Bargaining in the Shadow of Uncertainty*
Marina Agranov†

Hülya Eraslan‡

Chloe Tergiman§

This version: December 9, 2022

Abstract
In bargaining environments with stochastic future surplus, failing to delay agreement can beinefficient when the expected future surplus is sufficiently high. Theoretically, such inefficiencies
never arise under unanimity rule but can under majority rule. Using a laboratory experiment, wefind support for these predictions, both when the unanimity rule is predicted to be more efficientand when there should be no difference between the two rules. We also find large point prediction
deviations under the majority rule. We show these deviations can be explained by higher-thanpredicted egalitarian sharing and a lower risk of being excluded from future agreements.

* We are grateful to Nageeb Ali, Andrzej Baranski, Alessandra Casella, Matt Elliott, Natalie Lee, Steve Lehrer,

Salvatore Nunnari, Clemence Tricaud, and Christoph Vanberg for the helpful feedback. We also thank audiences at ETH Zurich, Heidelberg, 2021 NZAE Virtual PhD Workshop, Vanderbilt, Theory and Experiments
2022 (Center for Behavioral Institutional Design, NYUAD), UCLA, Padova, International Conference on Public Economic Theory (PET 2022). We benefited from thoughtful comments of referees and the editor. Eraslan
gratefully acknowledges support from National Science Foundation under grant SES-1730636.
† Division of the Humanities and Social Science, California Institute of Technology.
Email: magranov@hss.caltech.edu.
‡ Department of Economics, Rice University, NBER and Institute of Social and Economic Research, Osaka
University. Email: eraslan@rice.edu.
§ Smeal College of Business, Penn State University. Email: cjt16@psu.edu.

1Introduction

Since the seminal contribution of Baron and Ferejohn (1989), there has been an extensive
literature on experimental legislative bargaining (see the surveys of Palfrey (2015) andAgranov (2022)).1 These studies have focused on situations in which the size of a budget
to be distributed is a fixed quantity. While this represents a certain class of situations,
there are many cases in which the size of the surplus to be divided may change over time,and moreover players cannot write contracts addressing all future contingencies. This israther commonplace even in the financial sector where many creative contracts are available. One such example lies in Chapter 11 bankruptcies. Claimants need to decide onwhether to accept or reject proposals on how to restructure the firm. During this negotiation process, new information may arrive impacting the value of the firm, but nevertheless, the negotiated reorganization plans cannot, and in practice do not, reflect all thepossible contingencies that might affect the value of the bankrupt firm after it emerges
from bankruptcy (see Eraslan (2008)).
The bargaining rules in stochastic settings are far from uniform, in particular, there
is variation in the voting threshold required to reach an agreement. An example can befound in the realm of sovereign debt restructurings (see Benjamin and Wright (2019) andcitations therein). An interesting feature of sovereign debt negotiations is the cross country variations of the bargaining mechanism. Specifically, bonds governed by New Yorklaw require unanimous consent to change the core payment terms of the indenture, whilebonds governed by English law require a majority of the creditors to agree.2 This crosscountry variation in voting rules raises the question of the desirability of the unanimity
rule vis-à-vis the majority rule. In this paper, we use laboratory experiments to answer
this question in a multilateral bargaining setting with a stochastic surplus.
Our experimental design is grounded in the framework of Eraslan and Merlo (2002)
who theoretically study a stochastic model of multilateral bargaining with complete information and risk-neutral players focusing on the comparison between unanimity andmajority voting rules in terms of delays. They show that under unanimity rule, equilibrium outcomes are always efficient while under majority rule this need not to be the case.
Intuitively, in a stochastic environment, when the current surplus is smaller than the discounted value of the expected future surplus, it is optimal to delay agreement until a larger
surplus is realized. Under unanimity rule, the incentives of all players are aligned sinceeach player has veto power. But under majority rule, agreement on a proposal does not
require approval by all players. As such, some players receive nothing under the agreed1 The first experiments of Baron-Ferejohn model are McKelvey (1991), Frechette, Kagel, and Lehrer (2003),

and Diermeier and Morton (2005). For a meta-analysis, see Baranski and Morton (2021).
2 About 45% of the outstanding international sovereign bonds are governed by English law and about 52%
by New York law IMF (2020).

1

upon proposal in equilibrium.3 Given the risk of receiving no payment in the future, theplayers who are offered a positive share when the surplus is relatively small may be induced to accept the proposal if they expect to be excluded from future agreements when
the surplus is relatively large.
This mechanism explaining the potential inefficiency of the majority rule relative tothe unanimity rule does not seem to depend on the missing complexities of the real worldsuch as risk aversion. Indeed, we extend the theoretical analysis of Eraslan and Merlo(2002) and show that under risk aversion, continuation payoffs are weakly higher under
the unanimity rule, and if there is a delay under majority there is a delay under unanimity, but the converse is not true.4 Incorporating other real-life features in a theoreticalframework could be challenging. For example, introducing asymmetric information in a
multilateral bargaining model is difficult even with a deterministic surplus, and there are
only a handful of papers that do so, even then, in highly stylized settings (see Eraslan andEvdokimov (2019) for a survey).
As the name implies, a stochastic model of multilateral bargaining is an economic
model. As such it is “an artificial world that is designed to reveal connections that areotherwise hard to discern." Nonetheless, it is important to understand whether complexities left out of the model would change the key result on the different incentives affecting
the timing of agreement under different voting rules in a stochastic environment. One
way to address this robustness question is through lab experiments. Given the difficultyof fully controlling rationality, preferences, and information of the subjects, experiments
are well suited to study if these missing details matter. In this paper, this is precisely
what we do by experimentally comparing the unanimity and majority rules in a stochastic
bargaining framework in terms of delays and the distribution of surplus.
Our experimental design consists of a series of bargaining experiments in which a committee with three members allocates a budget for a given voting rule and stochastic process that governs the evolution of the budget size. Specifically, we run several treatmentsthat vary along two dimensions: (1) the voting rule which is either the simple majority(M) or the unanimity (U) rule; (2) the variation in the size of the budget: in the M48 andU48 treatments, the budget is $24 in the first bargaining round, but if an agreement is notreached, the budget can be either $24 or $48 with equal probabilities in each future round;in the M96 and U96 treatments, the budget is $24 in the first bargaining round, but if anagreement is not reached, the budget can be either $24 or $96 with equal probabilities ineach future round. In addition to our stochastic treatments, in order to tie our paper to
3 Their model normalizes the disagreement payoffs to zero for all players. Consequently, receiving zero
surplus is equivalent to receiving the disagreement payoff. In the debt restructuring application, all creditors
receive the same payment but different creditors might have different disagreement payoffs. As such, being
excluded from an agreement is analogous to receiving a payoff equal to or below the disagreement payoff.
4 See Proposition 1 in Appendix A.

2existing literature, we also run two deterministic ones, the M24 and U24 treatments, inwhich the size of the budget is fixed at 24. As we describe in detail in Section 3, the threeparameterizations are used to create three qualitatively different situations that differ interms of whether or not delays are predicted to happen:5 (a) both voting rules are expected
to result in immediate agreement (U24 and M24 treatments), (b) both voting rules are expected to result in delays (U96 and M96 treatments), and (c) only the unanimity rule isexpected to create delays (U48 treatment versus M48 treatment).
We find strong support for the main theoretical insight, namely, that in a stochastic
environment, the unanimity rule leads to more delays compared with the majority rule
when the expected future surplus is sufficiently higher than the current surplus,6 andeven when in the benchmark theory both rules should result in equal levels of efficientdelays. As predicted, in the 48 treatments, the unanimity rule results in small budgets
being delayed at a much higher rate than under the majority rule (80% versus 54% in the
U48 and M48 treatments respectively). This difference also holds in the 96 treatment (95%versus 75% in the U96 and M96 treatments respectively), despite the theoretical predictionthat both rules should result in equal levels of efficient delays. We note that in the U96
treatment, the levels of efficient delay are very close to the 100% point prediction.
The advantages of the unanimity rule in a stochastic environment that we documentin this paper stand in contrast to prior laboratory experimental findings that compared
the two voting rules in a deterministic setting, that is when the budget size is fixed. In
these situations, while the theory predicts no differences across voting rules in terms ofagreement, in practice the unanimity rule can result in more inefficient delays (Miller and
Vanberg (2013)).7 So although the drawback of the unanimity rule seems to exist in a
deterministic world, our experimental results show the advantages of the unanimity ruleover the majority rule in a stochastic one.
While the qualitative predictions of the model hold, we note deviations from point predictions, particularly in the Majority treatments. Indeed, subjects in the M48 treatment
delay too often (by over 50 percentage points), but delay not often enough in the M96 treatment (by 25 percentage points). What might account for this seeming puzzle? Data froma separate investment task as well as our theoretical contribution that extends the model
beyond risk-neutral players point to risk aversion as a likely explanation. While we findno difference across treatments in subjects’ risk attitudes, we do find that on average subjects are risk averse. This pushes subjects away from delays regardless of the voting rule,
5 These predictions are derived in the theoretical benchmark setting in which committee members have an
identical single date payoff function which is linear in their surplus share, and discount future bargaining
rounds at the rate of 20%. In the experiment, we implement discounting using the standard technique of
random termination which occurs at the end of each bargaining round (see Roth and Murnighan (1978)).
6 In what follows, we refer to delaying agreement when the budget is small as “efficient delays."
7 The subjects in our deterministic treatments exhibit behavior that aligns with these past findings.

3

but in the Majority treatment the impact is stronger because, in addition to the exogenous
risk of random termination, subjects face the endogenous risk of being left out of a futurecoalition. Thus risk aversion has a differential impact across voting rules and can explain
why subjects delay less under the majority rule compared with the unanimity one, evenin the 96 treatments. Regarding the M48 treatment, the risk-neutral benchmark alreadypredicts no delays. Instead, we observe delays in excess of 50%. What is then countering
the impact of risk aversion? As we discuss further below, in the laboratory, proposals areoften more egalitarian than predicted, and subjects often discuss fairness, even in the Majority treatments, lowering the risk of exclusion. This works towards increasing delaying
behavior. In short, the combination of risk attitudes, how big the size of the future budget
may be, and how likely one is to be excluded from a future winning coalition can jointly
explain the patterns in our data.
Next, we explore the origins of delays. In the game we analyze, delay can arise forone of two reasons: either the proposer does not submit a proposal, or, a proposal is submitted but does not receive the required number of votes. Regardless of the agreement
rule, the theory is silent regarding the origins of delays when delays are predicted. In theexperiments, we uncover interesting differences in the origins of delays for small budgets
between the two voting rules. For the 48 treatments, the proposers behave largely the sameunder the two agreement rules: 51% of proposers forgo making a small budget proposal
in the M48 treatment and 52% do so in the U48 treatment. The reason there are more
delays in U48 than in M48 rests almost entirely on the voting behavior of the committee
members: when a small-budget proposal reaches the floor in the M48 treatment, only 7%of them are rejected: voters go along with the proposal on the table. In the U48 treatment
this fraction is significantly higher and reaches 58%. Further, we find that the rejection ofthese inefficient proposals in the U48 treatment is due to a single vote in almost half of thecases, showing the importance of the voting rule itself in reaching efficient delays.
The explanation behind the higher levels of efficient delays in U96 relative to the M96
treatment is two-fold. In addition to delays arising due voters’ rejection of the proposals dividing the small budget more often in the U96 treatment compared with the M96
treatment (68% versus 15%), we see that proposers are significantly more likely to forgo
making a proposal under the unanimity rule than under the majority rule (83% versus
68%).
After establishing that efficient delays occur more often in the Unanimity treatments
than in the Majority treatments, we show that these findings are robust to controlling for
proposal types. Specifically, we find that equal split proposals dividing the small budget are more likely to fail under U48 than under M48 treatments (43% versus 4%) andthey are also more likely to fail under U96 than under M96 treatments (65% versus 3%).
These rates are even more striking when compared with the passage rates of equal split
4

