Common pool resource experinment paper(Economics Game Theory)

Int J Game Theory (1999) 28:241±252

999999

Strategic behavior of experienced subjects in a common pool resource game*

Claudia Keser1, Roy Gardner2

1 Institut fuÈr Statistik und mathematische Wirtschaftstheorie, UniversitaÈt Karlsruhe, Rechenzentrum, Zirkel 2, D-76128 Karlsruhe, Germany (e-mail: [email protected]) 2 Department of Economics and Workshop in Political Theory, Indiana University, Bloomington, Indiana 47405, USA (e-mail: [email protected])

Received May 1994/Final version August 1996

Abstract. This paper describes the results of an experiment applying the strategy method to analyze the behavior of subjects in an 8-player common pool resource (CPR) game. The CPR game consists of a constituent game played for 20 periods. The CPR game has a unique optimum and a unique subgame perfect equilibrium; the latter involves overinvestment in the appro- priation from the CPR. Sixteen students, all experienced in game theory, were recruited to play the CPR game over the course of 6 weeks. In the ®rst phase of the experiment, they played the CPR game on-line 3 times. In the second phase of the experiment, the tournament phase, they designed strategies which were then played against each other. At the aggregate level, subgame perfect equilibrium organizes the data fairly well. At the individual level, however, fewer than 5% of subjects play in accordance with the game equilibrium prediction.

Key words: Strategy method, common pool resources, Nash equilibrium, bounded rationality

* We are grateful to James Walker and Dean Dudley for their assistance in organizing the experiment. Thanks are also due to Wulf Albers, Nick Feltovich, Ron Harstad, Edgar Keser, Jack Knetsch, Bentley MacLeod, Mark Olson, Reinhard Selten, Bodo Vogt, and two anonymous referees for their valuable comments. This research would not have been possible without the generous support of the Workshop in Political Theory and Policy Analysis at Indiana University in Bloomington. Financial support from the Alexander von Humboldt Foundation (Transatlantic Cooperation Program), the National Science Foundation (Grant aSBR-9319835, the Commis- sion of the European Union (Human Capital & Mobility Fellowship), and the Center for Research in Experimental Economics and Political Decision Making (Tinbergen Institute) is gratefully acknowledged.

1. Introduction

A common pool resource (CPR) is a resource in which yield is subtract- able and exclusion is di½cult. Examples of CPRs include grazing commons, high seas ®sheries, and irrigation systems. CPRs often present a commons dilemma, where the pursuit of self interest by individuals leads to collective disaster ± the tragedy of the commons [Hardin, 1968]. Ostrom [1990] and Ostrom, Gardner, and Walker [1994] provide empirical evidence that the tragedy is avoidable. Similarly, McCay and Acheson [1987] and Monbiot [1994] present critical perspectives of Hardin's tragedy argument.

The formal modeling of CPRs starts with Gordon [1954], who examines the common property nature of the Canadian ®shing industry. His main concern is to demonstrate that the over®shing problem has its roots in the economic organization of the industry. He argues that users of a CPR will overinvest in appropriation from the resource, leading to the complete dissipation of economic rents from the resource and even its destruction. The present paper focuses on rent dissipation; for a study of destruction, see Walker and Gardner [1992].

In order to evaluate the various theoretical predictions for CPRs and the policy prescriptions that ¯ow from them, Ostrom, Gardner and Walker [1994] conducted a set of CPR experiments in which they varied the extent of communication possible among subjects. They found that subjects allowed to communicate often achieved nearly cooperative outcomes. If subjects were not allowed to communicate, then the aggregate outcome was best described by Nash equilibrium. However, no individual appeared to play a Nash equilibrium strategy. This result mirrors to some extent the vast literature on public goods experiments [Ledyard, 1995] and ultimatum games [Roth, 1995], where individuals again do not play a Nash equilibrium. In these latter cases the situation is even worse, in that the aggregate outcome is also not the Nash prediction. If individuals in a CPR game do not play Nash equilibrium, then policies based on that equilibrium's predictions are suspect .

The experiment conducted here explores further the question whether individuals play a Nash equilibrium in CPR games. The design is especially conducive to Nash equilibrium in two ways. First, all subjects come from a pool of honors undergraduate and graduate students who have had at least one semester of game theory. The subjects receive extensive experience in playing the game, with plenty of time for learning and re¯ection between plays. If any subject pool should be capable of ®nding a Nash equilibrium, it is this one. Second, we use the strategy method, introduced by Selten [1967] and subsequently used by Axelrod [1984], Selten, Mitzkewitz and Uhlich [1997], and Keser [1992] among others, to elicit directly the strategies used by subjects. In our experiment, subjects ®rst gain experience in playing the speci®c CPR game by playing it three times on-line at computer terminals. Then they design strategies for playing this game, one strategy each for each of 3 tournaments. In each tournament, a subject's strategy is played against all other submitted strategies, and strategies are then ranked and paid according to its average payo¨. Subjects revise their strategies after this feedback.

