Coming and going: Experiments on endogenous group sizes for excludable public goods

Journal of Public Economics 93 (2009) 336–351

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Journal of Public Economics

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Coming and going: Experiments on endogenous group sizes for excludablepublic goods☆

T.K. Ahn a, R. Mark Isaac b,1, Timothy C. Salmon b,⁎a Department of Public Administration, Korea University, Seoul 136-701, Republic of Koreab Department of Economics, Florida State University, Tallahassee, FL, 32306-2180, United States

a r t i c l e i n f o

☆ Theauthorswould like to thank theNational Sciencedebt of gratitude to Justin Esarey for programming the ethanks the financial support of the Korea Research Foun⁎ Corresponding author. Tel.: +1 850 644 7207; fax:

E-mail addresses: [email protected] (T.K. Ahn), mis1 Tel.: +1 850 644 7081; fax: +1 850 644 4535..2 Congestibility is an attribute of goods and services

(1959, 1969) added the dimension of excludability toexample). Both rivalousness and excludability vary ingoods and services exist in between Samuelson's pure

0047-2727/$ – see front matter © 2008 Elsevier B.V.doi:10.1016/j.jpubeco.2008.06.007

a b s t r a c t

Article history:Received 19 July 2007Received in revised form 25 June 2008Accepted 25 June 2008Available online 2 July 2008

When a public good is congestible, individuals wanting to provide the public good face challengesin forming groups of optimal size, selecting themembers of the group, and encouragingmembersto contribute for the public good. We conduct a series of experiments in which subjects formgroups using three different entry and exit rules. The experimental results are analyzed in terms ofgroup size, the level of public good provision, social efficiency, congestion and group stability. Wefind that entry restriction improves the average earnings for some individuals compared to freeentry/exit or restricted exit. For a given group size, individuals under the restricted entry rulecontributemore for the provision of the collective good. Also, for a givenaverage contribution levelof groupmembers, subjects under the restrictedentry rule suffer less fromthe congestionproblemand are better able to form groups of sizes closer to the optimal.

© 2008 Elsevier B.V. All rights reserved.

JEL classification:C92H41D85

Keywords:Public goodsCollective actionEntry and exit rulesGroup formationGroup size

1. Introduction

A congestible public good is neither purely private nor purely public and, thus, the optimal size of the group to jointly provideand consume its benefits is somewhere between one and the number of the entire potential users. When exclusion is feasible, acongestible public good can be provided as a club good.2 Collective action problems involving congestible public goods posemultiple related 50 challenges of which the primary one is in determining the size of the group.

The problem of optimal group size is extensively discussed by Buchanan (1965) and Olson (1965) with Buchanan focusingmostly on how the production technology determines the optimal size of the groupwhile Olson focuses more on how the behaviorof group members may limit the size and even how the behavior of group members may vary with the size of the group. Olson'smain argument regarding the relationship between group size and collective action was that the larger the group size the lessoptimal will be the provision of a collective good. Many scholars showed later that the relationship is not as simple as what Olson

Foundation (Grant#SES-0451970) for the researchsupport in funding these experiments.Weowea substantiaxperiments. We also thank the editor and two anonymous referees for their many useful suggestions. T.K. Ahndation and the Korea University.+1 850 644 [email protected] (R.M. Isaac), [email protected] (T.C. Salmon).

defined on the rivalrousness dimension of the standard two dimensional classification scheme. MusgraveSamuelson's (1954) rivalousness leading to the widely used 2×2 table (see Ostrom and Ostrom 1977, fordegree rather than being strictly dichotomous characteristics. Buchanan (1965) first pointed out that manyprivate goods and pure public goods and used the term “club good” to refer to them.

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337T.K. Ahn et al. / Journal of Public Economics 93 (2009) 336–351

argues but depends on the characteristics of the collective good such as the congestibility and the individual costs of contributiontoward the public good (Chamberlin, 1974; Esteben and Ray, 2001; Frohlich and Oppenheimer, 1970; Hardin, 1982; Oliver andMarwell, 1988). More recently Barham et al. (1997) point to the importance of both the factors of provision technology andindividual behavior in determining the optimal size of a group especially when the provision of the good is voluntary. The point isthat the optimal size of a group is a complicated mix of the underlying technology of the provision, how group members behaveand then how individual behavior in the group varies as the size and composition of the group vary.

To demonstrate the interaction between technology and individual behavior, consider the attributes of a community swimmingpool. Technologically, there is some capacity limit to the number of swimmers that can be supported in the pool before it gets toocrowded and the enjoyment of the pool is negatively impacted by more swimmers. The practical point at which this occurs, though,also depends on the behavior of those in the pool. A group of peoplewho are polite and considerate of others can likely grow relativelylargebefore thenegative externality becomes aproblem.However, evena small numberof peoplewhoact in a less consideratemannercan impair the enjoyment of the pool at a much lower group size. So a group attempting to provide a congestible public good for itsmembers faces this complex problem of determining and regulating the size of the group to achieve optimal levels of public goodproduction, which is a function of the technology of the good itself as well as the behavior of the group members.

When facedwith these issues, an important tool to a group is the ability todetermine its ownmembership.Ostrom(2005:260-1), forexample, argues that “groups' determining their ownmembership” is the first step toward “developing a greater trust and reciprocity.”While some groups may not be allowed to exclude others from consuming the public good (e.g. cities can not ban non-residents fromdriving on their roads), there are a variety of existing arrangements currently in use for exactly this purpose. Country clubs, cooperativeapartments, and medical practices, for example, have well specified rules for who can join, while neighborhood associations3 andreligious denominations have rules about who may leave. Ostrom (2005) calls these rules of entry and exit “boundary” rules.

Boundary rules define (1) who is eligible to enter and take a position in the group, (2) the process of selecting among potentialmembers, and (3) how an individual may leave (or must leave) the group.While these rules can be useful in regulating the size andcomposition of a group, they may lead to yet another complication for a group. That is, the behavior of groupmembers may changeas a function of the particular set of boundary rules chosen to govern group membership.

The questionwe focus on in this study is how different sets of boundary rules may affect the extent to which groups of efficientsizes are formed and the level of cooperation among group members in providing a congestible public good. To address this issue,we conduct a series of experiments in which 12 subjects form groups using different entry and exit rules and make decisions onthe level of contribution for a congestible public good. In our experiment group formation is fully endogenized in terms of groupsize and composition. The experiments are conducted under three different entry and exit rules: 1) free entry and free exit (FEE),2) restricted entry and free exit (RE), and 3) free entry and restricted exit (RX). When entry (exit) is restricted, an individual hopingto join (leave) a group must obtain the approval of a majority of group members.

To the extent that the subjects in our experiment can freely enter or freely exit a group, the experimental environment alsorelates to Tiebout's (1956) model of “voting with feet.” But, of course, the foot-voting in our experiment is limited by the rules ofentry and exit while Tiebout's original model assumes fully mobile consumer-voters. In addition, while several Tiebout modelsassume that groups determine the level of public goods provision by amajority rule (Epple et al., 1984; Goodspeed 1989; Epple andRomer, 1991; Nechyba, 1997; Fernandez and Rogerson, 1996, 1998, for example), our experimental design lets group compositionbe determined partially by majority rule (in RE and RX) and the provision level determined by a voluntary contributionmechanism. Ultimately whether our environment is seen as a specialization of a Tiebout model in regard to how the public good isprovided or as a broadened public good model with group formation, the ultimate issues to be examined are the same.