proposals when surplus is not stochastic. Both in the U24 and M24 treatments, equal-split
proposals almost always pass and there is no difference in the passage rates between these
two treatments (97% versus 99%).
The next set of results focuses on the distribution of payoffs among the players. We
show that under the unanimity rule, regardless of the size of the future budget, outcomesare more equal than under the majority rule. In fact, budgets are almost always equally
split among all committee members in the former case, while in the latter the resources
are shared by two of the three committee members between 39% and 57% of the timedepending on the treatment and the size of the budget at hand. Thus, not only does theunanimity rule lead to more efficient delays, it also results in a more equal allocation of
resources.
Finally, we explore the effect of communication on a decision to delay implementinga small budget. Under the unanimity rule, the only risk in delaying is the exogenous
termination of the game. Under the majority rule a second risk is present: the risk of being
excluded from a future budget proposal. We find that both of these elements are discussed
in both treatments. However, they mostly impact decisions in the Majority treatment andthe magnitude of this impact depends on the distribution of future surplus. In the M48
treatment, groups that discuss fairness and equality delay splitting the small budget more
often than those that do not engage in such discussions. This effect is consistent with there-assurance effect according to which conversations about fairness and equality reduce
the risk of being excluded from future coalitions. Such re-assurance plays no role in the
Unanimity treatment, in which three-way equal splits are considered a fait accompli. In
the M96 treatment, groups that discuss the risk of game termination due to discounting
are less likely to delay when the budget is small compared to groups that do not mention
such risk. This last effect points to the fragility of efficient decision-making under majority
voting rule.
Related literature. Our paper joins the experimental literature on legislative bargaining,which is extensive (see the surveys of Palfrey (2015) and Agranov (2022)). Therefore, we
focus our review on the subset of papers that compare the performance of various voting
rules. Miller and Vanberg (2013, 2015) study how different voting rules affect delay inbargaining in small and large committees. Contrary to the predictions of stationary equilibria, committees take longer to reach agreements under the unanimity rule comparedwith the majority rule. This points to a weakness of the unanimity rule when the setting
is deterministic in terms of future budget size.8
The papers discussed above consider bargaining with a fixed rather than a stochastic
8 Allowing for communication between bargainers can improve the outcomes of the unanimity rule asshown in Agranov and Tergiman (2019), however, it does not eliminate them.

5budgets.9 This is an important distinction since delay in bargaining with a fixed budget
leads to a sure loss of efficiency for impatient bargainers. The setup we study here, i.e.,bargaining with stochastic budgets is different since efficiency balances the two opposing
forces: the desire of the committee to wait for the realization of a large budget and thecost of such delay. The only paper that we are aware of that utilizes the Merlo and Wilson(1995) setting is Li and Houser (2020). The focus of their work is quite different from ours.
They study a two-person two-stage bargaining game in which the budget either expands
or shrinks deterministically between stages but the process determining the identity of the
proposer is stochastic. In contrast to our focus on the comparison of voting rules, Li andHouser (2020) compare outcomes in bargaining over gains and with that of bargaining
over losses. Their experimental results show that efficiency matters. In particular, themost common outcome in both cases is the equal split of the largest available resource.
Structure of the paper. The remaining paper is organized as follows. The first part ofSection 2 presents the setup and derives broad equilibrium predictions focusing on thecomparison of efficiency levels between the two voting rules. The second part of Section 2
outlines game parameters used in our experiments and specific predictions for each treatment. Section 3 details the experimental procedures. Section 4 presents our experimental
results, starting with our main result on the levels of efficient delays and their origins
across treatments, before describing the distribution of payoffs and the impact of communication chats between bargainers. Section 5 offers some conclusions. The appendicescontain proofs, experimental instructions, additional material regarding how conversations were coded, and extend some of the data analyses present in the main text.

2

Setup

In this section, we start by presenting the game we use to study the efficiency of majority
and unanimity voting rules. We then turn to the discussion of the main theoretical insights
regarding this environment, focusing on both qualitative and quantitative predictions.
Finally, we describe the parameters we use in our laboratory experiments and outline thetheoretical predictions for the chosen parameters.
The Game. A group of n (odd) members is charged with distributing a budget amongits members using a pre-specified voting rule denoted by q. The parameter q captures the
9 There also exists a small experimental literature on dynamic bargaining. Most related to our study isBattaglini, Nunnari, and Palfrey (2012). The authors present a legislative bargaining model in which public
and private goods are decided in each period over an infinite horizon. They show that the unanimity ruleleads to higher long-run public investment than the majority rule. Thus, like us, they find support for theunanimity rule from an efficiency perspective.

6

number of votes required to agree on an allocation with q = n indicating the unanimity
voting rule and q = n+1
2 indicating the majority voting rule. The bargaining happens overa series of (potentially infinite) bargaining rounds. Within each bargaining round, members use the standard Baron and Ferejohn (1989) bargaining protocol. In particular, atthe beginning of each bargaining round, all group members learn the budget size for theround, and one member is randomly selected to be the proposer. The proposer can either
submit an allocation, which is a vector of non-negative shares to all members that sums
up to the budget in the current round, or, alternatively, choose to bypass the round andmove straight to the next bargaining round without making a proposal. If an allocation is
submitted, then all group members vote on it. If the proposed allocation obtains at least
the required number of votes (q) then the allocation passes, the game is over, and all members collect the shares specified in the passed allocation. If, however, the allocation does
not get at least q votes, then it is defeated, and the group moves on to the next bargaining
round, which has the same structure as described above.
The uncertainty in this setup is captured by the stochastic process that governs therealization of the budget size as well as the selection of the proposer. In the first bargaining
round, the budget size is small and is denoted by y. In every bargaining round after the
first one, there is p percent chance that the budget will be small once more (y) and a (1-p)
percent chance that it will be large, ȳ, where ȳ > y.10 The group members have an identicalvon Neumann-Morgenstern stage utility function that is linear in the budget allocated tothem, and discount the future at a rate δ ∈ (0, 1). In the event that an agreement is neverreached, all group members receive a payoff of zero.
General Predictions. It is well known that in multilateral bargaining games like the one
considered here there is a continuum of subgame perfect equilibria even under the unanimity rule (e.g., Sutton, 1986). However, it has also been recognized that stationarity
is typically able to select a unique equilibrium (e.g., Baron and Ferejohn (1989); Merlo
and Wilson (1995)). Thus, we restrict attention to stationary subgame perfect (SSP) equilibria. In addition, given that the members are symmetric, we further restrict attention
to symmetric and pure SSP equilibria in which all members have the same continuation
payoff which we denote by vq , and, conditional on not being the proposer, each player is
included in a winning coalition with the same probability if an offer is made. This probaq−1
bility is equal to n−1 since each proposer needs to choose q − 1 other players to include in
his coalition out of the n − 1 remaining players.
To gain some intuition on the trade-offs that our setting encompasses, consider two
10 That the first bargaining round imposes a small budget is without loss of generality. Indeed, if instead thefirst round is also stochastic, then if a group obtains a large budget, an agreement is immediate, and for thosegroups who obtain a small budget, behavior is governed as described below.

7

cases:11(i) Delay is so costly that the members are willing to split the small budget instead of
waiting for the large budget to materialize. Assuming without loss of generality thata member accepts a proposal when indifferent,12 this happens when y −(q−1)vq ≥ vq .
In this case, the equilibrium continuation payoffs must satisfy:
vq =



py + (1 − p)ȳn−1 q−11p(y − (q − 1)δvq ) + (1 − p)(ȳ − (q − 1)δvq ) +
δvq =
.
n
n n−1
n

(1)

n−δqp

This case is possible if and only if ȳ ≤ c1 y where c1 = (1−p)δq , that is, when the large
budget is not too much larger than the small budget.
(ii) Delay is not so costly, and therefore, the members are willing to wait for the large
budget to materialize instead of reaching an agreement on the small budget. This
happens when y − (q − 1)δvq < δvq . In this case, the equilibrium continuation payoffs
must satisfy:
vq =





q−1
1
n−1
δvq .
pδvq + (1 − p)(ȳ − (q − 1)δvq ) +pδvq + (1 − p)
n
n
n−1
1−p

Collecting terms and rearranging, we obtain vq = ȳ n(1−δp) . This case is possible ifn(1−δp)and only if ȳ > c2 y where c2 = (1−p)δq , that is, when the large budget is sufficiently
larger than the small budget.
When the committee uses the unanimity voting rule, i.e. when q = n, we have c1 = c2 =1−δp. Consequently, either the necessary and sufficient condition for case (i) is satisfied,(1−p)δ
or the necessary and sufficient condition for case (ii) is satisfied, but they cannot be satisfied simultaneously. As such, the equilibrium is unique for all possible values of ȳ, y, δ andp under the unanimity rule. In this unique equilibrium, the committee never delays when1−δpthe value of large budget is relatively small, i.e., ȳ < (1−p)δ y and otherwise the committeedelays bargaining until the large budget is realized.13 However, when a committee uses
11 For more details, see Eraslan and Merlo (2002).
12 See footnote 7 in Eraslan and Merlo (2017) for an an explanation of why this assumption is without lossof generality.
13 Notice that the theory is silent regarding the mechanics of how delay in bargaining occurs. It can happen
in one of two ways. In the first, the proposer can choose to forgo making a proposal, in which case thecurrent bargaining round ends irrespective of the desires of the other committee members. In the second, theproposer can submit an allocation proposal, which can then be struck down by the committee members. In
that case the current bargaining round ends following other members’ behavior rather than because of theunilateral decision of the proposer. While in theory these two ways of inducing delay are identical in terms of
the eventual outcomes they produce, there may be large behavioral differences across them. Our experimental
design allows us to examine this point and investigate whether different voting rules tend to affect the ‘type’of the delay.

