A Common Pool Resource Game with Sequential ... - Springer Link

Experimental Economics, 6:91–114 (2003) c 2003 Economic Science Association


A Common Pool Resource Game with Sequential Decisions and Experimental Evidence LLUIS BRU AND SUSANA CABRERA University of M´alaga, Spain C. MONICA CAPRA∗ Washington and Lee University, Lexington, VA 24450, USA email: [email protected] ROSARIO GOMEZ University of M´alaga, Spain

Abstract We describe a common pool resource game in which players choose how much of the stock to extract in a sequential manner. There are two choices and one represents taking a larger proportion of the stock than the other. After a player makes a choice, the remaining stock grows at a constant rate. We consider a game with a finite number of alternating moves. It is shown that changes in the larger proportion of the stock that the players are allowed to take and the growth rate affect equilibrium, but have little effect on behavior in the laboratory. In addition to observing more cooperation than predicted, we observe that parameters that are strategically irrelevant affect behavior. The results of this research might help policy makers in developing adequate policies to prevent overexploitation of some natural renewable resources. Keywords: laboratory experiments, common pool resources, games JEL Classification: C73, C92

1.

Introduction

We present a common pool resource game where agents’ past decisions affect future decisions and profits through the changing size of the resource’s stock. In this game, each player decides whether to take a large share of the common stock or a small share in an alternating manner and for a finite number of rounds. Moreover, after a decision is made, the stock grows at an exogenously determined growth rate. The sequential nature of this game can be a good approach to some study natural renewable resource exploitation problems. For example, salmon return from salt water to freshwater to breed and the young migrate to salt water after they reach maturity. Each generation of salmon returns to spawn in exactly the same breeding places as the generation before it. This migration has produced some ∗Author

to whom correspondence should be addressed.

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conflicts between bordering countries such as Canada and the US over fishing rights since, when the fish are migrating south as adults, they are caught by American fishermen and, when these fish migrate north for spawning, they are caught by the Canadians. Similar problems appear in the case of straddling fishing stocks, i.e. those that move back and forth between a country’s Exclusive Economic Zone (EEZ) and the high seas. In its EEZ, a state has sovereign rights to apply conservation measures, whereas in high seas these rights do not apply. Examples of straddling stocks are turbot stocks in Canada and squid stocks in Patagonia.1,2 Finally, and away from renewable resources, joint research projects have similar features to the game discussed below; indeed, frequently partners in these projects act sequentially and, as the development process goes on, projects increase their value added and parties have the possibility to (partially or totally) hold-up on previous investments. Despite this and other examples, in this paper we will restrict ourselves to the fishery interpretation of our game. The traditional responses of governments to the problem of overexploitation of natural renewable resources like fish stocks have been either direct regulation of some of the inputs, such as the number and size of boats, or the restriction of fishing during certain seasons. These controls, however, lead to well-known problems of overuse of other unregulated inputs. Thus, more recently, governments have moved to a market-based method of conservation through the creation of property rights or quotas to fisheries in the form of shares on the total allowed amount of catches. However, the move to property rights is not without problems. First, regulation is still necessary since it needs monitoring and enforcement of the quotas, as fishermen have incentives to misreport their level of catches. In addition, the creation of quotas may also lead to distortions, as fishermen may still over-fish and then discard at sea lesser-value fish in order to keep the quota with high-valued species.3 In the case of salmon or straddling stocks, an additional problem arises from the difficulty of properly establishing property rights and from the fact that management of these resources implies that more than one country may have to agree on an exploitation policy. In this paper, we present a game in which players choose sequentially between a large and a small share of a common stock that grows at a constant rate. The two values that can be extracted, H and L, account for the different levels of exploitation that fishermen have available given both the natural conditions of the resource (its sustainability) and the limits imposed by direct regulation of inputs (i.e., quotas).4 Authorities may affect H through restrictions on total capacity of the industry (such as reducing fleet size). Indeed, regulation on the number and size of boats is relatively easy to implement, but these restrictions may lead to an overuse of other non-regulated inputs and, hence they may be an imperfect way to modify H . On the other hand, authorities can affect L by establishing a quota. Regardless, authorities would have to choose H and L in order to guarantee the resource’s long-term survival and, for this purpose, they must take into account the foreseeable response of players to these restrictions. Not surprisingly, the sub-game perfect equilibrium of this game depends on the specific values of the parameters. In particular, as H increases, the equilibrium changes from a lot of cooperation to cooperation at the beginning of the game only, and to no cooperation at all. However, the small share (L) has no effect on the equilibrium characterization. In

A COMMON POOL RESOURCE GAME WITH SEQUENTIAL DECISIONS

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contrast, in the laboratory, the effect of changes in the parameter values on behavior is quite different from what sub-game perfect equilibrium predicts. First, under some parameter values, we observe more cooperation than predicted, which implies less overexploitation than in the sub-game perfect equilibrium. Hence, some changes in parameters of the model may facilitate cooperation. Second, changes in the larger share of the stock (H ) that would lead to changes in the level of cooperation in the sub-game perfect equilibrium, have little effect on behavior in the laboratory. Third, changes in the lower share of the stock (L) that should not affect the level of cooperation in the sub-game perfect equilibrium, substantially modify behavior in the laboratory. Our work emphasizes the importance of design details on the workings of regulation. As we report below, players change their behavior in the laboratory depending on the values of the parameters of the game. Indeed, our results suggest that, in order to evaluate the impact of regulation on behavior, one ought to be skeptical about predictions that uniquely rely on theoretical concepts of equilibrium and pay little attention to experimental evidence. The game in this paper resembles the centipede game first introduced by Rosenthal (1981), but differs in that it does not end when a player takes the larger share of the common stock. Moreover, our game differs from other common pool resource games in that in the latter, agents simultaneously decide how much of a common good to exploit for a single period or repeatedly for several periods whereas, in our game, decisions are made sequentially. Finally, most of these games do not consider the effects of the resource’s natural growth and the dependence of the stock on players’ past decisions. Two exceptions are Walker and Gardner (1992) and Herr et al. (1997). In the first work, if agents simultaneously extract a share of the stock below some pre-determined quantity, they face the same game in the next period with certainty. If the sum of agents’ extractions belongs to an interval of quantities, they repeat their decisions in the next period with some positive probability. Finally, if the safe yield is surpassed, the resource is destroyed and the game ends. In this game, the effects of current actions on future actions and profits are extreme. In addition, if the resource is not destroyed, the stock size is always the same each period (i.e. the game is repeated) implying a very singular regeneration process of the resource.5 In Herr et al. (1997) time-independent and time-dependent appropriation externalities are considered for a common pool resource like a groundwater aquifer. The current extraction of one appropriator increases the extraction costs of others both in the current period and in future periods. Such externalities arise because pumping by one individual increases the depth-towater, and hence, cost of pumping of other appropriators. In their paper, current decisions affect future, but a natural growth rate of the resource is not considered. In the next section, we explain our game and derive the unique sub-game perfect equilibrium for a game with two players and a finite number of rounds. In Section 3, we present an experiment with the objective of comparing the theoretical results reached in Section 2 with the data. The experimental results are described in Section 4. Finally, Section 5 concludes. 2.

The game

Let  be the game with two players, 1 and 2, and N moves; N is a finite and even natural number. Player 1 chooses between H and L in moves 1, 3, . . . , N − 1, and player 2 chooses

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Initial stock: S1

1

Stock: (1-H)GS1

… Figure 1.