Oneobvious group formation toolwe leave outof ouranalysis is “forced exit,”or “expulsion.”Onemightquestion this choice since itwould be reasonable to expect that expulsion would be the most powerful group formation rule to boost cooperation among groupmembers. This issue has received previous investigation though as in Cinyabuguma et al. (2005) which reports that when there is athreat of expulsion the level of contribution for the provision of a public good rises to nearly 100% of the social optimum. Were we toinclude expulsion in our environment, there is every reason to expect similar results. Examining group formationwithout expulsion isstill aworthwhile exercise becausewhilemanygroups in the realworld use expulsion as an instrument to regulatemembership, thereare many other groups for which expulsion is either infeasible or too costly an option to exercise. Examples include forcing a city ortownship to de-annex from a larger unit, evicting a resident from a co-op apartment building, expelling a tenured facultymember andso forth. In these examples, expulsion is, if not physically impossible, often impractical, and the rules of entry and exit play a moreimportant role in determining group size andmembership. Consequently we designed our study to investigate themarginal effects oftheseperhaps lesspowerful tools of regulated entryandexit compared to abaseline conditionof free entryandexit to determine if theycan be used to increase or decrease pro-social behavior in settings for which exclusion is infeasible.4

The experimental results are analyzed in terms of five key variables: group size, contribution level, social efficiency, congestion,and group stability. The key result we will show is that subjects in the restricted entry treatment are able to achieve substantiallyhigher earnings than subjects in the other treatments. The subjects in the restricted entry treatment will generally form smaller

3 While most religious organizations may not restrict individual people from exiting, there are many current debates about whether the Episcopal andPresbyterian Churches in particular can restrict individual congregations from leaving. In certain “cult” religions individual exit is of course prohibited and insome countries with official religions, individual exit will also be restricted.

4 We note that we are not the first to examine group formation in games of this nature with a focus on mechanisms other than exclusion. Several other studiessuch as Coricelli et al. (2003), Gunthorsdottir et al. (2001) and Page et al. (2002) have experimentally examined various forms of less severe group formation rules,but with exogenously determined group size. Our intention in this study is to study the marginal effects of entry and exit restriction in a fully endogenized groupformation context.

338 T.K. Ahn et al. / Journal of Public Economics 93 (2009) 336–351

groups with higher contributions than groups of similar sizes in the other treatments. This in turn will lead to the fact that thegroups in the restricted entry treatment suffer less from any congestion problems than do the groups in the other treatments.

In the next section we present the details of our experimental design. In Section 3 we present a theoretical examination of theexperimental environment as well as some organizing benchmark predictions to use in evaluating the experimental data. InSection 4 we provide the experimental results and in Section 5 we conclude.

2. Experiment design

In each session of our experiment twelve subjects played a repeated public goods provision game with endogenous groupformation according to one of the three group formation rules. A total of twelve sessions were run, four sessions per treatment. Thesubjects knew that after an initial period of the game, the rest of the session would consist of a maximum of 25 periods. In caseswhere 25 periods could not be completed in two hours, including the time spent on instructions and exercises, the subjects weretold two periods before the final period when the session would end.

In each period, each subject belongs to a group of size n∈ {1,2,…,12} and makes a decision on how to divide 15 “tokens”between her own individual account and the group account in which she is a member. The exact group size is endogenouslydeterminedwithin the extremes of one and twelve in amanner wewill discuss later. The experimental design is constructed basedon four distinct features: 1. a congestible, as opposed to a pure, public good, 2. voluntary, as opposed to enforced or contractual,provision of the public good, 3. dynamic and endogenous, as opposed to fixed or exogenous, group formation, and 4. multipleinstitutions by which the group formation can occur. We will address each of these points in turn in the context of our design.

2.1. Congestible public good

The stage game, or the game played in each period, involves each of the subjects deciding how much of a total of 15 tokens tocontribute to the group account. Let xi denote the number of tokens individual i contributes to the group account. Let Gi representthe set of the other n−1 members in the group of player i. Given any choice of xi by subject i and vector of choices x− i by the groupmembers of Gi , the monetary payoff to i is determined by the payoff function defined in Eq. (1).

5 WeearningWe alsomake ahypothewe used

6 Estemade in

7 As tproceduseparatsubjectthe bansubjects

πi xi; x−ið Þ ¼ 2 15−xið Þ þ 6 xi þ ∑jaGi

xj

!−427

x3i −9ffiffiffiffiffiffi13

p

8� n−1ð Þ2: ð1Þ

e payoff function says that investment to the individual account returns 2 ECUs (Experimental Currency Units) for each token
Thleft in the individual account.5 Investments into the individual account have no direct cost, but of course there is the opportunitycost of foregoing investments to the group account. The returns from the group account are 6 xi þ ∑
jaGi

xj

� �which demonstrates the

public nature of the good as individual i receives returns in ECUs equal to 6 times his contribution level plus six times thecontribution level of every member of his group. There are two direct costs to the individual contained in the payoff function. First,there is a direct cost of contributing to the public good of 4

27 x3i . This is a convex function indicating that contributions become

increasingly more expensive the higher the contribution level. This could represent the notion that the first hour one spendsvolunteering to clean a neighborhood park is not very costly, but by hour 8 or 9, each hour becomes increasingly more onerous.6

The last term in the payoff function, 9ffiffiffiffi13

p8 � n−1ð Þ2, represents the second direct cost and is the group size cost, representing

the congestion aspect of the public good. This is a cost that a person pays simply from being a member of a group of size n. The costof being in a group is increasing in the number of members such that the kth member imposes more additional cost than does the(k−1)th member. The idea that congestion is increasingly costly with group size is a standard component in the description of acongestible public good and has therefore been incorporated into our design.7

2.2. Voluntary provision

In any given round, a subject chooses their own contribution level xi∈{0,1,2,...,15}. The provision of the public good is thereforepurely voluntary in the sense that no individual can directly affect the contribution of any other individual. While it would be

engaged in a variety of procedures to help subjects understand this payoff function. We provided hardcopies of extensive tables summarizing theirs from any combination of decisions they and their fellow group members might make that were designed to demonstrate all of the relevant trade-offsran them through an extensive on-screen help system explaining all stages of the decision process. During the experiment when a subject was asked tocontribution decision, the software included a test button that would allow the subject to enter a proposed contribution level for themselves and atical average for the rest of their group and then see all computations regarding the effect on their payoff and the payoff of others The instruction scripis available as a web supplement on the journal's website while screenshots and the payoff tables used are available from the authors upon request.ban and Ray (2001) provide a more detailed justification of assuming an increasing marginal cost of individual contribution when the contribution isfunds or in time.

his function indicates, it was possible in these sessions for a subject to lose money inside of a period. To deal with this we instituted a fairly standard set ores and subjects were fully informed about them at the start of the experiment. First we started subjects off with an initial balance of 100 ECUs (this wase from their show-up fee) which gave them an initial cushion of money to keep them from getting a negative balance early on. Second, in the event that adid lose so much money that their total earnings were negative, we would give them awarning and reset their balances to the 100 ECUmark the first timekruptcy occurred. If a subject were to go bankrupt twice, they would have been dismissed from the session receiving only their show-up fee. Only twowent bankrupt once while none went bankrupt twice.

.

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interesting to also investigate the nature of contract formation and how the existence of contracts could change behavior, ourinterest in this paper is in how well the group formation mechanisms alone will be able to help the subjects solve their socialdilemmas. Our results could therefore shed some light on the question of the degree to which enforceable contracts may or maynot be necessary or useful in this context.

2.3. Endogenous group formation

Theexperimentbeginswith a “preliminary round” inwhich the subjects are placed into groups by themselves (i.e.n=1) andplay thecontribution game once. The subjects are told that none of the other subjectswould ever be able to observe these choiceswhichmeansthat there is no possibility to use these choices as signals about the degree towhich a subject intends to contribute in the future rounds.This round is conducted in part to allow the subjects to practice with and become comfortable with the contribution mechanism butalso in part to allow us to observe their behavior when there was no tension between individually and socially optimal behavior.

