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Journal of Mathematical Psychology 42, 167�190 (1998)

Identifiability of Individual Contributions in aThreshold Public Goods Experiment*

Rachel Croson

The Wharton School of the University of Pennsylvania

and

Melanie Marks

The School of Business and Economics at Longwood College

This paper experimentally examines how information affects behavior in athreshold public goods game. Three treatments investigate how subjects reactto varying amounts of information about the contribution behavior of othergroup members. Results suggest that revealing anonymous information aboutothers' contributions leads to a significant decrease in contributions and anincrease in the variance of contributions. In contrast, when individual contri-butions are identified by subject number, average contributions increase andthe variance of contributions decreases significantly. � 1998 Academic Press

I. INTRODUCTION

In economics, pure public goods are defined as goods that are ``nonrival'' and``nonexcludable.'' Nonrivalry can be thought of as the jointly consumable nature ofa good, where consumption by one individual does not diminish the consumptionpossibilities of other individuals.1 Nonexcludability implies that once the public goodis provided, it is impossible (or prohibitively costly) to specifically prevent someindividuals from enjoying its benefits.2

Article No. MP971203

167 0022-2496�98 �25.00Copyright � 1998 by Academic Press

All rights of reproduction in any form reserved.

Correspondence and reprint requests should be sent to Rachel Croson, 1322 Steinberg Hall-DietrichHall, Department of OPIM, The Wharton School of the University of Pennsylvania, Philadelphia, PA19104-6366.

* The authors thank the Center on Philanthropy and the Lilly Foundation for funding this research,Lorin Hitt for econometrics guidance, and students at Longwood College for help in running theexperiment. Thanks also to participants in the Economic Science Association 1996 Fall meetings andtwo anonymous referees for their helpful comments. An application of this experiment to nonprofit sectorfundraising is addressed in a practitioner-oriented paper, Marks and Schansberg (1997).

1 This contrasts with the typical notion of a private good such as a hamburger, where consumptionby a first individual eliminates the possibility of consumption by others.

2 A standard example of a nonexcludable public good includes environmental cleanup since it is closeto impossible to exclude noncontributors from enjoying the benefits of clean air and water.

Recent fiscal stress has generated an interest among public officials in alternativeinstitutional arrangements for the delivery of public goods. Within the current politicalclimate, options which circumvent the ``T-word,'' taxes, are particularly attractive.Voluntary contribution institutions offer this feature, and thus represent a potentiallyviable option to supplement the tax-financed, majority voting collective choicemechanism for providing public goods. Perhaps fueled by the public policy interest innontax, decentralized methods of finance, the Public Finance literature in economicsreflects a rekindled theoretical interest in voluntary contribution models for providingpublic goods.

A member of the class of voluntary contributions mechanisms, the provision pointmechanism (PPM), is studied in this paper. In the PPM, a specific project andcost requirement are predetermined. Individuals impacted by the project submitbids stating their commitment to covering the project's costs. If the contributionsmeet or exceed the threshold, the public good is provided; otherwise the project is notprovided.3

One real world example of the PPM is the innovative Green Choice programproposed by the Niagara Mohawk Power Company of New York. The program is avoluntary partnership between Niagara Mohawk and local citizens who are concernedabout reducing air pollution in their communities. Consumers can opt tojoin the Green Choice program by committing to contribute an amount each monthabove and beyond their electric bills. Funds will be collected for a twelve-month periodand, if sufficient, will then be used to build an environmentally friendly energy projectand to plant trees.4 Because the provision of the energy project is ``all or nothing,'' theGreen Choice program is essentially a Provision Point process; if subscriptions fallshort of the necessary level, the program will be abandoned and the funds will bereturned to the contributors.5

The PPM has a set of efficient Nash equilibria (which will be characterized moreformally in Section III below). Roughly, an efficient Nash equilibria is comprised ofany vector of individual contributions that (1) exactly sums to the contributionthreshold and (2) does not involve any individual contributing an amount largerthan his benefits from the public good. Since there are generally many combinationsof individual contributions that could meet these conditions, the set of efficientNash equilibria can be very large and within this set, no equilibrium is better forall players than any other (the equilibria are not pareto-rankable).

While the PPM has the ability to privately provide public goods. The decisiongroup must agree not only to provide the project, but participants must agree onhow to divide the cost of the project. In the PPM, the task is one of coordination

168 CROSON AND MARKS

3 The PPM can include features to handle the refund of collected funds when the threshold is notreached, or the rebate of funds collected in excess of the threshold. The first issue is explored experimen-tally and theoretically in Isaac, Schmidtz and Walker (1989) and the second in Marks and Croson(forthcoming).

4 For a complete discussion of the Green Choice program; see Schultze (1995).5 As illustrated by the Green Choice program, the PPM is especially applicable when projects involve

a degree of technological lumpiness (for example, one half of a bridge or power plant is not useful), orin situations where decision makers are committed, for some reason, to a project of a particular scale.In these situations, a predetermined project size and an ``all or nothing'' provision is appropriate.

on one of the efficient equilibrium allocations. Do groups in the PPM successfullyselect and coordinate on one of the many sets of successful contributions? Or doesthe group's inability to coordinate result in nonprovision? Furthermore, are thereinstitutional features of the PPM that can be exploited to make provision morelikely? These questions suggest a role for experimentation. The experiment detailedin this paper makes two main contributions to our knowledge of the PPM. First,it provides additional evidence of the usefulness of the PPM for public goodsprovision in certain settings. Second, we investigate a potentially importantinstitutional detail of the PPM: the extent of information about individual contri-bution behavior provided to the participants. This experiment involves three treat-ments where the degree of knowledge about individual contribution behavior ismanipulated.