H

L

2

2

Stock: (1-L)GS1

H

L

H

L

1

1

1

1

…

…

…

…

…

…

…

The game.

between H and L in moves 2, 4, . . . , N . Figure 1 depicts this game. Let St be the level of stock in move t and S1 is the initial stock. In each round, a player must decide whether to take a large share of the stock, which we call decision H , or a small share of the stock, which we call decision L. After each decision the remaining stock grows at a constant rate equal to g. Define G ≡ 1 + g and assume 0 ≤ L < H < 1. Thus, the level of stock St+1 in move t + 1 is determined both by the stock level in move t and the action at taken in move t through the transition or reproductive function St+1 = f (St , at ) = (1 − at )GSt , where at ∈ {L , H }. Payoff in move t for any player is u(St , at ) = at St when the subject plays that move are equal to the sum of the stock and chooses at ,; the overall payoffs for each player  N /2−1 1 2 taken in each round without discounting, U = m=0 u(S N −2m−1 , a N −2m−1 ) and U =  N /2−1 m=0 u(S N −2m , a N −2m ) for players 1 and 2, respectively. In this game the action taken in a move affects both current payoffs (through the level of stock taken) and future payoffs (through its effect on future levels of the stock). In general, the stock level in move t is St = (1 − H )i (1 − L)t−i−1 G t−1 S1 , where i is the number of times that H has been chosen previously.6 Suppose, for instance, that the game has four rounds and that each player takes the larger share in each round. Then payoffs for players 1 and 2 are equal to U 1 = u(S1 , H ) + u(S3 , H ) = HS1 + HS3 = H (1 + (1 − H )2 G 2 )S1 and U 2 = u(S2 , H ) + u(S4 , H ) = HS2 + HS4 = H (1 + (1 − H )2 G 2 )(1 − H )GS1 , respectively. For any specific value of the parameters of the game, backward induction leads to a unique sub-game perfect equilibrium. As Propositions 1 and 2 below show, the specific actions that the sub-game perfect equilibrium prescribes in each move depend on the length of the game (i.e. the number of rounds that the game is played), the size of the large share, H , and the multiplier, G. Consequently, the size of the small share, L, does not play any role.7 More specifically, proposition 1 tells that, under backward reasoning, behavior will be “cooperative”, in the sense of taking small amounts of the stock all around the game until the last moves. The proof of this proposition and all other proofs are contained in Appendix A. Proposition 1. Assume (1 − H )H G 2 > 1. Under backward induction, the optimal strategies require playing L in any move before the last one, in which each player must choose H .

A COMMON POOL RESOURCE GAME WITH SEQUENTIAL DECISIONS

95

The equilibrium strategies in Proposition 1 (i.e., “play L in any move before the last one, in which play H ”) are not optimal when the other player does not play according to the backward induction reasoning. Consider for instance move N − 2, and assume that player 2 plays the next move as follows: he/she plays the opposite of what the other player has chosen in the former round. Then, it is better for player 1 to play H in move N − 2. But clearly, this is a sequence of moves where player 2 does not choose optimally at move N − 1. Proposition 1 shows that there may be values of the parameters G and H for which backward induction selects “cooperation” (i.e., always take a small share of the stock except in the last move) as the unique sub-game perfect equilibrium. Note also that G is restricted √ 2 ¯ , where H = 1− 1−4/G to take values√above 2, G > 2, while H must satisfy H < H < H 2 2 1+ 1−4/G and H = . 2 In some renewable natural resource examples such as fish and aquifers, high resource growth rates (G-values) may be realistic. For more general examples, the proof of Proposition 2 (see Appendix A) shows that for other values of G, some level of cooperation may still be the outcome of the sub-game perfect equilibrium. Proposition 2. Assume that (1 − H )H G 2 < 1. (i) If (1 − H )G 2 > 1 and the game is sufficiently large, then there exists a finite t-value, t < N , for which the sub-game perfect equilibrium of the game is to choose L in moves 1 through N − t and switch to H in moves N − t + 1 through N . (ii) Otherwise (i.e. when (1 − H )G 2 < 1), the sub-game perfect equilibrium of the game implies to always choose H, regardless of the move and the number of moves. The proofs of Propositions 1 and 2 show that backward induction always selects a unique Nash equilibrium, which is the unique sub-game perfect equilibrium of the game. The equilibrium and hence the path of play depends on, the total length of the game, the values of the parameters G and H , and varies from a lot of cooperation (when the parameters satisfy 1 < (1− H )HG2 ), to cooperation at the beginning of the game only (for intermediate values that satisfy (1 − H )HG2 < 1 < (1 − H )G 2 ) and then, to no cooperation at all (when they satisfy (1 − H )G 2 < 1). In any case, note that L never plays a role in the selection of the equilibrium, whereas the level of H plays a crucial role. Given that the sub-game perfect equilibrium might be a weak predictor of behavior in this game, we design an experiment to test behavior in a controlled environment and see under what conditions and to what extend people cooperate. The next section explains the details. 3.

The experiment

We analyze decisions of subjects in a controlled environment under four experimental treatments. In treatments 1 and 2, we analyze the effects of a change in the H -value, given an L-value. In these cases, the unique sub-game perfect equilibrium predicts full cooperation (Proposition 1). In treatment 1, the equilibrium coincides with the strategies that maximize the joint payoffs, but in treatment 2 the sum of payoffs is maximum when player 1 chooses

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Table 1. Sub-game perfect equilibria, joint payoffs maximizing outcomes, and individual earnings.

Treatment

Parameters

Equilibrium strategies and payoffs∗

Payoffs maximizing strategies and payoffs∗

1

H = 0.2; L = 0

LLLLHH (1620.00, 3888.00)

LLLLHH (1620.00, 3888.00)

2

H = 0.8; L = 0

LLLLHH (6480.00, 3888.00)

LLLLLH (0.00, 19440.00)

3

H = 0.9; L = 0

HHHHHH (98.83, 29.65)

LLLLLH (0.00, 21870.00)

4

H = 0.9; L = 0.6

HHHHHH (98.83, 29.65)

LLLLLH (270.82, 399.63)

∗ First

mover’s payoffs before the coma.