After the preliminary phase, subjects begin the part of the experiment with endogenous group formation. Themain phase startswith the subjects randomly assigned to groups labeled A–L such that each subject is again in a group by herself. In period 1, thesubjects choose a contribution level in those groups which means that they again make a choice in a group with n=1. But this timethe choice is viewable by others. In all of the subsequent periods (up to 25 periods), a period begins with the subjects being asked ifthey wish to switch groups. When asked this question they are able to view a screen showing the average contribution levels andgroup sizes of the 12 possible groups over the previous 5 periods. The contents of the screen include the number of subjects in eachgroup at the end of the previous period. This allows for subjects to send signals of various sorts through their choices beginning inperiod 1 of the main phase. If a subject indicates that she does wish to change groups, she is presented with a new screen. The newscreen is similar to the previous one but shows, along with the aggregate contribution level for each group for the past 5 periods,the number of subjects in each group who has chosen to remain in the group. Subjects are then allowed to choose which groupthey wish to enter. They are also allowed to re-enter into their group from the previous period even though they answered “yes” tothe question about changing groups. There is no cost of moving or trying to move to a new group. All choices are simultaneous.

2.4. Entry and exit rules

The group formation process takes place using one of three different entry/exit rules. The three treatments were Free Entry/Exit(FEE), Restricted Entry (RE) and Restricted Exit (RX).8 In the RE treatment, a subject choosing to leave a group is allowed to leave but heor shemust apply forentry in thegroupof their choice. Themembersof the target groupwhoremained in that group fromthepreviousperiod (i.e. thosewhoanswered “no” to the question aboutwhether or not theywished to change groups) are shown the choicehistoryof the applicant for the five previous rounds including his or her contribution level and size of group he or shewas in. The informationfor all applicants to a group is displayed simultaneously and the group members vote to approve or disapprove each applicantindividually. For anentryapplication to be approved,more than50%of the groupmembersmust vote to approve the application. If 50%or fewer vote in favor of the application, the applicant is returned to his or her previous group. An analogous process is used for thosechoosing to leave a group in the RX treatment. A subject in the RX treatment may choose which group to join and enter that groupfreely, but to be allowed to exit fromher current group the othermembers of the group fromwhich she is exitingmust vote to allow it.The groupmemberswhohave chosen to remain in the group see thefive-period choice history of the individual and vote to approve ordisapprove of the exit application using themajority rule. If the application is denied due to not receivingmore than 50% yes votes, thesubject stays in the group while if it is allowed the subject automatically enters the group he or she has selected to enter.9 Again, allchoices regarding moves were made simultaneously and then votes on those choices were also simultaneous.

Four experimental sessions were run for each treatment. Twelve subjects participated in each of the twelve sessions for a totalof 144 subjects. All sessions were conducted in a computer lab using software created with z-Tree (Fischbacher, 1999). Subjectswere paid a $10 show-up fee and ECUs translated into dollars at a rate of 1 ECU=$0.01. Average subject earnings were in the rangeof $20–25 in these sessions. Sessions lasted on average an hour and a half to two hours.

3. Benchmarks and conjectures

3.1. Stage game analysis: equilibria and social optima

3.1.1. Nash equilibriumThe stage game at the base of our experiment is a straightforward voluntary contributionmechanism. Once the subjects are in a

group, the size for that period is fixed and they must decide how many of their 15 tokens to contribute to the public good. If weassume a single shot interaction, the payoff structure we described above leads to a dominant strategy solution of contributing 3 to

8 We did not include a fourth treatment in which both entry and exit were restricted because we were most interested in the marginal effects of adding onetype of restriction or the other to the base of no restrictions. Also, a treatment involving restrictions on both entering and exiting a group would have imposedextra time pressures on the experiments causing us to change procedures from the other three treatments.

9 Note that no vote was necessary for someone who chooses to re-enter their group from the previous period in either the RE or RX treatments. Regardless ofthe outcome of the vote, they would still be in that group for the next period. Subjects who choose to possibly switch groups but then effectively remain in theirprior groups are not, however, allowed to vote on the applications of others entering or exiting their group.

Table 1Optimal investment amounts into group account depending on the size of the group

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

Individual optimum 3 3 3 3 3 3 3 3 3 3 3 3Earnings if all contribute equally 38 52 58 55 45 27 0 −35 −78 −129 −188 −255Group optimum 3 5 6 7 8 9 9 10 11 11 12 13Earnings if all contribute equally 38 57 78 97 113 127 136 143 145 142 136 124


the public good. This dominant strategy is not on the lower boundary of possible contributions due to the fact that the marginalbenefit to the subject is greater than the marginal cost for low levels of contribution. After the third token, contributing one moretoken costs more than the 4 ECUs it generates in net-return. This dominant strategy solution is valid regardless of the size of thegroup of which the subject is a member.10

We can add to the analysis of this basic stage game by considering a one period game that involves 12 players first decidingwhat group to join (ignoring for now any issues with the particulars of themechanism) and then deciding howmuch to contribute.This game is similar to that studied in Barham et al. (1997). For the contribution stage it is still the case that contributing 3 is thedominant strategy solution. Given that, the question is what sort of equilibria exist in regard to the formation of the groups? Thereare a variety of strategies which will lead to valid pure strategy equilibria in this case, but any such strategies must lead to one oftwo arrangements of subjects into groups. One possible equilibrium outcomewould involve the 12 players splitting themselves in4 groups with 3members each. The second possible outcomewould have the 12 players splitting into 3 groups of 4 members each.For any set of strategies that generated any other arrangement, there would necessarily be at least one player who could do betterby deviating to choose to join a different group.11

The argument behind this is straightforward. In the (3,3,3,3) arrangement assuming all subjects are contributing 3 tokens, thereturn to each is 57.78 ECUs. In the (4,4,4) arrangement the return to each player is 55.49. Any subject who chooses to deviate fromthe (3,3,3,3) arrangement would end up forming either a four-person or one-person group. The payoff for the four-person grouphas already been shown to be lower than that obtained in the three-person group and the earnings to being in a group of one are38 ECUs. From the (4,4,4) arrangement a subject could form a five-person group earning 45.10 ECUs or a one-person group both ofwhich generate lower earnings than staying in the four-person group.

From any arrangement other than (3,3,3,3) and (4,4,4), at least one subject can do better by deviating and either forming agroup by themselves (were they in a group of six or larger) or attaching themselves to a group that is smaller than three. There aremany possible sets of strategies that deliver either equilibrium arrangement of (3,3,3,3) and (4,4,4), but it is clear that no purestrategies that result in arrangements of any different partitions of subjects into groups can obtain in equilibrium. It is also clearthat of these two arrangements (3,3,3,3) Pareto-dominates (4,4,4).

3.1.2. Social optimumThe above analysis holds if the players are purely non-cooperative and self-interested. There is substantial prior evidence

suggesting that peoplemay have some form of other regarding preferences that may lead them to consider social efficiency in theirdecision making (see Camerer, 2003, for an extensive survey). To understand how such preferences might affect behavior, we canalso determine the socially optimal contribution levels and group sizes. The first level of analysis involves holding n fixed and thensolving Eq. (1) for the payoff maximizing contribution level assuming that xi=xj for all j in the group. The results of this calculationcan be found in Table 1. Note that while the individually optimal contribution level does not vary with the size of the group, thecontribution that maximizes the payoff for the group does vary with the size of the group. In fact, the group optimal contributionlevel increases almost linearly with group size.12 If players arewilling to contribute up to the level that is payoff maximizing for thegroup of which they are a member then this will allow them to form larger groups and increase their payoffs. As can be found byexamining Table 1, the group size that leads to the highest payoff for the members of that group is 9 giving each group member apayoff of 145 if they all contribute 11 tokens. This leaves the other 3 players doing best by forming a group of three and receiving apayoff of 78 each. The total payoff to society in that case is 9⁎145+3⁎78=1539. This turns out to be the arrangement of subjectsinto groups that is the socially optimal arrangement or the one that generates the highest sum of total payoffs for the twelvesubjects in a session.13 This socially optimal arrangement leads to an average payoff of 1539/12=128.25 which represents anincrease of 121% over the best equilibrium payoff, 58.