In the Group Only treatment, participants are informed only of aggregate statistics(total group contributions) and are not informed about individual behavior in anyway. With this limited information, a participant can determine what percentage of theaggregate contributions he provided, but has no information about the behavior of theremaining group members. The Individual��Anonymous treatment is designed to isolatethe effect of revealing information without identification of subjects. The informationabout individual contributions released in this treatment can communicate equitable orinequitable behavior in contributing. However, that behavior cannot be identified withany particular individual. In the Individual��Identifiable treatment, contributionsare revealed according to a commonly known subject number. By revealing contributionssuch that an individual's behavior is identifiable and can be followed throughoutthe experiment, the ability to form and maintain a reputation for ``doing one'sshare'' toward providing the public good is created. In this treatment, subjects haveboth information about individual contributions and the ability to identify whocontributed what.

This investigation sheds light on two questions. First, what is the effect of infor-mation about (equitable or inequitable) cost sharing in the public goods processes?A comparison between the Group Only and Individual��Anonymous treatments willbe used to answer this question (Section V). Second, does the addition of identifiabilityencourage individual contributions and therefore act to limit individual free-riding?A comparison between the Individual��Anonymous and Individual��Identifiabletreatments will be used to answer this question (Section VI).

II. PREVIOUS LITERATURE

Previous literature in this area comes from a number of different sources and ismotivated by many different social and economic problems. In this section we charac-terize a number of different types of public goods problems and mention previousexperiments which have manipulated information in each setting.

Continuous public goods have the characteristic that as more is contributed, theamount of public good available increases. In contrast, discrete public goods are all-or-nothing. If a threshold of contributions is reached, a fixed amount of public good

169IDENTIFYING INDIVIDUAL CONTRIBUTIONS

FIG. 1. A categorization of previous literature.

is provided; otherwise it is not.6 Continuous contributions have the characteristicthat each decision maker can choose his level of contribution from a continuum. Incontrast, with discrete contributions an individual either contributes or he does not;the decision is 0 or 1.

Although there are hybrids, much of the previous research can be classified inthis way. Figure 1 presents the assignment of various literatures in this area to thecategories outlined.

A. Continuous Public Goods�Discrete Contributions

Dawes (1980) reviews a number of experiments in this area. Three studies comparerates of cooperation under public disclosure of individual choices versus underanonymity. In a classic paper, Bixenstine, Levitt, and Wilson (1966), subjects playa six-person social dilemma game repeatedly. The authors find significantly higherlevels of cooperation when players are identified than when they make decisionsanonymously, after (but not before) a conversational break. A similar result is foundby Jerdee and Rosen (1974). In a slightly more recent article, Fox and Guyer (1978)have subjects play 30 repetitions of a four-player social dilemma game. The authorsreport significantly more cooperation in groups where decisions are public than ingroups where decisions are anonymous, independent of the existence of discussion.

B. Continuous Public Goods�Continuous Contributions

1. Volutionary contribution mechanism. The VCM is a continuous version of theprisoner's dilemma game. Reviews of VCM experiments can be found in Ledyard(1995) and Davis and Holt (1993, Chap. 6). The most relevant of the VCM experimentsto this study compare allocation levels under different informational conditions. Sell


6 One critical distinction is that the equilibrium outcomes of these games are very different. In continuouspublic goods games, there is typically a dominant strategy equilibrium of full free riding (contributingzero, or always defecting). In contrast, in discrete public goods games, there are multiple equilibria.Some are inefficient which involve free riding and underprovision of the public good. But some areefficient, which involve exact provision of the public good. Because of these different equilibria, a directcomparison of contribution levels across types of experiments is misleading. When a subject contributein a continuous public goods experiment, he is not playing his part of an equilibrium; in a discrete publicgoods experiment he may be.

and Wilson (1991) use four-person groups in a 10-round VCM. Over all 10 rounds,no significant difference in contribution level is found between groups who are toldindividual contributions and those who are told only group contributions. However,in the last five rounds, contributions are significantly higher under individual informa-tion than under total information. Similarly, Weimann (1994) compares contribu-tions of subjects in playing a five-player, 10-round VCM with different levels ofinformation. Using all the periods, no significant difference is found between themean behavior of subjects. In an attempt to reconcile these two results, Croson(forthcoming) examines four-person groups playing two 10-round VCMs. In thegroup treatment, subjects are told the total contributions of the rest of their group.In the individual treatment, subjects are told each individual's contribution. Resultssuggest that while mean contributions are not affected by this additional informa-tion, the variance of contributions is significantly higher in the latter treatment thanthe former. This increased variance could explain the differing results of the twoprevious studies.

2. Social loafing. In social loafing experiments, subjects are asked to producesome output, either individually or in a group, but are typically not compensatedas a function of their effort. The most relevant social loafing experiments to this studydemonstrate that when individual outputs are identifiable and outputs or effort levelscan be compared, the loafing effect is eliminated. Williams, Harkins, and Latane�(1981), show that subjects produce less noise when shouting together than when shoutingalone; however, when subjects in another treatment are hooked up with individualmicrophones but shout together, they produce as much noise as subjects who shoutalone. In a follow-up study, Harkins and Jackson (1985) ask subjects to think ofas many uses for a particular object as they can. As before, the group of subjectswhose outputs are identifiable think of as many uses for the object as the group whothink alone, while the set of subjects whose ideas are pooled, think of significantly fewer.