LLL and player 2 chooses LLH. In treatments 3 and 4, we compare behavior when a change in the L-value occurs, given an H value. In both cases, the equilibrium prediction is no cooperation (Proposition 2), whereas the sequence of decisions that maximize the sum of payoffs is LLLLLH (Table 1 contains a summary of the equilibrium strategies and payoff maximizing choices. Appendix B contains the payoff table for each experimental treatment). The G-value is taken to be 3 and the game has 6 moves. We ran eight sessions of the game; each session had four sections each with a six-move sequence that involved four pairs of subjects recruited from various departments at the University of Virginia. Each session of the experiment consisted of two different games, a bargaining game followed by this common pool resource game with sequential decisions. The sessions lasted less than two hours and average payoffs were $35 per person. After the bargaining game, the eight participants were divided into two groups of four people, the Red and the Blue. Each subject participated in one session only, maintaining the same role throughout the session. The subjects were numbered from one to eight and the matching procedure consisted of a reel circle, where the red players (1, 2, 3 and 4) were located on the outside of the circle, with the blue players (5, 6, 7, and 8) located on the inside. After each matching, the blue players were matched with the next red player in a counter clockwise direction. At the beginning of each section, the participants of the Red and the Blue groups shared a stock of an unspecified good, S, worth 100 cents that grew after each move at a constant rate. After the instructions were read aloud, the participants were asked to decide how much of the stock of the good to take in an alternating manner, with the Red participants choosing first, followed by the Blue participants. Each participant had two possible choices: to take H (a high percentage of the stock) or to take L (a low percentage of the stock). After the Red players made their choice, we collected their decision sheets and calculated how much of the stock remained, the remaining stock ((1− H )∗100) or ((1− L)∗100) tripled for the next move. Then we privately informed each Blue player about the decision of its Red partner and the remaining stock by writing these down on each Blue’s decision sheet. The Blue participants were then asked to make their choices (take H or L). After the Blue players made a choice, the remaining stock tripled for next move, we privately informed each Red

97

A COMMON POOL RESOURCE GAME WITH SEQUENTIAL DECISIONS Table 2. Sessions 1∗

Order of treatment sections by session. Section 1

Section 2

Section 3

Section 4

H = 0.9; L = 0

H = 0.9; L = 0.6

H = 0.2; L = 0

H = 0.8; L = 0

2∗

H = 0.9; L = 0

H = 0.9; L = 0.6

H = 0.8; L = 0

H = 0.2; L = 0

3

H = 0.2; L = 0

H = 0.8; L = 0

H = 0.2; L = 0

H = 0.8; L = 0

4

H = 0.8; L = 0

H = 0.2; L = 0

H = 0.8; L = 0

H = 0.2; L = 0

5

H = 0.9; L = 0.6

H = 0.9; L = 0

H = 0.9; L = 0.6

H = 0.9; L = 0

6

H = 0.9; L = 0

H = 0.9; L = 0.6

H = 0.9; L = 0

H = 0.9; L = 0.6

7

H = 0.8; L = 0

H = 0.2; L = 0

H = 0.8; L = 0

H = 0.2; L = 0

8

H = 0.9; L = 0.6

H = 0.9; L = 0

H = 0.9; L = 0.6

H = 0.9; L = 0

∗ The

last two sections of sessions 1 and 2 were ignored in the analysis of the data because of treatment-order effects.

player about its Blue partner’s decision and the remaining stock and so on for a total of six moves. In four of the eight sessions our treatment variable was the larger amount that the players could extract (H ), whereas in the other four sessions our treatment variable was the low amount that the players could extract (L). At the end of the session, the participants were paid in cash 40% of their cumulative earnings. Table 2 shows all treatment conditions per session. The numbers in each cell represent the high (H ) and the low (L) proportions of the common stock that the players could take in each move. Appendix C contains the instructions of the experiment described, with H = 0.9 and L = 0. The order of the treatments was changed in each session with the objective of controlling for order of treatment effects. In addition, to make sure that the participants understood the instructions, we asked them to work on an example with a sequence of H s and Ls of their choice. By working through an example, we made sure that the participants knew how to calculate their payoffs (see Appendix C for the instructions).

4. 4.1.

Experimental results Aggregate behavior

Figure 2 shows the percentage of H choices in each move under each treatment. Recall (from Table 1) that the equilibrium prediction is LLLLHH in treatments 1 and 2, whereas in the other two treatments the equilibrium strategies are HHHHHH. Even though the equilibrium prediction is completely different in treatments 1 and 2 from treatments 3 and 4, aggregate behavior follows a similar pattern in all treatments. The average percentage of L choices in moves 1 through 4 of treatments 1 and 2 is 66%, and 63% of players selected L in treatments 3 and 4. On the other hand, 100% of people chose H in move six, but we observe a percentage of L choices in move five in treatments 2, 3 and especially in treatment 4.8 This pattern of behavior is more consistent with the decisions that maximize

98

Figure 2.

BRU ET AL.

Percentage of Hs chosen by players in each move and treatment.

joint payoffs. Recall (from Table 1) that in treatment 1 the payoffs maximizing outcome coincides with the equilibrium prediction, whereas in treatments 2, 3 and 4, the social optimum is to choose LLLLLH. Table 3 shows the degree of efficiency of equilibrium and experimental results. Efficiency is measured as the percentage of maximum joint earnings extracted by the players. We can see which sequences of decisions are played with greater probability in each treatment by applying a simple one-tailed binomial test to each move and treatment to data shown in Table 4. The null hypothesis is that in each move there is no difference Table 3.

Efficiency of equilibrium and observed choices by treatment.

Parameters

Efficiency of equilibrium choices (%)

Efficiency of observed choices (%)

1

H = 0.2; L = 0

100.00

79.67

2

H = 0.8; L = 0

53.30

18.63

3

H = 0.9; L = 0

0.58

9.76

4

H = 0.9; L = 0.6

19.16

56.11

Treatment

Table 4.

Sequences with greater probability in each treatment. Treatment 1

LL (H or L)∗ (H or L) HH

Treatment 2

LL (H or L) (H or L) HH

Treatment 3

LL (H or L) HHH

Treatment 4

LLL (H or L) (H or L) H

∗ (H or L) means that the null hypothesis cannot be rejected at the predetermined level of significance.

A COMMON POOL RESOURCE GAME WITH SEQUENTIAL DECISIONS

99

between the probability of taking L and the probability of choosing H . The alternative hypothesis is that the probability of taking H is higher when we observe a higher number of H s. Similarly, when we observe a higher number of L decisions, the alternative hypothesis is that the probability of taking L is higher than the probability of taking H . Results from application of the test are shown in Table 4 for a 5 percent significance level. In order to have a more accurate idea on the existence of treatment effects, we analyzed the data across the treatments move by move. Table 5 shows the number of H and L decisions taken by individuals in each move under each treatment and the results from the application of a chi-square test for independent samples. As Table 5 shows, it can be concluded that there are no statistically significant differences across treatments for moves one, two, three and six.9 That is, the null hypothesis of no difference between the proportions of H s and Ls chosen in treatments cannot be rejected at a 5% significance level. Conversely, there are significant differences among treatments in move four and five for the same level of significance. In particular, for the fourth move of treatment 3, there is a statistically significant higher proportion of H s chosen than in the other treatments, and treatment 4 shows a higher proportion of L choices than in any the other treatments in move five.10 Combining the results from the tests, we observe that in all treatments players begin by taking a share L and finish taking the larger share H , regardless of the equilibrium prediction. Players’ choices are closer to the equilibrium prediction in treatments 1 and 2 than in treatments 3 and 4, where the equilibrium was never observed. In addition, the overall pattern of behavior seems to be more consistent with the payoffs maximizing outcomes than with the different equilibrium predictions (see Tables 1 and 3). In treatments 1 and 2 choices are statistically the same move by move, so that the large differences in the H -value do not affect the aggregate players’ behavior. Conversely, although the equilibrium prediction in treatments 3 and 4 is to choose H in all moves, treatment 4 shows a high level of L choices, which is broken only in the final move. When the L-value is high, behavior approximates joint payoffs maximizing outcomes. When the L-value decreases (like in treatment 3), players seem to be more impatient and we observe less cooperation. Thus, the L-value is affecting players’ behavior in these treatments even though it affects neither the equilibrium nor the social optimum. Next, we analyze the individual behavior with the purpose of looking for behavioral patterns that could be hidden in the analysis of aggregate data. In particular, we pay especial attention to the sequences of choices that players used more frequently in each treatment, and the payoffs that players obtained in these sequences. Evidence of imitative behavior and best-response behavior is also reported. 4.2.