8any other voting rule with q < n, multiple equilibria might be supported since c1 > c2 . Interestingly, if there is no delay under unanimity rule, the expected equilibrium payoffs do
not depend on the voting rule used by the committee, and only depend on the parametersof the stochastic process that governs the evolution of budget, i.e., (p, y, ȳ) and the discount factor δ. In this case, there is no-delay under the majority rule either, and because
the players are symmetric, the expected equilibrium payoffs are identical. Likewise, if
there exists a delay equilibrium under majority rule, then the expected equilibrium payoffin this equilibrium is equal to the (unique) expected equilibrium payoff under unanimity
rule by the symmetry of the players.
The analysis above focused on the case when the committee members are risk neutral.
In our framework, there are two sources of risk under the unanimity rule: the size of thebudget and the identity of the proposer. Both of these are exogenous risks. Under the
majority rule, however, there is an additional risk of not being included in the winningcoalition. This endogenous risk is at the heart of the result that there might be inefficientagreements under majority rule even when the committee members are risk neutral. The
inefficient equilibrium under majority arises when there is agreement on a proposal dividing the small budget even though it is efficient to wait. In this case, agreement when
the budget size is small arises as a result of the acceptance by members who are includedin the winning coalition. They do so to avoid the risk of being excluded from the winningcoalition in the future when the budget is large. As one would expect, this incentive becomes stronger when the committee members are risk-averse. An extension in Appendix
A illustrates this point.
Experimental Parameters. In choosing our experimental parameters, our goal was todesign the simplest experiment in which uncertainty about the size of the future budget would lead to different behavior in committees which use different voting rules, and,
consequently, to different efficiency levels. In particular, we focused on two voting rules
commonly used in practice, the majority and the unanimity voting rules, and designed six
treatments (three per voting rule), which we describe below.
In all treatments, groups of n = 3 members interacted over a (potentially infinite) number of bargaining rounds with a common discount factor of δ = 0.8.14 The budget size inthe first round was y = $24 in all treatments. The value of the large budget ȳ in ourstochastic treatments varied across treatments and was either ȳ = $48 or ȳ = $96. The
probability distribution over budget sizes was uniform and was the same in all bargaining
rounds after the first one and in all treatments. In addition to the stochastic treatments we
included two deterministic ones in which the budget was always equal to 24 to link our
14 In the next section we describe in detail how we implemented this potentially infinite horizon game inthe laboratory.

9

findings to past literature. We refer to the six treatments as M24, M48, M96, U24, U48,
and U96 indicating in the treatment label both the voting rule and the size of the large
budget, which are the only differences between treatments.
Predictions Given Parameters. The sizes of the large budgets in different treatments ensure that there exists a unique equilibrium in the Majority treatments. Indeed, for ourparameters, we obtain c1 = c2 = 1.5 for the Unanimity treatments and (c1 , c2 ) = (2.75, 2.25)
for the Majority treatments. Thus, for the Unanimity treatments, theory predicts that players would delay splitting the small budget both in U48 and U96, while the same behavior
is predicted only when the large budget is particularly large, as is the case in M96, for the
Majority treatments. Irrespective of the treatment, if members find themselves in a bargaining round with a large budget, they agree on an allocation immediately without any
further delay. These predictions are summarized in Table 1 in the Results section.

3

Experimental Procedures

All experimental sessions were conducted at the University of California, San Diego.15
Subjects were recruited from a database of undergraduate students. Twenty-four sessions
were conducted with 4 sessions per treatment and 12 subjects in each session, for a totalof 288 subjects. Each subject participated in only one session. Each experiment session
lasted about 90 minutes and average earnings were $25.7, including a $12 show-up fee.16
In each experimental session, subjects played twelve repetitions of the bargaining gamewith random re-matching between games, i.e., between repetitions. Before the beginningof each game, subjects were randomly divided into groups of three, i.e., each committee
consisted of three members. Each bargaining game mimicked the extensive-form gamedescribed in Section 2. Specifically, in all sessions, committees started with a small budgetof $24 in the first bargaining round.17 Each game had an unknown number of bargaining
rounds. At the beginning of a bargaining round, the committee members learned thesize of the budget for the round and one of the three committee members was randomly
selected to be the proposer. The proposer could either submit an allocation proposal (avector of non-negative payments to all three members that sums up to the budget in thecurrent round) or, alternatively, the proposer could choose to bypass the round and move
15 We are grateful to the UCSD Economics department for allowing us to use their experimental lab toconduct our experiments.
16 In Appendix B we present the instructions for the U96 treatment. The instructions for the other treatments
are similar except for the obvious differences in voting rule and the size of the potential future budget.
17 The purpose of this choice is two-fold. First, it allows us to normalize initial settings across groups andmore easily compare behavior in the different treatments. Second, when faced with a large budget, there
shouldn’t be any delays, rendering such rounds ineffective at asking whether voting rules and stochastic bargaining lead to inefficiencies.

10

straight to the next bargaining round without making a proposal. If the proposer chose tosubmit an allocation proposal, then all committee members observed it and voted on it. If
the allocation received the required number of votes, then the bargaining game was overand the committee members received the payoffs specified in the allocation that passed.
If the allocation did not receive the required number of votes, or if the proposer chose to
bypass the round, then the current bargaining round was over. In that case, there wasa 20% probability that the bargaining game was exogenously terminated, in which casepayoffs for all group members were zero. With the remaining 80% chance the bargaining
game continued to the next bargaining round where the proposer was once more randomly
selected. In other words, we implemented the discounting with random termination of
the game, which is one of the common ways to study infinitely repeated games in thelaboratory (see Frechette and Yuksel (2017)).
To help players recall the events in the current game, a table on the screen kept track of
all the information pertaining to the current game including the proposer ID, her choice(delay or allocation), and the votes of all group members in each bargaining round.18
Finally, after the proposer was selected at the beginning of a round but before she madeher choice, committee members had the opportunity to communicate with each other using an unrestricted chat tool. This chat tool allowed group members to send any message
they wanted to any subset of the group including private messages to just one member
and public messages that were delivered to all group members.19 There are two reasons
we allowed subjects to communicate with each other. First, it is hard to imagine a reallife dynamic bargaining environment in which members of the same committee cannot
communicate with each other. Second, communication in multilateral bargaining games
has been shown to bring the distribution of resources closer to theoretical predictions (seeAgranov and Tergiman (2014), Baranski and Kagel (2015), and Agranov and Tergiman(2019)), suggesting that communication channels are crucial to understanding how committees function.
At the end of the experiment, one of the twelve games that the subjects played was
randomly chosen for payment, and subjects were paid the amount they earned in this randomly selected game. In addition, after the twelve games, subjects played two investment
games that we use to evaluate subjects’ risk attitudes.20
18 We present the screenshots of the interface in Appendix C.
19 The chat option was on for at least 20 seconds and lasted until the proposer submitted her allocationor chose to delay. This 20-second period was implemented to make sure that if a group member wanted tocommunicate with others, she could do so and not be interrupted by the proposer who might otherwise havequickly entered the allocation and/or clicked delay. The instructions made the structure of the communication
process clear.
20 Appendix D includes the instructions for these investment games and presents summary statistics for thetwo tasks across treatments. We find no statistical differences across treatments in this game, which is why
we reject that treatment differences are due to differences in risk attitudes of our participants as measured by

11

4

Results

As the novelty of our paper lies in the stochastic nature of the budget size, our analysis
focuses on our four stochastic treatments, and we compare these outcomes with thosefrom our two deterministic treatments, namely M24 and U24, when relevant. We present
the results of our experiments in the following order. We begin with our main result
and study game dynamics, focusing on delays across treatments and the exploration theorigins of delays. Second, we document the distribution of resources within a committee
and the implied inequality levels across treatments. Finally, we explore the dialog between
committee members.
Approach to Data Analysis. We use all twelve repetitions of the bargaining game in our
analysis and data from all bargaining rounds, since we see limited learning across theserepetitions and rounds.21 All statistical tests are performed using regression analyses.
In all regressions, we cluster observations by session to account for the interdependencies that come from re-matching subjects between bargaining games. To compare averageoutcomes between treatments we regress the variable of interest on the constant and an
indicator for one of the two treatments. The p-values that we report are those associatedwith the estimated coefficient on the dummy for one of the treatments.
Finally, given that the theory predicts corner values for delay, i.e, no delay for small
budgets in the M48 and 100% delay for small budgets in the remaining three treatments,
any noisy behavior would necessarily produce outcomes which are not in line with thetheory. We therefore mainly focus on the qualitative predictions of the theory.

4.1

Game Dynamics

Frequency of delays. We start by describing the frequency of delays in our stochastic
treatments, which are shown in Table 1.
Delays when facing a small budget are frequent, and more so in committees under theunanimity voting rule than in committees under the majority voting rule: the fraction ofefficient delays is 80% in U48 compared with 54% in M48 (p < 0.001) and these numbers
are 95% and 73% (p < 0.001) for the M96 and U96 treatments, respectively.22
While our deterministic treatments show that disagreements also happen more frequently under unanimity than under the majority rule (26% in U24 and 10% in M24,
p < 0.001), we note that the gap between delays in the M48 and U48 treatments is 26
percentage points compared with the 16 of the deterministic treatments, indicating that
this game.
21 Learning and time trends are presented in Appendix F in Tables 8 and 9.
22 We replicate this table using only first-round data in Table 6 in Appendix F. The conclusions are nodifferent.

12

Table 1: Frequency of delays

48 treatment
Large budget
Prediction

Small budget
Prediction

96 treatment
Large budget
Prediction

Small budget
Prediction

Majority

Unanimity

Maj vs Un

2% (n=50)

9% (n=82)

p = 0.159

0%0%

54% (n=247)

80% (n=289)

0%

100%

4% (n=79)

3% (n=111)0%0%

73% (n=270)

95% (n=344)

100%

100%

p = 0.001

p = 0.772
p < 0.001

Notes: This table shows the total frequency of delays separately for small and large budgets at the group level.
This frequency encompasses cases in which the proposer chose to delay and those in which the proposal was
rejected by the committee. The last column shows the p-values from regression analyses comparing acrossvoting treatments for each row.

delaying per se in the stochastic treatments does not arise under the unanimity rule more
often simply because of behavioral frictions that make it inherently harder to reach agreements when everyone has to agree, but rather because of the stochastic budget interactedwith this voting rule. Also supporting this claim is the fact that statistical tests detect nodifference between the frequency of disagreement between the two voting rules when thebudget is large (p = 0.159 for M48 vs U48 and p = 0.772 for M96 vs U96).
Relative to the theoretical predictions, we note that while the unanimity rule produces
more delays than the majority rule in the 48 treatment as the theory predicts, the frequency of efficient delays in the M48 treatment for small budgets is substantial (morethan 50%) and, thus, should not be ignored. However, and perhaps surprisingly given behavior in the M48 treatment, we observe significantly fewer delays in the M96 committees
compared with the U96 committees, going against the theory that predicts equal levels of
efficient delays in these two treatments.
Although not predicted by the benchmark risk-neutral theory, data from a separate
investment task as well as our theoretical contribution that extends the model beyond riskneutral players point to risk aversion as a likely explanation for these patterns.23 While wefind no difference across treatments in subjects’ risk attitudes, we do find that on average
subjects are risk averse (see Table 5 in Appendix D). This pushes subjects away from delays
regardless of the voting rule. However, risk aversion has a differential impact across voting
23 See Appendices A and D.

13

rules: in the Majority treatment subjects face the endogenous risk of being left out ofa future coalition in addition to the exogenous risk of random termination if a delay isimplemented. This can explain why subjects delay less under the majority rule comparedwith the unanimity one, even in the 96 treatments. Regarding the M48 treatment, the riskneutral benchmark already predicts no delays. Instead we observe delays in excess of 50%.
As we discuss further below, in the laboratory, proposals are often more egalitarian than
predicted, and subjects often discuss fairness, even in the Majority treatments, lowering
the risk of exclusion. This force works towards increasing delaying behavior, pushingagainst the impact of risk aversion. In short, the combination of risk attitudes, how big thesize of the future budget may be, and how likely one is to be excluded from a future rulingcoalition can jointly explain the patterns in our data.
Figure 1: The proportion of small versus large budgets among budgets that are
implemented in stochastic treatments.