C. Discrete Public Goods�Discrete Contributions

Van de Kragt et al., (1983) first investigate provision in a discrete public goodgame, where individual contributions are all-or-nothing. In their canonical experiment,subjects are given a voucher worth 85 and organized into groups of seven. If threesubjects out of the seven (in another treatment, five subjects out of the seven)choose to contribute their voucher to the group, every group member receives abonus of 810. Later studies include Dawes et al. (1986) and Rapoport and Eshed-Levy(1989) who examine the impact of greed versus fear in these games. We know ofno previous experiment in this area which addresses the impact of individual oridentifiable information on play.

D. Discrete Public Goods�Continuous Contributions

Our paper investigates provision in a discrete public good game where individualcontributions are continuous. Previous papers with this same structure are Isaac,Schmidtz and Walker (1989) and Bagnoli and McKee (1991).7


7 Although the very first public goods experiments also involved a series of provision points (Marwell6 Ames, 1979, 1980, 1981).

Isaac, Schmidtz, and Walker (1989) modify the Isaac, Walker, and Thomas(1984) VCM framework by adding a provision point. Subjects are organized intogroups of four and the game is run both with and without returning contributionswhen the provision point is not met. The public good is provided in 250 of therounds (pooled over levels of provision points), although subjects are given infor-mation only about group decisions.

Bagnoli and McKee (1991) run a somewhat different version of the PPM game.In their experiments, the public good is provided in 86.70 of the rounds (pooledover homogeneous and heterogeneous valuations and endowments).8 In their experi-ment as in Isaac, Schmidtz, and Walker (1989), no information about individualcontributions is revealed.9 We know of no previous experiment in this area whichaddresses the impact of individual or identifiable information on play.

E. Summary

Although there has been active research in the area of public goods provision andsocial dilemmas, very little research has been done on the effect of individual and�oridentifiable information (and none in the area of discrete public goods). What wedo know comes from the continuous public goods literature. In social dilemma games(continuous public goods, discrete contributions) individual and identifiable informa-tion leads to higher cooperation rates than individual but anonymous information,in the presence of communication (Bixenstine, Levitt, 6 Wilson, 1966; Jerdee 6

Rosen, 1974) and even in its absence (Fox 6 Guyer, 1978). In the VCM (continuouspublic good, continuous contribution), individual but anonymous information hasno effect on subjects' average contributions (Sell 6 Wilson, 1991; Weiman, 1994;Croson, forthcoming) but significantly increases the variance of contributions (Croson,forthcoming). In social loafing experiments, identifiability significantly increases effortover individual anonymous information (Williams, Harkins, 6 Latane� , 1981; Harkins6 Jackson, 1985). To our knowledge, this is the first paper which examines iden-tifiability in the context of discrete public goods. However, a number of our hypotheses,described in Section IV, are drawn from this previous literature.


8 Although in a replication of Bagnoli and McKee, Mysker, Olson, and Williams (1997) foundsignificantly lower contributions when they added a record sheet to the procedures.

9 In more recent work, Rapoport and Suleiman (1993) investigate a threshold public good withcontinuous contributions and heterogeneous valuation for the public good in a repeated setting. In theirstudy, subjects are given no feedback between rounds, but are asked to make each decisions independentlybefore discovering the outcome of previous decisions. Croson and Marks (1996) examine the effects ofassignments (suggestions) and learning in these settings where information is group only. Marks and Croson(forthcoming) experimentally investigate the effect of rebates (how overcontributions are returned) alsounder group only information, while Marks and Croson (forthcoming) looks at the effect of information onheterogeneous valuations for the public good. Rondeau, Schultze, and Poe (1996) look at a one-shot settingswith incomplete information about the number of players, while Cadsby and Maynes (forthcoming) examinesubject pool effects.

III. THE GAME AND ITS EQUILIBRIA

A. The Model

Imagine that N individuals are concerned with providing a public good to theircommunity. Each individual has an endowment, Ei , consisting of a level of theprivate good, X. Individuals can choose to contribute any part of the private goodendowment towards provision of a public good, G. The public good is provided ina threshold fashion with a cost of K; if at least K units of the private good arecontributed, then the public good G is provided in quantity one, otherwise it is notprovided at all. Contributions in excess of K are not returned, although if fewerthan K units of the private good are contributed (and the provision-point is notreached) unused contributions are returned to their contributors.10

Each individual, i, has a utility function Ui (G, Xi). By specifying the utilityfunction in this way, it is assumed that individuals do not receive utility directlyfrom making a contribution themselves (i.e., there are no warm feelings from beingcharitable as in Andreoni, 1990). Rather, individuals care only about the total (publicand private) provision level. The utility function is further assumed to be of the linearform

Ui (G, X i)=Xi+viG,

where vi is individual i 's value for the public good, G.In the provision point mechanism, each individual is asked to anonymously submit

a message stating his or her contribution (consisting of some level of private good), _i ,towards the provision of the public good. In this case, _i=Ei&Xi , where Xi is theamount of the private good individual i chooses to retain for consumption. Each playerfirst chooses _i , then the public good is produced only if 7i_i�K. Otherwise contribu-tions _i are returned to the players and the public good is not provided. Individualsfinally receive their utility from the public good (if provided) and from privateconsumption.

