPE with the
following minimal winning coalition proposal strategies for all s ∈ ∆, where s1 ≥ s2 :
• Case A
s1 ≤ 1 −
3−δ
s,
3−2δ 2
s1 ≥
3−δ
s
3−2δ 2
:
xv = [1 − d2 , 0, d2 ], x1 = [dv , 1 − dv , 0], x2 = [dv , 0, 1 − dv ]
dv = s v −
d2 =
δs2
3 − 2δ
δ
(3 − δ)
s1 +
s2
3 − 2δ
(3 − 2δ)
37
• Case B
s1 > 1 −
3−δ
s,
3−2δ 2
s1 ≥
27−27δ+3δ 2 +δ 3
s2
(3−2δ)(3−δ)2
+
δ2
(3−δ)2
:
xv = [1 − d2 , 0, d2 ], x1 = [dv , 1 − dv , 0], x2 = [dv , 0, 1 − dv ]
dv = 0
d2 =
• Case C
x
v
s1 >
=
6−3δ
2(3−δ)
9 − 12δ + 3δ 2
δ
s2 +
2
(3 − 2δ)
(3 − 2δ)
− s2 , s 1 <
27−27δ+3δ 2 +δ 3
s2
(3−2δ)(3−δ)2





[1 − d2 , d2 , 0]
w/ Pr = 1 − µC
v




[1 − d2 , 0, d2 ]
w/ Pr = µC
v
+
δ2
(3−δ)2
:
, x1 = [dv , 1 − dv , 0], x2 = [dv , 0, 1 − dv ]
dv = 0
(−3 + δ)(3δ 2 − 12δ + 9)
(−3 + δ)(6δ − 4δ 2 )
(s
+
s
)
+
1
2
(−3 + 2δ)(δ 2 − 15δ + 18)
(−3 + 2δ)(δ 2 − 15δ + 18)
(δ 3 + 3δ 2 − 27δ + 27)s1 + (2δ 3 − 15δ 2 + 36δ − 27)s2 − 2δ 3 + 3δ 2
d2 =
µC
=
v
• Case D
x
v
=
h
δ (3δ 2 − 12δ + 9)(s1 + s2 ) + 6δ − 4δ 2
s1 ≤
6−3δ
2(3−δ)
− s2 , s 1 <
3−δ
s
3−2δ 2
i
:





[1 − d2 , d2 , 0]
w/ Pr = 1 − µD
v




[1 − d2 , 0, d2 ]
w/ Pr = µD
v
, x1 = [dv , 1 − dv , 0], x2 = [dv , 0, 1 − dv ]
δs2
3 − 2δ
(3 − δ)(9 − 6δ − δ 2 )
(3 − δ)(9 − 6δ + δ 2 )
=
s
+
s2
1
(3 − 2δ)(δ 2 − 15δ + 18)
(3 − 2δ)(δ 2 − 15δ + 18)
(δ 3 − 6δ 2 + 27δ − 27)s1 + (−δ 3 + 12δ 2 − 36δ + 27)s2
h
i
=
δ (δ 2 + 6δ − 9)s1 + (−δ 2 + 6δ − 9)s2
dv = s v −
d2
µD
v
It is tedious but straightforward to check that, if players play the proposal strategies in
cases A-D and these proposals pass, their continuation values are as follows:
38
• Case A
1
2−δ
1
−
s1 −
s2
1 − δ (3 − δ)(1 − δ)
(1 − δ)
(3 − δ)
(3 − 3δ + δ 2 )
s1 +
s2
v1 (s) =
2
(3 − δ) (1 − δ)
(3 − δ)2 (1 − δ)
(3 − 2δ)
(6 − 5δ + δ 2 )
v2 (s) =
s
+
s2
1
(3 − δ)2 (1 − δ)
(3 − δ)2 (1 − δ)
vv (s) =
(3)
(4)
(5)
• Case B
1
(3 − 4δ + δ 2 )
−
s2
(1 − δ)(3 − δ) (3 − 2δ)(1 − δ)(3 − δ)
(9 − 15δ + 9δ 2 − 2δ 3 )
(3δ − 4δ 2 + δ 3 )
s2 +
v1 (s) =
(3 − δ)2 (1 − δ)(3 − 2δ)
(3 − δ)2 (1 − δ)(3 − 2δ)
(3 − 2δ)
(3 − 4δ + δ 2 )
v2 (s) =
+
s2
(3 − δ)2 (1 − δ) (3 − δ)2 (1 − δ)
vv (s) =
• Case C
−9 − 7δ 2 + 15δ + δ 3
2δ 2 + 18 − δ 3 − 15δ
(s
+
s
)
+
1
2
(−1 + δ)(−3 + δ)(δ 2 − 15δ + 18)
(−1 + δ)(−3 + δ)(δ 2 − 15δ + 18)
6δ 3 − 33δ 2 + 54δ − 27
δ 4 − 13δ 3 + 48δ 2 − 63δ + 27
s1 −
s2 +
v1 (s) = −
(−1 + δ)(−3 + δ)(δ 2 − 15δ + 18)
(−1 + δ)(−3 + δ)(δ 2 − 15δ + 18)
−8δ 3 + 24δ 2 − 18δ
+
(−1 + δ)(−3 + δ)(δ 2 − 15δ + 18)
δ 4 − 13δ 3 + 48δ 2 − 63δ + 27
6δ 3 − 33δ 2 + 54δ − 27
v2 (s) = −
s2 −
s1 +
(−1 + δ)(−3 + δ)(δ 2 − 15δ + 18)
(−1 + δ)(−3 + δ)(δ 2 − 15δ + 18)
−8δ 3 + 24δ 2 − 18δ
+
(−1 + δ)(−3 + δ)(δ 2 − 15δ + 18)
vv (s) =
39
• Case D
δ 3 − 23δ 2 + 63δ − 45
δ 3 − 15δ 2 + 51δ − 45
s
+
s2 +
1
(−1 + δ)(−3 + δ)(δ 2 − 15δ + 18)
(−1 + δ)(−3 + δ)(δ 2 − 15δ + 18)
18δ 2 + 54 − δ 3 − 63δ
+
(−1 + δ)(−3 + δ)(δ 2 − 15δ + 18)
2δ 3 − 15δ 2 + 36δ − 27
δ 4 − 21δ 3 + 72δ 2 − 81δ + 27
s
−
s2
v1 (s) = −
1
(−1 + δ)(−3 + δ)(δ 2 − 15δ + 18)δ
(−1 + δ)(−3 + δ)(δ 2 − 15δ + 18)δ
−2δ 3 − 9δ 2 + 36δ − 27
δ 4 − 17δ 3 + 66δ 2 − 81δ + 27
v2 (s) = −
s
−
s2
1
(−1 + δ)(−3 + δ)(δ 2 − 15δ + 18)δ
(−1 + δ)(−3 + δ)(δ 2 − 15δ + 18)δ
vv (s) =
On the basis of these continuation values, we obtain players’ expected utility functions,
Ui (x) = xi +δVi (x). The reported demands are in accordance with Definition 1. In particular,
di , i = 1, 2 and dv can be easily derived from the following equations:
si + δVi (s) = di + δVi ([1 − di , di , 0])
sv + δVv (s) = dv + δVv ([dv , 1 − dv , 0])
The demands for non-veto player 1 are never part of a proposed allocation and have
therefore been omitted in the statement of the equilibrium proposal strategies above but we
will use them in the remainder of the proof. In cases C and D, the mixing of the veto player
is such that d1 = d2 . In the other two cases, d1 is as follows:
• Case A
s1 ≤ 1 −
3−δ
s,
3−2δ 2
s1 ≥
d1 =
• Case B
s1 > 1 −
3−δ
s,
3−2δ 2
3−δ
s
3−2δ 2
:
(4δ 2 − 12δ + 9)
(3δ − δ 2 )
s
+
s2
1
(3 − 2δ)2
(3 − 2δ)2
s1 ≥
27−27δ+3δ 2 +δ 3
s2
(3−2δ)(3−δ)2
+
δ2
(3−δ)2
:
(27 − 63δ + 51δ 2 − 17δ 3 + 2δ 4 )
(3δ 2 − 4δ 3 + δ 4 )
9δ − 15δ 2 + 9δ 3 − 2δ 4
d1 =
s1 +
s2 +
(3 − 2δ)3
(3 − 2δ)3
(3 − 2δ)3
40
Furthermore, all reported non-degenerate mixing probabilities are well defined. On the
basis of the expected utility functions, Ui , we can then construct equilibrium voting strategies, A∗i (s) = {x|Ui (x) ≥ Ui (s)}, i = v, 1, 2, for all s ∈ ∆. These voting strategies are
obviously stage-undominated. Then, to prove Proposition 1 it suffices to verify the optimality of proposal strategies. To do so, we make use of five lemmas. We seek to establish an
equilibrium with proposals that allocate a positive amount to at most one non-veto player.
Lemma 1 shows that the expected utility function for these proposals satisfies the following
continuity and monotonicity properties. Lemma 2 proves that minimal winning coalition
proposals are optimal among the set of feasible proposals in ∆. Lemma 3 establishes that
the equilibrium demands of the veto player and one non-veto player sum to less than unity
and that the demands of the two non-veto players are (weakly) ordered in accordance to
the ordering of allocations under the state s. Lemma 4 then establishes that the proposal
strategies for legislators i = v, 1, 2 in Proposition 1 maximize Ui (x) over all x ∈ W (s) ∩ ∆,
where W (s) is the set of all proposals that beat s in the voting stage. These proposals would
then maximize Ui (x) over all x ∈ W (s) if there is no x ∈ W (s) ∩ ∆/∆ that accrues i higher
utility. We establish that this is indeed the case in Lemma 5.
Lemma 1. Consider a symmetric Markov Perfect strategy profile with expected utility
Ui (s), s ∈ ∆, determined by the continuation values in equations (3)-(5). Then, for all
x = (x, 1 − x, 0) ∈ ∆ (a) Ui (x), i = v, 1, 2 is continuous and differentiable with respect to x,
(b) Uv (x) increases with x, while U1 (s) and U2 (s) does not increase with x.
Proof. An allocation x = (x, 1 − x, 0) ∈ ∆ belongs to case A in Proposition 2. Therefore
we can write Ui (x) = xi + δVi (x) as follows:
δ
δ(2 − δ)
−
(1 − x)
1 − δ (3 − δ)(1 − δ)
(3 − 3d + δ 2 )
U1 (x) = 1 − x + δ
(1 − x)
(3 − δ)2 (1 − δ)
(3 − 2δ)
U2 (x) = δ
(1 − x)
(3 − δ)2 (1 − δ)
Uv (x) = x +
41
(6)
(7)
(8)
Ui (x) is linear and continuous in x for i = v, 1, establishing part (a) of the Lemma. Regarding
part (b):
δ(2 − δ)
∂Uv (x)
= 1+
>0
∂x
(3 − δ)(1 − δ)
!
∂U1 (x)
(3 − 3d + δ 2 )
= − 1+δ
<0
∂x
(3 − δ)2 (1 − δ)
∂U2 (x)
(3 − 2δ)
= −δ
<0
∂x
(3 − δ)2 (1 − δ)
∂Uv (x)
∂x
> 0 for any δ ∈ [0, 1), since both the numerator and the denominator of
positive for any δ ∈ [0, 1);
denominator of
(3−3d+δ 2 )
(3−δ)2 (1−δ)
∂U1 (x)
∂x
δ(2−δ)
(3−δ)(1−δ)
are
< 0 for any δ ∈ [0, 1), since both the numerator and the
are positive for any δ ∈ [0, 1); and
since both the numerator and the denominator of
(3−2δ)
(3−δ)2 (1−δ)
∂U2 (x)
∂x
< 0 for any δ ∈ [0, 1),
are positive for any δ ∈ [0, 1).
By the definition of demands and the monotonicity established in part (b) of Lemma 1
we immediately deduce:
Lemma 2. Consider a symmetric Markov Perfect strategy profile with expected utility,
Ui (x), for x ∈ ∆, i = v, 1, 2, given by (6)-(8). Every minimal winning coalition proposal
of the veto player x(v, i, di (s)), i = {1, 2} is such that x(v, i, di (s)) ∈ arg max{Uv (x)|x ∈
∆, Ui (x) ≥ Ui (s)}; similarly, every minimal winning coalition proposal of a non-veto player
x(i, v, dv (s)), i = {1, 2} is such that x(i, v, di (s)) ∈ arg max{Ui (x)|x ∈ ∆, Uv (x) ≥ Uv (s)}.
Lemma 3.
For all s ∈ ∆, the demands reported in Proposition 1 are such that (a)
si ≥ sj ⇒ di ≥ dj , i.j = 1, 2, and (b) di + dv ≤ 1, i = 1, 2.
Proof. Part (a). Since we focus on the half of ∆ in which s1 ≥ s2 , we want to prove that
d1 ≥ d2 . In cases C and D the mixed strategy of the veto player is such that d1 = d2 , so we
focus on cases A and B.
42
• Case A:
(4δ 2 − 12δ + 9)
(3 − δ)
(3δ − δ 2 )
δ
s
+
s2
s
+
s
≥
1
1
2
(3 − 2δ)2
(3 − 2δ)2
3 − 2δ
(3 − 2δ)
3−δ
s1 ≥
s2
3 − 2δ
• Case B:
s1
(27 − 63δ + 51δ 2 − 17δ 3 + 2δ 4 )
(3δ 2 − 4δ 3 + δ 4 )
9δ − 15δ 2 + 9δ 3 − 2δ 4
s
+
s
+
1
2
(3 − 2δ)3
(3 − 2δ)3
(3 − 2δ)3
2
9 − 12δ + 3δ
δ
≥
s2 +
2
(3 − 2δ)
(3 − 2δ)
2
3
27 − 27δ + 3δ + δ
δ2
≥
s
+
2
(3 − 2δ)(3 − δ)2
(3 − δ)2
Part (b). Since we focus on the half of the ∆ in which s1 ≥ s2 , by part (a) of the same
Lemma, it is enough to prove that d1 + dv ≤ 1.
• Case A:
sv −
δs2
(4δ 2 − 12δ + 9)
(3δ − δ 2 )
+
s
+
s2 ≤ 1
1
(3 − 2δ)
(3 − 2δ)2
(3 − 2δ)2
δ2
s2 ≤ 1
sv + s1 +
(3 − 2δ)2
which holds for any δ ∈ [0, 1), because sv + s1 + s2 = 1 and
notice that
δ2
(3−2δ)2
δ2
(3−2δ)2
∈ [0, 1). To see this
is monotonically increasing in δ and is equal to 1 when δ = 1.
• Case B:
(27 − 63δ + 51δ 2 − 17δ 3 + 2δ 4 )s1 (3δ 2 − 4δ 3 + δ 4 )s2 9δ − 15δ 2 + 9δ 3 − 2δ 4
+
+
≤1
(3 − 2δ)3
(3 − 2δ)3
(3 − 2δ)3
43
Notice that
(27−63δ+51δ 2 −17δ 3 +2δ 4 )
(3−2δ)3
≥
(3δ 2 −4δ 3 +δ 4 )
(3−2δ)3
for any δ ∈ [0, 1), so the LHS has an upper
bound when s1 = 1 and s2 = 0. Therefore, we can prove the following inequality:
(27 − 63δ + 51δ 2 − 17δ 3 + 2δ 4 ) 9δ − 15δ 2 + 9δ 3 − 2δ 4
+
≤ 1
(3 − 2δ)3
(3 − 2δ)3
(3 − 2δ)3