Implemented Budgets. Aggregate outcomes in terms of the relative frequency of large
versus small budgets that are implemented among all budgets that are passed are a directconsequence of delaying behavior. These outcomes are shown in Figure 1. Among all
budgets that are passed, large budgets are adopted 56% of the time under U48 comparedwith 30% under M48 (p = 0.01); under U96 and M96 these numbers are 86% and 51%(p < 0.001), respectively. The conclusion is straight-forward: the unanimity rule leads tohigher levels of large budgets being distributed relative to the majority rule, both when
it is expected (as in the 48 treatments), but also when both rules should lead to the samelevels of large budgets passing (as in the 96 treatments).24
24 These same patterns persist if we look at more generalized outcomes that include games in which nobudgets were passed because of the random termination. In this case, the fraction of large budgets passed in

14

Origins of Delays. Why are small budgets are rejected? In the game we analyze, notimplementing a budget can come for one of two reasons: either the proposer decides toforgo submitting a proposal, and consequently moves the group to the next bargaining
round without a voting stage, or, a proposal is submitted but does not receive the required
number of votes. The figures presented in Table 1 do not distinguish these two scenarios.
Figure 2 unpacks these two situations.
Figure 2: Origins of delay for small budgets in treatments with uncertainty

Notes: This figure depicts the frequency of delays and rejected proposals in M48, M96, U48, and U96 treatments conditional on the budget being small.

Figure 2 uncovers interesting differences between the two voting rules. For the 48 treatments, there is no difference in delaying behavior on part of the proposers: the fraction ofdelays implemented by proposers is about 50% in both voting rules (p = 0.815). The reason there are more delays overall in U48 relative to M48 comes from the fact that under
the unanimity rule, when a proposer does submit a proposal, it is rejected with a muchhigher frequency than under the majority rule (58% in U48 versus 7% in M48, p < 0.001).
This points to the fact that departures from theory in the M48 treatment are largely due
to proposers’ choices, not voter behavior. In other words, voters tend to go along with theproposal when one is on the table.
In the 96 treatments, proposers are significantly more likely to forgo making a proposal
under the unanimity rule than under majority rule: 83% of the proposers delay in the U96
treatment and 68% in the M96 (p = 0.015).25 When proposals dividing the small budget
the U48 and M48 treatments are 39% versus 26%, respectively (p = 0.001). These numbers are 56% and 40%
when comparing the U96 and M96 treatments (p < 0.001). In the non-stochastic environment of the U24 andM24 treatments, these numbers are understandably 0 in both cases.
25 Individual propensities to delay show similar results and are presented in the Appendix F.

15

are submitted, they fail to pass in the U96 treatment close to 70% of the time, while they
fail to pass only 15% of the time in the M96 treatment (p = 0.001). That is, in the U96
treatment, in a large majority of groups, voters “correct" mistakes that proposers mightmake (submitting a small budget proposal). On the other hand, under the majority rule
voters fail to correct these mistakes. Given that in both treatments the risk of exogenous
termination is the same, as is the probability of a future large budget, differences in voting
behavior cannot be ascribed to these elements. On the other hand, the treatments do
differ in terms of the probability that a player who receives a positive share in a particular
proposal may be excluded from a future one: this risk exists in the Majority treatment,
while it is nonexistent in the Unanimity treatment (an element that is the result of thetheory but that also finds strong support in the data as shown in the next subsection).
While the unanimity rule is often seen as risky in the sense that one voter can impede thesmooth functioning of a committee, what the data in our experiment point to is that it also
only takes a single committee member to (potentially) reach a larger budget and correct
mistakes that stem from proposer behavior. Specifically, if the proposer fails to bypass
making a proposal, that single voter can rectify the situation. This is not the case underthe majority rule. Our data show that small budget proposals are rejected by a single
deciding vote in 47% and 74% of the cases in the U48 and U96 treatments, respectively.
Thus, this correction happens thanks to a single voter.26
Although the proposals dividing the small budget are not necessarily the same under
the two voting rules, this distinction is not crucial in understanding the aggregate statistics
on rejection rates. Indeed, we next show that proposals dividing the small budget are
rejected by voters not because of the proposals per se (for example, they offer voters too
little) but because of the possibility that the next bargaining round’s budget might be large.
In order to show this we focus on cross-treatment differences in terms of how identicalsmall budget proposals are treated by voters. The natural candidate for such a comparisonis the equal split allocation where all committee members receive an identical share (suchproposals are frequent in all treatments, see Table 2). Our data shows that the fraction ofpassage rates of small budget proposals that equally split the small budget among threemembers is significantly smaller in the Unanimity treatments than in the Majority treatments. Indeed, such proposals are much more likely to be accepted in the M48 compared
26 We also rule out that delays are more frequent under unanimity because subjects happen to experience

fewer random terminations due to the realizations of random draws. In fact, the data point to the opposite.
After small budgets were either delayed or rejected, committees in the U48 treatment experienced twice as
many delays due to discounting (random termination) as compared with the M48 treatment: 30.2% vs 15.1%(p < 0.001). Similarly, the fraction of random terminations in the U96 treatment is significantly higher thanthat in the M96 treatment: 34.4% vs 22.4% (p = 0.001). The qualitative differences hold even after conditioning on groups facing a delay at least once, or controlling for the number of bargaining rounds. In otherwords, players in the Unanimity treatments delay more often despite having been particularly unlucky andexperienced a higher likelihood of early termination. We refer the reader to Appendix F where we use theserandom termination fractions to explain the differences in average total earnings across treatments.

16

with the U48 treatment, 96% versus 57%, and this difference is statistically significant(p < 0.001). Similar behavior is observed in the 96 treatment, in which 97% of equal split
three-person coalitions splitting the small budget are accepted in the M96 treatment compared with only 35% in the U96 (p < 0.001). These results are even more striking giventhat these three-way equal split proposals of small budgets are almost always accepted inboth voting rules in the 24 treatment (99% in M24 and 97% in U24 with p = 0.398).
Result 1: Unanimity committees are more likely than majority committees to delay and attempt to reach large budgets, both when the unanimity rule is predicted to lead to more delays,
and when both rules should lead to identical levels of delays. As a result, large budgets are implemented more often under the unanimity rule than under the majority rule. When delays happen
in the majority committees, they primarily occur because proposers decide to forgo the current
stage and wait for the next stage. In the unanimity committees, in addition to proposers delaying in anticipation of a larger budget (and doing it more often, weakly speaking, than under the
majority rule), group members also often strike down the inefficient small-budget allocations at
a significantly higher rate than under the majority rule (both statistically and in magnitude)
leading to more efficient delays. These results are robust to controlling for proposal types.

4.2

Distribution of Resources within a Committee

We now turn to the question of how committees distribute resources among members
conditional on the budget size. In this section we focus on passed proposals, or in otherwords, the final payoffs of committee members.
Table 2 shows the frequencies of passed coalition types and the corresponding shares ofthe proposers depending on the size of the final budget appropriated. When committees
require a unanimous vote to pass a proposal, irrespective of the budget size, proposers
include all three members in the coalition and divide the resources equally among themin all but a few instances (see the last three rows of Table 2). However, when only a simple
majority is enough to pass a proposal, both two-person and three-person coalitions are
frequent. Moreover, many of the final allocations are equal splits within the coalitions,with the fraction of equal split being generally higher in the three- than in the two-person
coalitions.27,28 Tables 10 and 11 in the Appendix present frequencies of coalition types
in submitted proposals for small budgets and their likelihood of passing, i.e., obtaining
the required number of votes. These tables show that both two-person and three-person
coalitions are commonly proposed in the M48 and M96 treatments.29 Conditional on the
27 The p-values on Probit regressions are 0.039, 0.015, and 0.320 in the M24, M48, and M96 treatments,
respectively.
28 Generally speaking, the results from the U24 and M24 treatments are aligned with past work on the topic.
See Miller and Vanberg (2013), Agranov and Tergiman (2014) and Agranov and Tergiman (2019) for example.
29 Previous experimental work on the Baron-Ferejohn bargaining model suggest that the presence of all-17

Table 2: Distribution of resources within a coalition

M24
M48
M96
U24
U48
U96

Small Budget AppropriatedTwo person
Three person
Proposer’s
freq eq split
freq
eq split share (av)
44%
48%
56%
68%
10.7
57%
66%
43%
90%
11.0
44%
50%
56%
71%
10.4
0%
.
100%
92%
8.1
0%
.
100%
93%
8.1
0%
.
100%
100%
8.0

Large Budget AppropriatedTwo person
Three person
Proposer’s
freq eq split
freq
eq split share (av)
.
.
.
.
.
39%
53%
61%
100%
19.5
50%
53%
50%
74%
42.8
.
.
.
.
.
0%
.
100%
88%
16.2
0%
.
100%
97%
32.2

Notes: The table shows data for only proposals that passed, distinguishing between two-person and threeperson coalitions. For each coalition size, we list the frequency of observing it and how often these coalitions
divide resources exactly equally between its members.

coalition size, some proposals feature equal splits, while others unequal split of resources.
However, for each coalition size, equal split proposals have a higher chance of passing
compared to unequal ones.
The prominence of all-inclusive coalitions with large shares to all three members in the
M48 treatment explains in part why committees in this treatment are willing to delay until
they reach the large budget despite the theoretical prediction that they should appropriate
the small budget in the first bargaining round. Indeed, the theory predicts that only twoperson coalitions should be observed in this treatment, which makes waiting risky, as one
never knows whether one will be included in the future coalition or not. This uncertainty
is the main difference between the unanimity and the Majority treatments. Contrary tothese predictions, since in our data more than 60% of all coalitions are all-inclusive andin their majority divide shares in three equal parts among members of the majority committee, waiting for the big budget becomes less risky. Put differently, the presence and thefrequency of all-inclusive coalitions partially mitigates the uncertainty that members facein the majority committees but not in the unanimity ones. As a result, in the 48 treatment,
the empirical efficiency of majority committees is higher than theoretically predicted.
Figure 3 complements Table 2 by depicting the empirical CDFs of members’ payoffs(Panel (A)) and the empirical CDFs of the Gini coefficients (Panel (B)) in the 48 and the 96
treatments. For a fixed size of the future large budget (48 or 96), the majority committees
have much more variation in payoff distributions across committee members comparedwith the unanimity committees. This echoes the results presented in Table 2, in which
we show that majority committees feature both two- and three-person coalitions with either equal or unequal distribution of resources conditional on the coalition size, while theinclusive grand coalitions may be due to the understanding that proposers of rejected allocations face a lower
continuation value than non-proposers (see Agranov et al. (2020), Baranski and Morton (2021), and Lee andSethi (2021)

18

1
.8
.2
0

0

.2

ECDFs of GINI
.4
.6

ECDFs of Share
.4
.6

.8

1

Figure 3: Final payoffs of players

0

10

20

30

40

50

60

70

0

Share
M48
M96

.1

.2

.3

.4

.5

GINI
U48
U96

Panel (A): Empirical CDFs of Payoffs

M48
M96

U48
U96

Panel (B): Empirical CDFs of GINI coefficients

Notes: Panel (A) depicts the CDFs of players’ payoffs in each treatment, where zero payoff is associated withthe game termination or a zero share in the MWC. Panel (B) depicts empirical CDF’s of GINI coefficients. Here
we focus on games that did not terminate due to discounting.

unanimity committees naturally result in only three-person coalitions with mostly equal
splits among members. The voting rule also has a strong effect on the inequality of payoffs
within a committee, as the Gini coefficients show. In both 48 and 96 treatments, membersof unanimity committees all earn the same payoffs, while those in the majority committees
experience different payoffs, most of which come from two-person coalitions as describedin Table 2.
Result 2: Unanimity committees divide resources equally among their members irrespectiveof the budget size. Majority committees feature both two-person and three-person coalitions
and proposals often result in an equal split of resources within a coalition. The prominence of
three-person equal split coalitions in the M48 treatment reduces the endogenous risk of being
excluded in future winning coalitions which otherwise is present in two-person coalitions both
empirically and in theory. This reduction in endogenous risk provides a behavioral explanation
for the delays observed in the M48 treatment.