While the socially optimal arrangement is (9,3), this arrangement does have two drawbacks. First it results in inequity acrosssubjects. Nine subjects receive the “good” return of 145 while the three others must settle for around half of that at 78. Whowouldvoluntarily select into the lesser group? A more equitable arrangement would be to split the players equally into two groups of 6

10 This interior dominant strategy has the advantage over standard corner solutions in that it allow two sided errors so that both over- and under-contributionare possible. Further, having the solution be invariant to group size provides a firm benchmark in prediction and allows us to observe and compare changes incontribution behavior as group size changes.11 While there will exist many possible mixed strategy equilibria, we do not consider them.12 Notice that neither boundary, zero contribution or a contribution of 15 tokens, is ever individually optimal or group optimal.13 We do note that there are other configurations yielding very similar group payoff totals such as the following: 59 (6,6): 6×127+6×127=1524;(7,5)7×136+5×113=1517;(8,4):8×143+4×97=1532;(9,3): 9×145+3×78=1539;(10,2): 10×142+2×57=1534;(11,1): 11×136+1×38=1534;(12): 12×124=1488.

:


yielding a payoff of 127 to each subject and a total social surplus of 1524 which is just slightly smaller than the total from the (9,3)arrangement. A practical problem with the group size of 9 is that a player must be very trusting of his fellow group members toinduce him to contribute 11 tokens. The reason is that if his fellowgroupmembers choose instead to contribute 3 tokens, his payoffis −239. Thus investing 11 tokens is quite risky. If one is in a group of 6 and contributes the group optimal level of 9 and if all othersonly contribute 3, the high contributor's payoff is −57 which is also a bad outcome but it is less of a loss. Therefore in the (6,6)arrangement, it does not take as many groupmembers contributing above the individually optimal level to offset the potential lossa high contributor might sustain. Thus, while (9,3) is mathematically the socially efficient arrangement, it seems quite plausiblethat (6,6) may still be preferred by players of this game who have an interest in social efficiency because it achieves a moreequitable division of earnings without being too risky for high contributors.

3.2. Finitely repeated game with group formation

We have already identified two types of Nash equilibria of the stage game which involve all subjects contributing 3 tokens andassembling into group arrangements of (3,3,3,3) or (4,4,4). By moving to a finitely repeated version of this game we know thatsubgame perfect equilibria will exist inwhich these stage game equilibria are repeated in each period of the repeated game. Again,there are a large number of strategies in regard to the group formation process that would deliver these outcomes and be part of anequilibrium. The easiest such strategies would involve the subjects in period 2 perfectly separating into one of the two grouparrangements and then remaining there for the duration of the experiment. This could involve voting strategies of not allowing anyentry or exit after the initial groups are formed. It would also be possible to construct what might be referred to as nomadicequilibria in which subjects switch groups each round but keep the group partition always at (3,3,3,3) or (4,4,4). These equilibriawould require herculean coordination in practice but it is technically possible to derive them. Due to the fact that there are twopossible one-shot game equilibrium arrangements that are Pareto ranked, it is even possible to find equilibria that involvecontributions above the dominant strategy level for a time by employing punishment strategies. Because this would againinvolve incredible coordination of the subjects to coordinate on punishing deviators by adopting the (4,4,4) equilibrium in the lastperiod(s) rather than the Pareto dominant (3,3,3,3) equilibrium, it seems unlikely that these would emerge spontaneously in anexperiment without communication. Wewill therefore use as the benchmark equilibrium prediction the more plausible equilibriathat subjects will contribute three tokens per period and form into groups of size three or four.

Notice that the outcomes from the equilibria thatwehave described donot dependon the rules of the group formationprocess. That is,whether the group formation institution is FEE, RE or RX, the equilibria described above will continue to exist. Given that all subjects aregoing to contribute 3 tokens in the contribution phase, the composition of a group is irrelevant and subjects should either vote to allowothers in or out of their group in amanner so as tomaintain a group size of 3 or 4. There is again a large set of strategies thatwould achievetheseoutcomesbut therearenoother reasonableoutcomes that canbesupportedasequilibria. This is an importantpointbecause if playerscontribute at the dominant strategy level then theoretically the group formation mechanisms should have no impact on the outcomes.

The socially optimal arrangement of subjects into groups also does not change depending on the group formation mechanism.Regardless of themechanism, the socially optimal outcomewould involve subjects in the repeated game sorting into one group of nineand one group of three with subjects in both groups contributing at their group optimal levels. Of course, if subjects pursue fairness aswell as social efficiency, the group formation outcome would be two groups of six as discussed before. If subjects contribute at a levelabove the dominant strategy level, then the group formation institutionsmay have an impact as groups have to be concerned about thesize of thegroup relative to the average level of contributions. In theRE institution subjectsmayonlyallowhigher contributing applicantsin their group while in the RX institution they may not allow high contributing members to exit. If subjects pursue such goals in theirvoting behavior, this could lead to feedback effects on the future contributions of subjects. Itmaywell be that certain institutions are ableto fostermaintaineddeviations fromthebenchmarkequilibriumwhileothersdonot. This is thekeypointweare interested inexamining.

3.3. Conjectures

While the subjects in our experiments have a stage game dominant strategy of contributing three tokens in the contribution stage,there is room to substantially improve their earnings depending on how the groups are formed and the degree to which groupmembers cooperate. We seek to understand how the mechanism of group formation might help or hinder the degree of cooperation.

There are prior experimental studies of public goods provision in which various forms of entry and exit rules are used. Forminggroups based on mutual preferences (Page et al., 2002; Coricelli, et al., 2003) and allowing members to expel other members by amajority vote (Cinyabuguma et al., 2005) have been found to increase the level of contribution toward the public good. In theseexperiments, however, the endogeneity in group formation is somewhat limited in the sense that the size of the group is eitherfixed orcan only decrease. Ehrhart and Keser (1999), in an experiment similar to our FEE treatment, find that allowing individuals to movefreely between groups does not enhance cooperation.What theyfind instead is a persistent instability caused as free-riders chase aftercontributors who are “on the run” to form a cooperative group.14 Ahn et al. (2008) investigate the same set of three entry/exit rules

14 The public good environment created by Ehrhart and Keser utilized the standard linear provision function with a fixed marginal cost — thus the individualincome-maximizing contribution level is zero. But the marginal return from a unit contribution changes such that being in a large group can be harmful tocontributors if there are too many free-riders. The authors do not explicitly address, nor is it clear from the experimental design, whether the experiment relatesto a congestible public good.


with fully endogenous group formation, but the experiment utilizes a pure public good and, thus, the subjects have an overwhelm-ing incentive to form the largest possible group. The study finds that entry restriction allows high contributors to teach lowcontributors to be concerned about social efficiency by repeatedly denying them entry into the high contributing group until the lowcontributor learns to become a high contributor. Further, the subjects who are taught about social efficiency keep their contributionshigh after they successfully join a cooperative group, even though the other group members have no other means of coercingsuch behavior. Thus, entry restriction has an educational effect of teaching low contributors to become high contributors. However,entry restriction also resulted in smaller groups and, thus, the positive externalities of a large group were not fully utilized. Theaverage earnings of the subjects were actually lower when entry was restricted than when exit was restricted or when both entryand exit were free. Another interesting finding is that exit restriction may have frustrated high contributors when their attempts todepart from a low contributing group were voted down. Some individuals exhibited a perverse pattern of behavior in which theylowered their contribution level even below the individual optimum, probably to obtain an exit or to punish the low contributors.