≤ 1
(3 − 2δ)3
• Case C:
(−3 + δ)(6δ − 4δ 2 )
(−3 + δ)(3δ 2 − 12δ + 9)
(s1 + s2 ) +
≤ 1
(−3 + 2δ)(δ 2 − 15δ + 18)
(−3 + 2δ)(δ 2 − 15δ + 18)
(δ − 6)(−3 + 2δ)
(s1 + s2 ) ≤
(−3 + δ)2
which holds for any δ ∈ [0, 1), since sv + s1 + s2 = 1 and
To see this notice that
(δ−6)(−3+2δ)
(−3+δ)2
(δ−6)(−3+2δ)
(−3+δ)2
≤ 1 for any δ ∈ [0, 1).
is monotonically decreasing in δ and it is equal to 5/4
when δ = 0.
• Case D:
(1 − s1 − s2 ) −
δs2
(3 − δ)(9 − 6δ − δ 2 )
(3 − δ)(9 − 6δ + δ 2 )
−
s
−
s2 ≤ 1
1
3 − 2δ (3 − 2δ)(δ 2 − 15δ + 18)
(3 − 2δ)(δ 2 − 15δ + 18)
−30δ 2 + 54δ − 27 + 3δ 3
−9δ 2 + 36δ − 27
−
s1 −
s2 ≤ 0
(−3 + 2δ)(δ 2 − 15δ + 18)
(−3 + 2δ)(δ 2 − 15δ + 18)
which holds for any δ ∈ [0, 1] because the coefficients of s1 and s2 in the right hand side
are always non-positive: they are strictly increasing in δ and are equal to 0 for δ = 1.
We now show that equilibrium proposals are optimal over feasible alternatives in ∆.
Lemma 4. µi [z|s] > 0 ⇒ z ∈ arg max{Ui (x)|x ∈ W (s) ∩ ∆}, for all z, s ∈ ∆.
Proof. All equilibrium proposals take the form of minimal winning coalition proposals:
x(v, j, dj (x)) when the veto player is proposing and x(j, v, dv (x)) when a non-veto player is
proposing. Also, whenever µv [x(v, 1, d1 )|s] > 0 and µv [x(v, 2, d2 )|s] > 0, we have d1 = d2
44
so that Uv (x(v, 1, d1 )) = Uv (x(v, 2, d2 )). Thus, in view of Lemma 2 it suffices to show that
if µi [x(i, j, dj)|s] = 1, then Ui (x(i, j, dj )) = Ui (x(i, h, dh )), h 6= i, j, i.e. proposer i has no
incentive to coalesce with player h instead of j. This is immediate for a non-veto player,
since only coalescing with the veto player guarantees the possibility to change the state. To
show that - for the veto player - if µv [x(v, j, dj)|s] = 1, then Uv (x(v, j, dj )) = Uv (x(v, h, dh )),
j 6= h, it suffices to show dh ≥ dj by part (b) of Lemma 1. In Proposition 1 we have s1 ≥ s2 ,
(by part (a) of Lemma 3) d1 ≥ d2 , and when d1 6= d2 , we have µv [x(v, 1, d1 )|s] = 0 which
gives the desired result.
We conclude the proof by showing that optimum proposal strategies cannot belong in
∆/∆. In particular, we show that if an alternative in ∆/∆ beats the status quo by majority
rule, then for any player i we can find another alternative in ∆ that is also majority preferred
to the status quo and improves i’s utility.
Lemma 5. Assume x ∈ W (s) ∩ ∆/∆; then for any i = v, 1, 2 there exists y ∈ W (s) ∩ ∆
such that Ui (y) ≥ Ui (s).
Proof. Consider first the veto player, i = v. Let x ∈ W (s) ∩ ∆/∆. Consider first
the case x ∈ A∗v (s). Then, x is weakly preferred to s by v and at least one i, i = 1, 2.
Now set y = x(v, j, dj (x)), where dj (x) is the applicable demand from Proposition 1. We
have Uj (x(v, j, dj (x))) ≥ Uj (x), by the definition of demand. From part (b) of Lemma
3 have dv (x) + dj (x) ≤ 1 and as a result xv (v, j, dj (x)) = 1 − dj (x) ≥ dv (x); hence,
Uv (x(v, j, dj (x))) ≥ Uv (x), which follows from the weak monotonicity in part (b) of Lemma
1. Thus, y = x(v, j, dj (x)) ∈ W (s) (because is supported by v and j), and we have completed the proof for this case. Now consider the case x 6∈ A∗v (s), i.e. Uv (s) > Uv (x).
Part (a) of Lemma 3 ensures that dv (s) + dj (s) ≤ 1, hence proposal y = x(v, j, dj (s)) has
xv (v, j, dj (s)) = 1 − dj (s) ≥ dv (s). Then Uv (y) ≥ Uv (s) > Uv (x), and y ∈ W (s) ∩ ∆.
Now consider a non veto player, i = 1, 2. Let x ∈ W (s)∩∆/∆. Consider first the case x ∈
A∗i (s). Then, x is weakly preferred to s by v and (at least) i. Now set y = x(i, v, dv (x)), where
dv (x) is the applicable demand from Proposition 1. We have Uv (x(i, v, dv (x))) ≥ Uv (x), by
45
the definition of demand. From part (b) of Lemma 3 have dv (x) + di (x) ≤ 1 and as a result
xi (i, v, dv (x)) = 1 − dv (x) ≥ di (x); hence, Ui (x(i, v, dv (x))) ≥ Ui (x), which follows from the
weak monotonicity in part (b) of Lemma 1. Thus, y = x(i, v, dv (x)) ∈ W (s) ∩ ∆ (because is
supported by v and i), and we have completed the proof for this case. Finally, consider the
case x 6∈ A∗i (s), i.e. Ui (s) > Ui (x). Part (a) of Lemma 3 ensures that dv (s) + di (s) ≤ 1, hence
proposal y = x(i, v, dv (s)) has xi (i, v, dv (s)) = 1−dv (s) ≥ di (s). Then Ui (y) ≥ Ui (s) > Ui (x),
and y ∈ W (s) ∩ ∆, which completes the proof.
As a result of Lemmas 4 and 5, equilibrium proposals are optima over the entire range
of feasible alternatives which completes the proof.
Proof of Proposition 2
The result derives from the features of the MPE characterized in the proof of Proposition 1.
In this MPE, once we reach allocations in the absorbing set ∆, which happens after at most
one period, the veto player is able to increase his share whenever he has the power to propose,
and keeps a constant share when not proposing. For any ε and any starting allocation s0 ,
there exists a number of proposals by the veto player—which depends on δ—such that the
veto player’s allocation in the status quo will be at least 1 − ε for all subsequent periods.
Let this number of proposals be n∗ (ε, δ, s0 ). Since each player has a positive probability
of proposing in each period, the probability that in infinitely many periods the veto player
proposes less than n∗ (ε, δ, s0 ) is zero.
Proof of Proposition 3
This result follows directly from the equilibrium demand of the poorer non-veto player in
the absorbing set ∆, dnv (s, δ) =
δ
s .
3−2δ nv
When δ = 0, this demand is zero. This means
that, when the status quo is in ∆—a set that is reached in at most one period—the poorer
non-veto supports any proposal by the veto player. The veto player can thus pass his ideal
outcome as soon s ∈ ∆ and he proposes. On the other hand, when δ ∈ (0, 1), this is not
46
possible, and the poorer non-veto player always demands a positive share of the dollar to
support any allocation that makes the veto player richer. The convergence in this case is
only asymptotic as the non-veto player’s demand is always positive as long as the allocation
to the richer non-veto is positive, that is as long as the poorer veto player does not have the
whole dollar in the status quo.25
Proof of Proposition 4
As for Proposition 1, we focus on the allocations in which s1 ≥ s2 . The other cases are symmetric. Consider the following equilibrium proposal strategies (all supported by a minimal
winning coalition) and demands (as defined in the proof of Proposition 1):
• CASE A: s1 ≤ 1 −
2−δ(1−pv )
2−δ(1+pv ) s2 ;
s1 ≥
2−δ(1−pv )
2−δ(1+pv ) s2
A
1
A
A
2
A
A
xv = [1 − dA
2 , 0, d2 ], x = [dv , 1 − dv , 0], x = [dv , 0, 1 − dv ]
2pv δ
s2
2 − (1 + pv )δ
2 − δ(1 − pv )
δ(1 − pv )
s1 +
s2
2 − δ(1 + pv )
2 − δ(1 + pv )
−p2v δ 2 − δ 2 − 2pv δ + 2δ + 2pv δ 2
−4pv δ + 4 + 2pv δ 2 − 4δ + p2v δ 2 + δ 2
s
+
s2
1
(2 − δ(1 + pv ))2
(2 − δ(1 + pv ))2
dA
= sv −
v
dA
=
2
dA
=
1
25
Notice that when the initial division of the dollar—which is assumed to be exogenous—assigns the
whole dollar to the veto player, then the status quo will never be changed and the veto player gets the whole
dollar in every period.
47
2−δ(1−pv )
• CASE B: s1 > 1− 2−δ(1+p
s2 ; s1 ≥
v)
p3v δ 3 −2p2v δ 3 +pv δ 3 +p2v δ 2 −2pv δ 2 +δ 2 −4δ+4
p3v δ 3 −pv δ 3 −2p2v δ 2 +2pv δ 2
s
+
2
(2−(1+pv )δ)(2−(1−pv )δ)(1−pv δ)
(2−(1+pv )δ)(2−(1−pv )δ)(1−pv δ)
B
1
B
B
2
B
B
xv = [1 − dB
2 , 0, d2 ], x = [dv , 1 − dv , 0], x = [dv , 0, 1 − dv ]
dB
= 0
v
p2v δ 2 − δ 2 − 2pv δ + 2δ
−2pv δ 2 + 2δ 2 + 2pv δ − 6δ + 4
s
+
2
(2 − δ(1 + pv ))2
(2 − δ(1 + pv ))2
16δ − 8 + 2p2v δ 2 + 8pv δ − 16pv δ 2 − 10δ 2 − 2p3v δ 3 + 2δ 3 − 2p2v δ 3 + 10pv δ 3 − 2pv δ 4 + 2p3v δ 4
=
s1 + ...
(−2 + pv δ + δ)3
6pv δ 3 + 2p3v δ 3 − 8p2v δ 3 + 4p2v δ 2 − 4pv δ 2 + 4p2v δ 4 − 2pv δ 4 − 2p3v δ 4
+
s2 + ...
(−2 + pδ + δ)3
−δ 3 − 7pv δ 3 + 5p2v δ 3 + 4pv δ 2 + 4pv δ − 8p2v δ 2 + 4δ 2 + 2pv δ 4 − 4δ + 3p3v δ 3 − 2p3v δ 4
+
(−2 + pδ + δ)3
dB
=
2
dB
1
• CASE C: s1 >
x
v
dC
v
dC
1
µC
v
=
2−δ
2−δ(1−pv ) −s2 ; s1
<
p3v δ 3 −2p2v δ 3 +pv δ 3 +p2v δ 2 −2pv δ 2 +δ 2 −4δ+4
p3v δ 3 −pv δ 3 −2p2v δ 2 +2pv δ 2
s2 + (2−(1+p
(2−(1+pv )δ)(2−(1−pv )δ)(1−pv δ)
v )δ)(2−(1−pv )δ)(1−pv δ)