Individual behavior

Table 6 shows the possible strategy sequences and the number of couples who played each sequence under each experimental treatment. It shows the sequences sorted in descending

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Table 5. Observed number of decisions in each treatment and application of the chi-square test for independent samples.

Treatments

1 (H = 0.2)

2 (H = 0.8)

3 (H = 0.9)

4 (H = 0.9; L = 0.6)

Total

25

Move 1 H

8

6

5

6

L

18

20

23

22

83

Total

26

26

28

28

108

23

d f = 3; α = 0.05; χ 2 = 1.3; H0 is not rejected Move 2 H

4

7

7

5

L

22

19

21

23

85

Total

26

26

28

28

108

43

d f = 3; α = 0.05; χ 2 = 1.5; H0 is not rejected Move 3 H

9

12

14

8

L

17

14

14

20

65

Total

26

26

28

28

108

23

15

62

d f = 3; α = 0.05; χ 2 = 3.4; H0 is not rejected Move 4 H

12

12

L

14

14

5

13

46

Total

26

26

28

28

108

92

d f = 3; α = 0.05; χ 2 = 9.8; H0 is rejected Move 5 H

26

24

24

18

L

0

2

4

10

16

26

26

28

28

108

107

Total

d f = 3; α = 0.05; χ 2 = 15.6; H0 is rejected Move 6 H

26

26

28

27

L

0

0

0

1

1

26

26

28

28

108

Total

d f = 3; α = 0.05; χ 2 = 2.5; H0 is not rejected H0 : the proportion of subjects choosing H and L is the same in each treatment. H1 : the proportion of subjects choosing H and L differs across treatments.

order, starting from the most frequently chosen. Individual payoffs for the sequences most often played are also shown. The complete payoff matrix for each treatment is in Appendix B. We observe that most couples played the sequence LLLLHH. In addition, this is the sequence most used in the first and second treatment (11 and 6 couples, respectively).

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A COMMON POOL RESOURCE GAME WITH SEQUENTIAL DECISIONS

Table 6. Strategies, number of couples that play each strategy, payoff of the sequences most often played, and joint payoffs maximizing outcomes∗ . Strategy sequence

H = 0.2, L = 0 Treatment 1

H = 0.8, L = 0 Treatment 2

H = 0.9, L = 0 Treatment 3

H = 0.9, L = 0.6 Treatment 4

LLLLHH

11 (1620; 3888)

6 (6480; 3888)

3 (7290; 2187)

3 (333; 232)

23

LLLHHH

3 (1296; 650)

3 (1296; 2938)

4 (729; 2649)

8 (193; 242)

18

LLHHHH

2 (1217; 2920)

5 (979; 588)

7 (883; 265)

1 (201; 114)

15

LLHLHH

1 (1476; 3110)

2 (2016; 778)

1 (1539; 219)

1 (236; 112)

5

LHLHHH

0

0

4 (93; 150)

0

4

HLHHHH

2 (993; 2336)

0

2 (178; 26)

0

4

HLLHHH

2 (1057; 2920)

1

1

0

4

HHLHHH

1

0

1

1

3

LHHHHH

0

1

1

1

3

LLLLLH

0

0 (0; 19440)

0 (0; 21870)

3 (271; 400)

3

LHHLHH

1

1

0

1

3

LLHHLH

0

0

2

0

2

HHHHHH

2 (798.75, 1917)

0

0

0

2

HLLLLH

0

0

0

2

2

LLHLLH

0

0

1

1

2

HLHHLH

0

0

0

2

2

HHLLHH

0

2

0

0

2

HHHLHH

0

2

0

0

2

HHLHLH

0

0

1

0

1

LHHLLH

0

0

0

1

1

HLHLHH

1

0

0

0

1

LLLHLH

0

0

0

1

1

HLLLHH

0

0

0

1

1

LHLLLL

0

0

0

1

1

HLHLLH

0

1

0

0

1

LHLLHH

0

1

0

0

1

HHHHLH

0

1

0

0

1

26

26

28

28

108

Total ∗ Joint

payoffs maximizing outcomes in bold.

Sum

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BRU ET AL.

The second sequence most played was LLLHHH and it was the sequence of decisions most often used in the fourth treatment. Finally, the sequence LLHHHH is the third most often used and it was the most frequently played in the third treatment. Note that these sequences of strategies have two things in common: (1) L choices in the two first moves and (2) H choices in the last two moves. We observe that the sequence most often played in the first treatment corresponds to the equilibrium, which coincides with the joint payoffs maximizing outcome; 11 of the 26 couples reached this optimum social and equilibrium outcome (22 players of the 52 subjects or a 42.31% of the players). A total number of 27 players (51.92%) played their dominant strategy LLH. In the second treatment, the number of subjects who played the equilibrium sequence is lower than in the first treatment. Indeed, only six couples reached the equilibrium outcome, which represents 23.08% of players. Seven more individuals selected their dominant strategy in the second treatment, which means that only 36.54% of subjects behaved rationally.11 Hence, we observe that in the first treatment, where the H value is the lowest (H = 0.2), people behaved more “rationally.” On the other hand, no couple chose the joint payoffs maximizing outcome in treatment 2 (i.e., when H = 0.8). Note that, in this case, players 1 would have obtained payoffs of 0 cents, whereas in the sequence most often played (i.e., the equilibrium prediction) these players obtained payoffs of 6,480 cents. The individual data show a difference in behavior induced by the H -value that we were not able to observe at an aggregate level of the analysis. Choosing LLH is the dominant strategy in both treatments, but the greater the H -value, the higher the number of subjects selecting H , which results in a deviation from their optimal decision. In the context of fisheries, individual data suggest that, when the capacity level or H -value is high with respect to the quota or L-value, subjects cannot overcome the “temptation” of taking the large share before the final move. For a given quota the effect that high capacity on choice results in lower payoffs. In the first treatment, when capacity is low (H = 0.2), the equilibrium prediction was the sequence most frequently played, and it resulted in 324 cents and 237.6 cents more for players 1 and 2, respectively than the second sequence most frequently played, where player 2 deviates in move 4 choosing to extract the large share. In the second treatment, when capacity is large (H = 0.8), the difference in payoffs between the most frequently played sequence (i.e., the equilibrium outcome) and the second most frequently played (LLHHHH) is 5,500 and 3,300 cents for players 1 and 2, respectively. It is interesting to note that, even though deviating in the second treatment is more expensive than in treatment 1, people did it more frequently, so that the explanation based on the “temptation” that a high H -value induces more H choices seems to be a good and intuitive explanation of behavior in treatment 2.12 In the third and fourth treatments, the individual analysis of data confirms our results in the previous subsection. Indeed, no couple followed the equilibrium behavior, which was to take H in all moves. In these two treatments players do not maximize their earnings playing the Nash equilibrium. In particular, the joint payoff maximizing strategy in both cases is to play LLLLLH supporting them payoffs of (0.00, 21,870.00) for treatment 3 and (270.82, 399.63) for treatment 4. However, only three couples in treatment 4 played the optimal strategy. Note that, in those outcomes, player 2 would obtain

A COMMON POOL RESOURCE GAME WITH SEQUENTIAL DECISIONS

103

zero cents in treatment 3 in absence of any mechanism to share earnings, against the payoffs of 882.90 cents attained in the most often played sequence. In treatment 4, the strategy most often played was LLLHHH with payoffs of (193.06, 241.52). In any case, the sequences of decisions most often played in these treatments support higher payoffs for both players than the Nash.13 In addition, in the fourth treatment, when the quota is high in relation with capacity, we observe high cooperation, especially by the first players.14

5.