Crosson et al. (2004), utilizing a ‘social poker’ experimental paradigm, study whether individuals form efficient sizes of groups.In their study, the group formation is based on unanimity rule. They find that the subjects often form groups larger than what isindividually or socially optimal and take more of the club good than is agreed upon among the club members.

If we combine our theoretical analysis above with what we learn from these previous experimental studies, we can form aseries of conjectures and benchmark predictions regarding what we expect to see as outcomes of our experiments. We organizeour conjectures around the five key variables of our interest.

1. Contribution: The lower bound prediction is that subjects will contribute at the stage game dominant strategy of 3. The upperbound prediction is that subjects will contribute up to the socially optimal level. Neither extreme is empirically likely so thequestion is whether there are differences due to the institutions in how close subjects get to either boundary. In the REtreatment, small groups of contributors could get together into a group and exclude low contributors. This would lead to aprediction that the RE treatment should give the greatest likelihood of observing high contributing groups. Note that this doesnot necessarily imply higher overall contributions due to the fact that the other groups of the excluded subjects may becontributing at a low level and therefore driving down the sessionwide average. Previous literature suggests that ability to exit agroupmay increase cooperation (Orbell et al., 1984; Schuessler, 1989; Vanberg and Congleton, 1992) which suggests that the RXtreatment may generate the lowest contributions.

2. Group Size: Our theoretical analysis suggests that if subjects are contributing at the stage game dominant strategy, then theyshould be in groups of size three or four. If subjects are more cooperative, they may be able to support larger groups up to amaximum of nine in a single group. If the RE rules allow for greater cooperation, then it would be reasonable to expect thatgroups of size six to nine could bemore likely to form, at least if groups form efficiently as a function of contribution levels. Sincethe RX and FEE treatments allow free entry it would be quite easy for large groups to form even when they are not desirable.Consequently, it is unclear as to what the a priori prediction would be in regard to the relative differences in groups' sizesbetween treatments. We should expect at least some groups above the equilibrium level in the RE treatment but it is not clearwhether the other treatments will yield larger or smaller groups.

3. Efficiency: If the other predictions are accurate and the RE rules generate higher contributions in groups of intermediate sizes,while the other rules yield groups too large given the level of contributions, then the RE treatment should produce moreefficient outcomes. This is the opposite of the result found in Ahn et al. (2008) where the large low contributing groups in RXand FEE earned more than the smaller high contributing groups in RE. It is due to the differing nature of the public good in thecurrent environment, i.e., the fact that the public good is congestible, that we expect to produce the different result.

4. Congestion: It is clearly expected that congestion should be lower in the RE treatment. This is due to the fact that the groups inthe RE treatment can exclude additional members if the group is becoming congested as well as our expectation that thecontributions as a function of group size may be higher in the RE treatment. In the other two treatments, group members haveno way to keep additional group members from entering even if the group size is already too large given the contribution levelof the members. Due to the added difficulty of subjects leaving a group in the RX treatment, it might be expected that the RXtreatment could end up as the most congested treatment.

5. Stability: Groups may be more stable when entry is restricted compared to the other two conditions if subjects are able togenerate cooperation. The reason is because then subjects would either not wish to move groups or because low contributorswould not be allowed to move into the high contributing group. Exit restriction may also lead to substantial stability due to thedifficulty in leaving a group. Consequently we expect the FEE treatment to be most unstable.

4. Results

In this section, we present the experimental results to examine the impact of the entry and exit rules on the five variables ofinterest. Before we analyze the results in detail, we first present the overall summary statistics to allow a simple comparison acrossthe treatments. Table 2 shows the average scores for the five variables at the session level (called observations 1 to 4) as well as theaverages at the treatment level. To emphasize the differences across sessions due to the institutions, we have calculated thesesummary statistics by removing the first five periods of data from consideration. During these first few periods the groupformation rules have not hadmuch time to influence outcomes so we are concentrating on the portion of the sessions inwhich therules may have had an effect. The data underlying all of the subsequent figures also have the first five periods of data removed butany panel regression results are based on the full data set because the panel methods allow us to deal with this issuewithout losing

Table 2Average group size and per period earnings over the 4 sessions conducted for each treatment

Obs 1 Obs 2 Obs 3 Obs 4 Over All

Free entry/exit (FEE) Contribution 4.87 3.68 3.50 4.06 4.03Contribution % 0.82 0.63 0.70 0.69 0.71Group size 4.00 3.69 2.86 4.21 3.69Earning 63.96 48.82 53.48 46.84 53.27Congestion −0.55 −0.04 −0.61 .18 −0.25Stability 83 / 83 74 / 74 98 / 98 123 / 123 94.5 / 94.5

Restricted entry (RE) Contribution 5.27 4.91 4.15 4.17 4.63Contribution % 0.88 0.89 0.81 0.77 0.84Group size 3.84 3.58 3.48 3.2 3.53Earning 82.46 77.76 58.02 56.09 68.58Congestion −1.06 −0.94 −0.49 −0.88 −0.84Stability 21 / 48 33 / 68 24 / 65 39 / 77 29.25 / 64.5

Restricted Exit (RX) Contribution 4.02 3.95 3.59 4.48 4.01Contribution % 0.76 .63 .64 .74 0.69Group size 3.62 4.14 3.81 3.93 3.88Earning 52.91 56.82 45.32 67.92 55.74Congestion −0.22 0.24 0.24 −0.26 0.00Stability 40 / 81 32 / 57 21 / 81 14 / 43 26.75 / 65.50


data. In the table, Contribution is simply the average number of tokens contributed to the group account. Contribution % is theabsolute contribution divided by group optimal level of contribution given the size of the group in which a contribution is made.For example, contributing 5 tokens in a group of 6 would return 5/9=0.56 because the group optimal level of contribution in agroup of size 6 is 9. The variable Group Size gives the average group size in a treatment calculated by using the groups as the unit ofobservation.15 Earning reports the average earnings in ECUs per period for the individuals. Congestion is a measure indicating theaverage number of subjects in a group above the optimal number the group should have given its current level of averagecontribution. A negative value indicates that given the current contributions, there are too few group members while a positivevalue indicates that there are toomany groupmembers and, thus, the group is congested.16 The Stability row provides two differentmeasures of group stability. They are listed as the number of total successful moves per session and the number of attemptedmoves per session. We present these summary statistics so that the reader can start to form some notion of the structure of thedata, but we would urge the reader use caution in making comparisons of variables across treatments at this level of aggregation.As we will show, the important differences between treatments become clear only when the data is analyzed in a conditionalmanner that deals with some of the complexities in the experimental environment.

4.1. Group size

As explained in the previous section, the equilibrium prediction based on our experimental design regarding group size is 3 or4. Subjects could increase their earnings substantially if they form larger and cooperative groups, but forming larger groupsinvolves the risk of obtaining much smaller earnings if they do not raise their contribution levels correspondingly. Table 2 showsthat the group sizes on average fall between three and four in all three treatments. The RX treatment has the largest average groupsize (3.88), followed by the FEE treatment (3.69) and the RE treatment (3.53). The differences in the means and medians of groupsize between the three treatments are small and mostly insignificant,17 but looking only at the averages hides some potentiallyimportant differences in the distribution of group sizes. To compare the distribution of group sizes across treatments Fig. 1 showsdensity plots of the proportion of groups of each size for each treatment. A visual examination shows that the RE treatmentgenerates a large number of single member groups,18 relatively few two- and three-member groups and a lot of groups with five orsix people. The FEE and RX treatments generate comparatively fewer groups of size one but more of sizes two to four. The RXtreatment, on the other hand, generates a relatively larger number of large groups than the other two treatments. The local peaks inthe FEE and RX treatments at group sizes of three and four correspond to our predictions based upon our benchmark equilibrium