B. Equilibria of the Game

With any parameter values consistent with the public goods problem underdiscussion,11 and with common knowledge of K and each vi , in this game thereexists a continuum of efficient Nash equilibria in which the players collectively contri-bute exactly enough to achieve the provision point and provide the public good. These


10 The first condition corresponds to the No Rebate treatment of Marks and Croson (forthcoming). Thesecond corresponds to the Refund treatment of Isaac, Schmidtz, and Walker (1989) and the Money-BackGuarantee of Dawes et al. (1986). The rebate and refund condition was chosen for their simplicity and com-parability to previous experiments.

11 For example, for there to be a public good, the value to the entire community of the provision of thegood must be larger than the cost. Thus we need 7vi�K. Similarly, each individual must haveenough of an endowment so as to make his contribution feasible. A sufficient (but not necessary) conditionis Ei�vi \i.

equilibria have the same group contribution level; they are distinct from one anotheronly in the cost-sharing rule used to divide the contributions toward the public goodamong the participants. These equilibria consist of all possible vectors of allocationssuch that:

(1) 7i _i=K (the public good is exactly provided)

(2) _i�vi \i (no individual contributes more to the public good than hisindividual value).

In addition, there is a continuum of inefficient equilibria of this stage game. Aninefficient equilibrium outcome is such that the provision point is not met, the publicgood is not provided, and no single participant can rationally increase his contributionto reach the threshold.

C. The Experimental Game

1. Experimental design and parameters. The experimental game which implementsthis model involves five players, each of whom receives 55 tokens (Ei) which they canallocate to a GROUP or to a PRIVATE account. The PRIVATE account modelsindividual and personal consumption of resources. Each token allocated to thePRIVATE account earns 1/ for its contributor. The GROUP account models volun-tary contribution to the public good. If at least 125 (K) tokens are allocated tothe GROUP account each player receives an additional bonus payment (vi=50/)regardless of their allocation decision.12 Tokens above 125 contributed to the GROUPaccount are not returned. If fewer than 125 are allocated to the GROUP account, alltokens submitted to the GROUP account are returned to their contributors and areretroactively invested in their contributor's PRIVATE account at 1/ per token. Allof the parameter values are were publically announced and thus commonly knownto all participants.

Each group of five subjects plays the game 25 times with the same partners. Oncean individual commits to a contribution, he is not given the opportunity to reevaluateand resubmit a different contribution before the outcome of the round is announced.13

Notice that the players in this game are all symmetric. Later we will refer to theequality of contributions (and thus payoffs when the public good is provided) asbeing an equitable outcome. Clearly, this assumption relies on the symmetry of thegame.

2. Equilibria. As mentioned above, the set of efficient equilibria of the stagegame involve each group of five subjects collectively allocating exactly 125 tokensto the GROUP account. Additionally, this stage game has a unique symmetric efficient


12 Thus a subject's profit is maximized when the provision point is reached by his group, but hecontributes nothing toward that effort. He then earns the 50/ bonus payment plus 55/ from investinghis tokens in the private account.

13 This experimental procedure is typical of provision point mechanism games and is based on theconceptualization of Bagnoli and Lipman (1989) and implementation of Bagnoli and McKee (1991) andIsaac, Schmidtz, and Walker (1989).

equilibrium in which each of the five members of the group allocates 25 tokens to theGROUP account. The stage game also has a continuum of inefficient Nash equilibriain which the public good is not provided.

In the inefficient equilibria, somewhere between 0 and 92 tokens are allocated to thegroup account, but no player unilaterally wants to supplement the account to achievethe provision point. Thus all tokens are returned to the players who are then indif-ferent between any unsuccessful allocation.14 Only twelve instances of this inefficientequilibrium were observed out of the 325 group decisions.15 As a result, our subse-quent analysis will focus on the efficient equilibrium outcomes reached by the groups.

Notice that the efficient and inefficient equilibria are characterized for one-shotstage games on which the experiment was based. In fact, subjects played each stagegame repeatedly (25 times) with the same group. The finite repetition of a stagegame with multiple equilibria has multiple equilibria itself which are not charac-terized here.

3. Experimental implementation. Experimental instructions were written so thatthey conform to language developed in the classic public goods paper, Isaac,Walker, and Thomas (1984). All language referring to ``investments'' or ``contribu-tions'' was intentionally removed and replaced by words such as ``allocation'' oftokens. Common information was established by reading instructions out loud andusing an overhead projector to display results. Subjects were asked to complete aquiz in order to ensure that everyone understood how to calculate their earningsand a session did not begin until everyone had correctly completed the quiz.

All sessions in this experiment involved subjects recruited from undergraduateclasses at Longwood College. Each session involved one group of five subjects. Byhaving only one group in a room, individuals could identify who was in theirgroup, but they were not allowed to communicate in any way.16 Subjects were paidtheir earnings in this experiment privately and in cash at the end of the session. Theaverage length of the session was 2 hours and average earnings of the participantswere around 817.

4. Implementation of treatments. In all three treatments, subjects were seated atdesks with subject numbers prominently displayed on them. In the Group Onlytreatment, individual allocations to the GROUP account were not revealed inany way and subjects were only informed of the aggregate allocation to the publicgood. At the end of a period in the Group Only treatment, the Fig. 2 overhead wascompleted and displayed.