Our main result is as follows. In both the on-line and tournament phases, the aggregate outcomes are described fairly well by Nash equilibrium. At the same time, we observe almost no subjects (fewer than 5%) playing a Nash equilibrium strategy. This is especially striking in light of the experimental

242 C. Keser, R. Gardner

design chosen. These results, together with those in the literature, suggest the need for a theory of individual behavior in CPRs di¨erent from Nash equilibrium.

The paper is organized as follows. The following section presents the CPR game and solves for its Nash equilibrium and three other proposed solutions. Section 3 describes the organization of the experiment. The results of the on-line and tournament phases of the experiment are presented in section 4, while section 5 concludes the paper.

2. The CPR game

The CPR game is a twenty-fold repetition of a constituent game, which is itself a symmetric, 8-player, non-cooperative game. Each player i is endowed with 25 tokens, which can be allocated to a CPR (denoted xi) or to a safe alternative (denoted 25ÿ xi), where xi A f0;1; . . . ;25g. The safe alternative yields a payo¨ of 5 cents per token. The payo¨ to i from investment in the CPR depends both on xi and on aggregate group investment in the CPR, Sxj. The group payo¨ in cents to the total investment in the CPR is given by the production function

F�Sxj�� 23Sxj ÿ 0:125�Sxj�2 �1� Note that F is a concave function with F�0�� 0, F 0�0� > 5, and F 0�200� < 0. (1) implies that initial investments in the CPR pay better than the safe alternative, but that large investments in the CPR pay less than the safe alternative. At the optimal level of CPR investment, individuals invest some but not all their tokens in the CPR.

Each player i receives a fraction of the group payo¨ to the total investment in the CPR proportional to xi. Thus, the payo¨ ui to a player i from investing xi in the CPR and �25ÿ xi� in the safe activity depends both on his own and the other players' investment in the CPR and is given by

ui�x�� 125 if xi � 0 5�25ÿ xi���xi=Sxj�F�Sxj� if 0 < xi U 25

� �2�

where x ��x1; . . . ; x8� is the vector of individual investments in the CPR. The constituent game is repeated 20 times by the same group of 8 players.

The players have complete information about the game ± the payo¨ function for each player, the values of all parameters, and the endpoint. In the begin- ning of each period, the 8 players simultaneously and independently make their investment decisions. Whatever the history of the previous periods, the decision situation for each player is always the same, i.e. the one described by the constituent game. At the end of each period, each player is informed about the total amount of tokens invested in the CPR by the entire group and about his or her payo¨. No information is provided about the individual decisions of the other 7 players.

Consider four solution concepts for the CPR game, one of which is a game equilibrium solution (symmetric subgame perfect equilibrium), and three of which are not game equilibrium solutions (optimum, rent dissipation, and naive optimum). To compute the game-theoretic solution, consider ®rst the

Strategic behavior of experienced subjects in a common pool resource game 243

constituent game. A Nash equilibrium is a pro®le x� � �x�1; . . . ; x�8� such that for each player i

x�i A arg max xi

ui�xi; xÿi� �3�

where xÿi is the vector of all strategies excluding player i's. Applying the ®rst order conditions to (2), one has that player i's best reply

function is given by

ri�xÿi�� 25 if Sj 6�i xj < 94 72ÿ 0:5Sj 6�i xj if 94 U Sj 6�ixj U 144 0 if Sj 6�i xj > 144

8< : �4�

The Nash equilibrium satis®es

x�i � ri�x�ÿi� for all i: �5�

Solving (5) simultaneously for all players i, one ®nds x�i � 16 as the unique symmetric Nash equilibrium.1 That there is a symmetric equilibrium follows from Selten [1973]. Using a backward induction argument as in Selten [1978], one can see that the unique symmetric subgame perfect equilibrium2 of the CPR game is to play the symmetric Nash equilibrium each period. The group investment in the CPR is 128 tokens, 16 per player. From (2), each player earns 157 cents each period. Contrast this to the safe payo¨ of 125 cents each period.

The optimal solution to the constituent CPR game maximizes total group payo¨, given by

Suj�x�� 5�200ÿ Sxj�� F�Sxj� �6�

Maximizing (6) over Sxj, one ®nds that the optimal solution is a group investment of 72 tokens, 9 by each player, paying each player 206 cents. The optimal solution to the CPR is to play the optimal solution each of 20 periods. Notice that player earnings at subgame perfect equilibrium are only (157/206) � 76% of those at optimum.