4.3

Communication and Bargaining Dynamics.

For each session, two independent coders read through all the conversations. At the bargaining round level they identified whether a conversation pertained to one of eleven different content categories in the Majority treatments, and one of eight different content
categories in the Unanimity treatments.30 Agreement across our coders was very high,30 These categories are described in Appendix E. There are more categories in the Majority treatment because
we also recorded whether there were conversations related to minimum winning coalitions and the division19

ranging from 80.5% to 99.5%, depending on the question and the treatment.31 In our
analyses below, a conversation is counted as having broached a particular topic only if
both coders agreed that such content occurred.
The vast majority of groups engage in meaningful conversations at least once over thecourse of each bargaining game: 97.9%, 92.7%, 96.3% and 91.7% of groups for the M48,
M96, U48, and U96 treatments, respectively.32
Since the main feature of our setup is the uncertainty about the size of the futurebudget, we focus on it and explore the link between the content of the chats and bargaining outcomes.33 Specifically, we ask whether subjects discuss the potential future budget
size and the risk associated with delaying, and whether these conversations affect the frequency of delays across voting treatments.
Table 3: Topics of conversation across treatments
48 treatment
Size of future budget
Support for delaying
Talk about equality/fairness
Threats to vote no if not equal
Threats to vote no if small budget
Risk of game termination
Talk about equality within winning coalition
Talk about unequal split within winning coalition
96 treatment
Size of future budget
Support for delaying
Talk about equality/fairness
Threats to vote no if not equal
Threats to vote no if small budget
Risk of game termination
Talk about equality within winning coalition
Talk about unequal split within winning coalition

Majority

Unanimity

Majority vs Unanimity

70.6%
57.2%
49.2%
2.1%
0.0%
6.3%
72.2%
23.7%

78.2%
59.0%
74.3%
6.7%
1.7%
41.4%
n.a.
n.a.

p = 0.423
p = 0.894
p = 0.012
p = 0.228
n.a.
p < 0.001
n.a.
n.a.

88.6%
75.8%
43.8%
6.9%
6.2%
13.3%
63.8%
53.9%

82.7%
75.5%
54.4%
11.3%
9.3%
20.3%
n.a.
n.a.

p = 0.431
p = 0.971
p = 0.369
p = 0.474
p = 0.580
p = 0.229
n.a.
n.a.

Notes: The table reports the fraction of chats that contain discussions of topics specified in each row. We use‘n/a’ indicates chat categories which are not relevant to the unanimity voting rule. The last column depicts
the p-values comparing the frequencies between the majority and the unanimity voting rules for each row.

Table 3 shows the fraction of groups which engage in various topics of conversation.
There are many similarities in how often topics are discussed across treatments. For the 48
of resources within those. The coders were not privy to our research questions and their payment depended
on how well their coding matched up with each other.
31 Further, the average Cohen Kappa score across all sessions was 0.78 for the majority sessions and 0.73 forthe unanimity ones.
32 A conversation is meaningful if group members discuss the game or anything pertaining to how to play
the game.
33 We refer the reader to the survey of Agranov (2022) for a detailed discussion of the effects of communication in experimental bargaining games with majority and unanimity voting rules.

20

treatments, the main difference in chat content between two voting rules is how frequently
subjects discuss equality and fairness issues as well as the risk of game termination; bothof these topics are more frequently raised in groups that use the unanimity voting rulethan those that use the majority voting rule. At the same time, in the 96 treatments, weobserve no significant differences in the topics of discussion. Nevertheless, the interestingquestion is whether conversations about specific game features explain variation in delays
across voting rules, which is what we explore next.34
Table 4: Impact of chats on delay
Delay in 48 treatment

Delay in 96 treatment

0.35 (0.31)-1.28∗∗ (0.44)
0.19 (0.21)-0.15 (0.46)-0.38∗ (0.26)
1.04∗∗ (0.35)
yes
329
94
0.1037

1.58∗∗ (0.40)-1.21∗∗ (0.45)
0.28 (0.45)-1.36∗∗ (0.53)-0.15 (0.34)
0.42 (0.43)
yes
297
94
0.1434

Constant
Majority voting rule
Chat about risk of game termination
Majority x Chat about risk of termination
Chat about equality
Majority x Chat about equality
Session dummies
Number of observations
Number of participants
Pseudo R-sq

Notes: The table presents the coefficients from panel probit regressions of delaying on treatment dummies,
conversation topics and their interactions with voting rules. Standard errors are clustered at the participant
level. To control for interdependencies of observations that come from the same session, we include session
fixed effects. Observations are restricted to the second half of the bargaining games (and the first bargaining
round within those).∗∗ and ∗ indicate p < 0.05 and p < 0.10, respectively.

Table 4 presents regression analyses that explore the link between the content of conversations and delay frequencies across voting rules, separately for 48 and 96 treatments.
These regressions show a few interesting patterns. First, regardless of the distribution of
future surplus, conversations that precede formal bargaining correlate with the frequency
of delays in the Majority treatments but not in the Unanimity treatment.35
Second, different types of conversations have different effects on delay frequencies depending on the distribution of future surplus. In the M48 treatment, proposers delay
significantly more often when group members discuss equality and fairness topics. This
34 We note also that the content of conversations changes as one moves from the 48 to 96 treatments, condi-

tional on a voting rule. (See Table 12 in Appendix F.) In particular, for the majority voting rule, increase inpotential future budget is associated with more talks about the future budget (p = 0.001), delaying (p = 0.093),
risk of game termination (p = 0.025), and splitting resources unequally within minimum winning coalition(p = 0.025). Similarly, for the unanimity voting rule, an increase in potential future budget is associated withmore conversations about delays, even though the increase is not statistically significant (p = 0.201) and fewer
conversations about both the risk of game termination (p = 0.005) and equality within a group (p = 0.017).
35 This can be seen by the statistically significant interaction effects between an indicator for the Majority
treatment and the content of the conversation.

21

effect is consistent with the intuition that chats about equality reduce the risk associatedwith being excluded from the minimum winning coalition in the Majority treatment. This
provides group members ‘reassurance’ that they will benefit from future budgets and results in more delays in anticipation of a large budget. The same reassurance effect is notpresent in the Unanimity treatment, since the vast majority of all allocations feature threeway splits of the resources. This may explain why discussions about equality have very
little effect on proposers’ behavior in the Unanimity treatment.
At the same time, in the M96 treatment, proposers who have been privy to discussions
about the risk of game termination delay less often than those who have not discussed thisissue. Interestingly, this effect is only present in the Majority treatment, despite the factthat the same risk is present in the Unanimity treatment.
Result 3: Group members engage in meaningful conversations before making bargaining decisions and discuss both issues of fairness and equality among committee members as well as
the risk of game termination due to delays. Conversations’ content is only significantly correlated with proposers’ behavior in the Majority treatments. In the M48 treatment, conversations
about equality may reduce the risk of being excluded from a future coalition, and, as a result,
may generate more delays. Contrary to that, in the M96 treatment, conversations about gametermination are associated with less delays.

5

Concluding Remarks

In this paper we experimentally explore the performance of majority and unanimity voting
rules in multilateral bargaining environments with uncertainty. In our setting, the budget
to be allocated changes stochastically over time. Because the size of a future budget can belarge relative to the present one, delays can be efficient even under complete information
with impatient bargainers. However, whether it is an equilibrium to wait for larger budget
to be divided depends on the voting rule: if it is worth delaying the decision under the
majority rule, it is also true in the unanimity rule, but the opposite direction does not hold.
Furthermore, under the unanimity rule, the outcome is always efficient. In other words,
the unanimity rule achieves weakly higher efficiency levels compared with the majority
rule.
Using the level of efficient delays as the criterion along which we rank the two voting
rules, we find that the unanimity rule leads to better outcomes. Indeed, efficient delays
occur more frequently under the unanimity rule than under the majority rule, even when
the theory predicts that both rules should lead to equally efficient outcomes. The difference in the voting rules is especially large in magnitude when reaching agreements too
soon entails large efficiency losses. Under the majority voting rule, both current proposers

22and coalition partners vote in favor of small budget proposals, perhaps out of fear of being excluded from the coalition in the future. Indeed, under the majority rule, minimum
winning coalitions do not include all the committee members and our results highlight its
failure to lead to efficient outcomes even when the potential future budget is so large that
efficiency is theoretically predicted. On the contrary, under the unanimity voting rule,
this fear is absent as no proposal can pass without unanimous agreement, which guarantees that all members will be allocated a positive share of the surplus.
While the unanimity rule has been viewed with suspicion because it gives “a minority anegative upon the majority" (Hamilton, The Federalist 22), our results show little support
for the hypothesis that under unanimity a minority imposes costly delays on the majority.

References
Agranov, M. (2022). Legislative bargaining experiments. In E. Karagozoglu and K. Hyndman (Eds.), Bargaining: Current Trends and Future Directions.
Agranov, M., C. Cotton, and C. Tergiman (2020). Persistence of power: Repeated multilateral bargaining with endogenous agenda setting authority. Journal of Public Economics 184.
Agranov, M. and C. Tergiman (2014). Communication in multilateral bargaining. Journalof Public Economics 118, 75–85.
Agranov, M. and C. Tergiman (2019). Communication in bargaining games with unanimity. Experimental Economics 22(2), 350–368.
Baranski, A. and J. K. Kagel (2015). Communication in legislative bargaining. Journal of
the Economic Science Association 1.1, 59–71.
Baranski, A. and R. Morton (2021). The determinants of multilateral bargaining: A comprehensive analysis of Baron and Ferejohn majoritarian bargaining experiments. Experimental Economics, 1–30.
Baron, D. P. and J. A. Ferejohn (1989). Bargaining in legislatures. American Political Science
Review 83, 1181–1206.
Battaglini, M., S. Nunnari, and T. R. Palfrey (2012). Legislative bargaining and the dynamics of public investment. American Political Science Review 106(2), 407–429.
Benjamin, D. and M. L. Wright (2019). Deconstructing delays in sovereign debt restructuring. Oxford Economic Papers 71(2), 382–404.