In the Individual��Anonymous treatment, individuals were informed of the aggregategroup allocation level as well as the vector of individual allocations to the GROUP


14 An example of these equilibria is a vector of allocations to the group account like(19, 19, 18, 18, 18). Here 92 tokens have been allocated, but since the value of the public good to eachplayer is only 50, no one wants to contribute an additional 33 tokens to unilaterally supplement thegroup account to achieve 125. No equilibrium of this sort exists in which 94�124 tokens are contributed(when 93 tokens are contributed some players are indifferent between supplementing and not).

15 Interestingly, 11 of these 12 instances occurred in the Individual��Anonymous treatment.16 Experiments were run by three individuals. One was responsible for monitoring the subjects to

prevent communication��verbal or nonverbal. Students were warned that actions such as talking aloud,sighing, and looking at others would result in their dismissal from the experiment.

FIG. 2. Group Only overhead.

FIG. 3. Individual��Anonymous overhead.

account. However, individual allocations were posted anonymously in randomorder so that no particular subject's allocation could be determined. To make thisclear to subjects, individual allocations were copied onto small slips of paper whichwere put into a box and were then drawn out randomly and posted in this order.At the end of a period in the Individual��Anonymous treatment, information wasrevealed on an overhead as shown in Fig. 3.

In the Individual��Identifiable treatment, individual allocations to the GROUPaccount were posted according to (commonly known) subject number. At the endof a period in this treatment, information was revealed on an overhead as shownin Fig. 4.

In each of the three treatments, group sizes of five were used and five replicationsof each treatment were performed. Sessions differed only in the information revealedconcerning individual allocations to the GROUP account.

FIG. 4. Individual��Identifiable overhead.


IV. HYPOTHESES

In this section we present a number of different hypotheses about play in the PPMgame under our three treatments. Some come from game-theoretic predictions ofNash equilibrium play, some from previous experimental results in other contexts,and some from our own intuition. The first subsection presents hypotheses whichaddress all three treatments, the second, hypotheses about the effect of anonymousinformation, and the third, hypotheses about the effect of identifiability.

A. All Three Treatments

One hypothesis we would like to test is the extent to which the Nash equilibriumpredictions help to organize the data in this game. Our first hypothesis is simplythat groups will coordinate on a Nash equilibrium outcome in all three treatments.

HA1 : Group contributions will be one of the set of efficient Nash equilibria ineach period of this game in all three treatments.

In practice, subjects in experiments rarely hit upon the Nash equilibrium outcomedirectly. However, we often observe behavior which converges toward the equilibriumprediction. Thus our second hypothesis involves the convergence of group contribu-tions toward the efficient Nash equilibrium amount over time.

HA2 : Groups contributions will converge toward the efficient Nash equili-brium level of 125 over time in all three treatments.

B. Hypotheses about Anonymous Information: Group Only vsIndividual��Anonymous

Previous results in the PPM literature suggest that the Nash equilibrium hasonly limited power to explain outcomes; groups often fail to coordinate. A competing(and perhaps more likely) hypothesis stems from the idea that to play a Nash equilibriainvolves subjects solving a difficult coordination problem. We suspect that thisproblem will be easier to solve as the amount of information available grows. Thuswe suggest

HB1 : Group contributions will be closer to the efficient Nash equilibrium levelof 125 in the Individual��Anonymous treatment than in the Group Only treatment.17

In addition, we suspect the additional information available to subjects in theIndividual��Anonymous treatment will cause them to converge toward the Nashequilibrium level of 125 faster than in the Group Only treatment. This leads to

HB2 : Group contributions will converge toward the efficient Nash equilibriumlevel of 125 faster in the Individual��Anonymous treatment than in the GroupOnly treatment.


17 As one anonymous referee notes, being ``close'' to 125 does not imply that the public good is provided.However, it does signal that the group is closer to successfully coordinating on the efficient Nash equilibriumoutcome. It is this coordination with which we are primarily concerned.

We know that in continuous public goods games, increasing the information aboutindividual decisions available to the subjects often increases the level of cooperation(contribution). If this extends to a threshold public goods setting, then these resultssuggest

HB3 : Group contributions will be higher in the Individual��Anonymous treatmentthan in the Group Only treatment.

Finally, we suspect that giving subjects information about individual contribu-tions will lead them to focus on the fairness component of the game. The game hasa unique symmetric efficient equilibrium in which each of the five members of the groupallocates 25 tokens to the GROUP account; this solution might be considered themost ``fair'' in that it equalizes everyone's earnings in the experiment. One measureof the fairness of an outcome is the variance of individual contributions. We suspectthat under Individual��Anonymous information, subjects will be more focused onfairness, and will thus have contributions which are closer together than underGroup Only information.18

HB4 : The variance of individual contributions within each group will be lowerin the Individual��Anonymous treatment than in the Group Only treatment.

C. Hypotheses about Identifiability: Individual��Anonymous vsIndividual��Identifiable

As in the previous subsection, we suspect that the coordination problem inherentin this game will be easier to solve as the amount of information available grows.Thus we suggest

HC1 : Group contributions will be closer to the efficient Nash equilibrium levelof 125 in the Individual��Identifiable treatment than in the Individual��Anonymoustreatment

HC2 : Group contributions will converge toward the efficient Nash equilibriumlevel of 125 faster in the Individual��Identifiable treatment than in the Individual��Anonymous treatment.

Previous literature from continuous public goods games suggests that as subjectsare given more information, they contribute (cooperate) more. Thus,

HC3 : Group contributions will be higher in the Individual��Identifiable treatmentthan in the Individual��Anonymous treatment.