Conclusion

We study a common pool resource game with sequential decisions that can be used to model exploitation of some renewable resources. The game resembles the centipede game in the sense that players alternate in deciding whether to take a large share (H ) or a small share (L) of a common stock that grows at a constant rate, but unlike the centipede game, this game does not end when a player takes the larger share. A feature of this game is that the sub-game perfect equilibrium is susceptible to changes in the size of the larger share that players are allowed to take in each decision round and to the resource’s natural growth rate. Indeed, equilibrium can go from cooperation in all moves to no cooperation at all. In the context of renewable resource exploitation, L could be related directly to a quota defining how much one is allowed to catch, and H could represent the extraction capacity. Theoretically, for a fixed natural resource growth rate, only changes in the capacity will affect the amount of the resource taken in equilibrium. To test the theoretical predictions of our game, we analyze four treatments with different parameters and compare the predictions with the behavior observed in laboratory experiments. The experiments consist of eight experimental sessions with eight subjects recruited from a variety of departments at the University of Virginia. Each session has two treatments and each game has a total of six moves. The aggregate data show that behavior is not affected by changes in relevant parameters, as theory predicts. In addition, move-to-move analysis of the data shows evidence that changes in the L value, which is strategically irrelevant, affects behavior. Combining the aggregate and individual data analysis two results are obtained. First, subjects pay special attention to the difference between the H and L values. A high H -value (with respect to the L) seems to imply that an important fraction of players are not able to resist the “temptation” of selecting it, even when it is irrational to do so. Second, decisions respond to joint payoff maximization, imitation, or best-response behavior rather than to equilibrium. In the context of natural resource management, our results suggest that the closer the capacity is to the quota, the higher is the tendency not to surpass the quota. Consequently, the magnitude of the extraction capacity with respect to the quota is important in guaranteeing the survival of the renewable resource. This implication is relevant because policy makers can both impose quotas and target capacity. When capacity is too large relative to the quota, the fleet tends not to comply with the quota. Thus, policies directed to reducing capacity are more effective.

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Appendix A: Proofs Proposition 1 Proof: Moves N and N − 1 are the last moves for player 2 and 1, respectively. For any node of the game reached in move N , player 2 chooses H in the last move. The payoff in any final node is u(S N , a N ) = a N S N , where the stock level in this final move S N depends on the number of times i that H has been previously played as follows: S N = (1 − H )i (1 − L) N −1−i G N −1 S1 . Playing H is strictly better for any level of stock S N than playing L, since u(S N , H ) = HS N > u(S N , L) = LS N for H > L. Similarly player 1 chooses H in his/her last move for any stock level S N −1 since u(S N −1 , H ) = HS N −1 > u(S N −1 , L) = LS N −1 as long as H > L. We move backwards to move N − 2. The level of stock is S N −2 = (1 − H )i (1 − L) N −3−i G N −3 S1 , where again i is the number of times that H has been played previously. Now the action a N −2 affects not only the payoff in period N − 2, but also payoff in the last move through its effect on the stock level. The level of stock in move N will be S N = (1 − H )2 G 2 S N −2 if H is chosen and S N = (1 − L)(1 − H )G 2 S N −2 if player chooses L. Thus player must compare payoffs S N −2 {L + (1 − L)(1 − H )G 2 H } and S N −2 {H + (1 − H )2 G 2 H }. Player 2 chooses L if L +(1− L)(1− H )G 2 H > H +(1− H )2 G 2 H , independently of the path followed before, and thus for any node of the game reached in move N −2. The former inequality amounts to H (1− H )G 2 > 1 or H (1− H )−1/G 2 > 0, which is true by assumption. Hence, for any node in move N −2, player 2 chooses to take L. A similar reasoning shows that for any sub-game reached in move N −3, player 1 chooses L if L + (1 − L)2 G 2 H ≥ H + (1 − L)(1 − H )G 2 H , that is, if H (1 − L)G 2 ≥ 1. But L < H . Thus H (1 − L)G 2 > H (1 − H )G 2 > 1. For previous moves of the game, the proof goes by induction. Let m be the number of remaining moves before taking the final decision for a player.15 Assume that playing L is sub-game perfect for both players later on in moves m − 1, m − 2, etc, except for the last move (i.e. when there does not remain any more moves, m = 0), where both play H . Payoffs from playing L are Um2 (L) = S N −2m {L + (1 − L)2 G 2 L + · · · + (1 − L)2(m−1) G 2(m−1) L + (1 − L)2m−1 (1 − H )G 2m H }   1 − (1 − L)2m G 2m 2m−1 2m = S N −2m L + (1 − L) (1 − H )G H 1 − (1 − L)2 G 2 whereas payoffs from playing H are Um2 (H ) = S N −2m {H + (1 − L)(1 − H )G 2 L + · · · + (1 − L)2(m−1)−1 (1 − H )G 2(m−1) L + (1 − L)2(m−1) (1 − H )2 G 2m H }  H−L 1 − H 1 − (1 − L)2m G 2m = S N −2m + L 1−L 1−L 1 − (1 − L)2 G 2  2(m−1) 2 2m + (1 − L) (1 − H ) G H

A COMMON POOL RESOURCE GAME WITH SEQUENTIAL DECISIONS

105

Player 2 chooses L if: Um2 (L) > Um2 (H )

  1 − (1 − L)2m G 2m H−L 2m−1 2m ⇔ S N −2m L − 1 + (1 − L) (1 − H )G H > 0 1−L 1 − (1 − L)2 G 2 1 − (1 − L)2m G 2m ⇔ X m2 ≡ L − 1 + (1 − L)2m−1 (1 − H )G 2m H > 0 1 − (1 − L)2 G 2

(Note that for m = 1, (1 − L){(1 − H )G 2m H − 1} > 0 ⇔ (1 − H )G 2m H > 1. But this is just what we assume about H ). Assume that this is indeed satisfied for m − 1, 2 2 2 Um−1 (L) > Um−1 (H ) ⇔ X m−1

≡L

1 − (1 − L)2(m−1) G 2(m−1) − 1 + (1 − L)2(m−1)−1 (1 − H )G 2(m−1) H > 0. 1 − (1 − L)2 G 2

but note that X m2 is increasing in m: 2 X m2 − X m−1 = L(1 − L)2(m−1) G 2(m−1)

+ (1 − L)2(m−1)−1 (1 − H )G 2(m−1) H {(1 − L)2 G 2 − 1} > 0. Thus it is optimal for player 2 to choose L in move N − 2m. Following a similar argument, player 1 chooses L in move N − 2m − 1, i.e. when m moves remain before the final decision H , if  H−L 1 − (1 − L)2m G 2m L 1−L 1 − (1 − L)2 G 2  2m 2m − 1 + (1 − L) G H > 0

Um1 (L) > Um1 (H ) ⇔ S N −2m−1

⇔ X m1 ≡ L

1 − (1 − L)2m G 2m − 1 + (1 − L)2m G 2m H > 0 1 − (1 − L)2 G 2

It can be shown that X m1 > X m2 > 0, thus it is also optimal for player 1 to choose L instead of H in any move before the last one.