23

Diermeier, D. and R. Morton (2005). Experiments in majoritarian bargaining. In Social
choice and strategic decisions, pp. 201–226. Springer.
Eraslan, H. (2008). Corporate bankruptcy reorganizations: Estimates from a bargaining
model. International Economic Review 49(2), 659–681.
Eraslan, H. and K. S. Evdokimov (2019). Legislative and multilateral bargaining. Annual
Review of Economics 11, 443–472.
Eraslan, H. and A. Merlo (2002). Majority rule in a stochastic model of bargaining. Journalof Economic Theory 103, 31–48.
Evdokimov, K. S. (2020). Uniqueness of equilibrium payoffs in the stochastic model ofbargaining. Economics Letters 188, 108931.
Frechette, G. and S. Yuksel (2017). Infinitely repeated games in the laboratory: Four perspectives on discounting and random termination. Experimental Economics 20.2, 279–
308.
Frechette, G. R., J. H. Kagel, and S. F. Lehrer (2003). Bargaining in legislatures: An experimental investigation of open versus closed amendment rules. American Political Science
Review 97(2), 221–232.
IMF (2020). The international architecture for resolving sovereign debt involving privatesector creditors—recent developments, challenges, and reform options.
Lee, N. and R. Sethi (2021). Agenda-setting power in legislative bargaining.
Li, S. and D. Houser (2020). Stochastic bargaining in the lab. working paper.
McKelvey, R. D. (1991). An experimental test of a stochastic game model of committee
bargaining. Laboratory research in political economy, 139–168.
Merlo, A. and C. Wilson (1995). A stochastic model of sequential bargaining with complete
information. Econometrica 63, 371–399.
Merlo, A. and C. Wilson (1998). Efficient delays in a stochastic model of bargaining. Economic Theory 11(1), 39–55.
Miller, L. and C. Vanberg (2013). Decision costs in legislative bargaining: an experimental
analysis. Public Choice 155(3-4), 373–394.
Miller, L. and C. Vanberg (2015). Group size and decision rules in legislative bargaining.
European Journal of Political Economy 37, 288–302.

24

Palfrey, T. R. (2015). Experiments in political economy. handbook of experimental economics, vol. 2.
Roth, A. E. and J. K. Murnighan (1978). Equilibrium behavior and repeated play of theprisoner’s dilemma. Journal of Mathematical Psychology 17, 189–198.

25

A

Effects of risk attitudes

Suppose now the payoff of a committee member from receiving a surplus of x is givenby u(x) where u is a strictly increasing concave function with u(0) = 0. As before, let
vq denote the continuation payoff of a committee member under voting rule q. Thus, toinduce acceptance from another committee member, the proposer needs to offer a surplusof u −1 (δvq ) to that committee member. The proposer needs (q − 1) votes in addition to
her vote for the proposal to be accepted. Thus, if the proposer makes an offer that will
be accepted when the budget is y, she receives a share of y − (q − 1)u −1 (δvq ). If u(y − (q −1)u −1 (δvq )) > δvq , then the proposer is strictly better of making a proposal that will be
accepted. If u(y − (q − 1)u −1 (δvq )) < δvq , then the proposer is strictly better of delaying (ormaking an offer that will not be accepted, which is payoff equivalent). Let αy,q (v) ∈ [0, 1]
denote the probability that the proposer makes an offer that will be accepted when thebudget is y and the continuation payoff is v. From the discussion above, we have



 1
αy,q (v) =  0

if y > qu −1 (δv)
if y < qu −1 (δv).

(2)

When there is a delay, all players receive their continuation payoffs. When there isan agreement, conditional on not being the proposer each player receive their discounted
continuation payoff with the same probability. In equilibrium, the continuation payoff vq
must be consistent with the strategies. Thus, we have
vq =

X


py

y



n
o n−1 q−1
1
−1
max u(y − (q − 1)u (δvq )), δvq +
αy,q (vq )+ (1 − αy,q (vq )) δvq (3)nnn−1where py is the probability that the budget size is y.
As before, assume there are two possible budget sizes, i.e. y ∈ {y, ȳ} with ȳ > y, py = p
and pȳ = 1 − p. It is straightforward to see that there can be no equilibrium in which
there is a delay with strictly positive probability when the budget size is large, i.e. there
cannot be any equilibrium with αȳ,q < 0. If it were the case, u(ȳ − (q − 1)u −1 (δvq )) ≤ δvq and
αy,q = 1. Substituting in (3), we obtain vq = 0. But since u is strictly increasing, we obtaina contradiction to (2). In light of this observation, equation (3) can be written asvq = pno n−1 q−1
1
max u(y − (q − 1)u −1 (δvq )), δvq +
αy,q (vq )+ (1 − αy,q (vq )) δvq
n
n
n−1


q−1
1
+(1 − p) u(ȳ − (q − 1)u −1 (δvq )) +
δvq .
n
n

(4)

But notice that αy,q (.) can take any value between 0 and 1 when the proposer is indifferent between delaying and making an offer that will be accepted when the small budget is26

realized. As a consequence, the operator whose fixed points characterize the equilibrium
continuation payoffs is a correspondence, not a function. Intuitively, except under unanimity rule, whether the proposer delays or makes an offer that will be accepted makes adifference in the continuation payoff of the proposers since those who are excluded fromthe winning coalition receive a payoff of zero in the event of an agreement but receive theircontinuation payoffs when there is a delay. Define the functions fq and gq asno n−1  q−1
1−1+ (1 − α) δv
fq (v, α) = max u(y − (q − 1)u (δv)), δv +
α
n
n
n−1
and
gq (v) =q−11u(ȳ − (q − 1)u −1 (δv)) +δv,nn

and define the correspondence Γ as
Γ (v) = v 0 ∈ R : ∃α ∈ [0, 1] such that α = αy,q (v) and v 0 = pfq (v, α) + (1 − p)gq (v) .

(5)

It follows that vq is an equilibrium continuation payoff if and only if vq ∈ Γ (vq ).36
Since u(0) = 0 and u is strictly increasing, equilibrium continuation payoffs must be
strictly positive: vq > 0 for all q. Otherwise, if vq = 0, then the right hand side of (4) isstrictly positive while the left-hand side is zero.
The existence of equilibrium can be established using Kakutani’s Fixed Point Theorem
along the lines of Theorem 3 in Eraslan and Merlo (2002) which proves existence when
the players are risk neutral. Eraslan and Merlo (2002) also show that there may be multiple equilibria under majority rule but equilibrium is unique under unanimity rule when
the players are risk neutral (Merlo and Wilson (1998)). Once risk aversion is allowed,
the equilibrium payoffs need not be unique even under unanimity rule however. Merlo
and Wilson (1995) show that the equilibrium payoffs are unique, provided that the utility functions satisfy a contraction property which is trivially satisfied when players arerisk-neutral or when the cake process is deterministic. Given the difficulty of verifying
this condition when players are risk-averse and the cake process is stochastic, Evdokimov(2020) provides another sufficient condition that is easy to verify. In particular, he showsthat this sufficient condition is satisfied when the utility functions are isoelastic with astrictly positive minimum bound.
Even when the equilibrium is not unique, our main result is robust:
Proposition 1. If vq is an equilibrium continuation payoff for the q-quota game, then there
exists an equilibrium continuation payoff vn for the unanimity game such that vn ≥ vq .
36 The arguments above establish that if v ∈ Γ (v ), then v is a stationary subgame perfect continuationqqq

payoff. The converse follows using the one-shot deviation principle.

27

Proposition 1 shows that equilibrium continuation payoffs are weakly higher underunanimity rule in the sense that for any equilibrium in the q-quote game, there exists anequilibrium in the unanimity game with a higher equilibrium continuation payoff. One
implication of this is that the unanimity rule is (weakly) more efficient. Also, note that any
player (including the proposer) votes to accept the proposal only if she receives at leasther continuation payoff. Thus, a second implication of the proposition is that, if there isan equilibrium with delay under majority rule, there is also an equilibrium with delay
under unanimity rule but not vice versa.
Proof of Proposition 1: We use the following lemma throughout the proof.
Lemma 1. If b > max{c, a} > 0, thenu(a)u(b)−u(c)
< a .
b−c

Proof. Since u is a strictly increasing concave function with u(0) = 0, we have u(b)/b <u(b)−u(c)u(a)/a. Likewise we have u(b)/b < u(c)/c. The latter inequality implies that b−c <
u(b
k
b .
Since the equilibrium continuation payoffs must be strictly positive, it suffices to showthat fn (v, α) ≥ fq (v, α) for any v > 0 and α ∈ [0, 1] with α = αy,q (v) and gn (v) ≥ gq (v) for any
v > 0. Note that fn (v, α) ≥ fq (v, α) if and only ifononα(n − q)δv ≥ max u(y − (q − 1)u −1 (δv)), δv − max u(y − (n − 1)u −1 (δv)), δv .

(6)

There are three cases to consider:
Case 1: u(y − (n − 1)u −1 (δv)) ≥ δv. Then, since v > 0 and u is strictly increasing, we haveu(y − (q − 1)u −1 (δv)) > δv, α = 1 and (6) reduces to(n − q)δv ≥ u(y − (q − 1)u −1 (δv)) − u(y − (n − 1)u −1 (δv)).
Let a = u −1 (δv), b = y − (q − 1)a, and c = y − (n − 1)a. Then the above inequality is equivalentto u(a)/a ≥ [u(b) − u(c)]/(b − c). It is satisfied by Lemma 1.
Case 2: u(y − (q − 1)u −1 (δv)) < δv. Then α = 0 and (6) is trivially satisfied.
Case 3: u(y − (n − 1)u −1 (δv)) < δv ≤ u(y − (q − 1)u −1 (δv)). Then (6) reduces toα(n − q)δv ≥ u(y − (q − 1)u −1 (δv)) − δv.
If the right hand side is zero, then the inequality is trivially satisfied. Otherwise, α must
be equal to 1 and the inequality is satisfied sinceu(y − (q − 1)u −1 (δv)) − δv < u(y − (q − 1)u −1 (δv)) − u(y − (n − 1)u −1 (δv)) ≤ (n − q)δv

28

where the first inequality is satisfied by the conditions defining Case 3, and the second
inequality is satisfied by Lemma 1 by choosing a, b and c as in Case 1.
It remains to show that gn (v) ≥ gq (v) for any v > 0. The proof of this is identical to theproof of Case 1 above.
k