C

 [1 − dC
2 , d2 , 0]
w/ Pr = 1 − µC
v


C

 [1 − dC
2 , 0, d2 ]
w/ Pr = µC
v
C
2
C
C
, x1 = [dC
v , 1 − dv , 0], x = [dv , 0, 1 − dv ]
= 0
(pv δ − 1)(−pv δ 2 + δ 2 + pv δ − 3δ + 2)
(pv δ − 1)(p2v δ 2 + 2δ − δ 2 − 2pv δ)
(s
+
s
)
+
1
2
(pv δ + δ − 2)(p2v δ 2 − 2pv δ − δ + 2)
(pv δ + δ − 2)(p2v δ 2 − 2pv δ − δ + 2)
(p3v δ 3 − 2p2v δ 3 + pv δ 3 + p2v δ 2 − 2pv δ 2 + δ 2 − 4δ + 4)s1
=
+
2pv δ((−pv δ 2 + δ 2 + pv δ − 3δ + 2)s1 + (−pv δ 2 + δ 2 + pv δ − 3δ + 2)s2 + p2v δ 2 + 2δ − δ 2 − 2pv δ)
(−p3v δ 3 + pv δ 3 + p2v δ 2 − 4pv δ 2 − δ 2 + 4pv δ + 4δ − 4)s2
+
+
2pv δ((−pv δ 2 + δ 2 + pv δ − 3δ + 2)s1 + (−pv δ 2 + δ 2 + pv δ − 3δ + 2)s2 + p2v δ 2 + 2δ − δ 2 − 2pv δ)
p3v δ 3 − pv δ 3 − 2p2v δ 2 + 2pv δ 2
+
2pv δ((−pv δ 2 + δ 2 + pv δ − 3δ + 2)s1 + (−pv δ 2 + δ 2 + pv δ − 3δ + 2)s2 + p2v δ 2 + 2δ − δ 2 − 2pv δ)
= dC
2 =
48
• CASE D: s1 ≤
x
v
=
2−δ
2−δ(1−pv )
− s2 ; s1 <
2−δ(1−pv )
2−δ(1+pv ) s2