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Proposition 2 Proof: When (1 − H )HG2 < 1 and one move remains before the final decision, m = 1, both players choose H instead of L. Assume that the sub-game perfect equilibrium implies to play H when m − 1 moves remain for each player. For m, to play L gives payoffs to player 2 Um2 (L) = S N −2m {L + (1 − L)(1 − H )G 2 H + · · · + (1 − L)(1 − H )2(m−1)−1 G 2(m−1) H + (1 − L)(1 − H )2m−1 G 2m H }   1 − L 1 − (1 − H )2(m+1) G 2(m+1) H−L = S N −2m − H 1− H 1 − (1 − H )2 G 2 1− H and to play H gives payoffs  Um2 (H ) = S N −2m H + (1 − H )2 G 2 H + · · · + (1 − H )2(m−1) G 2(m−1) H  + (1 − H )2m G 2m H = S N −2m H

1 − (1 − H )2(m+1) G 2(m+1) 1 − (1 − H )2 G 2

Thus player 2 chooses L if

Um2 (L)

>

Um2 (H )

  H−L 1 − (1 − H )2(m+1) G 2(m+1) ⇔ S N −2m −1 >0 H 1− H 1 − (1 − H )2 G 2 1 − (1 − H )2(m+1) G 2(m+1) ⇔ Z m2 ≡ H −1>0 1 − (1 − H )2 G 2

If (1 − H )G > 1, Z m2 is strictly increasing in m and limm→∞ Z m2 = ∞. So one can find a finite m-value for which the optimal decision for player 2 is choosing L when m 1 moves remain. If (1 − H )G < 1 then Z m2 goes to H 1−(1−H − 1. This last term is )2 G 2 2 positive if and only if (1 − H )G > 1. In this case, one can again find a finite m-value for which the optimal decision for player 2 is choosing L when it remains m moves; otherwise (i.e., when (1 − H )G 2 < 1) and assuming perfectness, it is never optimal for player 2 to play L . In the reasoning above, it has been assumed that player 1 plays H in move m −1 whenever player 2 plays H in move m − 1. This is indeed the case as both players compare the same payoffs when it is sub-game perfect to play H from move m − 1 onwards. Further, it is straightforward to see that, if it is optimal for player 2 to play L when m moves remain, then it is also optimal for player 1 to play L just before him/her.

0.00 180.00 20.00

0.00 0.00 720.00 80.00

224.00 0.00

483.20 0.00

HHH 483.20

224.00

1376.00 0.00 1376.00

80.00 0.00

2016.00 0.00 2016.00

720.00 0.00

0.00

180.00 3542.40

144.00

20.00 3542.40

20.00

345.60

345.60 964.64

135.20

0.00 6048.00

0.00 3948.00

HLH

155.52

777.60

777.60

3888.00

777.60

3888.00

275.84

224.00

339.20

80.00

979.20

720.00

86.40

86.40

432.00

432.00

432.00

432.00 587.52

275.84

224.00

339.20

117.50

241.92

587.52

80.00 1209.60

979.20

720.00 1209.60

144.00 3170.40 20.00 3158.40

48.00

48.00

0.00 4128.00

964.64 2038.66

135.20 2536.32

48.00 1056.80 2536.32

48.00

60.00 1180.80 2548.32

60.00

0.00 240.00

80.00

160.64

108.80

48.00

48.00

48.00

48.00

403.20 240.00

144.00 240.00

339.20

0.00 492.00

HHL

0.00 3602.40

HHH

160.64

108.80

339.20

80.00

403.20

79.10

203.52

203.52

825.60

395.52

144.00 1017.60

119.17

108.80

119.17

108.80

131.84

80.00

195.84

144.00

259.20

71.50

96.38

165.50

289.92

357.50

481.92

827.52

0.00 1449.60

798.75 1917.00

135.20 2315.14

849.44 2384.26

20.00 2881.92

973.44 2396.26

144.00 2893.92

(Continued on next page.)

65.28

65.28

131.84 134.40

80.00 134.40

195.84 326.40

144.00 326.40

259.20 672.00

0.00 672.00

798.75 324.48

135.20 324.48

849.44 393.60

20.00 393.60

973.44 405.60

144.00 405.60

60.00 1296.00 3170.40 1036.80 492.00 1036.80 2980.32

60.00

Treatment 2 (H = 0.8, L = 0)

993.44 2336.26

164.00 2833.92

432.00 1056.80 2920.32 1056.80

432.00

432.00 1216.80 2920.32 1180.80

432.00

0.00 2160.00

993.44

164.00

0.00 4428.00

540.00 1296.00 3650.40 1296.00

540.00

HLL

Treatment 1 (H = 0.2, L = 0)

LHH

Player 2

3888.00 1296.00 2160.00 1296.00 2937.60 1296.00 240.00 1296.00 1017.60

0.00 19440.00

6480.00 0.00 6480.00

HHL

HLH

HLL

LHH

LHL

LLH

20.00

3110.40 1056.80

3888.00

2488.32

LLL

180.00

3110.40 1216.80

3888.00

3110.40

Player 1

0.00

LHL

3888.00 1296.00

4860.00

164.00

164.00 0.00

1316.00 0.00 1316.00

20.00 0.00

1476.00 0.00 1476.00

180.00 0.00

1620.00 0.00 1620.00

0.00 0.00

LLH

HHH 1200.80 0.00 1200.80

HHL

HLH

HLL

LHH

LHL

LLH

LLL

Player 1

LLL

Appendix B: The matrix payoffs for each treatment

A COMMON POOL RESOURCE GAME WITH SEQUENTIAL DECISIONS

107

171.00

243.90

HHL

HHH

270.82

333.02

220.70

236.26

142.70

158.26

130.18

134.06

LLL

LLH

LHL

LHH

HLL

HLH

HHL

HHH

Player 1

90.00

819.00

1539.00

LHH

HLH

810.00

LHL

HLL

0.00

7290.00

LLH

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

26.81

33.81

53.25

81.24

107.25

135.24

213.00

324.98

LLL

LLL

Player 1

(Continued.)