B

Instructions for U96 treatment

This is an experiment in the economics of decision making. The instructions are simple,
and if you follow them carefully and make good decisions you may earn a CONSIDERABLE AMOUNT OF MONEY which will be PAID TO YOU IN CASH at the end of theexperiment. In addition to what you will earn in the experiment, you will get a $12 participation fee if you complete the experiment.
In this experiment you will play 12 Matches. At the start of each Match you will berandomly divided into groups of 3 members each. In any Match you will not know theidentity of the subjects you are matched with and your group-members will not know
your identity. At the start of each Match, each member of the group will be assigned an ID
number (from 1 to 3), which is displayed on the top of the screen. Since ID numbers will
be randomly assigned prior to the start of each Match, all members are likely to have their
ID numbers vary between Matches. In addition, since you will be randomly re-matched toform new groups of 3 at the start of each Match, it is impossible to identify subjects using
their ID numbers.
Each Match consists of one or more Rounds. Your ID number will stay the same during
all the Rounds of a Match. However, once the Match is over, you will be randomly rematched to form new groups of 3 members each and you will be assigned a (potentially)
NEW ID. Please make sure you know your ID number when making your decisions.
In each Match, each group will decide how to split a sum of money (the “budget”).
One of the 3 members in your group will be randomly chosen to be the proposer. Eachmember has the same chance of being selected to be the proposer. The proposer can takeone of two actions. The proposer can either submit an “allocation proposal” of how to split
the budget among the 3 members, or the proposer can hit a “delay” button. In the firstRound of a Match, the budget available to be split will be 24 dollars. We will describe thebudget available in other Rounds as well as what happens if the proposer chooses to hit
the “delay” button shortly.
Suppose the proposer chooses to make an allocation proposal. After the allocation proposal is submitted, it will be posted on your computer screens with the allocation to you
and the other members clearly indicated. You will then have to decide whether to accept or
reject the allocation proposal. Allocation proposals will be voted up or down (accepted or
rejected) by unanimity rule. That is, if all three members approve the allocation proposal,
29the match ends and the earnings from this match are given by the approved allocation
proposal. If at least one of three members rejects the allocation proposal, it is voted down.
If the allocation proposal is voted down (that is at least one member of your group
votes against it), then one of two things can happen:• With 20% chance the Match ends and all members of your group will earn 0 dollars
for this Match.
• With 80% chance you move on to the next Round of this Match. In this case, oneof the 3 members in your group will be randomly chosen to be the proposer for thisround. After the proposer has been chosen, he will have the choice between hitting
the “delay” button, or making an allocation proposal on how to split the budget.
However, budget will either be 24 dollars or 96 dollars, with 50/50 chance of each.
In other words, there is 50% chance that the proposer in Round 2 will be dividing 24
dollars between group members and 50% chance that the proposer in Round 2 will
be dividing 96 dollars. The proposer in Round 2 and all group members will know
the size of the budget available for division before making any decisions. If the
proposer submits an allocation proposal and it is voted down, then again with 20%
chance the Match ends and all members of your group will earn 0 dollars for thisMatch, and with 80% chance you will move on to the next Round of this Match. If
the group moves on to the next Round, then, again, one of the 3 group-members will
be randomly chosen to either hit the “delay” button, or make an allocation proposal
on how to split budget among the 3 members with each member equally likely to bechosen as a proposer. The budget size will either be 24 dollars or 96 dollars, with
50/50 chance of each. In fact, for all Rounds after the first Round, the budget will
either be 24 dollars or 96 dollars, with 50/50 chance of each. This process repeats
itself until a Match ends, either because of the 20% chance it ends between Rounds,
or because an allocation proposal has passed.
Recall that instead of submitting an allocation proposal, a proposer can choose to “delay.” If a proposer chooses “delay”, then the group goes through the same stages as if a
proposal is rejected. That is, if a proposer chooses “delay” then with 20% chance the
Match ends and all members receive 0 dollars for this Match. With 80% chance the groupmoves on to the next Round within the Match, one member of your group is randomly
chosen to be the next proposer and the amount of money to split is either 24 dollars or 96
dollars with 50/50 chance of each etc.
To summarize, in any given round, if an allocation proposal is rejected, or if the proposer chooses “delay,” then with 20% chance the Match ends and members of the groupearn 0 dollars for this Match. With 80% chance a new Round starts, one member of yourgroup is randomly chosen to be the proposer and the budget to be split is either 24 or 96
30

dollars, each with 50/50 chance. This continues until a Match ends, either because of the
20% chance it ends between rounds, or because an allocation proposal passes.
Communication: In each Round, after one voter is selected to propose a split but before
he/she submits his/her allocation proposal, members of a group will have the opportunity
to communicate with each other using chat boxes. The communication is structured asfollows. On the top of the screen, each member of the group will be told her ID number.
You will also know the ID number of the member who is currently selected to make aproposal. Below you will see three boxes, in which you will see all messages sent to either
all members of your group or to you personally. You will not see the chat messages thatare sent privately to other members. If you would like to send the message that will bedelivered to the entire group, please type your message underneath the first chat box and
hit SEND. If you would like to send a private member of your group, please type yourmessage underneath the chat box that indicates the chat with that member and hit SEND.
There is a 20 second period of time at the start of each Round during which the proposer cannot submit his/her allocation or choose delay. During this time, any person in
the group can choose to use the chat function on his/her screen. The chat option will beavailable as soon as the Round starts, and for at least 20 seconds. The chat option will become unavailable when the proposer either submits his allocation proposal or hits delay.
You are not to communicate in any other way with any other subject while the experimentis in progress. This is important to the validity of the study.
Remember that in each Match subjects are randomly matched into groups and ID numbers of the group-members are randomly assigned. Thus, while your ID number stays thesame during all the Rounds in a Match, your ID number is likely to vary from Match to
Match, and therefore it is impossible to identify your group-members using your ID number.
At the conclusion of the experiment we will randomly select one of the 12 Matches
to count for payment. The $12 participation fee will be added to your earnings in that
randomly selected Match.
Review. Let’s summarize the main points:
1. The experiment will consist of 12 Matches. There may be several Rounds in each
Match.
2. Prior to each Match, you will be randomly divided into groups of 3 members each.
Each subject in a group will be assigned an ID number.
3. At the start of each Match, in Round 1, one subject in your group will be randomly
selected to be a proposer in this Round. The proposer can choose either to submit
31

an allocation proposal or to delay. The size of the budget in Round 1 is 24 dollars.
Before the proposer chooses his/her action, all members of the group can use thechat box to communicate with each other. You may send public messages that will
be delivered to all members of your group as well private messages that will bedelivered to specific members of your group.
4. Proposals to each member must be greater than or equal to 0 dollars.
5. If all 3 members accept the allocation proposal, the Match ends.
6. If one or more members reject the allocation proposal, or if the proposer chose to hit
the “delay” button, then one of two things can happen:• With 20% chance the Match ends and all members of the group earn 0 dollars.
• With 80% chance the Match continues. In this case, one member of the groupwill be randomly selected to be the proposer in Round 2. The budget available
for division in Round 2 will be either 24 or 96 dollars, each with 50/50 chance.
The proposer can choose either to delay or to submit an allocation proposal,
etc. . .
7. The process in step 6 repeats itself until a Match is over, either because of the 20%
rule, or because an allocation proposal has passed. At the end of the experiment, thecomputer will randomly select one of the 12 Matches you played, and your earnings
in this selected match will be paid to you in cash together with the participation feeof $12.
Are there any questions?

C

Screenshots for U96 treatment

Before starting the experiment, we will show you a few screenshots so that you can familiarize yourself with the interface. After that, we will start the experiment, in which
you will play 12 Matches. Please note that the numbers and decisions from the screenshots below are just examples and are not meant to indicate what you should do in thisexperiment.
The screenshot in Figure 4 is a typical screenshot that proposers see.

32

Figure 4: Screenshot of the Proposer

33

Please take a look at the bottom part of the screen depicted in Figure 5:

Figure 5: Bottom Part of Screenshot of the Proposer
Notice that there are three boxes labeled with the ID numbers of the members. This iswhere the proposer writes his/her allocation, corresponding to the amounts to members
1, 2 and 3, respectively. The proposer is the only member of the group who can choose tosubmit an allocation or “delay”. When you are done choosing an allocation, hit submit. If
you choose to “delay”, hit Delay.
Let’s look at the rest of the screen. On the top left side you will be able to see the historyof the current Match depicted in Figure 6:

Figure 6: History of Current Match
Take a moment to look at that. It will show you the budget size for each Round of the
Match, the ID number of the proposer for that Round, and once the proposal has been
submitted votes have taken place you will see those too. If the proposer chose to “delay”
then you will see “DELAY” in the space under “proposal.”
Below the Match-history box, you will see the chat boxes depicted in Figure 7. The
left chat box shows the group conversations, while the middle and the right box show theprivate conversations with the other two members. Below each chat box are the boxes you
will use to send messages if you choose to do so.

34

Figure 7: Chat Box
Below in Figure 8 is a screenshot of the non-proposers. It is identical to the proposer
screens except for the right hand side since only proposer can choose to submit an allocation or “delay.”35

Figure 8: Screenshot of the Non-Proposer36

Below is another example of a within Match history (see Figure 9). In this particular
example, the proposer in Round 1 was member 1 and he/she chose to Delay. The match
continued to the next round. In Round 2, the proposer was member 2, the budget was 96
dollars and the proposer chose to submit an allocation according to which member 1 gets
2 dollars, she (member 2) gets 50 dollars and member 3 gets 44 dollars. The proposal was
rejected since it didn’t not receive all 3 yes votes. The match continued to the next round.
In Round 3, member 2 was again randomly chosen to be the proposer and she submitted
another allocation according to which member 1 gets 90 dollars, member 2 (herself) gets
6 dollars and member 3 gets 0 dollars. Notice that in this table you can always see the sizeof the budget as well as who was proposer in every round, what action they took (propose
an allocation or “delay”) and the results of the votes.

Figure 9: Match History
If a proposer submits an allocation, all members of the group see the screen like theone in Figure 10. The proposal is clearly indicated, and your payoff if the proposal isapproved is highlighted in red. You can then vote yes or no to the proposal. Please notethat the numbers here are just examples and are not meant to indicate what you should doin this experiment.
The proposer for this round was member 1.
The proposer chose [2 22 O]. which is displayed below.
Your payoff is shown in red
Allocation Proposal-

Member1
2

Member 2
22

Member 30

Please click the button below corresponding to your vote on this proposal and click Next:
Yese No

Figure 10: Voting Screen

After members vote all members see the screen like the one in Figure 11:37

Figure 11: Summary of Votes
Your earnings are always highlighted in red. If the match randomly ends because of
the 20% rule you, you will see the messages shown in Figure 12 on the right hand side ofyour screen.

Figure 12: Termination Message
Are there any questions?

D

Investment Tasks

Investment Task 1. You are endowed with 200 tokens (or $2) that you can choose to keep
or invest in a risky project. Tokens that are not invested in the risky project are yours to
keep.
The risky project has 50% chance of success:• If the project is successful, you will receive 2.5 times the amount you chose to invest.
• If the project is unsuccessful, you will lose the amount invested.
Please choose how many tokens you want to invest in the risky project. Note that you
can pick any number between 0 and 200, including 0 or 200.
Investment Task 2. You are endowed with 200 tokens (or $2) that you can choose to keep
or invest in a risky project. Tokens that are not invested in the risky project are yours to
keep.
The risky project has 40% chance of success:38• If the project is successful, you will receive 3 times the amount you chose to invest.
• If the project is unsuccessful, you will lose the amount invested.
Please choose how many tokens you want to invest in the risky project. Note that you
can pick any number between 0 and 200, including 0 or 200.
In the experiment, one of the two investment tasks was randomly chosen to count for
payment.
Table 5 presents summary statistics for decisions in Investment Tasks across treatments. There are no statistical differences across treatments in this game. We therefore
reject that treatment differences are due to differences towards risk as measured in thisgame.
Table 5: Behavior in investment tasks across treatments.

