The evolution of strategies in a repeated trust game

Journal of Economic Behavior & Organization Vol. 55 (2004) 553–573

The evolution of strategies in a repeated trust game Jim Engle-Warnicka , Robert L. Slonimb,∗ a

Department of Economics, McGill University, 855 Sherbrooke St. West, Montreal, Que., Canada H3A 2T7 Economics Department, Weatherhead School of Management, Case Western Reserve University, 11119 Bellflower Road, Cleveland, OH 44106, USA

b

Received 19 May 2003; received in revised form 30 September 2003; accepted 13 November 2003 Available online 30 July 2004

Abstract We report results from a trust-game experiment comparing behavior in an institution in which relationships end at a definite time, terminating concern for the future, with behavior in an institution in which relationships end at an indefinite time, inducing concern for the future. Although the level of trust was the same in both institutions when subjects were inexperienced, it fell in the definite but not indefinite institution as subjects gained experience. The divergence in efficiency can be explained by the institutions’ initial effect on repeated-game strategies and by the evolution of these strategies over time in a best response manner. © 2004 Elsevier B.V. All rights reserved. JEL classification: C72; C91 Keywords: Trust; Repeated games; Experiments; Repeated-game strategies

1. Introduction Most contracts are incomplete and require some degree of trust to obtain an efficient outcome. Trust, as a lubricant of the social system (Arrow, 1974), can increase institutional efficiencies (Fukuyama, 1995; Knack and Keefer, 1997; Putnam et al., 1993).1 However, ∗

Corresponding author. Tel.: +1 216 368 5811; fax: +1 216 368 5093. E-mail addresses: [email protected] (J. Engle-Warnick), [email protected] (R.L. Slonim). 1 Carpenter et al. (2004) offer an example of an attempt to make this statement more precise by correlating laboratory behavior with the type of survey data found in this literature. 0167-2681/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jebo.2003.11.008

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institutions may also affect trust. For example, Bohnet et al. (2001) find that low and high probabilities of contract enforcement lead to more trust than medium enforcement levels. As another example, Fehr and G¨achter (2002) find that imposing explicit incentives can crowd out reciprocity. In this paper, we examine a “trust” game in which subjects make binary choices. Each subject starts with US$ 0.40. One subject, the Trustor, may send all or none of his money to a second subject, the Trustee. If the Trustor does not send, the game ends. If he sends, his US$ 0.40 is doubled and given to the Trustee who may return either US$ 0.60 or nothing.2 We refer to the send and return actions as trust and reciprocity.3 We examine two institutions (i.e., two sets of rules of the game) that determine how many rounds subjects play this game with the same partner. In the “finite” institution subjects have the same partner for five consecutive rounds. In the “indefinite” institution the number of rounds that subjects have the same partner is determined stochastically: after every round a random draw determines whether subjects play the game again with the same partner or with a new partner. In both institutions, subjects play the repeated game many times.4 The Trustor’s send action in the repeated game is consistent with the definition of trust proposed by Mayer et al. (1995, p. 712) that trust is “the willingness of a party to be vulnerable to the actions of another party based on the expectation that the other will perform a particular action important to the Trustor, irrespective of the ability to monitor or control that other party.” Consistent with the repeated games we study, their definition stresses that trust is the willingness to engage in risk taking behavior and that Trustors may possibly monitor and respond to Trustee actions.5 In the finite institution, trust behavior may occur, but it must do so despite game-theoretic predictions that assume that players are only concerned about their own monetary payoff. Trust in this institution requires people to make themselves vulnerable to others, knowing that it is not possible to punish untrustworthiness sufficiently because the relationship will be terminated too soon. In the indefinite institution, trust behavior may occur for whatever reason it occurs in the finite game, but may also occur as a result of strategic play of equilibrium strategies. Trust, as part of an equilibrium strategy, can be achieved through concern for the future, i.e., even without institutions having contract enforcement mechanisms (Ostrom, 1998; Greif, 1993).6

2 Berg et al. (1995, p. 126) state that their similar game satisfies the following conditions for trust: “(1) Placing trust in the Trustee puts the Trustor at risk; (2) relative to the set of possible actions, the Trustee’s decision benefits the Trustor at a cost to the Trustee; and (3) both Trustor and Trustee are made better off from the transaction compared to the outcome which would have occurred if the Trustor had not entrusted the Trustee.” 3 Cox (2004) defines a measure of trust and reciprocity as the difference between the actions observed in a oneshot trust game and those observed in an analogous one-shot dictator game, thus controlling for other-regarding preferences. We abstract from this issue and refer to actions in the trust game itself as trust and reciprocity. 4 Bereby-Meyer and Roth (2003) and Selten and Stoeker (1986) also experimentally examine repeated supergames. 5 The game does not, however, separate trust from risk: Bohnet and Zeckhauser (2004) hypothesize and demonstrate that trust involves more than risk since trust, but not risk, may lead to betrayal. 6 Our experimental evidence is consistent with two types of trust found in the social capital literature. Generalized trust is trust that is extended to a population consisting of unknown people. Strategic trust is trust extended to a particular person based on a history of experiences with her. Generalized trust may be operative as subjects are