15 Consider the two possible partitions of 12 individuals: (6,6) and (11,1). By averaging over groups we get 6 for both partitions. An alternative way of doing itwould be to use each individual subject as the unit of observation. Doing so would give us an average of 6 for the first case but 10.17 for the second. This alternateapproach, however, has the disadvantage of overweighting the presence of large groups. Both approaches have their drawbacks, and we have chosen to use thegroup as the unit of analysis in this case.16 One should not interpret this measure as a naive prescriptive variable though. A negative value, for example, means the group should grow only if the newmember(s) contribute at the average level of the existing members. An alternative prescription of a negative (positive) value of the congestion measure is that themembers should decrease (increase) their contributions. We intend this though just as a snapshot measure of congestion that allows us to easily compare acrossgroup sizes.17 The only significant difference in a two-sample Wilcoxon rank-sum (Mann-Whitney) test is between the RE and RX treatments if one consider the individual-period as the unit of observation (p=0.01). Using each period or each session as an observation results in statistically insignificant differences in all three pairedcomparisons.18 This is due largely to the fact that in the RE treatment, groups aggregated slower in the first few periods as some subjects who attempted to move into othergroups were denied. This led to some individuals staying in their initial groups longer than in the other treatments.

Fig. 1. Density plots of the distributions of group sizes for each treatment for all periods except 1–5.

Fig. 2. Graphs depicting average contributions plotted against group size and broken out by treatments.


prediction. The RE treatment local peak, in the range of five to six subjects per group, is more in linewith our prediction based uponthe equitable version of the socially efficient group partition. Further, we see very few groups larger than nine in any treatmentwhich our analysis says are generally unsupportable.

4.2. Contributions

Table 2 shows that the average contribution levels are higher than the individual optimal level of three tokens in all threetreatments. Further, it appears that the contribution levels are highest in the RE treatment. As noted, however, it is difficult tomake much of that difference due to the fact that there are factors that influence contributions other than the different entry andexit rules. In particular, if group sizes differ across treatments, then this could be driving any differences in average levels ofcontributions. If subjects have a concern for social efficiency then contributions should rise as group size rises. It is important thento have a clear picture as to whether in the RE treatment, subjects are contributing at a higher level for a given group size orwhether they are just forming larger groups. Fig. 2 shows the average contribution levels across the three treatments according tothe size of groups. What appears to be the case is that contributions in the FEE and RX treatments are approximately the sameacross group sizes but contributions in the RE treatment are on average higher for moderately sized groups of four to six members.To provide a statistical verification of this point, Table 3 presents the results of a random effects panel regression of contributions

Table 3Results of a random effects panel regression of contributions on the main determinants

Coeff p-value

Constant 1.37 b0.001RE 0.37 b0.001RX −0.03 0.733Own t−1 0.43 b0.001Group optimal 0.20 b0.001Period −0.03 b0.001Num obs (groups) 3288(144)R2 (overall) 0.285


on the main determinants of contributions. These determinants include dummy variables for whether the group formation rule isRE or RX, leaving FEE as the reference, along with the individual's own contribution level from the previous period, the period andwhat the group optimal contribution level would be for that person in that period given the size of their current group. We do notinclude group size as it is almost perfectly colinear with the Group Optimal contribution. By including either variable, we aredetermining if subjects increase their contributions as group size increases. We include Group Optimal rather than group size as itis the more theoretically appropriate measure. All of these variables are significant except the dummy variable for the RXtreatment. The coefficient on the RE dummy variable is positive and significant indicating that holding all of these other thingsconstant, individuals in the RE treatment do on average contribute .37 tokens more than in the other two treatments. We also seethe standard result found in most public good games which is that contributions do decay over time, but the effect is quite small ason average contributions drop by 0.03 tokens per round or .75 tokens across the entire experiment.

4.3. Earnings and efficiency

In our experiment, subjects' earnings depend on the size of the groups they form and the level of contributions in those groups.We have already seen that groups of sizes four to six are more likely to form in the RE treatment and the subjects in the REtreatment contribute more in those moderately sized groups, than those in the other two treatments. This leads to significantconsequences in regards to earnings. As Table 2 shows, the average per period earning is the highest in the RE treatment (68.58)followed by the RX treatment (55.74) and the FEE treatment (53.27). Recall that in the stage gameNash equilibrium predictions, thesubjects' earning is 57.78 ECUs if they form group of size three or 55.49 ECUs if they form groups of size four. The average earningsin the RX and FEE treatments are very close to the equilibrium prediction while that in the RE treatment is somewhat greater.

Fig. 3 shows the average per period per subject earning conditional on the size of the group to which a subject belongs. Thegeneral indication is that subjects in groups of size 4–6 earn more in the RE treatment than the other two and this likely accountsfor the average earnings difference. In a group of size six, the individuals in the RE treatment earned close to 90 ECUs per periodwhile those in the other two treatments earned around 60 ECUs. This is certainly a substantial difference. Another interesting pointto observe from the figure is a very clear indication why large groups were not very common. The average earning per groupmember for groups larger than nine was negative and significantly so. The average earnings for different group sizes are importantin understanding how the treatments effect the distribution of earnings as shown in Fig. 4. While the overall average earnings maynot demonstrate much of a difference, the key point is that the Restricted Entry treatment allows for a greater maximum earnings

Fig. 3. Average earnings of group members by group size broken out by treatment.

Fig. 4. Histograms of average per period earnings by individuals in each treatment.


as can be clearly seen from both the rightward shift of the RE distribution and the fatter right tail, as compared to either FEE or RX.Kolmogorov–Smirnov tests of the equality of the distributions reveal that there is no significant difference between the FEE and RXdistributions (p-value=0.852) but there is a significant difference between FEE and RE as well as RE and RX (p-value=0.001 and0.018). We can also demonstrate this by examining the average of the earnings from the highest earning groups in each period bytreatment. In the FEE and RX treatments, the highest earning groups averaged 69.27 and 67.80 ECUs respectively. In the REtreatment, the average earnings of the highest earning groups was 86.60 ECUs per period. Notice that in the FEE and RX treatments,the best groups are not doing much better than the average groups while in the RE the best group is doing significantly better thanaverage. The average earnings of the lowest earning group was not different across treatments as it was 35.57, 30.91 and 35.51 forthe FEE, RE and RX treatments. So the indication is that the RE treatment allows for the existence and survival of high contributing/earning groups while the FEE and RX treatments do not. This improves the earnings for those groups but the lowest earnings arenot too different and this pulls the overall average earnings down to look more similar across treatments. Further evidence on theearnings differences across treatments can be found in Table 4, which shows two random effects panel regressions of per periodearnings of individuals on their main determinants. The first specification, the basemodel, regresses individual earnings per periodon treatment dummies, own contribution and group size split into two different variables. Fig. 3 makes it quite clear that the effectof group size on earnings changes after a group size of six. Consequently, we have used one variable that represents the size of thegroup if the group size is less than or equal to six (coded 0 otherwise) and a second variable that codes the group size if it is greaterthan six (coded 0 otherwise). This second variable for group size if the size is greater than six requires using another dummyvariable, labeled Big Group, that is set to 1 if the group is bigger than six. This set of variables allows us to deal with the slope changeand its corresponding intercept shift in a convenient manner.19 We present this specification primarily to demonstrate the impactof treatment on earnings. The second specification examines the interactions between treatments and contributions and groupsize. The results from the base model show that on average subjects in the RE sessions earned 12.06 ECUsmore per period, holdingall else equal, and this comports well with the average difference in earnings shown in Table 2. Over 25 periods of the experiment,that adds up to substantially higher earnings.20 By examining the fully interacted model we can break down the impact ofgroup size and contributions on earnings more carefully. In the FEE and RX treatments, marginal contributions decrease earningsby about 7 ECUs on average per additional token contributed. In the RE treatment, the effective coefficient on contributionsis −7.29+2.72=−4.57 leading to a more moderate decrease on earnings from an additional contributed token. Perhaps the moreimportant point is the impact of group size on earnings. In all treatments increasing the group size increases earnings so long asthe group has less than 6members. The rate of increase in the RE treatment, however, is more than twice that of the other treatments(i.e., the effective coefficient for RE is 5.43+6.73=12.16 which is 2.24 times the base coefficient of 5.43). For groups of size greater than

19 We have also conducted these regressions assuming the other possible breakpoint of n=7 instead of n=6 to check the robustness of our choice. The fit levelsare about the same but are slightly better on both regressions using the breakpoint of n=6. The interpretation of the coefficients remains the same.20 With the exchange rate of 1 ECU=$0.01 this indicates that subjects in the RE treatment earned on average 25⁎$0.1231=$3. 077 5 more per subject.