D

 [1 − dD
2 , d2 , 0]
w/ Pr = 1 − µD
v


D

 [1 − dD
2 , 0, d2 ]
w/ Pr = µD
v
D
2
D
D
, x1 = [dD
v , 1 − dv , 0], x = [dv , 0, 1 − dv ]
2pv δ
s2
2 − (1 + pv )δ
(pv δ − 1)(2p2v δ 3 − 2pv δ 3 − 3p2v δ 2 + pv δ 2 − 2δ 2 + 3pv δ + 7δ − 6)
= dD
s1 +
2 =
(−3 + 2δ)(pv δ + δ − 2)(p2v δ 2 − 2pv δ − δ + 2)
(pv δ − 1)(p2v δ 3 − 2pv δ 3 + δ 3 − 3p2v δ 2 + 3pv δ 2 − 4δ 2 + 3pv δ + 7δ − 6)
s2
+
(−3 + 2δ)(pv δ + δ − 2)(p2v δ 2 − 2pv δ − δ + 2)
(4p3v δ 4 − 4p2v δ 4 − 6p3v δ 3 + 4p2v δ 3 + 2δ 3 + 3p2v δ 2 − 11δ 2 + 20δ − 12)s1
=
+
T
(−p3v δ 4 + pv δ 4 − 6pv δ 3 − 2δ 3 + 3p2v δ 2 + 14pv δ 2 + 11δ 2 − 12pv δ − 20δ + 12)s2
+
T
dD
= sv −
v
dD
1
µD
v
where T = 2pv δ[(2p2v δ 3 − 2pv δ 3 − 3p2v δ 2 + pv δ 2 − 2δ 2 + 3pv δ + 7δ − 6)s1 +
D
+ (p2v δ 3 − 2pv δ 3 + δ 3 − 3p2v δ 2 + 3pv δ 2 − 4δ 2 + 3pv δ + 7δ − 6)s2 ], and µC
v , µv are the probabilities
that the veto player coalesces with non-veto player 2 in cases C, and D respectively. These are
C
D
D
well defined probability in [0,1] such that dC
1 = d2 and d1 = d2 , or such that s1 + δv1 (s, µv , d2 ) =
s2 + δv2 (s, µv , d2 ).
Remember that the veto player proposes with probability pv and each non-veto player with
probability (1 − pv )/2. If proposers use the proposal strategies above and these proposals pass,
players’ continuation values in cases A-D are as follows:
• Case A
vv (s) =
v1 (s) =
v2 (s) =
1
(1 − pv )(2 − δ)
1
−
s1 −
s2
(1 − δ) (1 − δ)[2 − δ(1 − pv )]
(1 − δ)
(1 − pv )[2 − δ − δpv (3 − 2δ)]
(1 − pv )
s1 +
s2
2(1 − δ)(1 − pv δ)[2 − (1 − pv )δ]
2(1 − δ)(1 − pv δ)
p2v δ − δ − 2pv + 2
−2p2v δ 2 + 2pv δ 2 + p2v δ − 4pv δ − δ + 2pv + 2
s1 +
s2
2(1 − δ)(1 − pv δ)[2 − (1 − pv )δ]
2(1 − δ)(1 − pv δ)[2 − (1 − pv )δ]
49
• Case B
2pv (pv δ 2 + 3δ − δ 2 − pv δ − 2)
2pv (−pv δ + 2 − δ)
s2 +
(pv δ + 2 − δ)(−1 + δ)(−2 + pv δ + δ)
(pv δ + 2 − δ)(−1 + δ)(−2 + pv δ + δ)
(−2pv δ 3 + 2p2v δ 3 − 2p2v δ 2 + 6pv δ 2 − 4pv δ)(−1 + pv )
v1 (s) = −
s2 +
2(−2 + pv δ + δ)(−1 + δ)(pv δ + 2 − δ)(pv δ − 1)
(−4 + 2pv δ 3 + 2p2v δ 3 − 8pv δ 2 + 8pv δ − 3p2v δ 2 − δ 2 + 4δ)(−1 + pv )
−
2(−2 + pv δ + δ)(−1 + δ)(pv δ + 2 − δ)(pv δ − 1)
2
2
2p δ − 2pv δ 2 − 2p2v δ + 6pv δ − 4pv
−p2v δ + δ + 2pv − 2
v2 (s) = − v
s2 −
2(−1 + δ)(pv δ + 2 − δ)(pv δ − 1)
2(pv δ + 2 − δ)(−1 + δ)(pv δ − 1)
vv (s) =
• Case C
(p2v δ 3 + 2pv δ 2 − pv δ 3 − p2v δ 2 − pv δ − 3δ + δ 2 + 2)pv
(s1 + s2 ) +
(−1 + δ)(pv δ + 2 − δ)(p2v δ 2 − 2pv δ − δ + 2)
(2δ − p2v δ 3 + pv δ 3 − p2v δ 2 − pv δ 2 + 4pv δ − 4)pv
(−1 + δ)(pv δ + 2 − δ)(p2v δ 2 − 2pv δ − δ + 2)
−4 + δ 3 − 5δ 2 + 8δ − 2p2v δ 4 + 2p3v δ 4 − 2p3v δ 3 + 3p2v δ 3 + 2pv δ 3 − p2v δ 2 − 6pv δ 2 + 4pv δ
v1 (s) = −
s1 +
2δ(p2v δ 2 − 2pv δ − δ + 2)(pv δ + 2 − δ)(−1 + δ)
4 − δ 3 + 5δ 2 − 8δ − p2v δ 2 + p2v δ 3
s2 +
−
2δ(p2v δ 2 − 2pv δ − δ + 2)(pv δ + 2 − δ)(−1 + δ)
−4pv δ + 4δ + 4p2v δ 2 − 4δ 2 + δ 3 + pv δ 3 − p2v δ 3 − p3v δ 3
−
2δ(p2v δ 2 − 2pv δ − δ + 2)(pv δ + 2 − δ)(−1 + δ)
4 − δ 3 + 5δ 2 − 8δ − p2v δ 2 + p2v δ 3
v2 (s) = −
s1
2δ(p2v δ 2 − 2pv δ − δ + 2)(pv δ + 2 − δ)(−1 + δ)
−4 + δ 3 − 5δ 2 + 8δ − 2p2v δ 4 + 2p3v δ 4 − 2p3v δ 3 + 3p2v δ 3 + 2pv δ 3 − p2v δ 2 − 6pv δ 2 + 4pv δ
−
s2
2δ(p2v δ 2 − 2pv δ − δ + 2)(pv δ + 2 − δ)(−1 + δ)
−4pv δ + 4δ + 4p2v δ 2 − 4δ 2 + δ 3 + pv δ 3 − p2v δ 3 − p3v δ 3
−
2δ(p2v δ 2 − 2pv δ − δ + 2)(pv δ + 2 − δ)(−1 + δ)
vv (s) =
50
• Case D
H
s1 +
− 2pv δ − δ + 2)(−3 + 2δ)
I
s2 +
+
2
(−1 + δ)(pv δ + 2 − δ)(pv δ 2 − 2pv δ − δ + 2)(−3 + 2δ)
12 − 6pv δ − 20δ + 11δ 2 + 7pv δ 2 + 3p3v δ 3 − 2δ 3 − 2pv δ 3 − 3p2v δ 3 + 2p2v δ 4 − 2p3v δ 4
+
(−1 + δ)(pv δ + 2 − δ)(p2v δ 2 − 2pv δ − δ + 2)(−3 + 2δ)
J
v1 (s) = −
s1
2
2
2δ(pv δ − 2pv δ − δ + 2)(−3 + 2δ)(pv δ + 2 − δ)(−1 + δ)
K
s2
−
2
2
2δ(pv δ − 2pv δ − δ + 2)(−3 + 2δ)(pv δ + 2 − δ)(−1 + δ)
L
v2 (s) = −
s1
2δ(p2v δ 2 − 2pv δ − δ + 2)(−3 + 2δ)(pv δ + 2 − δ)(−1 + δ)
M
−
s2
2
2
2δ(pv δ − 2pv δ − δ + 2)(−3 + 2δ)(pv δ + 2 − δ)(−1 + δ)
vv (s) =
(−1 + δ)(pv δ + 2 −
δ)(p2v δ 2
where H = −11δ 2 + 20δ + 2δ 3 + 6pv − 12 + 6p3v δ 2 − 9p2v δ + 5p2v δ 3 + 4pv δ 3 + 3p2v δ 2 − 12pv δ 2 +
5pv δ + 2p3v δ 4 − 2p2v δ 4 − 7p3v δ 3 ), I = 6pv + 16δ + 6p3v δ 2 − 9p2v δ + 6p2v δ 3 + 3pv δ 3 − p2v δ 2 − 12pv δ 2 + 9pv δ +
2p3v δ 4 − 2p2v δ 4 − 6p3v δ 3 − 12 + δ 3 − 7δ 2 , J = −12p3v δ 4 + 10p2v δ 4 + 9p3v δ 3 − 9p2v δ 2 − 21pv δ 3 + 18pv δ 2 +
12 + 6δ 4 pv + 4p3v δ 5 − 4p2v δ 5 − 24δ 3 + 51δ 2 − 44δ + 4δ 4 , K = −p3v δ 4 + δ 4 pv − 3pv δ 3 − 9p2v δ 2 − 4pv δ 2 +
12pv δ + 2p2v δ 3 + p2v δ 4 + 3p3v δ 3 + 6δ 3 − 15δ 2 − δ 4 − 12 + 20δ, L = −2p3v δ 4 + 3p3v δ 3 + 6p2v δ 3 − 9p2v δ 2 −
3pv δ 3 − 8pv δ 2 + 12pv δ − 12 + 2δ 4 pv + 2δ 3 − 11δ 2 + 20δ, and M = −11p3v δ 4 + 5δ 4 pv − 21pv δ 3 − 9p2v δ 2 +
22pv δ 2 − 4p2v δ 3 + 11p2v δ 4 + 9p3v δ 3 + 4p3v δ 5 − 4p2v δ 5 − 20δ 3 + 47δ 2 + 3δ 4 + 12 − 44δ.
One can show that these equilibrium strategies and the associated value functions are part of
a symmetric MPE, using the same strategy employed in the proof of Proposition 1. The only
difference with the proof of Proposition 1 is in the proof of Lemma 3 (b). With heterogenous
recognition probabilities, dv + d2 is not always less than or equal to 1 when pv ∈ (0.5, 1). This
condition is what determines the bounds on δ or on (s1 + s2 ) in the statement of Proposition 4. In
particular, for δ, the binding case is the allocation where s1 = s2 = 0.5. This is the case in which
non-veto players are most demanding, as it can be proven by inspection of d2 in the four cases above.
√
1+3pv − 1+6pv −7p2v
C
C
Setting s1 = s2 = 0.5 and solving for dv + d2 ≤ 1, we obtain the bound δ < δ =
.
4p2
v
Figure 10 shows the space of (pv , δ) for which the MPE in Proposition 4 exists.
51
δ
pv
Figure 10: Existence of MPE from Proposition 4. The shaded area represents the pairs of δ and pv
for which the MPE does not exist.
Proof of Proposition 5
This result follows directly from the equilibrium demand of the poorer non-veto player in the
absorbing set ∆, that is, dnv (s, δ) =
δ(1−pv )
2−δ(1+pv ) snv .
When δ = 0, this demand is zero. This means
that, when the status quo is in ∆—a set that is reached in at most one period—the poorer nonveto supports any proposal by the veto player. The veto player can thus pass his ideal outcome
as soon s ∈ ∆ and he proposes. On the other hand, when δ ∈ (0, 1), this is not possible, and the
poorer non-veto player always demands a positive share of the dollar to support any allocation that
makes the veto player richer. The convergence in this case is only asymptotic and the speed of
the convergence is inversely related to dnv (s, δ), which is strictly increasing in δ for any pv ∈ (0, 1),
strictly increasing in snv for any pv ∈ (0, 1) and any δ ∈ (0, 1], strictly decreasing in pv for any
δ ∈ (0, 1).
Proof of Proposition 6
Veto Players are Decisive: q = 0
When the coalition of veto players is decisive (that is, q = 0) we can consider a more general
setup where we relax the two assumptions employed in this section: each veto player proposes with
52
chance
pv
k ;
each non-veto player proposes with chance
1−pv
n−k ;
and the space of possible agreements
is composed of all vectors of non-negative transfers to the n legislators which sum to 1. We can
prove the following result:
Proposition 8. In the game with n legislators and k < n decisive veto players, there exists
a symmetric MPE in which, irrespective of the discount factor, the number of veto and non-veto
players, the initial division of the dollar and the distribution of agenda setting powers, we transition
to an absorbing state where the k veto players get the whole pie as soon as one veto player proposes.
In the absorbing state, the share to each veto player is strictly larger than his starting share, unless
s0 is an absorbing state or δ = 0.
The result of Proposition 6 for q = 0 is a special case of the more general result in Proposition
8 above, for the case where pv = 1. Below, we prove Proposition 8.
In this game an allocation is s = [sv1 , sv2 , . . . , svk , s1 , s2 , . . . , sn−k ], where svi , i = 1, 2, . . . , k,
denotes the share to a veto player and si , i = 1, 2, . . . , (n−k) denotes the share to a non-veto player.
The unique minimal winning coalition is composed of all veto players. The result of Proposition 8
follows from the existence of a symmetric MPE where veto players propose a positive allocation to
the members of the minimal winning coalition and non-veto players propose a positive allocation
to these members and to themselves. In particular, consider the following proposal strategies for
all status quo policies s ∈ ∆:
• When the proposer is veto player vi, she offers dvvj (s) = svj +
δpv
k(1−δ+pv δ)
of the other (k − 1) veto players, 0 to all non-veto players, and 1 −