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In the data, we find that the level of trust is indistinguishable between the two institutions when subjects are inexperienced. However, as subjects gain experience, the level of trust decreases in the finite game but does not change in the indefinite game. Further, regardless of experience or institution, trust declines across rounds within the repeated game but resets when new relationships begin. The decline across rounds and the reset at the start of new relationships is consistent with existing results in other finitely repeated games with the potential for cooperative behavior (e.g., Andreoni, 1988; Camerer and Weigelt, 1988; Selten and Stoeker, 1986). The divergence in the level of trust between the two institutions is a new finding. This paper makes two contributions to our knowledge about trust. First, although the level of trust is indistinguishable between the finite and indefinite games when players are inexperienced, suggesting that the unobserved reason for trust behavior may also be similar, in contrast we find evidence of almost entirely distinct repeated-game strategies across the institutions: Trustors in the finite and indefinite game use distinct strategies to play the repeated-game. Using an inference procedure developed in EngleWarnick and Slonim (2002), we infer Trustor strategies in the finite game that contain many behaviors including na¨ıve, punishment, and endgame behaviors. In the indefinite game we infer only one strategy. This strategy considers the indefinite future relationship length; in every round it threatens to permanently stop trusting if the Trustee is untrustworthy.7 Second, we find that although the level of trust falls in the finite but not indefinite game, suggesting that behavior evolves differently in the two institutions, in contrast we find that strategies evolve in a similar best response manner for both player types in both institutions.8 When there is consideration for the future (the indefinite game) trust remains high, and when there is not consideration for the future (the finite game) trust collapses. The paper proceeds with a description of the experimental procedures and then describes how we infer repeated-game strategies from the data. Results and the conclusion follow.

2. Experimental design Subjects were randomly and anonymously chosen to be a Trustor or a Trustee for an entire session (the instructions referred to the Trustor and Trustee as Player A and B, respectively). Trustors and Trustees were randomly and anonymously paired to play each repeated trust randomly and anonymously paired with another person drawn randomly from her population, and strategic trust may be operative during the course of the fixed pairings within the supergames. Uslaner (2002) and references therein discuss various definitions of types of trust. 7 This strategy exacts the maximum punishment for betrayal of trust in the repeated game. Bohnet and Zeckhauser find in a one-shot trust game that Trustors demand a higher “minimum acceptable probability” for the good outcome when a second subject plays the role of the Trustee than when nature assumes the role and they attribute the difference to betrayal costs. 8 Thus, we isolate determinants of trust levels in our repeated games as best responses to opponent strategies. Other papers in this volume study other determinants of trust, such as social distance (Buchan and Croson, 2004), socio-economic background (G¨achter et al., 2004) and gender and race (Eckel and Wilson, 2004).

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game. Within a repeated game the person with whom a subject was paired stayed the same. Across consecutive repeated games subjects were always paired with a different partner. Every round of the repeated game began with all subjects receiving US$ 0.40. To start each round, Trustors chose between the actions Send and Don’t Send. If the Trustor chose Don’t Send, the round ended with each participant pocketing US$ 0.40. If a Trustor chose Send, then his US$ 0.40 was doubled and given to the Trustee. The Trustee then chose between the actions Return and Keep. If the Trustee chose Return the pair split the US$ 1.20 evenly, and if the Trustee chose Keep then she earned US$ 1.20 and the Trustor earned nothing.9 Subjects only knew the outcomes for the games they participated in. In the finite treatment every repeated game had five rounds. In the indefinite treatment each round was the last round with a fixed probability of 0.20 (i.e., with probability 0.80 the person with whom a subject was paired stayed the same, and with probability 0.20 a subject was randomly and anonymously paired with a different person to start a new repeated game). After reach round, the subjects learned whether the round was the last in the relationship or whether the relationship continued. All subjects experienced the same realizations of supergame (i.e., repeated-game) lengths, which were drawn in advance from the stated distribution. All procedures were common knowledge except that subjects were unaware of the total number of repeated games. Subjects were recruited for two-hour sessions and told that they would play many games. The longest session took 110 min and the average time was 90 min. Subjects played fifty supergames in every session. The average realized supergame length (i.e., number of rounds in the repeated game) in the indefinite treatment had 5.2 rounds, which is slightly but insignificantly greater than the expected length of 5.0. The shortest and longest supergames had 1 and 16 rounds, respectively. We ran four indefinite and four finite game sessions. There were three sessions with 20 subjects (one in the indefinite and two in the finite treatment), three sessions with 18 subjects (one in the indefinite and two in the finite treatment), and two sessions with 16 subjects (both in the indefinite treatment). The sessions were run at the University of Pittsburgh Economics Laboratory. Subjects received a US$ 5.00 participation fee plus their earnings from six supergames that were randomly selected at the end of each session. The finite and indefinite institutions contrast the two cases in which subjects within a supergame do and do not have concern regarding the future. Pairing subjects anonymously provides a tough test for the evolution of trust since subjects were unable to build a reputation across the repeated games.10 Having the same expected length across both institutions (five rounds) controls for the possibility that different expected lengths induce different behavior. We examine fifty supergames so that subjects had ample opportunity to gain experience. In the stage game, the subgame perfect equilibrium is for the Trustor to play Don’t Send and for the Trustee to play Keep if the Trustor plays Send. Playing these actions in each round is the unique subgame perfect equilibrium of the finitely repeated game and is 9