Finally, we again expect that as the level of information available in the experimentincreases, subjects will be more concerned about the equity of the distribution of contri-butions in their group. Thus we predict that the variance of individual contributionswithin each group will decrease as the level of information increases, suggesting


18 Since this is a symmetric game, equal contributions imply equal earnings whether or not the publicgood is provided. If the public good is provided, unequal contributions imply unequal earnings. Thusa stronger concern with equity should move individual contributions closer together.

HC4 : The variance of individual contributions within each group will be lowerin the Individual��Identifiable treatment than in the Individual��Anonymous treatment.

V. RESULTS: THE EFFECT OF ANONYMOUS INFORMATION

Figures 5, 6, and 7 show group contributions for each treatment. Group contri-butions are clearly clustered around the efficient equilibrium outcome of 125 andseem to converge toward it over the course of the experiment. As in previous experiments,the data depicted in our figures enable us to reject our initial hypothesis (HA1) thatgroups will routinely play the Nash equilibrium outcome of 125 tokens. Such outcomeswere rarely observed.

In this section we discuss the effect of anonymous information by comparing theoutcomes of the Group Only and the Individual��Anonymous treatments. We findthe addition of anonymous individual information hinders coordination in thisgame, contrary to our initial hypotheses. In Section VI below, we discuss the effectof identifiability by comparing the Individual��Anonymous and the Individual��Identifiable treatments. We find the addition of identifiability helps coordination inthis game, consistent with our hypotheses.

To isolate the effect of information about individual contributions on contributiondecisions, we compare the Group Only treatment with the Individual��Anonymoustreatment. As noted earlier, in the Group Only treatment subjects are informed onlyabout the total allocation to the GROUP account. In the Individual��Anonymoustreatment, after each period subjects learned the entire distribution of individualcontributions. However, these contributions were anonymous in the sense that nocontribution could be tied to any particular individual. The main result from thiscomparison is that (contrary to our hypotheses) the additional information aboutindividual contributions impedes group coordination and provision of the publicgood. Table 1 provides some summary statistics which show this general result.

From this table we can see that average contributions and proportion of successfulprovisions were lower under Individual��Anonymous information than under GroupOnly information.

A. Group Contributions

One important outcome which differed between the treatments was that of group-level contributions. To test for statistical differences in group contributions betweenthe two treatments we ran an OLS random effects regression as below. Many of theregressions reported will involve the same set of independent variables.

Contributionit=:0+:1 Ind Anon+:2 Periodt

+:3 Periodt* Ind Anon+7i{1:i Groupi+=i .

The independent variables are a dummy representing the Individual��Anonymoustreatment, the period number and the period number times the treatment dummy(to capture trends in contributions over time in each treatment), a dummy for eachgroup except one per treatment (to capture fixed group effects), and an error term.



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TABLE 1

Summary Statistics: Group Only vs Individual��Anonymous

Group Only Individual�Anonymous

Proportion of successful provisions 55.20 39.20

Average group contribution 124.03 118.13

In a random effects regression, a separate error term designed to capture group-specificvariance is estimated for each group. See Greene (1990) for a complete description ofrandom effects regressions.

The dependent variable of interest in this subsection is group i 's contribution inperiod t. The results from the regression are presented in Table 2. Results from thegroup dummies are suppressed for ease of presentation.

As Table 2 shows, group contributions were significantly lower under Individual��Anonymous information than they were under Group Only information.19 This resultflatly contradicts our hypothesis HB3 , that increased (anonymous) information wouldlead to higher contributions. This analysis also demonstrates a small but significantincrease in contributions over time under Group Only information, and a marginallysignificant increase over time under Individual��Anonymous information, neither ofwhich were hypothesized.

B. On the Distance from Equilibrium Play

Another measure of interest is the distance between group contributions and theefficient equilibrium outcome of 125 tokens. This can be thought of as a measureof efficiency; deviations below 125 represent inefficient allocations because the publicgood is not provided while deviations above 125 represent inefficient allocationsbecause the extra resources are wasted.20 The regression to test differences betweentreatments in this measure uses the absolute difference between a group's contributionin any period from 125 as a dependent variable and the same independent measuresas above (see Table 3).

The regression shows that groups in the Individual��Anonymous treatment makecontributions significantly farther away from the efficient equilibrium contributionlevel of 125 than groups in the Group Only condition.21 This result is not consistent


19 As one anonymous referee points out, the coefficients above may be influenced by the particulargroups which were chosen to be excluded from the group dummy variables. A linear hypothesis testusing the coefficients from the regression above and comparing the difference between the means of eachtreatment yields a similar result as the regression, with an estimate of &4.093 and a t-statistic of &5.091( p<0.0000). A simple comparison of the two means again yields a similar estimate of &5.904 and at-statistic of &4.317 ( p<0.0001).

20 Although, as one anonymous referee pointed out, these deviations are differentially inefficient.Deviations over 125 only cost 1/ per token to the group; deviations below cost significantly more, sincethe public good is not provided.

21 A linear hypothesis test using the coefficients from the regression above and comparing the dif-ference between the means of each treatment yields a similar result as the regression, with an estimateof 3.882 and a t-statistic of 6.440 ( p<0.0000). A simple comparison of the two means again yields asimilar estimate of 4.208 and a t-statistic of 3.810 ( p<0.0002).