1−
(k−1)δpv
k(1−δ+pv δ)
P
n−k
i=1 si
Pn−k
i=1
v
j6=i dvj
P
si to each
(s) = svi +
to herself.
• When the proposer is non-veto player i, he offers dnv
vi = svi , i = 1, 2, . . . , k, to the k veto
players, 0 to all other non-veto players, and 1 −
P
i svi
=
Pn−k
i=1
si to herself.
If these proposals pass, allocations which do not give anything to any non-veto players are
absorbing states and one of these allocations is reached as soon as one veto player proposes. Remember that a veto player is selected to propose with probability pv /k and a non-veto player with
probability (1 − pv )/(n − k). Therefore, if legislators play the proposal strategies above and these
53
proposals pass, legislators’ continuation values for allocation s ∈ ∆ are as follows:
pv 1 − j6=i dvvj (s)
pv (k − 1) dvv1
vvi (s) = (1 − pv ) [sv1 + δvv1 (s)] +
+
k
1−δ
k
1−δ
"
P
#
n−k
si
svi
pv i=1
+
(1 − δ) (1 − δ)k(1 − δ + pv δ)
P
=
"
vi (s) =
P
n−k
(1 − pv )
X
1 − pv n−k
i=1 si
si + δvi (s) =
n − k i=1
(n − k) − δ(1 − pv )
#
On the basis of these continuation values, we obtain veto players’ expected utility functions,
Uvi (x) = xvi + δvvi (x). The reported demands for veto players as a function of the proposer’s
type are in accordance with Definition 2. In particular, dvvi and dnv
vi , i = 1, 2, . . . , k, can be easily
derived from the following equations:
svi + δvvi svi ,
svi + δvvi svi ,
n−k
X
i=1
n−k
X
!
si
=
dvvi
1−δ
=
dnv
vi
!
si
i=1
+ δvvi
dnv
vi ,
n−k
X
!
si
i=1
By the definition of demands, supporting the proposals outlined above is an equilibrium voting
strategy. Finally, we need to prove that those proposal strategies are optimal, given the continuation values. First, note that the expected utility a veto player derives from a policy x is an
increasing and linear function of what x assigns to her and of what x assigns to all non-veto players. Second, since
p
k(1−δ+pv δ)
∈ [pv /k, 1/k] is always smaller than 1, the optimal proposal for player
vi is the one that maximizes xvi , subject to being approved, that is, subject to giving the other
veto players at least dvvj . The unique proposal that maximizes this objective function subject to
this constraint is the one that gives exactly dvvj to the other veto player, 0 to the non-veto players,
and the remainder to the proposer.
Veto Players are not Decisive: q ∈ 0, 23 m
i
h
i
In this case, a proposal passes if it receives the support of the k veto players, plus at least q ∈ 1, 23 m
non-veto legislators. Remember that we denote with snv = (m − m)sm the total share of resources
54
allocated to non-veto players in allocation s. The results of Proposition 6 follow from the existence
of a symmetric MPE with the following minimal winning coalition proposal strategies for all feasible
allocations:
• Case A m ≥ q:
δ
– The proposing veto player offers dA
vi = svi + k snv to the other (k − 1) veto players, the
remainder to himself, and 0 to everybody else. The proposal passes with the support of
the veto players and the poor non-veto players, (who are indifferent between the status
quo and the proposal because dA
m = 0). At the beginning of the following period, the
status quo allocation is such that m = m > q. This means that the new status quo
belongs to case A.
• Case B q > m ≥ 2q − m:
δ (m−m)−2(q−m)
– The proposing veto player offers dB
v = svi + k (m−m)−δ(q−m) snv to the other (k − 1) veto
players, dB
m =
1
(m−m)−δ(q−m) snv
to (q − m) randomly chosen rich non-veto players, the
remainder to himself, and 0 to everybody else. The proposal passes with the support
of the veto players, the (q − m) rich veto players who are offered a positive amount and
the poor non-veto players (who are indifferent between the status quo and the proposal
because dB
m = 0). At the beginning of the following period, the status quo allocation
st+1 is such that mt+1 = mt + (m − q). Since mt ≥ 2q − m, this means that mt+1 ≥ q,
or that the new status quo belongs to case A.
• Case C m < 2q − m:
2
2
δ (m−q) (1−δ)
– The proposing veto player offers dC
v = svi + k [m−m−δ(q−m)]2 snv to the other (k − 1) veto
players, dC
m =
(q−m)−δ(2q−m−m)
(q−m)[m−m−δ(q−m)] snv
to (q − m) randomly chosen rich non-veto players,
the reminder to himself, and 0 to everybody else. The proposal passes with the support
of the veto players, the (q − m) rich veto players who are offered a positive amount and
the poor non-veto players (who are indifferent between the status quo and the proposal
because dC
m = 0). At the beginning of the following period, the status quo allocation
st+1 is such that mt+1 = mt +(m−q). Since mt < 2q −m, we have mt+1 < q. Moreover,
55
since q < 23 m, we have mt+1 ≥ 2q − m. This means that the new status quo belongs to
case B.
If players play the proposal strategies in cases A-C and these proposals pass, their continuation
values are as follows:
• Case A
vvi (s) =
svi
snv
+
1 − δ (1 − δ)k
vm (s) = 0
vm (s) = 0
• Case B
vvi (s) =
svi
(m − q)
+
snv
1 − δ (1 − δ)k[m − m − δ(q − m)]
vm (s) = 0
vm (s) =
snv
(q − m)
(m − m) (m − m) − δ(q − m)
• Case C
vvi (s) =
svi
(m − q)[(1 − δ)(m − m) + δ 2 (m − q)]
+
snv
1−δ
(1 − δ)k[m − m − δ(q − m)]2
vm (s) = 0
vm (s) =
(q − m)
snv
(m − m) (m − m) − δ(q − m)
On the basis of these continuation values, we obtain players’ expected utility functions, Uvi (x) =
xvi + δvvi (x), Um (x) = 0 + δvm (x), and Um (x) = xm + δvm (x). The reported demands are in
B
C
A
B
C
B
C
accordance with Definition 2. In particular, dA
vi , dvi , dvi , i = 1, 2, . . . , k, dm , dm , dm , dm , and dm
56
can be derived from the following indifference conditions:
A
svi + δvvi
(svi , snv ) =
dA
vi
1−δ
B
A
B
B
svi + δvvi
(svi , snv ) = dB
vi + δvvi dvi , (q − m)dm
C
B
C
C
svi + δvvi
(svi , snv ) = dC
vi + δvvi dvi , (q − m)dm
A
A
= dA
0 + δvm
m + δvm
B
A
0 + δvm
= dB
m + δvm
C
B
0 + δvm
= dC
m + δvm
B
A
sm + δvm
(snv , m) = dB
m + δvm
C
B
C
(snv , m) = dC
sm + δvm
m + δvm (q − m)dm , m + (m − q)
By the definition of demands, supporting the proposals outlined above is an equilibrium voting
strategy. Finally, we need to prove that those proposal strategies are optimal, given the continuation
values. In case A, Uvi (x) is a linear and positive function of both xvi and xnv . Since
1
1−δ
≥
δ
k(1−δ)
for any k ≥ 1, and δ ∈ [0, 1], Uvi (x) is maximized when xvi is as large as possible. Similarly, in
cases B and C, Uvi (x) is a linear and positive function of both xvi and xnv . Since the coefficient of
xvi ,
1
1−δ ,
is greater than or equal to the coefficient of xnv for any k ≥ 1, δ ∈ [0, 1], m ≥ 1, m ≤ m,
and q < m, Uvi (x) is maximized when xvi is as large as possible. This means that the expected
utility of the proposing veto player is maximized when xvi is as large as possible. The proposal
strategies above are the acceptable proposals which give the largest possible share to the proposer
(because they make the agents of a minimal winning coalition barely indifferent between accepting
and rejecting).
Proof of Proposition 7
This result in part (a) follows from the equilibrium demand of poor non-veto players which demand
zero to support any allocation. This means that, when the status quo is such that m > q, the
proposing veto player can expropriate the non-veto players completely with the support of the poor
non-veto players and the other veto players. When in the initial status quo m0 < q, the proposer
57
cannot extract completely the rich non-veto players because he needs the support of (q − m0 ) rich
non-veto players who demand a positive allocation. However, the number of cheap coalition partners
at the beginning of the second period will be larger: m1 = m − (q − m0 ) = m0 + (m − q) > m0 ,
which holds ∀q < m. When q > m0 ≥ 2q − m, we have m1 > q. In this case, in the second
period the proposer does not need the support of any rich non-veto player and we converge to the
absorbing state where non-veto players have zero resources. When m0 < 2q − m, we have m1 < q
so, in the second period, the proposer needs to muster the costly support of (q − m1 ) rich non-veto
players. However, since q < 32 m, we have m2 ≥ q: in the third period the proposer does not need
the support of any rich non-veto player and we converge to the absorbing state where non-veto
players have zero resources.
The result in part (b) follows from investigating the sum of all non-veto players’ value functions
evaluated at the initial status quo, s0 . Since the value of any initial allocation to poor non-veto
players is 0 and the value to each non-veto player is the same, this sum is given by (m − m)vm (s0 )
or by:
X
0
0
vi (s ) = (m − m)vm (s ) =
i∈nv