134.06

130.18

158.26

142.70

236.26

220.70

333.02

270.82

243.90

171.00

819.00

90.00

1539.00

810.00

7290.00

0.00

27.98

38.48

57.92

99.91

111.92

153.91

231.67

399.63

21.87

218.70

218.70

2187.00

218.70

2187.00

2187.00

21870.00

LLH

125.32

124.34

123.26

119.38

201.26

197.38

193.06

177.50

178.29

171.00

162.90

90.00

882.90

810.00

729.00

0.00

28.30

30.05

59.21

66.21

113.21

120.21

236.85

264.84

24.30

24.30

243.00

243.00

243.00

243.00

2430.00

2430.00

LHL

HLL

26.49

46.17

264.87

461.70

264.87

461.70

2648.70

4617.00

0.00

105.39

98.10

162.90

90.00

153.90

81.00

729.00

125.32

124.34

123.26

119.38

201.26

197.38

193.06

177.50

28.59

31.22

60.38

70.88

114.38

124.88

241.52

283.51

101.02

100.04

107.06

103.18

104.06

100.18

128.26

112.70

29.20

30.95

35.81

42.81

116.81

123.81

143.25

171.24

27.00

27.00

27.00

27.00

270.00

270.00

270.00

270.00

Treatment 4 (H = 0.9, L = 0.6)

178.29

171.00

162.90

90.00

882.90

810.00

729.00

0.00

Treatment 3 (H = 0.9, L = 0)

LHH

Player 2

101.02

100.04

107.06

103.18

104.06

100.18

128.26

112.70

105.39

98.10

162.90

90.00

153.90

81.00

729.00

0.00

29.49

32.12

36.98

47.48

117.98

128.48

147.92

189.91

29.19

48.87

48.87

245.70

291.87

488.70

488.70

2457.00

HLH

98.83

98.59

98.32

97.34

95.32

94.34

93.26

89.38

98.83

98.10

97.29

90.00

88.29

81.00

72.90

0.00

29.58

30.01

37.30

39.05

118.30

120.05

149.21

156.21

29.43

29.43

51.30

51.30

294.30

294.30

513.00

513.00

HHL

98.83

98.59

98.32

97.34

95.32

94.34

93.26

89.38

98.83

98.10

97.29

90.00

88.29

81.00

72.90

0.00

29.65

30.30

37.59

40.22

118.59

121.22

150.38

160.88

29.65

31.62

53.49

73.17

296.49

316.17

534.87

731.70

HHH

108 BRU ET AL.

A COMMON POOL RESOURCE GAME WITH SEQUENTIAL DECISIONS

109

Appendix C: Instructions Instructions (Section 1)

Red/Blue

This part of the experiment has four sections. We will start by dividing the participants into two groups: the Red and the Blue. Your group is shown on the top right side of these instructions. At the beginning of each of the four sections, the participants of the Red group will be matched with a participant of the Blue group for a sequence of six moves. Thus, each of you will face a different partner in each of the four sections, but will keep the same partner during the six-move sequence. At the beginning of each section, the participants of the Red and Blue groups share a stock of an unspecified good that grows at a constant rate. Participants will be asked to decide how much of the stock of the good to take in an alternating manner with the Red participants choosing first, followed by the Blue participants. There are two possible choices of how much to take in each move of a sequence of moves; the choices are H or L. Earnings Your earnings for this exercise will partly depend on your choice and the choice of the person you are matched with. The initial stock of the good is worth 100 cents. In each move, a choice of H represents a take of 90% of the stock in that move. A choice of L represents a take of 60% of the stock in that move. Once a player has made the decision of how much to take of the stock (90% or 60%), the remaining stock (10% or 40%) will triple for the next move. Thus, the participant who chooses in the next move will face a stock that is three times the remaining size of the stock. For example, if in move 1 the red player chooses H, he or she keeps 90% of the stock or 90 cents (90% of 100) and leaves for the next move the remaining 10% or 10 cents. The remaining amount triples for move 2, thus in move 2 the stock is equal to 3 ∗ 10 = 30 cents. On the other hand, if player Red chooses L, he or she keeps 60% of the stock or 60 cents (60% of 100) and the remaining 40% or 40 cents triples for the next move, thus in move 2 the stock is equal to 3 ∗ 40 = 240 cents. In the second move, player Blue has to decide to take H (90%) or to take L (60%) from the new stock, whose size depends on the choice the Red player made in the previous period. This process continues for a total of 6 moves. The cumulative earnings for the Red player are equal to the sum of the earnings in moves 1, 3, and 5. The cumulative earnings for the Blue player are equal to the sum of the earnings in moves 2, 4, and 6. In this part of the experiment you will receive in cash 30% of your cumulative earnings. Summary Initial stock = 100 cents Player Red begins and is followed by player Blue The possible choices in each move of the six-move sequence are: To take H = 90% of the stock, or

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BRU ET AL.

To take L = 60% of the stock Stock for next move = 3*(initial stock in move − stock taken); The cumulative earnings for the Red player are the sum of the earnings in moves 1, 3, and 5 The cumulative earnings for the Blue player are the sum of the earnings in moves 2, 4, and 6. Your cash payment is 30% of your cumulative earnings. Test To see how your earnings are determined in the first three moves, please fill in the blanks with any choice you want and calculate the earnings and the stock for the next period. Recall that a choice of H takes 90% of the stock and a choice of L takes 60% of the stock. This is just an example and will have no effect on your earnings. Red chooses next move is 3* =

. Thus makes Red

% of 100 cents) in move 1. Stock for the

cents.

. Thus makes Blue chooses next move is 3* = cents. Red chooses move is 3* =

cents (

. Thus makes

cents (

cents (

% of

% of

cents) in move 2. Stock for the

cents) in move 3. Stock for the next

cents.

The earnings for moves 4, 5, and 6 and the stock for moves 5 and 6 are calculated in similar manner. Record of results Now each of you should examine the record sheet, which is on the last sheet attached to these instructions. Your identification number and the identification number of the person you are match with for the six-move sequence are written on the top left of this sheet. Now, please look at the columns of your record table. Going from left to right, you will see columns for “Move,” “Stock (in cents),” “Your choice (H or L),” “Other’s choice (H or L),” and “Your earnings (in cents).” At the beginning of move 1, the Red participants must make a choice and record it on their record table; we will then collect the decisions and write down their earnings and the stock for the next period on their record table. Then, Reds’ decisions and the remaining stock for move 2 will be reported to the Blue participants. The Blue participants must make their choice in move 2; we will then collect their decisions and write down the earnings and the stock for move 3 on their record table. Blues’ decisions and the remaining stock for move 4 will be reported back to the Red participants, and so on. Once the six-move sequence finishes, you may calculate your cumulative earnings and write them down in the space provided. Your cash earnings will equal 30% of your cumulative earnings in this section. Finally, another sequence of six moves will follow this one.

111

A COMMON POOL RESOURCE GAME WITH SEQUENTIAL DECISIONS

At this time, if you have any questions about the procedures described, please raise your hand. ------------------(next sheet)----------Record Sheet (for the Red participants) Your ID: Other’s ID: Record Table Move 1

Stock (in cents)

Your choice (H or L)

Other’s choice (H or L)

Your earnings (in cents)

100

2 3 4 5 6 Cumulative earnings:

Instructions (Section 2)


Red/Blue

This six-move sequence is similar to the previous one. At the beginning of this sequence, each participant of the Red group will be matched with a different participant of the Blue group. Again, the initial stock of the good is worth 100 cents. However, in this sequence a choice of L represents a take of 0% of the stock, whereas a take of H represents a take of 90% of the stock. Once a player has made the decision of how much to take of the stock (90% or 0%), the remaining stock (10% or 100%) will triple for the next move. As before, the Red participants begin. The cumulative earnings for Red are calculated by adding the earnings of moves 1, 3 and 5. The cumulative earnings for Blue are calculated by adding the earnings of moves 2, 4 and 6. You will be paid 30% of your cumulative earnings in this section. ------------------(next sheet)----------Instructions (Section 3) This six-move sequence is similar to the previous one. At the beginning of this sequence, each participant of the Red group will be matched with a different participant of the Blue


Note that a record sheet with a record table was attached to every instruction sheet.