TABLE 2

Group Contributions: Group Only vs Individual��Anonymous

Term Estimate Std error t-ratio p-value

Intercept 118.416 1.294 &91.50 .0000IndAnon &4.844 1.294 &3.74 .0002Periodt .205 .087 2.35 .0194Periodt *IndAnon .146 .087 1.67 .0959

Note. R2, 0.247617; R2 adj, 0.212843; Observations, 250.

with hypothesis HB1 , which suggested that more information would aid groups incoordinating and lead to play closer to the equilibrium. In both treatments, however,this distance decreases with experience. Consistent with hypothesis HA2 , groupcontributions in both treatments converge toward the efficient equilibrium amountover time.

In addition, it is straightforward to see that contributions in the Group Onlyconditions converge to the equilibrium outcome of 125 faster than do contributionsunder Individual��Anonymous information (t=&2.06, p=0.0404). This refuteshypothesis HB2 .

C. On the Equality of the Outcomes

A final measure of interest between the two treatments is the equality of thedistribution of individual contributions. Our hypothesis HB4 suggests that whenindividual contributions are reported to the subjects, subjects will focus on theequity of the contributions and will themselves contribute more equally. Thus thevariance of individual contributions within a group will be smaller. In fact, this isnot what was observed. In the Individual��Anonymous treatment the variance ofindividual contributions within each group was significantly higher than the varianceof individual contributions under Group Only information. Table 4 reports theresults of a random effects regression where the dependent variable is the standarddeviation of individual contributions within each group in each period.

As can be seen in Table 4, the standard deviation of individual contributionswithin each group was significantly larger for groups in the Individual��Anonymous

TABLE 3

Absolute Difference from 125: Group Only vs Individual��Anonymous


Intercept 11.964 .097 12.33 .0000IndAnon 3.852 .097 3.97 .0000Periodt &.325 .065 &4.97 .0000Periodt *IndAnon &.134 .065 &2.06 .0405



TABLE 4

Standard Deviations of Individual Contributions within Groups: Group Only vsIndividual��Anonymous


Intercept 6.507 .339 19.17 .0000IndAnon 1.052 .339 3.10 .0022Periodt &.039 .023 &1.71 .0088Periodt *IndAnon .067 .023 &2.93 .0037


treatment than those in the Group Only treatment.22 In both treatments, this variancebetween individual contributions decreased over time (although only marginally inGroup Only).

D. Summary of Group Only versus Individual��Anonymous

Our hypotheses originally suggested that anonymous information about individualcontributions would aid coordination and cooperation, leading to higher contribu-tions, closer-to-equilibrium outcomes and lower variance of individual contributionswithin groups. However, the data from this comparison suggest that, in contrast,the additional anonymous information presented to the subjects hindered publicgoods provision in this game. Group contributions were significantly lower andfarther from the efficient equilibrium level under Individual��Anonymous informationthan under Group Only information. In addition, the variance of individual contribu-tions within each group was significantly higher under Individual��Anonymous thanunder Group Only information.

VI. RESULTS: THE EFFECT OF IDENTIFIABILITY

To isolate the effect of identifiability we compare the two treatments in whichsubjects are given information about individual contributions after each round ofplay. In the Individual��Anonymous treatment contributions are revealed anonymouslyso that subjects' contributions could not be linked to them. In the Individual��Identifiabletreatment, contributions were revealed by subject number. Thus an individual's contribu-tions could be ``tracked'' over time. Table 5 provides some summary statistics fromthese treatments.

The main result from this comparison is that making individual contributionsidentifiable significantly increases group contributions and thus the proportion ofsuccessful provisions.


22 A linear hypothesis test using the coefficients from the regression above and comparing the dif-ference between the two means of each treatment yields a similar result as the regression, with anestimate of 1.240 and a t-statistic of 5.882 ( p<0.0000).

TABLE 5

Summary Statistics: Individual��Anonymous and Individual��Identifiable

Individual�Anonymous Individual��Identifiable

Proportion of successful provisions 39.20 52.80

Average group contribution 118.13 123.99

A. Group Contributions

The first outcome which differed between the treatments was that of group contri-butions. The same regression as before is run, with the treatment dummy representingthe Individual��Identifiable treatment. Table 6 reports the results.

Group contributions were significantly higher when individual contributions wereidentified than when they were not.23 This result is consistent with hypothesis HC3 .The same increase in contributions over time is observed in the Individual��Anonymoustreatment as before, while no significant effect is observed in the Individual��Identifiabletreatment.

B. On the Distance from Equilibrium Play

A second measure of interest is the distance group contributions were from theefficient equilibrium outcome of 125 tokens. Table 7 reports the results from thisregression.

Groups in the Individual��Identifiable treatment make contributions significantlycloser to the efficient equilibrium contribution level of 125 than groups in theIndividual��Anonymous treatment, consistent with our hypothesis HC1 that moreinformation is better.24 In both treatments, this distance decreases significantly withexperience, thus group contributions in both treatments converge toward the efficientequilibrium amount over time, consistent with HA2 .

In addition, convergence toward the efficient Nash equilibrium outcome of125 is significantly faster under Individual��Identifiable information than underIndividual��Anonymous information (t=4.497, p=0.0000), consistent with hypo-thesis HC2 .

C. On the Equality of the Outcomes

The final measure of interest between these two treatments is the equality of thedistribution of individual contributions. In the Individual��Anonymous treatmentthe variance of individual contributions within each group was significantly higher


23 A linear hypothesis test using the coefficients from the regression above and comparing the dif-ference between the two means of each treatment yields a similar result as the regression, with anestimate of 4.345 and a t-statistic of 5.345 ( p<0.0000). A simple comparison of the two means againyields a similar estimate of 5.864 and a t-statistic of 4.273 ( p<0.0001).