Table 4Random effects regressions of earnings per period on its main determinants

Base model Full interactions

Coeff p-value Coeff p-value

Constant 57.30 b0.001 73.31 b0.001RE 12.06 b0.001 −23.38 b0.001RX 1.48 0.554 −5.09 0.299Own contrib −5.78 b0.001 −7.29 b0.001Own⁎RE – – 2.72 b0.001Own⁎RX – – 0.83 0.265

Group size (n≤6) 8.21 b0.001 5.43 b0.001Group⁎RE – – 6.73 b0.001Group⁎RX – – 0.76 0.415

Group size (nN6) 125.89 b0.001 117.55 b0.001Group size (nN6) −14.31 b0.001 −13.94 b0.001Group⁎RE – – 1.07 0.100Group⁎RX – – 0.53 0.412

Period −0.46 b0.001 −0.50 b0.001Num obs (groups) 3432(144) 3432(144)R2 (overall) 0.21 0.24


6, additional group members cause a decrease in earnings, though the decrease might be slightly less severe in the RE treatment.Earnings are slightly decreasing as the experiment progresses as indicated by the negative coefficient on period.

It is possible to run an alternative regression to those presented in Table 4 to address the question as to whether those subjectswho contributed more on average had higher earnings than those who contributed less. This can be done by running a cross-section regression that uses each subject as an observation and regresses the total earnings of each subject on contributions andthe interaction variables between contribution and treatments. We used the FEE as the baseline and included two dummyvariables, one for the RE and one for the RX treatment. The base coefficient on average contributions, which is the total effect forthe FEE treatment, is −99 and significantly different from 0 (p=0.003). The coefficient on the interaction term betweencontribution and RE treatment is 136 and significant (p=0.005) while the coefficient on the interaction term between contributionand RX is 36 and not significant (p=0.48). The results indicate that there is an institutional difference between RE and FEE inthat higher contributions at a minimum hurt a subject less in the RE. To determine the overall effect, however, we must look atthe combined effect of the base coefficient with the coefficient on the interaction terms. The combined coefficient is 37 for theRE (−99+136) and not significantly different from 0 (p=0.29). The combined coefficient for the RX is −63 (−99+36) and at theborder of significance (p=0.105). A literal interpretation would be that one more token contribution per period results in 99 ECUsless total earnings in FEE, 63 ECUs less in RX, and 37 ECUs more in RE. Because the test result is not significant for the combinedeffect of the RE treatment, we cannot conclude that contribution increases earnings in the RE treatment. Instead, we can say with ahigh level of certainty that contribution did not decrease earnings in the RE. On the other hand, the results show that contributionaffected earnings negatively in the FEE and RX treatments.21

These results seem to allow an institutional understanding of the evolution of cooperation. In most public goods and othersocial dilemma experiments, free-riders earn larger payoffs than the cooperators. Evolutionary theories suggest that the typeswho are materially more successful are more likely to increase their proportions in a population. (Axelrod, 1981, 1986; Gintis,2004.) The fact that there are significant proportions of cooperators in the naturally occurring situations as well as in the labsuggests that humans have developed institutions that favors cooperators, or at least support a mixed population of cooperatorsand free-riders in an evolutionary equilibrium. (Boyd and Richerson, 1985; Henrich, 2004a,b; Orbell et al., 2004; Sethi, 1996). Ourresults indicate that entry restriction could be one of the institutions that support a mixed population, if not unambiguously favorcooperators.

4.4. Congestion

Amismatch between the group size and contribution level of a group results in inefficiencies when a public good is congestible.A group is congested (or crowded) if it has more members than what is optimal given the level of contribution by the individualsin that group. Thus, a more cooperative group can be large yet not congested while a smaller group could be congested ifindividuals in that group contribute too little. This takes into account both the technological and the behavioral aspects ofcongestion discussed in the introduction. To see the level of congestion we constructed a congestion measure by calculating theoptimal number of members each group should have had given the average level of contribution in the group in that period. Thecongestion measure is the difference between the actual number in the group and the number the group “should” have had giventhe average level of contributions bymembers of the group. Notice that this measure could be positive or negative. A positive value

21 The p-values are for two-tailed tests of the effects of linear combination of coefficients, using the LINCOM command in STATA. Total number of observationsin the regression is 144 because each subject is used as an observation. Full regression results are available upon request.

Fig. 5. The average of the congestion index for each group size broken out by treatment.


means the group is congested and has too many members given their level of contributions. A negative value means that thegroup is undersized and could support more members if they came in and contributed at the average of the existing groupmembers. Fig. 5 shows the extent to which groups of each size are congested in the three treatments. Recall that the individuallyoptimal contribution level is 3 and the corresponding optimal group size is also three. Therefore, if the subjects followed thisdominant action the congestion measure would simply be n−3. The actual congestion measures are substantially lower than n−3.As expected, larger groups suffer more from congestion as the congestion becomes positive beginning around a group size of 5.Groups in the RE treatment generally maintain this property of being less congested than in the two other treatments until thegroup size reaches 7 members.

The similarity across treatments shown in Fig. 5 might make the average congestion numbers shown in Table 2 appear odd. InTable 2 it is shown that the FEE and RX treatments have an average congestion index close to 0 (−0.25 and 0 respectively) while theRE average is close to −1 (−0.84 to be precise). The prime reason for these differences in average congestion levels is the differencesin group sizes. The average of the congestion index is lower for the RE treatment in part because the RE groups are slightly lesscongested but also because the number of large and highly congested groups is relatively small in that treatment. Fig. 6 shows thismore clearly. The figure shows the average congestion level in each treatment by period. The RX line is clearly above the RE line asis the FEE for most of the experiment. The reason for this is that the subjects in the RE treatment kept the large and very congestedgroups from forming early on in the experiment while those in the FEE and RX treatments failed to do so. The fact that the averagecongestion in the FEE and RX treatments is approximately 0 might be taken as an indication that subjects in those sessions aredoing a good job of solving the group formation problem, but this is not the case. Relatively few of the groups in those treatments

Fig. 6. Average congestion level by treatment over time.

Fig. 7. Attempted and successful group changes.


form with a zero congestion index but rather there are a number of large and highly congested groups and then there are somesmaller groups that are too small. On average these cancel out to yield the average of around 0. Again, the reason the RE treatmentis delivering different results is due to the ability of subjects in those sessions to stop the large and overly congested groups fromforming while at the same time contributing at the level close to group optimal in groups of moderately large sizes of four to six.