The economic rent from a CPR, denoted R�Sxj�, is the di¨erence between CPR production and the opportunity cost of CPR investment, 5Sxj, given by

R�Sxj�� F�Sxj�ÿ 5Sxj �7�

This rent is maximized at the optimal solution. The rent dissipation solution for the constituent game occurs when (7) equals zero. The root of (7) is a

1 One can check that the corner solutions xi � 0 or 25 do not satisfy (5). The above derivation ignores the restriction that xi be integer valued. When this restriction is imposed, no other sym- metric equilibria are created; however, over a thousand asymmetric equilibria are created. These take the following form. Let k be an integer from 1 to 4. Let k players invest 15, 8±2k players invest 16, and k players invest 17. The result is an equilibrium. Since these asymmetric equilibria di¨er very little from the symmetric equilibrium, we focus exclusively on the latter. 2 We use the term ``Nash equilibrium'' hereafter to denote this equilibrium.

244 C. Keser, R. Gardner

group investment of 144 tokens in the CPR, 18 by each player. At the rent dissipation solution, each player earns 125 cents. The rent dissipation solution for the CPR game dissipates the rent in each period. This is the worst of the 4 solutions from the viewpoint of the players.

The ®nal solution considered is the naive optimum, where the group payo¨ from total investment in the CPR is maximized, ignoring opportunity cost. As such, it can be thought of as a boundedly rational version of the optimal so- lution. The naive optimum is achieved by maximizing (1) with respect to total investment in the CPR, which is achieved at a group investment of 92 token. At the naive optimum, the marginal return from investment in the CPR is zero.

Table 1 gives an overview of the main characteristics of the four solutions presented above for the constituent game. This overview includes group and individual investments in the CPR (the latter assuming symmetry), individual payo¨s (in cents), and the realized percentage of the optimal rent, or rent ef- ®ciency. The latter measures how e¨ectively the CPR is being utilized.

3. Organization of the experiment

The subjects were 16 volunteers, recruited from graduate or honors under- graduate economics courses at Indiana University, Bloomington. Each subject had taken a course in game theory. Subjects were informed that the experi- ment would last from mid-January to late February, 1993, and that they would have to complete the experiment. The experiment was organized in two phases. In the ®rst phase, the on-line phase, subjects received written instructions and played the CPR game three times at computer terminals. In the second phase, the tournament phase, subjects designed strategies for playing the CPR game. In each of the three on-line sessions, subjects were randomly divided into groups of 8 to play the CPR game once. During play, no communication between subjects was allowed. Subject interaction was

Table 1. Theoretical solutions of the constituent game (ignoring the integer constraint)

solution characteristic CPR group investment

CPR individual investment

(if symmetry)

individual total payo¨

(cents)

rent e½ciency

%

safe payo¨ xi � 0 Ei 0 0 125 0

group optimum dF�Sxj�

d Sxj � 5 72 9 206 100

naive optimum dF�Sxj�

d Sxj � 0 92 11.5 200 92

Nash equilibrium qui�x�

qxi � 0 Ei 128 16 157 40

complete rent dissipation

F�Sxj� Sxj

� 5 144 18 125 0

Strategic behavior of experienced subjects in a common pool resource game 245

restricted to the simultaneous investment decisions via computer terminals.3 Immediately after each on-line session, subjects were paid in private in cash one half of their individual earnings. The on-line phase lasted two weeks.

At the completion of the on-line phase, subjects entered the tournament phase, which lasted four weeks. At the beginning of this phase, subjects were instructed in the design of strategies in ¯ow chart format for playing the CPR game. Subjects had one week to design a strategy. In addition, the experimenters were available at any time to answer individual questions. Once strategies had been completed and submitted, they were translated into com- puter code (Turbo Pascal). The strategies then competed against one another in a computerized tournament. The tournament was designed so that there were 70 plays in total, with each strategy being involved in 35 plays. The grouping of strategies was done as follows. Each subject was randomly matched with another so that if subjects 1 and 2 were matched the strategies of subjects 1 and 2 were together in each play. This reduced the total number of di¨erent groupings from 12,870 (16 choose 8) to a more manageable number, 70 (8 choose 4).4

In each tournament round, the success of a strategy was measured by its average earnings over the 35 plays. After each round, each subject received a ranked ordering of all subjects' average earnings, by subject number, which was kept anonymous. Each subject also received a record of his or her5 indi- vidual investments, group investments in the CPR, and his or her individual earnings for all 20 periods of each of the 35 plays. Prior to the next tour- nament, each subject had the opportunity to revise and resubmit his or her strategy, and fresh pairings were drawn. Compensation for the ®rst two tournaments equaled one half of a subject's average earnings for that tournament, but compensation for the ®nal tournament equaled the full amount of a subject's average earnings. Subjects were also paid $5 for commenting on their strategy submitted between tournaments.