Treatments
M24
U24
M48
U48
M96
U96

Investment Task 1
mean
median121.5
100(p = 0.544)(p = 1.00)
111.7
100

Investment Task 2
mean (st dev)
median107.5
100(p = 0.983)(p = 1.000)
107.8
100

125.4
(p = 0.364)
113.5

100
(p = 1.00)
100

100.8
(p = 0.671)
108.1

100
(p = 1.000)
100

117.4
(p = 0.593)
125.6

100
(p = 1.00)
100

99.0
(p = 0.252)
118.4

100
(p = 1.000)
100

Notes: The p-values are the result of OLS and quantile regressions with clustering at the session level.

E

Coding the free-form communication

For 48 and 96 treatments, the coders were asked to code conversations using the categories
listed below. The last three categories are for Majority treatments only:
1. Is there any discussion relevant to the experiment (budget, how to split it, whether todelay or not, how often the game is terminated, what is fair, anything that happened
in other rounds, clarifications on the experiment etc. . . )? Yes/No
2. Is there any talk about delay/big pie/big budget, anything about the fact that thebudget can be “big”? Yes/No
3. Is there any talk risk of game being terminated? Yes/No

39

4. Is there any talk about equality and fairness among all group members? Yes/No
5. Are there threats to vote no if not equal division? Yes/No
6. Are there threats to vote no if small budget? Yes/No
7. Other threats? Yes/No
8. Is there any indication of support to wait for a big pie or delay? Yes/No
9. Is there a discussion of a minimum winning coalition, i.e., excluding one member
and colluding among two members to divide the budget just among them? Yes/No
10. Is there a conversation about splitting budget equally between you and me? Yes/No
11. Is there a conversation about dividing resources unequally within minimum winningcoalition? Yes/No

F

Additional Analysis

Delays in first bargaining rounds only. Table 6 mirrors Table 1 in the main text, but
focused only on the first stage of each game where the budget is small by design.
Table 6: Frequency of delays

48 treatment
Small budget
Prediction

96 treatment
Small budget
Prediction

Majority

Unanimity

Maj vs Un

52% (n=192)

83% (n=192)

p = 0.001

0%

100%

76% (n=192)

96% (n=192)

100%

100%

p < 0.001

Notes: This table shows the total frequency of delays for small budgets at the group level in the first bargaining
stage. This frequency encompasses cases in which the proposer chose to delay and those in which the proposal
was rejected by the committee. The last column shows the p-values from regression analyses comparing acrossvoting treatments for each row.

Total committee earnings. Table 7 show the predicted earnings and the average number of dollars that were distributed among committee members in each treatment withstandard errors in parentheses.

40

Table 7: Predicted and observed total earnings of a committee, by treatment

Predictions
Majority
Unanimity
Experiment
Majority
Unanimity
Majority vs Unanimity

24 treatment

48 treatment

96 treatment

24
24

24
32

64
64

23.63 (0.24)
22.75 (0.52)
p = 0.143

26.50 (0.82)
26.13 (1.43)
p = 0.813

47.13 (0.69)
56.25 (1.25)
p < 0.001

24 vs 48

48 vs 96p = 0.008
p = 0.048

p <0.001
p <0.001

Notes: We present the averages of total number of dollars appropriated by committees in expectation as wellas in the experiment in each treatment. Robust standard errors are calculated based on regressions withclustering at the session level and presented in the parenthesis. The last line reports statistical tests comparing
the two voting treatments obtained using regression analysis. The last two columns report statistical tests
comparing the average surpluses in different treatment with the same voting rule.

Table 7 shows that higher expected future surplus translates into higher average earnings of a committee. Indeed, for each voting rule separately, committees appropriate significantly higher surpluses in the 48 treatment compared with the 24 treatment and in the
96 treatment compared with the 48 treatment (see the last two columns with p-values).
Moreover, comparing the average surplus across voting rule for a fixed size of the future expected surplus, we note that two opposing forces determine this ranking. Under the
majority rule, committees pass small budgets more often and large budgets less often thancommittees under unanimity rule, which gives unanimity rule committees an advantage
compared with majority rule committees. On the other hand, as we described in Footnote
26, subjects under the unanimity rule were “unlucky" (statistically speaking) and experienced higher termination rates (controlling for delaying) compared with the majority rule.
Empirically, in the M48 and U48 treatments, these two forces happen to exactly offset eachother, which is why we observe similar average earnings in the two voting rules, despite thefact that subjects in the Unanimity rule choosing to delay small budgets significantly more
often. In the M96 and U96 treatments on the other hand, where the gains from waiting toreach a round with a higher budget are the highest, the first force dominates the second,
and the unanimity rule outperforms the majority rule.
Delays at the individual level. Figure 13 depicts the histograms of individual propensity to delay for each participant when he or she was selected to be the proposer conditional on the budget being small. These are average frequencies of delays per person acrossall twelve games played in a session.
The Figure 13 reveals that there is noticeable heterogeneity in individual propensity
to delay in each treatment. However, despite this heterogeneity, the comparison across41

Fraction
.2
0

0

.1

.1

Fraction

.2

.3

.3

.4

Figure 13: Individual propensity to delay, by treatment

0

.2

.4
.6
Fraction of Delays
U48

.8

1

0

.2

M48

.4
.6
Fraction of Delays
U96

Panel (A): 48 treatment

.8

1

M96

Panel (B): 96 treatment

Notes: For each participant, we compute the frequency with which she delayed splitting the small budget
when she was selected to be a proposer over the course of the entire experiment.

voting rules is similar to those at the aggregate level. Indeed, proposers are more likely todelay splitting the small budget in the Unanimity treatment as compared with the Majority
treatment. This effect is statistically significant for the 96 treatment (p = 0.0074) but notfor the 48 treatment (p = 0.7183).
Learning. Table 8 below compares the fraction of small budget proposals that are passed
in each treatment, separating the data between the first and second halves of the experiment. In both halves of the experiment, small budgets are much more likely to pass in
the Majority treatments than in the Unanimity treatments that feature stochastic futurebudgets.
Table 8: Fraction of small budget proposals passed in the two halves of the experiment.
Treatment

First Half

Second Half

M48

93.7%
(p < 0.001)
40.9%

93.2%
(p = 0.002)
43.2%

87.8%
(p < 0.001)
34.4%

81.1%
(p = 0.026)
28.0%

U48
M96
U96

Notes: For each budget distribution we compare the outcomes in the Majority and Unanimity treatments using
regression analysis, in which we regress the variable of interest on the constant and an indicator for one of
the treatments, while clustering standard errors by session. We report the p-value associated with estimatedcoefficient on the dummy for one of the treatments.

In Table 9 we replicate material from the main text, but breaking it down by firstand second halves of the game as well as by bargaining round. We focus on the rejection
42

Table 9: Rejection rates in the first and second half of the experiment by bargaining
round
overall

Small budgets
1st round 2nd round>3rd round

Large budgets
overall51.2%
p < 0.001
81.3%

50.0%
p = 0.003
83.3%

52.9%
p = 0.005
86.1%

62.5%
p = 0.656
50.0%

0.0%
.
9.8%

67.4%
p < 0.001
93.5%

65.6%
p < 0.001
93.8%

70.8%
p = 0.061
92.5%

75.0%
p = 0.004
94.1%

3.2%
p = 0.968
3.4%

56.4%
p = 0.025
77.9%

54.2%
p = 0.007
82.3%

68.2%
p = 0.587
75.0%

50.0%
p = 0.851
53.9%

3.9%
p = 0.465
7.3%

78.3%
p < 0.001
96.0%

85.4%
p = 0.002
99.0%

61.3%
p < 0.001
97.8%

63.4%
p = 0.178
84.4%

4.2%
p = 0.477
1.9%

first half
M48
U48
M96
U96
second half
M48
U48
M96
U96

rates of small budgets in line with our analysis in the main text. We note no fundamental differences: small budgets are more likely to be rejected in the Unanimity treatments
compared with the Majority ones (this is also generally true if we break it down by bargaining rounds). There is no cross-treatment differences in how large budgets are treated.
This aligns with the conclusions obtained when grouping the data from all games and all
bargaining rounds together as we did in the main text.

Proposed Allocations for Small Budgets. Table 10 shows which types of proposals are
made when the budget to be split is of size 24 in each treatment.
In all of the Majority treatments, the modal proposal is an equal split among all threemembers of the group (this fraction is between 34.9% and 37.7%), though roughly halfof the proposals are of size 2, and the other half of size 3. In the Unanimity treatments,
a substantial majority of proposals provide an equal split of resources among all threemembers of the group. Cross-treatment differences in terms of proposal types have strong
implications on inequality within groups.
Table 11 shows how frequently small budget proposals are accepted in each treatment,by the type of proposal. In the Majority treatments, regardless of the type of small budget
proposal, a large majority pass (the fraction ranges from 66.7% to 100%). Strikingly, thesefractions remain high even when delaying is an equilibrium, as in the M96 treatment. In
the Unanimity treatment, however, the fraction of small budget proposals that pass range
from 0% to 96.6%, and, in line with the theoretical predictions, far fewer of these proposals
43

Table 10: Distribution of proposal types in submitted allocations (small budget
proposals).
Treatment

Coalition Size 2
Equal Splits
Unequal Split

Coalition Size 3
Equal Split
Unequal Split

M24

20.0%

25.4%

35.6%

19.0%

M48

35.3%

23.0%

37.7%

4.1%

M96

20.9%

27.9%

34.9%

16.3%

U24

2.4%

1.6%

70.73%

25.2%

U48

0%

0.7%

68.6%

30.7%

U96

0%

1.8%

89.5%

8.8%

Notes: Equal split coalitions of size 2 are proposals in which two members receive the exact same amount while the third
receives nothing. Equal split coalitions of size 3 are proposals in which all three members receive the exact same amount.

pass when the cost of early agreement is high, as in the U48 and U96 treatments.
Table 11: Fraction of accepted proposals dividing the small budget.
Treatment

Equal Splits

Coalition Size 2
Unequal Splits

Coalition Size 3
Equal Splits
Unequal Splits

M24

97.6%

82.7%

98.6%

87.2%

M48

100%

78.6%

95.7%

100%

M96

88.9%

66.7%

96.7%

85.7%

U24

na

na

96.6%

22.6%

U48

na

0%

57.3%

9.3%

U96

na

na

35.3%

0%

Notes: Equal split coalitions of size 2 are proposals in which two members receive the exact same amount while the third
receives nothing. Equal split coalitions of size 3 are proposals in which all three members receive the exact same amount.
We report data for which we have at least 10 observations.

Conversation Topics Across Treatments Within a Voting Rule. Table 12 below shows
the statistical tests comparing frequencies of conversation topics across budget size within
the same voting rule.

44

Table 12: Topics of conversation across treatments
Size of Future Budget
Support for Delaying
Talk about equality/fairness
Threats to vote no if not equal
Threats to vote no if small budget
Risk of Game Termination
Talk about equality within MWC
Talk about unequal split within MWC

M48 vs M96
p = 0.001
p = 0.093
p = 0.681
p = 0.120
n.a.
p = 0.025
p = 0.333
p = 0.025

U48 vs U96
p = 0.679
p = 0.201
p = 0.017
p = 0.533
p = 0.102
p = 0.005
n.a.
n.a.

Notes: We report the p-values comparing the frequencies of conversation topics across treatments withdifferent potential budget sizes conditional on the voting rule.

45