A version of this game with a larger strategy space was originally experimentally studied by Berg et al. Camerer (2003) provides a survey of trust experiments. 10 Keser (2002) finds greater levels of trust develop as the Trustee’s greater reputation is made available to the Trustors.

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one of many possible equilibrium strategies of the indefinitely repeated game. By the Folk Theorem of Repeated Games, the minimum discount factor required in the indefinite game to sustain trust in any equilibrium is 0.75 (e.g., see Fudenberg and Maskin, 1986).11 Given the discount rate of 0.80 used in the experiment, trust is thus sustainable in equilibria in the indefinite game, so trust may be the result of strategic behavior in the indefinite (but not finite) game. 3. Inferring repeated-game strategies from actions The better we understand the repeated game strategies that subjects use, the better we understand the observable trust and reciprocity actions. The difficulty in determining the strategies that subjects use is that we observe actions and not the strategies that generate them. To infer repeated game strategies from actions, we use a strategy model and procedure; Engle-Warnick and Slonim (2002) provide the details.12 To model repeated-game strategies we use finite automata. Finite automata are decision rules that indicate the actions to play (e.g., Send and Don’t Send for the Trustor and Return and Keep for the Trustee) in response to a sequence of actions taken by a partner. They are often used in economic theory to explore issues related to complexity, and they can represent many behaviors. For instance, they can represent naive behavior (e.g., unconditionally always trust), waiting or counting behavior (e.g., reciprocate for three rounds, then stop reciprocating forever), and punishment behavior (e.g., do not trust for two rounds whenever a partner is untrustworthy).13 Restricting the strategy space to finite automata does not qualitatively change the set of repeated-game equilibrium outcomes (e.g., Abreu and Rubinstein, 1988; Binmore and Samuelson, 1992). To determine the (finite automata) strategies that generate the data, we start with the same set of automata that Engle-Warnick and Slonim (2002) use. This set contains 18 strategies for the Trustors and 32 strategies for the Trustees. These strategies represent a rich set of possible behaviors, including equilibrium strategies. From this set, we find a subset of these strategies that maximizes the goodness of fit of the data from half of the sessions.14 We then verify the goodness of fit of these inferred “best fitting” strategies on the remaining statistically independent sample of sessions.15 The inference procedure simultaneously 11 This is verified by specifying that the Trustee always plays Return, and the Trustor plays Send in the first round, Send if the Trustee plays Return, and Don’t Send forever if the Trustee plays Keep in any round. 12 For complementary methods for getting at underlying strategies see the success of “Tit for Tat” in computer tournaments in Axelrod (1984), direct strategy elicitation using the “strategy method” in Selten et al. (1997), and the collection of attentional data in Costa-Gomes et al. (2001). 13 The finite automata we use are, strictly speaking, not game-theoretic strategies since they are unable to respond to contingencies that arise if they make incorrect decisions. 14 Since the procedure determines the number of strategies (automata) that are required to describe the data, one concern is that of over fitting the data: adding a strategy weakly increases the number of observations fit by a set of strategies. To avoid over fitting the data, the procedure imposes a penalty cost proportional to the number of strategies inferred: increasing the number of inferred strategies only occurs if the marginal goodness of fit of adding a strategy exceeds this penalty cost. 15 An automaton fits a supergame if it replicates a subject’s decision when played against the actions of the subject’s partner. A set of automata fits a set of supergames if each supergame is fit by at least one automaton in the set.