4. Results

The main result of the experiment, that subjects do not play the predicted CPR game equilibrium, can be seen in both phases. For the on-line phase of the experiment, a broad overview must su½ce.6 As seen in ®gures 1a, b, c observed group investment in the CPR is generally in between the game equilibrium prediction and the point of complete rent dissipation. The aggregate numbers behind ®gures 1a,b,c are shown in summary table 2. Ag- gregate investment in the CPR gets closer to the game equilibrium prediction with each play. Note in particular that the standard deviation is falling monotonically from 9 to 4. At the individual level, however, data are not supportive of the game equilibrium prediction. Indeed, as seen in ®gure 2 which depicts the individual investment decisions over all three plays, fewer than 5% of subjects ever chose the predicted value of 16 tokens. All subject

3 The software used James Walker's CPR program on Novanet. 4 Later we ran the tournaments using all 12,870 con®gurations. The results obtained were essentially the same as those found during the experiment. 5 Four subjects were women. 6 More comprehensive data are available from the authors upon request.

246 C. Keser, R. Gardner

Fig. 1a. Group investment in the CPR in the ®rst on-line session

Fig. 1b. Group investment in the CPR in the second on-line session

Fig. 1c. Group investment in the CPR in the third on-line session

Strategic behavior of experienced subjects in a common pool resource game 247

payo¨s lie between the safe alternative, $25, and the optimum, $41.20. A majority of subjects earned less than the game equilibrium payo¨, $31.40. These results are consistent with those of the previous studies (Ostrom, Gardner, and Walker [1994]), and so can be considered a replication of them.

For the tournament phase of the experiment, we ®rst give a broad over- view, and then discuss in further detail broad features of the individual strat- egies submitted. Figure 3 shows the time series of average group investment in the CPR for the three tournaments; also refer to summary table 2. Average group investment is highest (142) in the ®rst tournament, and it exceeds the game equilibrium level after the ®rst period. Average group investment in the second tournament is lowest (120), and it never exceeds the game equilibrium level after the ®rst period. Average group investment in the third tournament (139) is in between the average group investment levels in the ®rst two tour- naments. Notice in contrast to ®gure 1 the almost complete absence of puls- ing. This is a law of large number e¨ect, caused by averaging over 70 plays each. One striking result from the tournament phase is the greater dispersion in average individual payo¨s than in the on-line phase. This ¯uctuation is driven by the ¯uctuation in average group investment, which is again greater than in the on-line phase. Note that in the ®rst tournament the average payo¨ ($24.96) is worse than the safe alternative ($25).

We now turn to individual strategies submitted in the tournament phase. We classify strategies in the following way. An open loop strategy is one in which the decisions are not contingent on previous outcomes. It prescribes a ®xed sequence of decisions. An example of an open loop strategy is the symmetric subgame perfect equilibrium, which prescribes investing 16 tokens in any period, regardless of previous outcomes. A closed loop strategy makes decisions depend on previous outcomes. Table 3 gives an overview of the

Fig. 2. Individual investment decisions in the three on-line sessions

248 C. Keser, R. Gardner

types of strategies observed during this phase. Note ®rst of all that only one subject submitted the symmetric subgame perfect equilibrium strategy. The great majority of strategies submitted (34/48) were closed loop. All but one of these closed loop strategies are casuistic, in that they make decisions based on

Fig. 3. Time paths of average group investment in the three tournaments

Table 2. Overall results in the ®rst phase (I) and the second phase (II) of the experiment

experiment round

average group

investment

standard deviation

average rent

e½ciency

average payo¨

actual payo¨/ optimum

actual payo¨/game equilibrium

I-1 135 9 22% 28.54 69% 91%

I-2 134 7 25% 28.86 70% 92%

I-3 131 4 32% 30.15 73% 96%

II-1 142 0% 24.96 61% 79%

II-2 120 53% 33.63 82% 107%

II-3 139 13% 27.11 66% 86%

7 The lone exception estimated group investment from past investment, and then made the best response to the estimate.

Strategic behavior of experienced subjects in a common pool resource game 249

a small number of case distinctions [Selten, 1990].7 The case distinctions are principally based on the observed group investment in the previous period. With these case distinctions the casuistic strategies typically distinguish between a low, median, and high group investment. Several strategies refer to the naive optimum to locate the region of low investment. The region of high investment is typically around the point of complete rent dissipation. Addi- tional case distinctions often occur in this region.