0,
if q < m0

0

0

 m−mq−m
0 −δ(q−m0 ) snv ,
if q > m0
When q < m0 , this value is constant. When q > m0 , this value is increasing in q, δ and s0nv and
decreasing in m.
The result in part (c) follows from the equilibrium demand of veto players:
are strictly positive for any k ≥ 1, δ ∈ (0, 1), and q < m;
δ (m−m)−2(q−m)
k (m−m)−δ(q−m)
δ
k
and
δ (m−q)2 (1−δ)2
k [m−m−δ(q−m)]2
(the demand in case B)
is strictly positive for any δ ∈ (0, 1), q < m, and m < q (which is true in case B). This means
that each veto player gets a positive fraction of the resources expropriated from non-veto players
regardless of the identity of the proposer.
58
Equilibrium Selection
The legislative game studied in this paper is an infinite horizon dynamic game with a plethora of
subgame perfect equilibria and, thus, an equilibrium-selection issue. As standard in the literature
on dynamic legislative bargaining, I do not consider equilibria involving stage-dominated or nonstationary strategies.26 Even so, it is still possible that other Markovian equilibria of this game
exist. In this Section, I characterize sufficient conditions for full appropriation by the veto player to
be the unique equilibrium outcome, and show that the equilibrium identified in Section 3 is robust
to changes in the legislators’ degree of patience.
Before we state the sufficient conditions for uniqueness of the equilibrium outcome, we need
two additional definitions:
Definition 2. (Continuity) An MPE is continuous if the continuation value functions, Vi ,
induced by equilibrium strategies are continuous.
Definition 3. (Weak Monotonicity) An MPE of this game is weakly monotonic if the value
functions associated to this equilibrium for each non-veto player are strictly increasing in the share
assigned to that player, keeping the share assigned to the veto player constant.
Definition 2 is a standard continuity assumption. While continuity of equilibrium strategies
and value functions is generally not assured in dynamic legislative bargaining games this is an
appealing characteristic of the MPE characterized in Section 3, as proved in Proposition 11 below.
Regarding the weak monotonicity property in Definition 3, notice that it is much less stringent than
a regular monotonicity condition on equilibrium value functions, satisfied when the value functions
of the dynamic game are strictly increasing in the share to oneself (similarly to the utilities of the
stage game). Weak monotonicity is implied by regular monotonicity, but the reverse is not true:
weak monotonicity requires Vi to be strictly increasing in xi only along the ray where the resources
allocated to the veto player do not change, but does not restrict the change in the value function
when moving in other directions in ∆. In other words, an MPE satisfies the weak monotonicity
condition if a forward-looking non-veto player is always worse off when transferring some of his
26
In large legislatures, non-stationary strategies that depend on the history are implausible because of
legislators turnout, extraordinary commitment, coordination, and/or communications requirements.
59
share to the other non-veto player.
For example, the MPE characterized in Section 3 satisfies regular monotonicity only for δ low
enough.27 For δ high enough, however, the value functions associated with this MPE are not
necessarily increasing in the share to oneself and they can be decreasing when both the share to
oneself and the share to the veto player increase at the same time. The MPE in Section 3, however,
satisfies weak monotonicity for any δ.
We are now ready to state Proposition 9:
Proposition 9. Full appropriation of the dollar by the veto player is the unique absorbing state
of the dynamic legislative bargaining game with veto power in any continuous and weakly monotonic
Markov perfect Nash equilibrium.
Proof. We want to show that x = (xv , x1 , x2 ) is not an absorbing policy, unless x = (1, 0, 0). It is
straightforward to prove that (1,0,0) is absorbing. We can show that no other policy x 6= (1, 0, 0) is
absorbing by contradiction. Assume x 6= (1, 0, 0) is absorbing. Then EUv (x), the expected utility
of the veto player from x, is xv /(1 − δ). First, notice that the veto player would be better off with a
different policy y, such that yv > xv . This is because, moving to y the veto player can guarantee to
himself an expected utility of at least yv /(1 − δ) (by vetoing any future proposed change). To run
into a contradiction and show the desired result, we therefore need to show that the veto player,
when proposing, can always find a non-veto player who is at least indifferent between the status quo
policy x and a new policy y such that yv > xv . Assume that, in x, x2 > x1 ≥ 0 (this is without loss
of generality, because we can switch labels for all other policies different than (1,0,0)). Consider the
policy x0 = (xv , x1 + x2 , 0). By the weak monotonicity condition, this policy is strictly preferred to
x by non-veto player 1 (the veto player gets the same amount and he gets more). By continuity of
the value functions, there always exists > 0 such that EU1 of the policy x00 = (xv +, x1 +x2 −, 0)
is weakly greater than EU1 (x). This means that the veto player is able to and willing to move to a
different policy, when proposing (an event that happens with positive probability), and that, thus,
6=(1, 0, 0) cannot be absorbing.
27
For δ = 0, the value functions of the infinite horizon game retain the same properties of the stage game
utilities, and are therefore strictly increasing in the share to oneself. Since the value functions of the MPE
characterized in Section 3 are continuous in δ, regular monotonicity holds for δ low enough.
60
Proposition 9 states that full appropriation of the endowment by the veto player is the unique
long run equilibrium outcome of this dynamic game in a class of Markovian equilibria. The intuition
behind Proposition 9 is that, given continuity and weak monotonicity of the value functions, a
proposing veto player will always be willing and able to pass a policy that gives him a higher share
of the endowment, unless he already holds the whole dollar.
Proposition 10 shows that the same intuition holds in any subgame perfect equilibrium of this
game if the discount factor is sufficiently low.
Proposition 10. If δ < 3/7, full appropriation of the dollar by the veto player is the unique
absorbing state of the dynamic legislative bargaining game with veto power in any subgame perfect
equilibrium.
Proof. We want to show that x = (xv , x1 , x2 ) is not an absorbing policy, unless x = (1, 0, 0). It is
straightforward to prove that (1,0,0) is absorbing. We can show that no other policy x 6= (1, 0, 0) is
absorbing by contradiction. Assume x 6= (1, 0, 0) is absorbing. Then EUv (x), the expected utility
of the veto player from x, is xv /(1 − δ). First, notice that the veto player would be better off with a
different policy y, such that yv > xv . This is because, moving to y the veto player can guarantee to
himself an expected utility of at least yv /(1 − δ) (by vetoing any future proposed change). To run
into a contradiction and show the desired result, we therefore need to show that the veto player,
when proposing, can always find a non-veto player who is at least indifferent between the status
quo policy x and a new policy y such that yv > xv . Assume that, in x, x2 > x1 ≥ 0 (this is without
loss of generality, because we can switch labels for all other policies different than (1,0,0)). From
the reasoning above, the lower bound of Vv (x), the continuation value of the veto player, in any
SPE is xv /(1 − δ). We can find the lower bound of Vi (x), the continuation value of non-veto player
i = {1, 2}, in any SPE:
1
xi +
3
Vi (x) ≥
2
1
3
δxi +
3
1
3
2
δ xi + . . . +
t
1
3
δ t−1 xi + . . . =
xi
= V2 (x)
3−δ
This is what a non-veto player can get if he proposes and implements the status quo allocation as
long as he is the proposer (that is, in the history where i is the proposer in every following period).
The upper bound of V2 (x) in any SPE is given by the infinite stream of all available resources minus
61
the lower bounds of the continuation values of the two other players, that is:
V2 (x) ≤
1
1 − x1 − x2
x1
−
−
1−δ
1−δ
3−δ
which simplifies to:
V2 (x) ≤
2x1 + (3 − δ)x2
= V2 (x)
(1 − δ)(3 − δ)
Consider now the policy x0 = (xv , x1 + x2 , 0). Non-veto player 2 strictly prefers x0 to x and vote in
favor of this proposal in any SPE as long as:
x02 + δV2 (x0 ) > x2 + δV2 (x)
This inequality is satisfied if:
δ<
3x1
5x1 + 2x2
The LHS is decreasing in x2 . This means that the binding policy is (0, 1/2, 1/2) and that the
inequality is satisfied for all x 6= (1, 0, 0) as long as δ < 3/7.
By continuity of the value functions, there always exists > 0 such that EU2 of the policy
x00 = (xv + , 0, x1 + x2 − ) is weakly greater than EU2 (x). This means that, if δ < 3/7, the veto
player is able to and willing to move to a different policy, when proposing (an event that happens
with positive probability), and that, thus, 6=(1, 0, 0) cannot be absorbing.
In addition to generating the unique equilibrium outcome from a reasonable class of equilibria,
the MPE characterized in Section 3 is robust to changes in the legislators’ time horizon and degree
of patience. First, regarding the time horizon, it can be shown that any finite horizon version
of this dynamic legislative bargaining game has a unique subgame perfect equilibrium in stage
undominated voting strategies. While the characterization of this equilibrium for the T-period
problem is cumbersome, it can be proven that, in this equilibrium, only a minimal winning coalition
of legislators receives a positive share of the pie, the veto player never accepts a reduction to his
62
allocation (with the possible exception of the first period, if the initial status quo gives a positive
allocation to everybody), and the veto player can always increase his allocation when proposing,
with his coalition partner demanding a premium to go along with this proposal.28 In this sense,
the MPE characterized in Section 3 is the limit of the unique subgame perfect equilibrium of the
finite game as the number of periods played goes to infinity.
Second, regarding the robustness to legislators’ discount factors, this MPE is well behaved as
shown by Propositions 1 and Proposition 11 below: it exists for any δ ∈ [0, 1) and for any s ∈ ∆,
and it is continuous in δ and s, meaning that a small change in the status quo implies a small change
in proposal strategies and, by extension, to the equilibrium transition probabilities. An immediate
implication of the continuity of transition probabilities is the fact that continuation values and
expected utility are continuous. This MPE, moreover, converges smoothly to the unique SPE of
the stage game as δ → 0.
Proposition 11. The continuation value functions, Vi , and the expected utility functions, Ui ,
induced by the equilibrium in Proposition 1 are continuous.
Proof: The result of Proposition 11 follows once we establish that the proposal strategies in the
equilibrium from Proposition 1 are weakly continuous in the status quo s, i.e. that in equilibrium
a small change in the status quo implies a small change in proposal strategies and, by extension,
to the equilibrium transition probabilities.
Lemma 6. The equilibrium proposal strategies µ∗i in the proof of Proposition 1 are such that
for every s ∈ ∆ and every sequence sn ∈ ∆ with sn → s, µ∗i [·|sn ] converges weakly to µ∗i [·|s].
Proof: The equilibrium is such that µ∗i [·|s] i = 1, 2 has mass on only one point x(i, v, dv (s))
and that µ∗v [·|s] has mass on at most two points x(v, 1, d1 (s)), and x(v, 2, d2 (s)). It suffices to show
that these proposals (when played with positive probability) and associated mixing probabilities are
continuous in s (see Kalandrakis 2004). Continuity holds in the interior of Cases A-D in Proposition
1, so it remains to check the boundaries of these cases. In order to distinguish the various applicable
w
functional forms we shall write dw
i and µv [·|s] where w = {A, B, C, D} identifies the case for which