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determines the number of strategies and the specific strategies that are consistent with and describe the data. The results section will describe the behavior of the inferred best-fitting automata strategies. 4. Results Fig. 1 shows the actions that subjects chose across rounds and supergames in the finite and indefinite games. The figure shows the proportion of times that Trustors play Send and Trustees play Return (where the proportion of times that Trustees play Return is conditional on Trustors playing Send). Each figure shows the proportions averaging over 10 consecutive supergame intervals. Within each interval the figure shows the proportions for each round. In the finite game we show behavior for all five rounds. In the indefinite game we show behavior for the first seven rounds since there are not many supergames with more than seven rounds in any interval. Fig. 1 indicates that the proportion of Send actions decreases dramatically across rounds.16 For instance, during supergames 11–20 of the finite game Trustors play Send 95 percent of the time in round 1 but only 10 percent of the time in round 5. The smallest decrease from the first to fifth round for any interval in the finite game is nearly 50 percent. A similar though less steep decline occurs for the indefinite game Trustors; the decline from the first to the fifth round for them is between 35 and 50 percent across every interval.17 We present the main results by stating each finding and then providing the evidence. Result 1. The aggregate efficiency of both institutions is the same when subjects are inexperienced. Table 1 shows how often Trustors and Trustees play Send and Return in the first round of the first and last supergames. It shows that in the first round of the first supergame, Trustors play Send 9 percent less often in the finite than indefinite game, and Trustees play Return 4 percent more often in the finite than indefinite game. To test whether the finite and indefinite institutions significantly affect inexperienced players actions, we estimate the following logit models: • Model 1: Trustor Regression: Sendi = f(a + βfini ) • Model 2: Trustee Regression: Returni = f(a + βfini ) where Sendi = 1 if Trustor i plays Send and 0 otherwise, Returni = 1 if Trustee i plays Return, 0 if she plays Keep and “missing” if the Trustor does not play Send, f( ) is the logit 16 This decrease within repeated games has been observed in other social dilemmas with different matching mechanisms. See, for example, Clark and Sefton (2001) and the references therein. 17 The proportion of Return actions taken by Trustees also decreases across rounds during the first block of supergames, but the magnitude of this decrease falls across supergames. In fact, after the first ten supergames, the Trustees play Return at an increasing rate across some rounds. This increase is likely due to selection bias since the Trustee actions are observed only after the Trustors play Send. This selection bias occurs if the Trustors can use information from an earlier round to condition their choice to play Send with the Trustees who are more likely to play Return.

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A: Finite Games 1.00

Trustor

Proportion of Send and Return Actions

0.90

Trustee

0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 SG 1-10

SG 11-20

SG 21-30

1 2 3 4 5

1 2 3 4 5

SG 31-40

SG 41-50

0.00 1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

B: Indefinite Games

Proportion of Send and Return Actions

1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20

Trustor

Trustee

0.10 SG 1-10

SG 11-20

SG 21-30

SG 31-40

SG 41-50

0.00 1 2 3 4 5 6 7

1 2 3 4 5 6 7

1 2 3 4 5 6 7

1 2 3 4 5 6 7

1 2 3 4 5 6 7

Fig. 1. Proportion of send and return actions by intervals of 10 supergames (SG).

function and fini = 1 if subject i is in the finite game and 0 otherwise. We find that actions are not significantly different across the institution when subjects are inexperienced, so we cannot reject β = 0 (P > 0.20 for both player types). Since the level of trust is unaffected by the institution, efficiency is also unaffected (since the outcome is efficient if and only if Trustors play Send). Result 1 indicates that the finite and indefinite institution did not affect the initial level of trust when subjects were inexperienced. This result might suggest, therefore, that strategic

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Table 1 Initial actions (in proportions) in the first and last supergame Player action

First supergame

Last supergame

Trustor send

Trustee return

Trustor send

Trustee return

Finite Indefinite P-value

0.82 (n = 38) 0.91 (n = 35) >0.20

0.81 (n = 31) 0.77 (n = 32) >0.40

0.53 (n = 38) 0.89 (n = 35) 0.07

0.90 (n = 20) 0.87 (n = 31) >0.40

Note: The P-value is the value for the coefficient on institution, which measures the difference between finite and indefinite games.

repeated-game behavior across the two institutions was also unaffected when subjects were inexperienced. However, as we will show below (Result 4), strategic behavior was quite distinct when subjects were inexperienced. Indeed, one of the central findings of this paper is that the institutions affected initial strategic behavior despite observable similar initial levels of trust. Result 2. The aggregate efficiency of the institutions diverges as subjects gain experience. Fig. 1 shows a striking change as subjects gain experience; Trustors in the finite but not indefinite game decrease the rate they play Send. For instance, in every round the finite game Trustors play Send less often in the last than (round 1:92 percent versus 58 percent; round 2:88 percent versus 43 percent; round 3:83 percent versus 32 percent; round 4:67 percent versus 16 percent; round 5:27 percent versus 9 percent). Less dramatic, the finite game Trustees directionally play Return less often across supergames while the indefinite game Trustees play Return more often. For instance, in every round the indefinite game Trustees play Return more often in the last than first 10 supergames (round 1:81 percent versus 92 percent; round 2:83 percent versus 90 percent; round 3:67 percent versus 84 percent; round 4:61 percent versus 75 percent; round 5:50 percent versus 70 percent; round 6:66 percent versus 81 percent; round 7:50 percent versus 84 percent). Estimating Model 1 and Model 2 for subject choices in the first round of the last supergame (and controlling for session effects since subject choices are no longer independent observations within session), we find that the Trustee actions remain insignificantly different across the institutions (P > 0.40), but the Trustors are significantly less likely to Send in the finite than indefinite game. While the finite game was insignificantly (9 percent) less efficient than the indefinite game in the first round of the first supergame, by the last supergame the finite game institution becomes significantly (36 percent) less efficient: the Trustors are becoming less trusting in the finite than indefinite game. Consistent with this reduced efficiency, Fig. 2 shows that the average payoff per round decreases steadily in the finite game for both player types as they gain experience. The Trustees are financially hurt the most by the decrease in trust; from the first to the last 10 supergames the Trustees lose 10 cents per round (or in expected value they earn US$ 5 less in the last than first 10 supergames), whereas the Trustors only lose on average 4 cents per round. In the indefinite game, the Trustors actually gain 4 cents per round while the Trustees lose 7 cents per round. In sum, the finite game players lose nearly five times as much from the first to last ten supergames (US$ 7 in expected value) as the indefinite game players