4.5. Group stability

Another important aspect of the group formation process is the stability of the groups. In our experiment, there is no explicitcost of changing groups so any stability we see is simply due to either the cohesiveness of the group or to the limitations deliveredby the majority rule voting systems. There are a number of possible measures of group stability of which we utilize two: the totalnumber of attempted moves between groups per period and the total number of actual moves per period. In the FEE treatment,these two measures are obviously identical. In the RE and RX cases, they might not be if other subjects are voting to deny entry orexit to the applicants. Table 2 gave an overview of the total number of attempted and successful moves per session. The resultswere that there were many more attempts and successes in the FEE compared to the other two treatments while the other twotreatments show roughly comparable numbers of attempts and successes. Fig. 7 gives another more detailed look at this byshowing the number of attempted vs. successful moves per period broken out by the three treatments. The data underlying thatfigure consider only attempted/successful moves in the RE and RX treatments when a vote occurs. There are many other attemptsnot reported in that figure (all successful of course) for which no vote was required because the subjects were applying to enter agroupwith nomembers in the RE treatment or to exit a groupwith no other members in RX. Note that since therewere 12 subjectsper session and 4 sessions per treatment there was a potential maximum of moves per period if every subject in all sessions of atreatment tried to move between groups. This figure shows the relationship between attempted and successful moves over timesquite clearly. Note that the figure also shows another interesting point that in both the RE and RX treatments applicants are beingdenied even in the last periods of the experiment. Tables 5 and 6 show additional details regarding the success of applications toenter or exit a group. The first table contains some general summary statistics on the number of yes vs. no votes as well as thesuccess rates of applications in both groups while the latter contains the outcome of fixed effects logit regressions regarding thedeterminants of a successful application to enter or exit a group. In general, the variables that should matter do and do so inpredictable ways. In the RE treatment an applicant's probability of acceptance into a group is rising in their own recentcontributions, but decreasing in the average level of contributions of the group and decreasing in the congestion of the group.Positive values for the congestionmeasure mean that there aremore members than are sustainable so the higher is the congestion

Table 5Summaries of votes and success rates of applications in RE and RX treatments

Votes Outcomes of applications

RE RX RE RX

Yes 431 46.4% 267 39.1% Success 119 43.9% 62 25.7%No 497 53.6% 415 60.9% Failure 152 56.1% 179 74.3%Total 928 100% 682 100% Total 271 100% 262 100%

Table 6Fixed effects logit regressions regarding the determinants of a successful application to enter or exit a group

RE RX

Coeff p-value Coeff p-value

Applicant contribution t−1 0.66 0.025 −1.14 0.002Applicant contribution t−2 0.22 0.543 −0.19 0.506Num of voters in relevant group 1.15 0.012 1.47 b0.001Avg contrib of relevant group t−1 −2.74 0.002 −1.16 0.107Avg contrib of relevant group t−2 −0.79 0.031 1.99 0.015Congestion t−1 −1.29 0.023 −2.06 0.001Period −0.28 0.005 0.02 0.746Num obs (groups) 138 (26) 136 (23)ln L −15.86 −15.68


index, the less desirable it is to admit any newmembers. For the RX treatment, a candidate's probability of being allowed to leave agroup is decreasing in the candidates contributions, increasing in the number of groupmembers and the (past) contributions of thegroup. The potentially odd result is that in the RX treatment the more congested is the group, the less likely is exit to be approved.The expected result would be the opposite. The most likely cause of this problem is the fact that these regressions suffer from anendogeneity problem in regard to the congestion measure for which there is no clear remedy. The issue is that there are very fewapplications to get into congested groups in the RE treatment and very few applications to leave uncongested groups in the RXtreatment. Since we do not observe the counterfactual cases of what would the outcomes be were subjects to try to enter/exit theother groups there is a bias to the interpretation of these coefficients. Because the congestion measure seems to correlate moststrongly with the decision to enter/exit a group the endogeneity problemmainly loads on this coefficient though it could certainlybe said to impact the estimates regarding the past contributions of the relevant group. We therefore provide this regression as anindicator of how things are likely to matter acknowledging that inference from it may be problematic.

5. Conclusion

Many real word collective action problems involve congestible public goods. The individuals who can benefit from thesegoods face a challenge in forming a group of proper size and providing a desirable level of the public good. Because enforceablecontracts are not always feasible, individuals, even in a group inwhich they share the benefits exclusively of the outsiders, still facethe problem of voluntarily providing the good among themselves. When the public good is congestible and the contributionvoluntary, the optimal group size is not exogenously fixed, but a function of the extent to which the group members cooperatewith each other to provide the public good. While all groups will eventually suffer the problem of congestion as the group getslarger and larger, a cooperative group will suffer congestion less and later while being able to enjoy the positive externalities of apublic good.

In this paper we utilize a series of experiments to study how the boundary rules, the rules of entry and exit, effect the extent towhich individuals form groups of optimal sizes and cooperate with each other to provide a congestible public good. We find that,when entry requires approval of the majority of groupmembers, the applicants tend to raise their contributions and the voters usethe boundary rules to carefully select new members. The average contribution and the earnings were much higher when entry isrestricted and the groups tend to maintain optimal sizes given the level of contribution. Both free entry/exit and restricted exitrules resulted in significantly lower contribution and earnings and higher levels of congestion compared to restricted entry rule.

The results from the Restricted Entry mechanism are important for a variety of reasons. The first element of that is to note thatwe have shown that restricted entry leads to both increased contributions and increased efficiency in the case of congestible publicgoods. This is not what was found for non-congestible public goods. As prior research in Ahn et al. (2008) showed in the case ofpure public goods, restricted entry enhanced cooperation at the cost of efficiency due to smaller group sizes limiting the scope ofthe positive externality. Further, the avenue through which these higher efficiencies are achieved is also interesting as subjectswho may initially be low contributors are taught to become reliable high contributors by having their applications denied untilthey adjust their behavior. One can think of this as a “training” effect or a situation in which a candidate is required to provethemselves to obtain entry. The fact that this requirement alone achieves the efficiency increase without an expulsion threatshould a candidate backslide after entry is good news for the many organizations for which entry restriction is easier thanexpulsion. There are a broad range of organizations such as Co-op apartments, neighborhood organizations, country clubs orfraternal organizations, as well as many employers such as Universities which find themselves in the situation of typically havingquite strong power to block applicants from joining them but very weak or difficult to exercise power to eject low performingmembers. Our results suggest that organizations with such restrictions may still be able to generate high social efficiency througheffective use of their entry restriction powers.

While many experimental studies on collective action try to explain the higher-than-expected levels of cooperation amongsubjects we also show that not only does the restricted entry result in higher average earnings for everyone, but it changes thedynamics of a social dilemma such that the contributors are no longer disadvantaged in terms of their earnings compared to thefree-riders. This again is the opposite of the result found in prior research with pure public goods. The fact that restricted entryraises earnings for everyone might be the reasonwhy many groups have restrictions on entry especially when a congestible public


good is involved. The fact that contributors can do better than free-riders under restricted entry suggests an institutionalmechanism by which cooperators in social dilemmas might survive and perhaps even thrive.

An obvious area for further inquiry is the question of why these mechanisms support different behavior. The key issue is thedegree to which the preferences for participants to engage in other regarding behavior interact with the mechanisms. Onepossibility is that themechanisms differ in theway that they provide frames for participants to engage in other regarding behaviors(whether altruism, reciprocity, fairness, etc.) and so they consequently trigger such preferences in a different manner. A secondpossibility is that the mechanisms differ in how they teach individuals to engage in behavior consistent with other regardingpreferences. This could take the form of teaching subjects how to express such preferences that they already possess or perhapseven altering the nature of the nature of whatever other regarding preferences the individual may possess. We see some evidencefor both possibilities in our data. For example, being denied exit from a group may frame a condition leading to spiteful behaviorfrom those denied exit. On the other hand, being denied entrymay serve to teach pro-social behavior to potential entrants which isallowing some subjects to express their pro-social preferences in away not possible in the othermechanismswhile simultaneouslyperhaps teaching others to hold similar preferences. The subtle nature of these issues are quite complex and should be the subjectof further study.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.jpubeco.2008.06.007.

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Coming and going: Experiments on endogenous group sizes for excludable public goods

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