the respective functional form applies.
28
The equilibrium analysis for T=2 is available from the author.
63
• Boundary of Cases A and B: at the boundary (as in the interior of the two cases) we have
B
µA
v [x(v, 1, d2 )|s] = µv [x(v, 1, d2 )|s] = 0; at the boundary we have s1 = 1 −
3−δ
3−2δ s2 ,
then:
dA
= dB
v
v =0
9 − 12δ − 3δ 2
s2
(3 − 2δ)2
9 − 12δ − 3δ 2
δ
= dB
=
s2 +
2
(3 − 2δ)2
(3 − 2δ)
dA
= dB
1
1 =1−
dA
2
• Boundary of Cases B and C: at the boundary we have s1 =
27−27δ+3δ 2 +δ 3
s
(3−2δ)(3−δ)2 2
+
δ2
;
(3−δ)2
then:
C
µB
v [x(v, 1, d2 )|s] = µv [x(v, 1, d2 )|s] = 1
dB
= dC
v
v =0
9 − 12δ + 3δ 2
δ
s2 +
(3 − 2δ)2
(3 − 2δ)
9 − 12δ − 3δ 2
δ
= dC
=
s2 +
2
2
(3 − 2δ)
(3 − 2δ)
dB
= dC
1
1 =
dB
2
• Boundary of Cases C and D: at the boundary we have s1 = 1 −
3−δ
3−2δ s2 ;
then:
D
µC
v [x(v, 1, d2 )|s] = µv [x(v, 1, d2 )|s] =
=
(−36δ 3 + 3δ 4 + 153δ 2 − 270δ + 162)s2 + 15δ 3 − 2δ 4 − 72δ 2 + 135δ − 81
h
dC
v
= dD
v =0
dC
1
C
D
= dD
1 = d2 = d2 =
=
i
(−12δ 2 + 3δ 3 + 9δ)s2 − 9δ 2 − 2δ 3 + 36δ − 27 δ
(−3 + δ)(−12δ 2 + 3δ 3 + 9δ)
(−3 + δ)(−2δ 3 − 9δ 2 + 36δ − 27)
s
+
2
(2δ − 3)2 (δ 2 − 15δ + 18)
(2δ − 3)2 (δ 2 − 15δ + 18)
• Boundary of Cases D and A: at the boundary we have s1 =
3−δ
3−2δ s2 ;
then:
A
µD
v [x(v, 1, d2 )|s] = µv [x(v, 1, d2 )|s] = 1
dD
= dA
v
v = sv −
δs2
3 − 2δ
D
A
dD
= dA
1
1 = d2 = d2 =
64
(3 − δ)2
s2
(3 − 2δ)2
8.1
Additional Tables
Round
2.61* 13.21*
3.16*
2.53*
(0.22) (0.85)
(0.57)
(0.24)
Constant
22.68* 17.90* 23.37*
20.64*
(0.55) (0.87)
(1.05)
(1.18)
Sample
All
δ = 0 δ = 0.50 δ = 0.75
Pseudo-R2
0.0186 0.0184 0.0221
0.0255
Observations 2788
640
1144
1004
Table 5: Share to veto player: Tobit estimates. Observations are group-rounds and include the
initial status quo exogenously assigned by the computer (coded as group outcome in round 0).
Standard errors clustered by groups composed of same subjects in parentheses. * significant at the
1% level.
65
Non-Markovian SPNE with Gridlock
I propose strategy profiles such that the initial allocation can be supported as the outcome of a
Subgame Perfect Nash Equilibrium (SPNE) and, thus, there is no convergence to full expropriation
by the veto player. This SPNE exists as long as the discount factor is high enough and the two
non-veto players receive enough. In particular, I want to prove that:
Proposition 12. For any s ∈ ∆ such that minj=1,2 sj ≥ 1/4 , there is a δ(s) such that for
δ > δ(s) the initial division of the dollar can be supported as the outcome of a Subgame Perfect
Nash Equilibrium of the game.
The idea behind the proof is the following: if a non-veto player accepts a proposal that expropriates the other non-veto player, we switch to a punishment phase in which we reverse to the MPE
characterized above. The discount factor needed to support this outcome depends on the share
granted to the two non-veto legislators at the beginning of the game: the lower the allocation an
agent receives in the initial status quo, the more profitable a deviation.
Proof. To support the initial allocation s0 as the outcome of a Subgame Perfect Nash Equilibrium,
employ the following strategy configuration:
1. whenever a member is recognized, he proposes the status quo allocation s0 and everyone
supports it;
2. if a proposer deviates by proposing z 6= s0 , every non-veto player j votes against the proposal;
3. if a non-veto player j deviates by voting contrary to the strategies above, from the following
period on we reverse to the MPE equilibrium proposal and voting strategies characterized in
Section 3.
The strategies for the punishment phase are clearly a SPNE as shown in the proof of Proposition
1 (MPE being one of the many SPNEs of this game). We need to show that, under certain conditions
on s0 and δ, the non-veto players have no profitable deviation from the equilibrium strategy on the
equilibrium path. The payoff to a non-veto player if she follows the equilibrium strategy is:
j
VEQ
(s) =
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sj
1−δ
The payoff to deviating and proposing or voting in favor an allocation z 6= s0 is given by:
j
j
VDEV
(x) = xj + δvM
P E (x)
j
where vM
P E (x) is the value function from the MPE characterized in the proof of Proposition 1.
The most profitable deviation when proposing is a proposal that assigns the whole dollar to oneself
(if this is in the acceptance set of the veto player). Similarly, the most profitable deviation when
voting is to accept a veto player’s proposal that assigns the whole dollar to oneself. In both cases
the expected utility from the deviation is as follows (assuming the deviator is agent 2):
j
(0, 1, 0) = 1 + δ
VDEV
3 − 3δ + δ 2
(3 − δ)2 (1 − δ)
Equilibrium strategies are preferable to the most profitable deviation when this condition holds:
sj
1−δ
sj
3 − 3δ + δ 2
(3 − δ)2 (1 − δ)
(3 − 2δ)2
(3 − δ)2
≥ 1+δ
≥
Since this condition has to hold for both non-veto players, we conclude that an equilibrium where
the initial status quo is never changed can be supported by a SPNE if the following condition holds:
min s0i ≥
i=1,2
(3 − 2δ)2
(3 − δ)2
The right-hand side is a linear and decreasing function of δ, and it is equal to 1 when δ = 0 and to
1/4 when δ = 1. This means that there exists a discount factor for which the proposed strategies
can support the initial status quo allocation forever, only as long both non-veto player have at least
1/4 of the dollar each at the beginning of the game.
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Experimental Instructions
Thank you for agreeing to participate in this experiment. During the experiment we require your
complete, undistracted attention and ask that you follow instructions carefully. Please turn off your
cell phones. Do not open other applications on your computer, chat with other students, or engage
in other distracting activities, such as reading books, doing homework, etc. You will be paid for
your participation in cash, at the end of the experiment. Different participants may earn different
amounts. What you earn depends partly on your decisions, partly on the decisions of others, and
partly on chance. It is important that you do not talk or in any way try to communicate with other
participants during the experiments.
Following the instructions, there will be a practice session and a short comprehension quiz. All
questions on the quiz must be answered correctly before continuing to the paid session. At the end
you will be paid in private and you are under no obligation to tell others how much you earned.
Your earnings are denominated in FRANCS which will be converted to dollars at the rate of 60
FRANCS to 1 DOLLAR.
This an experiment in committee decision making. The experiment will take place over a
sequence of 10 matches. We begin the match by dividing you into 4 committees of 3 members
each. Each of you is assigned to exactly one of these committees. You will be given a temporary
Committee Member Number (either 1, 2 or 3) and you are not told the identity of the other
members of your committee. One of the members of your committee is selected at random by the
computer to be the Veto Player for this committee. The Committee Member Number of the Veto
Player will be displayed on your computer. For example, if you are Committee Member Number 1
and the Veto Player for this committee in this match is Committee Member Number 1, then you
are the Veto Player in your committee in this match. In each match, your committee will make
budget decisions over a sequence of several rounds.
In each round, your committee has a budget of 60 francs. Your committee must decide how
to divide this budget into private allocations A1, A2, and A3, in integer amounts. These private
allocations A1, A2, and A3 have all to be greater than or equal to 0 and must add up to exactly
60. If your committee budget decision is (A1, A2, A3), then A1 francs go directly to member 1’s
earnings, A2 francs go to member 2’s earnings, and A3 francs go to member 3’s earnings.
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Here is the procedure for how your committee makes budget decisions. At the beginning of the
first round, the computer randomly selects an initial budget decision (A1, A2, A3) and displays
it on your computer as what we call the Standing Budget. Next, each of you makes a provisional
proposal for an alternative budget decision you would like your committee to consider. (You may
propose the Standing Budget itself if you wish.) Your proposal can be any budget decision—
that is, any three non-negative numbers (including 0s) that add up to exactly 60. After all three
members of your committee have chosen provisional proposals, one of these provisional proposals
is selected at random by the computer to be the Proposed Budget. The Proposed Budget will be
displayed on your computer, along with the number of the Committee Member who proposed it.
The committee then conducts a vote between the Standing Budget and the Proposed Budget. The
Proposed Budget passes only if the Veto Player and at least one other committee member vote
in its favor. If the Veto Player votes against the Proposed Budget, the Standing Budget wins. If
the Veto Player votes in favor of the Proposed Budget but the two other committee members vote
against it, the Standing Budget wins. Your earnings in this round are determined by your private
allocation in whichever budget decision wins in the voting stage.
One important aspect of your committee’s budget decision is that it is inertial. That is, the
budget decision that prevails in round 1 becomes the Standing Budget in round 2 and will thus
determine the private allocations in round 2 if your committee does not agree on a different budget
decision. Every round, the budget decision of your committee determines both your earnings in
this round and the Standing Budget for the following round.
The total number of rounds in a match will depend on the rolling of a fair 8-sided die. When
the first round ends, we roll it to decide whether to move on to the second round. If the die comes
up a 1 or a 2 we do not go on to round 2, and the match is over. We will describe in a moment
what happens after a match is over. If the die comes up a 3, 4, 5, 6, 7, or 8, we continue to the
next round. In round 2, your Committee Member Number, the members of your committee and
the identity of the Veto Player all stay the same. Round 2 proceeds just as round 1, with the
exception that the Standing Budget in round 2 is whatever the committee decision was in round
1. Therefore, if the original Standing Budget won the voting stage in round 1, this continues as
the Standing Budget in round 2. But if the Proposed Budget in round 1 won the voting stage,
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then it replaces the original Standing Budget and becomes the new Standing Budget for round 2.
The proposal and voting process then follows the same rules as round 1. Once again, each member
types in a proposal, the computer then randomly selects one of them to be the Proposed Budget
and a vote is taken between the round 2 Standing Budget and the Proposed Budget. After round
2 is over, we roll the 8-sided die again to determine whether to move on to a third round. We
continue to more rounds, until a 1 or a 2 is rolled at the end of a round and the match ends. It is
important to remember that your Committee Member Number, the members of your committee,
and the identity of the Veto Player all stay the same in all rounds of the match. In round T, the
Standing Budget is always whatever the committee decision was in round T-1.
After the first match ends, we move to match 2. In this new match, you are reshuffled randomly
into 4 new committees of 3 members each. Your assigned a new Committee Member Number (1,
2, or 3). The computer randomly selects a Standing Budget for each committee for round 1, and
randomly selects a Veto Player for each committee. The match then proceeds the same way as
match 1. This continues for 10 matches. After match 10, the experiment is over. Your total
earnings for the experiment are the sum of your earnings over all rounds and all matches.
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