Perhaps what is most striking about the strategies is their stability. Half of all subjects submitted a closed loop strategy in each round, and these with only minor modi®cations. Four more subjects submitted an open loop strat- egy in each round, again with only minor modi®cations. The remaining four subjects switched between the two types. These subjects are the driving force behind the ¯uctuations observed in this phase.

Besides the data contained in the strategies themselves, we also collected data in the form of subjects' commentaries about their strategy and the experiment in general between tournament rounds.8 From the commentaries, it becomes clear that subjects took their decision task seriously, and that they were aware of the possibility of high payo¨s in the game. There was dis- agreement, however, over the identity of the optimum, with several subjects aiming for the naive optimum instead. Two main types of attitude toward strategy emerge. Four of the subjects are proactive. They express the intention of dominating the CPR, making large investments and forcing the rest of the players to adjust theirs downward. Ten of the subjects are reactive. They express the intention of reacting in their investment to what has happened previously in the CPR. We can distinguish between two variants within the reactive type. One variant simply imitates the group behavior observed in the previous period. The other variant invests more when group investment is small; less, when group investment is large. These attitudes also appear quite stable over the tournament phase. All the evidence from the strategies and from the commentaries on them suggests that the subjects by and large had reached a behavioral equilibrium by the end of the experiment, which equi- librium was de®nitely not a game equilibrium.

7. Conclusion

In this experiment, we applied the strategy method to a CPR game. After replicating existing results in on-line play, we conducted three tournaments

8 Writing the commentaries was strictly voluntary. Highlights of these commentaries are avail- able from the authors upon request.

Table 3. Classi®cation of strategies in phase II

tournament round a subgame perfect equilibrium strategies

a open loop strategies (in total)

a closed loop strategies

II-1 1 3 13

II-2 0 6 10

II-3 0 5 11

250 C. Keser, R. Gardner

using strategies submitted by our subjects. Our sixteen subjects, who were well paid (average earnings, $125) and highly motivated, performed the tasks we set for them over a 6 week period in a timely and serious manner. These sub- jects, honors undergraduates and graduate students, had all taken a semester of game theory prior to the experiment. Our main result is that, even though game equilibrium organizes the data at the aggregate level, fewer than 5% of the subjects behave in accord with that equilibrium.

In his book Fun and Games, Binmore [1992] gives two reasons for why we might expect to observe Nash equilibrium play. The ®rst, the eductive reason, says that players will reason their way to equilibrium; the second, the evolu- tive, says that subjects will ®nd their way to equilibrium by some sort of adaptive or learning process. Both these aspects were present in the design of this experiment. The subjects had all been tested on solving for the symmetric Nash equilibrium of a CPR game in their game theory course (the eductive aspect), and the subjects had 6 weeks and repeated plays of the CPR game during which to re¯ect and learn (the evolutive aspect). From this standpoint, our main result, that subjects rarely play a Nash equilibrium of the CPR game, is even more striking. Here was a design where one could reasonably expect to observe Nash equilibrium, but the predictions of that equilibrium failed badly at the individual level. What is needed for the further analysis of strategic behavior of experienced subjects in a CPR game is a theory other than Nash equilibrium.

One possibility is a cooperative theory. What is interesting about our results is that, contrary to those in other strategy experiments (Selten, Mitzkewitz and Uhlich, 1996; Keser, 1992), we observed no attempt by sub- jects to actively achieve cooperation. This might be for two reasons, which can be gleaned from the subjects' commentaries. First, it is not clear to the sub- jects at exactly which contribution level cooperation should take place. This is the problem created by the naive optimum. Second, subjects see no possibility of in¯uencing the behavior of others in the 8-player CPR game. The major concern of the subjects is to avoid negative rents. We observe in large two types of subjects, the proactive type and the reactive type. The main di¨erence between the two types is that the proactive type believes that the others will `equilibrate' to avoid negative rents, while the reactive type is the one that is indeed equilibrating. The way these two types interact leads to the behavioral equilibrium observed in the simulation phase.

One way to proceed from here in the analysis would be to study a game model with the above two player types, as found in our experiment. Once player types are determined for each player, and a player's type is known only to that player himself, one has an incomplete information game with a plethora of Bayes-Nash equilibria. In particular, some of these equilibria might well describe our results, to the extent that players' types are stable. Further research, both theoretical and experimental, on the distribution of player types in CPR games is needed before we understand the predictive failure of Nash equilibrium observed here.

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