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Average Earnings Per Round $0.65

Indefinite-Trustors Indefinite-Trustees Finite-Trustors Finite-Trustees

$0.60

$0.55

$0.50

$0.45

$0.40 SG 1-10

SG 11-20

SG 21-30

SG 31-40

SG 41-50

Fig. 2. Average earnings per round by supergame (SG) interval.

(US$ 1.50 in expected value). Determining why the Trustors in the finite but not indefinite game become less trusting is now addressed. Result 2 indicates that the finite and indefinite institutions lead to a divergence in the level of trust as subjects gained experience. This result might suggest, therefore, that strategic repeated-game behavior across the two institutions was also diverging as subjects gained experience. However, as we will show below (Result 5), strategic behavior was evolving in a similar best response manner as subjects gained experience. However, the best response behavior, in combination with the distinct institutional details, leads to different levels of trust evolving as subjects gained experience. Result 3. Experienced Trustors are more likely to terminate relationships with an exclusively trusting and reciprocating history in the finite than indefinite game. However, inexperienced Trustors and Trustees and experienced Trustees are equally likely to terminate relationships in the finite and indefinite games. Fig. 3 presents the proportion of times that the players deviate from a mutually trusting (i.e., Send–Return–Send–Return–etc.) relationship. We refer to the proportion of actions that deviate from this mutually trusting relationship as hazard rates. We again show the proportions both across supergame intervals and rounds within each interval. These figures show that in virtually every round of every interval less than half the subjects stop trusting or stop reciprocating if the relationship has a mutually trusting history. In other words, the most common choice when a relationship has a mutually trusting history is to maintain the trusting relationship. We thus first examine choices while relationships are mutually trusting since this history represents a high frequency event in the data.

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Hazard Rates (Proportion Trustees Don't Send and Trustors Send given mutually trusting relationship)

A: Finite Games 0.70

SG 1-10

SG 11-20

SG 21-30

SG 31-40

SG 41-50

0.60

Trustor

Trustee

0.50 0.40 0.30 0.20 0.10 0.00

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

B: Indefinite Games Hazard Rates (Proportion Trustees Don't Send and Trustors Send given mutually trusting relationship)

0.50

SG 1-10

SG 11-20

SG 21-30

SG 31-40

SG 41-50

0.40

Trustor

Trustee

0.30

0.20

0.10

0.00

1 2 3 4 5 6 7

1 2 3 4 5 6 7

1 2 3 4 5 6 7

1 2 3 4 5 6 7

1 2 3 4 5 6 7

Fig. 3. Hazard rates for relationship histories with full trust and reciprocity by intervals of 10 supergames (SG).

Panel 3A shows a fairly remarkable change in the hazard rate for both the finite game Trustors and Trustees as they gain experience. During the first twenty supergames the hazard rate for both players is less than 10 percent in the first round and rises steeply through the fourth round. However, by the last 10 supergames, the first round hazard rate for the Trustors is over 40 percent, and although the hazard rates for the Trustors appear quite different across the supergames, by the last thirty supergames the hazard rates for the Trustees appear to