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group. Again, the initial stock of the good is worth 100 cents. However, in this sequence a choice of L represents a take of 60% of the stock, as in the first case, whereas a take of H represents a take of 90% of the stock. Once a player has made the decision of how much to take of the stock (90% or 60%), the remaining stock (10% or 40%) will triple for the next move. As before, the Red participants begin. The cumulative earnings for Red are calculated by adding the earnings of moves 1, 3 and 5. The cumulative earnings for Blue are calculated by adding the earnings of moves 2, 4 and 6. You will be paid 30% of your cumulative earnings in this section. ------------------(next sheet)-----------

Instructions (Section 4) This six-move sequence is similar to the previous one. At the beginning of this sequence, each participant of the Red group will be matched with a different participant of the Blue group. Again, the initial stock of the good is worth 100 cents. However, in this sequence a choice of L represents a take of 0% of the stock, as in the second case, whereas a take of H represents a take of 90% of the stock. Once a player has made the decision of how much to take of the stock (90% or 0%), the remaining stock (10% or 100%) will triple for the next move. As before, the Red participants begin. The cumulative earnings for Red are calculated by adding the earnings of moves 1, 3 and 5. The cumulative earnings for Blue are calculated by adding the earnings of moves 2, 4 and 6. You will be paid 30% of your cumulative earnings in this section.

Acknowledgment We would like to thank John Turner, Francisco Mancera, and especially Charles Holt for their help. Suggestions made by two anonymous referees and the editor of the review, Arthur Schram, contributed substantially to the improvement of a previous version. Research support of the University of Arkansas and Spain’s Ministry of Education under grant PB981402 are gratefully acknowledged.

Notes 1. The juridical regime that applies to this example was established by the 1982 UN Convention on the Law of the Sea. See Munro (2000). 2. Another example of a renewable common pool resource are aquifers, because the extraction decision of a party affects future decisions of the others in a sequential manner. 3. See Grafton (2000). 4. We are aware our assumption that the resource growth rate is exogenously given are oversimplifications of reality. More realistically, the amount of the stock extracted each period will determine its growth rate.

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5. Let S be the initial stock of the common resource and the stock at the beginning of each period in which the game is played. Let X be the sum of quantities withdrawn by the agents in a period. Then, at the end of that period, the remaining stock is S–X . If the game is repeated for the next period, the stock size is again S, so S/(S–X ) multiplies the remaining stock. 6. We could add a discount rate δ redefining G = (1 + g)δ; then St would be the discounted value of the remaining stock at t, St = δ t−1 (1 − H )i (1 − L)t−i−1 (1 + g)t−1 S1 . 7. The iterated deletion of weakly dominated strategies also leads to the unique perfect equilibrium of the game. 8. All subjects chose H in move six, except one person in treatment 4. 9. Note that in move 3 of the fourth treatment, when the move is considered as an isolated move with no relation with other moves or treatments, L is chosen with a greater probability than H . The chi-square test suggests, however, that there are no significant differences among treatments in this move. 10. In order to know which treatments are significantly different in moves 4 and 5, we used additive partitions of the 2 × 4 contingency tables for these moves, and modifying the χ 2 -values for partitions it can be concluded that in the fourth move treatments 1, 2 and 4 are not significantly different, but they together differ from treatment 3 for a low significance level of 0.01 Then, in treatment 3 for the fourth move, there is a statistically significant higher proportion of H s chosen than in the other treatments. The modified χ 2 -value is 9.5. On the other hand, the fifth move of treatment 4 exhibits significant differences with respect to treatments 1, 2 and 3 for a very low significance level of 0.001. In particular, this treatment shows a higher proportion of L choices than any of the other treatments. In this case, the corresponding modified χ 2 -value is 13.1 (see Siegel and Castellan, 1988). 11. In treatment 1, a binomial test cannot reject the null hypothesis that the dominant and a non-dominant strategy are chosen with the same probability. In treatment 2, we reject the null hypothesis in favor of the alternative that non-dominant strategies are chosen with higher probability at the 0.05 significance level. 12. This finding is at odds with McKelvey and Palfrey (1995a, b) and other experimental results—among them; Capra et al. (1999)—that suggest that the chances of making “mistakes” are greater, the smaller the loss implied by deviating from an equilibrium is. 13. In treatments 3 and 4, a chi-square test rejects the null hypothesis that there are no differences in the proportion of all possible sequences chosen by each player. However, a partitioned chi-square test shows that we cannot reject the hypothesis that the two most frequently chosen sequences (LHH and LLH) were selected in the same proportion. 14. Another explanation of individual behavior could come from best-response. In this game, to play LLH is the individual dominant strategy in treatments 1 and 2. Then, in these cases, to cooperate by taking the low share of the stock in all moves except in the last one is the individual best response to any potential or observed behavior of the rival. In treatments 3 and 4 there are no dominant strategies, but it can be seen how the best response to any sequence of decisions observed is to choose L in moves 1 through 4, whereas the best response in moves 5 and 6 is to take H , independently of what happened in the past. For example, if we look at the payoffs matrix for treatment 3 in Appendix C, we can see how the best response of player 2 in move 2 to a previous L-choice by player 1 is to choose L, because player 2 has potential earnings of 21,870 cents in this case, if the sequence LLLLLH is played. If player 1 chose H , the best response of player 2 is also to take L, with potential payoffs of 2,187 cents if the outcome HLLLLH is reached. To illustrate, suppose that player 2 observed the sequence HHH. The best response is to take L in move 4, because it has potential gain of 48.7 cents which is greater than the corresponding move of taking H. 15. Player Move Remaining moves

1

2

1

2

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N − 2m − 1 N − 2m N − 2(m − 1) N − 2(m − 1) − 1 . . . N − 3 N − 2 N − 1 N m

m−1

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References Capra, C.M., Goeree, J.K., Gomez, R., and Holt, C.A. (1999). “Anomalous Behavior in a Traveler’s Dilemma?” American Economic Review. 89, 678–690. Grafton, R.Q. (2000). “Performance of and Prospects for Rights-Based Fisheries Management in Atlantic Canada.” In B.L. Crowley (ed.), Taking Ownership: Property Rights and Fishery Management on the Atlantic Coast, AIMS. Herr, A., Gardner, R., and Walker, M. (1997). “An Experimental Study of Time-Independent and Time-Dependent Externalities in the Commons.” Games and Economic Behavior. 19, 77–96. Mckelvey, R.D. and Palfrey, T.R. (1995a). “Quantal Response Equilibria for Normal Form Games.” Games and Economic Behavior. 10, 6–38. Mckelvey, R.D. and Palfrey, T.R. (1995b). “Quantal Response Equilibria for Extensive Form Games.” Caltech Working Papers. Munro, G.R. (2000). “The Management of Transboundary Fishery Resources and Property Rights.” In B.L. Crowley (ed.), Taking Ownership: Property Rights and Fishery Management on the Atlantic Coast, AIMS. Rosenthal, R. (1981). “Games of Perfect Information, Predatory Pricing, and the Chain Store Paradox.” Journal of Economic Theory. 25, 92–100. Siegel, S. and Castellan, N.J. (1988). Nonparametric Statistics for the Behavioral Sciences. McGraw-Hill Inc. 2nd ed. Walker, J.M. and Gardner, R. (1992). “Probabilistic Destruction of Common Pool Resources: Experimental Evidence.” The Economic Journal. 102, 1149–1161.