24 A linear hypothesis test using the coefficients from the regression above and comparing the dif-ference between the two means of each treatment yields a similar result as the regression, with anestimate of &4.854 and a t-statistic of &7.863 ( p<0.0000). A simple comparison of the two means againyields a similar estimate of &4.968 and a t-statistic of &4.360 ( p<0.0001).

TABLE 6

Group Contributions: Individual��Anonymous vs Individual��Identifiable


Intercept 119.430 1.309 91.27 .0000IndIdent 5.858 1.309 4.48 .0000Periodt .225 .088 2.56 .0112Periodt *IndIdent .125 .088 1.42 .1556


TABLE 7

Absolute Difference from 125: Individual��Anonymous vs Individual��Identifiable


Intercept 11.668 .994 11.74 .0000IndIdent &4.148 .994 &4.17 .0000Periodt &.128 .067 &1.92 .0567Periodt *IndIdent &.331 .067 &4.95 .0000


TABLE 8

Standard Deviations of Individual Contributions within Groups: Individual��Anonymous vsIndividual��Identifiable


Intercept 6.027 .353 17.06 .0000IndIdent &1.532 .353 &4.34 .0000Periodt &.049 .024 &2.06 .0402Periodt *IndIdent &.057 .04 &2.40 .0173


than the variance of individual contributions when contributions were identified(see Table 8).

The standard deviation of individual contributions within each group was significantlysmaller for groups in the Individual��Identifiable treatment than those in theIndividual��Anonymous treatment, consistent with our hypothesis HC4 .25 In both


25 A linear hypothesis test using the coefficients from the regression above and comparing the dif-ference between the two means of each treatment yields a similar result as the regression, with anestimate of &2.535 and a t-statistic of &11.554 ( p<0.0000). A simple comparison of the two meansagain yields a similar estimate of &1.790 and a t-statistic of &3.861 ( p<0.0001).

treatments, this variance between individual contributions decreased somewhatover time.

D. Summary of Individual��Anonymous vs Individual��Identifiable

Our hypotheses originally suggested that making individual contributions iden-tifiable would lead to higher contributions, less deviation from the equilibriumcontribution level, and individual contributions which were closer together. Thedata from this experiment support all of these hypotheses. Group contributions weresignificantly higher and closer to the efficient equilibrium level in the Individual��Identifiable treatment than in the Individual��Anonymous treatment, and the varianceof individual contributions within each group was significantly lower.

The differences between these two treatments can be interpreted as the effect ofidentifiability. In both, contributions are revealed, but in the Individual��Identifiabletreatment, subjects are able to link contributions to each individual. Perhaps theaddition of identifiability imposes an additional cost for shirking against the group.There are no significant differences between group contributions, distance from equi-libria or variance of contributions within groups between the Group Only treatmentand the Individual��Identifiable treatment ( p=0.3128, p=0.6582, p=0.0678 fromthree regressions similar to those described above). Whatever is lost by providingindividual information is (almost) exactly recovered by making that informationidentifiable.

VII. DISCUSSION AND CONCLUSIONS

Given the existence of multiple efficient Nash equilibria in the PPM, public goodprovision is essentially a coordination game where the socially optimal outcome isto provide without wasting public resources through over-contribution. We conjec-tured that a richer informational environment which provided information aboutindividual contribution behavior would aid groups in coordinating; this turned outto be only somewhat true.

We investigated three treatments which varied in the amount of informationavailable to the participants about the individual behavior of their counterparts. Inthe first treatment, Group Only, subjects were informed of the aggregate groupcontribution to the public good. In a second treatment, Individual��Anonymous,subjects were provided with the distribution of individual contributions in a randomorder. Contributions were thus anonymous, but the degree of similarity of contribu-tions could be determined. Contrary to our conjecture, this anonymous contributioninformation actually impeded coordination. The overall success of the mechanismdecreased as public good provision fell from 550 to 390. Results of random effectsregressions show that the aggregate contribution level decreased, and the distance fromthe equilibrium contribution level and variance of individual contributions increased.

We speculate that under Individual��Anonymous information, inequities whichexist between individual contributions may be more salient. When it becomes obviousthat other contributors are giving less, subjects may reduce their own contribution,leading to lower contribution levels and success rates.


A third treatment, Individual��Identifiable, investigated the impact of makingindividual contributions identifiable rather than reporting them anonymously. Whatwas lost by revealing contributions anonymously was almost fully regained whenindividual contributions became identifiable. Success rates increased to almost 530

in this last treatment, and contributions were significantly higher, while distancefrom equilibrium outcomes and variance of individual contributions significantlylower than in the Individual��Anonymous treatment.

We speculate that when contributions are identifiable, the tendency to reduce one'sown contribution in response to information about the spread of contributions may bebalanced by considerations of reputation within the group. Interestingly, providingidentifiable information is not significantly different than providing only groupinformation. This is consistent with our observation that both types of informationstructures are observed in real fundraising campaigns.26

This study provides several important results. As also shown in previous studies,the efficacy of the PPM is sensitive to the specific institutional features includedin the design. Results of this experiment clearly show that the dissemination ofinformation (or not) can be an important aspect of the PPM design. In creatinginstitutions for the voluntary provision of public goods, policy-makers will want toconsider impacts of the announcement of individual contributions.

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Received December 22, 1997


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