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converge in a qualitative sense: they appear normal shaped across rounds with a peak during the third and fourth rounds. In contrast with the finite game volatility, Panel 3B shows a remarkably constant pattern of hazard rates across supergames in the indefinite game. The Trustors are most likely to deviate from a mutually trusting relationship in the first round than in any other round. In all but the first round, the Trustors never deviate from the trusting action more than 10 percent of the time, and often deviate less than 5 percent of the time. In contrast, the likelihood that Trustees deviate is greatest in the third through sixth rounds and is greater for Trustees than Trustors in all but the first round of all the intervals. In other words, the Trustees are the major culprits causing mutually trusting relationships to fail in the indefinite but not finite game. The “hazard strategies” provide a useful starting point for examining strategic behavior since they measure the event that triggers the efficiency loss either for the current round (if the Trustors deviate first) or for future rounds (if the Trustee deviates first and the Trustor consequently stops playing Send).18 To model hazard strategies, we assume players deviate from the mutually trusting relationship with a probabilistic distribution conditional on the round and unobservable individual characteristics. To test whether there is any difference between the finite and indefinite game, we estimate the survival rate (i.e., one minus the hazard rate) for each player type. Fig. 4 depicts the cumulative survival rate for the first and last supergame for each institution. The figures show that the cumulative finite game survival rate in the last supergame is in every round less than the other three survival rates. To examine differences in survival rates across the games, we estimate the following survival models: • Model 3: Trustors Regression: Sendi = D(fini ) • Model 4: Trustees Regression: Returni = D(fini ) where the indicator variables Sendi , Returni , and fini are defined above and D( ) is the survival rate distribution. We estimate survival rate models for each player type and for the first and last supergame separately using several distributional assumptions.19 Table 2 provides the survival rate regression results from the distribution that best fit the data using the AIC criterion for model selection. The table confirms what is visible in Fig. 4. First, there is no statistical difference in the median survival rate between the finite and indefinite game during the first supergame for either the Trustor or Trustee (P > 0.20). However, the median survival rate for the Trustors but not Trustees during the last supergame is significantly shorter in the finite than indefinite game (P < 0.01). Thus, the hazard rate model results suggest, similar to Result 1, that the institutions do not affect trust for inexperienced players and that, similar to Result 2, experienced players are less trusting in the finite than indefinite game. As we now show, however, we find that the repeated-game strategies of the inexperienced Trustors are distinct across the two 18 “Hazard strategies” are not true strategies as they do not provide a plan of action for any possible contingency; they only specify a plan to break a mutually trusting relationship. 19 We used five distributions (D) for the survival rate: D = exponential, weibull, log-normal, log-logistic and generalized gamma. All the estimated distributions provide the same conclusion regarding the effect that the finite and indefinite games have on the survival rate.

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A: Survival Rate In The First Supergame 1.00 Finite Game

0.90

Indefinite Game 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 Rd 1: Trustor

Rd 1: Trustee

Rd 2: Trustor

Rd 2: Trustee

Rd 3: Trustor

Rd 3: Trustee

Rd 4: Trustor

Rd 4: Trustee

Rd 5: Trustor

Rd 5: Trustee

B: Survival Rate In The Last (50th) Supergame 1.00 0.90

Finite Game

0.80

Indefinite Game

0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 Rd 1: Trustor

Rd 1: Trustee

Rd 2: Trustor

Rd 2: Trustee

Rd 3: Trustor

Rd 3: Trustee

Rd 4: Trustor

Rd 4: Trustee

Rd 5: Trustor

Rd 5: Trustee

Fig. 4. The probability of trust and reciprocity surviving by round in the first and last supergame.

institutions and that these strategies evolve in a similar best response manner as subjects gain experience. Result 4. Inexperienced Trustors use different strategies in the indefinite and finite institutions. Result 5. Although different in the two institutions, the strategies evolve in a similar (best response) manner for Trustors and Trustees in both institutions.

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Table 2 Survival model results for supergames with trust/reciprocity history Player

First supergame

Last supergame

Trustor

Trustee

Trustor

Trustee

Number of observations Rounds at risk

73 240

63 222

73 221

51 189

Parameter estimates Finite indicator variable Constant

−0.22 2.33∗∗

−0.06 1.91∗∗

−1.28∗∗ 2.14∗∗

−0.15 1.44∗∗

Distribution having lowest AIC Log-likelihood Akaike’s AIC

Log-normal −52.50 113.00

Generalize gamma −66.59 143.18

Log-normal −68.52 145.04

Log-logistic −49.34 106.68

Note: A negative and significant parameter estimate for “Finite” indicates the players’ survival rate for trust or reciprocity is shorter in the finite games than in the indefinite games. ∗∗ Significance at the 1 percent level.

Despite the previous analysis that shows inexperienced subject actions are similar across institutions, we now show that repeated-game strategies consistent with inexperienced Trustor actions are entirely different across the institutions. Further, despite the previous analysis that suggests behavior diverges across the institutions as subjects gain experience, we also show that strategies evolve in a similar best response manner for Trustors and Trustees in both institutions and that this best response behavior can explain the divergence in trust across the institutions. 4.1. The inferred strategies Table 3 lists the inferred Trustor and Trustee strategies during each interval of the finite and indefinite games.20 The lower half of Table 3 describes these strategies. Reading the Trustor strategies from top to bottom, they exhibit the following behavior: the first four strategies, called Period Counters, conditionally play Send for the first four, three, two, and one rounds before playing Don’t Send for the duration of the game; the next two strategies, called Never Send and Always Send, unconditionally always play the same action; the final two strategies, called Good and Grim Trigger, play Send in round 1 and continue to play Send in subsequent rounds conditional on the opponent’s action, and play Don’t Send forever upon seeing the opponent’s other action. The inferred Trustee strategies have the following behavior: the first four strategies, also called Period Counters, conditionally play Return for four, three, two and one rounds before playing Keep for the duration of the supergame; the next two strategies, called Always Keep and Always Return, unconditionally always play the same action; the next strategy plays keep in the first round; the last strategy plays Return the first time Trustor plays Send, no matter what round this occurs, and then plays Keep thereafter. 20

Given only 50 repeated-game observations per subject and the results (below) indicating that the inferred strategies change as subjects gain experience, we are unable to infer meaningfully individual subjects’ strategies.

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Table 3 Repeated-game strategies inferred from subject actions Supergame interval Trustor

Five-period counter four-period counter Three-period counter Two-period counter Never send Always send Good trigger Grim trigger

Trustee

1–10

11–20

21–30

F∗ F∗

F∗ F∗ F∗

F F∗

F∗

F F∗ F∗ F∗ F F∗

I∗

I∗

I∗

31–40

41–50

Automata strategies

1–10 F∗ I, F∗

F∗

F∗

F∗ I∗ , F

I∗ , F

Five-period counter Four-period counter Three-period counter Two-period counter Always keep Always return Keep first round Return once only

F∗

Description of inferred strategies Five-period counter Send in rounds 1–4 then Don’t Send, Don’t Send always after Trustee Keeps Four-period counter Send in rounds 1–3, then Don’t Send, Don’t Send always after Trustee Keeps Three-period counter Send in rounds 1–2, then Don’t Send, Don’t Send always after Trustee Keeps Two-period counter Send in round 1, then Don’t Send, Don’t Send always after Trustee Keeps Never send Don’t Send in all rounds Always send Send in all rounds Good trigger Send until Trustee Returns then Don’t Send Grim trigger Send until Trustee Keeps then Don’t Send

I∗ , F I I

11–20

21–30

31–40

41–50

I, F∗ F∗ F∗

I F∗ F∗

I, F F

I∗ , F

I∗ , F I

I, F∗ I, F∗ F∗ F∗ I∗ , F

F I∗ , F

I

Description of inferred strategies Five-period counter Return in rounds 1–4 then Keep, Keep always after Trustor Doesn’t Send Four-period counter Return in rounds 1–3 then Keep, Keep always after Trustor Doesn’t Send Three-period counter Return in rounds 1–2 then Keep, Keep always after Trustor Doesn’t Send Two-period counter Return in round 1 then Keep, Keep always after Trustor Doesn’t Send Always keep Keep in all rounds Always return Return in all rounds Keep first round Keep in round 1 Return once only Return first time Trustor Sends, then Keep

Note: This table lists the strategies inferred in supergame intervals by player type. A letter “F” in a cell indicates a strategy inferred in the finite games, a letter “I” in a cell indicates a strategy inferred in the indefinite games, and an asterisk “*” indicates the inferred strategy is a best response to an opponent’s inferred strategy. There were ten uninferred Trustor strategies and 24 uninferred Trustee strategies.

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Automata strategies

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Table 4 Strategies as best responses to opponent play (in proportions) Treatment

Trustor

Trustee

Finite

Indefinite

Finite

Indefinite

Available strategies Inferred strategies P-value

0.23 (n = 90) 0.70 (n = 20) <0.001

0.17 (n = 90) 1.00 (n = 5) <0.001

0.45 (n = 160) 0.63 (n = 19) 0.034

0.03 (n = 160) 0.28 (n = 15) <0.001

Note: The P-value is the value for a binomial test of the null hypothesis that the proportion of best response strategies inferred is equal to the proportion of best response strategies available for inference; rejection indicates that inferred strategies are disproportionately best responses to inferred opponent strategies.

Table 3 shows that the strategies that subjects use include a rich heterogeneous set of behaviors. We infer strategies that count, conditionally punish, and play n¨aively for both player types. Not only are strategies different across the institutions, but Trustors also use only one strategy in the indefinite game, but eight in the finite game. Many strategies are inferred at different intervals in the finite game for both Trustors and Trustees. We now discuss the evolution of strategic behavior. 4.2. The inferred strategies as best responses We want to know if the strategies we infer are in some way best responses to the empirical distribution of opponent play. To explore this question, we say that a strategy is a best response to an opponent’s inferred strategy if, when played against the opponent’s strategy, it produces an expected payoff that is at least as high as any other available strategy. Thus, we are looking at a constrained best response where the strategy space is constrained to the set of available strategies. To identify best response strategies, we play all the available strategies against each inferred opponent strategy and note the strategies that produce the highest expected payoff. For each interval, we define the opponents’ strategies as those inferred in the same or previous interval.21 Fig. 5 presents the percent of available and inferred strategies that are best responses to opponent strategies for both player types and games for each interval. For instance, Panel A shows that for the finite game, Trustors during the first ten supergames, 17 percent (3/18) of the available strategies were best responses while 75 percent (3/4) of the inferred strategies were best responses. Fig. 5 shows that the inferred strategies for Trustors and Trustees in the finite and indefinite games are disproportionately best responses to opponent play; across all supergames, player types and games, the percent of inferred strategies that are best responses is almost always greater than the percent of available strategies that are best responses. Aggregating across supergames for each player type and game, Table 4 shows that we infer 47, 18, 83 and 25 percent more best response strategies than if we randomly picked available strategies for the finite game Trustors and Trustees and indefinite game Trustors 21

For instance, the opponents’ strategies used to assess whether the Trustor strategies are best responses during Supergames 31–40 are the inferred Trustees’ strategies during Supergames 31–40 and Supergames 21–30.

568

All Available Strategies Inferred Strategies

A: Trustor Finite Games

Percentage Of Strategies That Are Best Responses To Inferred Opponent Strategies

90%

Percentage Of Strategies That Are Best Responses To Inferred Opponent Strategies

100%

90%

70% 60% 50% 40% 30% 20% 10% 0%

80% 70% 60% 50% 40% 30% 20% 10% 0%

1-10

11-20

21-30

31-40

41-50

All SGs

1-10

11-20

21-30 31-40 41-50 Supergame (SG) Intervals

Supergame (SG) Intervals

C: Trustor Indefinite Games

All Available Strategies Inferred Strategies

D: Trustee Indefinite Games 90%

Percentage Of Strategies That Are Best Responses To Inferred Opponent Strategies

90%

Percentage Of Strategies That Are Best Responses To Inferred Opponent Strategies

100%

100%

80% 70% 60% 50% 40% 30% 20% 10%

All SGs

All Available Strategies Inferred Strategies

80% 70% 60% 50% 40% 30% 20% 10% 0%

0% 1-10

11-20

21-30 31-40 41-50 Supergame (SG) Intervals

All SGs

1-10

11-20

21-30

Fig. 5. Strategies as best response to opponent play (in percentages).

31-40

41-50

Supergame (SG) Intervals

All SGs

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100%

80%

All Available Strategies Inferred Strategies

B: Trustee Finite Games

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Table 5 Number of inferred strategies emerging and/or disappearing Player type

Trustor

Finite Indefinite

12 0

Trustee 7 8

Note: This table lists the number of inferred strategies that emerge or disappear during the finite and indefinite sessions, by player type.

and Trustees, respectively. For each player type and game we reject the null hypothesis that we infer strategies that are randomly chosen from the available strategy set in favor of inferring strategies that are best responses (binomial test: P < 0.05 for finite game Trustees and P < 0.001 for finite game Trastors and both indefinite game player types). Thus, the inferred strategies are consistent with our notion of constrained best response dynamics in both treatments for both player types. 4.3. The evolution of the inferred strategies Table 5 shows the frequency that strategies emerge and disappear, where emergence is defined as being inferred in an interval after not being inferred in the previous one, and disappearance is defined as not being inferred in an interval after being inferred in the previous one. Across the players and games, there are 27 instances where strategies emerge and disappear. A test for whether these changes are equally distributed across the finite and indefinite Trustors and Trustees is rejected (␹2 1 : P < 0.002). Table 5 shows that the Trustor strategies emerge and disappear less often in the indefinite than finite game, but the Trustee strategies change equally often across the institutions. Thus, Trustors and Trustees are best responding to opponent play in both institutions, and this best response behavior leads to more changes in the strategies that the Trustors use in the finite than indefinite game. We now describe how the strategies inferred across the supergame intervals imply that trust unravels in the finite but not indefinite game. Starting with the finite game, Table 3 shows that we infer four Trustor strategies during the first interval (Supergames 1–10). Table 3 also indicates (with asterisks) that three of these strategies are best responses to the inferred Trustee strategies. These best response strategies trust for the first four rounds if they play against any of the inferred Trustee strategies. During this interval we also infer three Trustee strategies, and two are best responses to the inferred Trustor strategies. These two strategies reciprocate trust for the first three rounds if they play against the inferred Trustor strategies. During the second interval (Supergames 11–20), all four inferred Trustor strategies are best responses to the inferred Trustee strategies; however, only one of these strategies will trust for the first four rounds if it plays against any of the inferred Trustee strategies. Thus, among the inferred best response strategies, less trust will occur during the second than first interval. During this interval, we also infer four Trustee strategies, three of which are best responses. Only one of these best response strategies will reciprocate trust for three rounds. Thus, even after 20 supergames, trust and reciprocity begin to unravel. By the last 20 supergames, trust and reciprocity unravel much further. In the fourth interval (Supergames 31–40), we infer three Trustor strategies that are best responses to

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inferred Trustee strategies, and none of these strategies will trust past the second round (and only one will trust even in the second round) if they play an inferred Trustee strategy. During this interval, we also infer four Trustee strategies that are best responses to Trustor strategies, and only one reciprocates trust past the second round. In the last interval, when subjects have the most experience, we only infer one Trustor strategy that is a best response to the inferred Trustee strategies, and this strategy (Never Send) will never trust when playing with any Trustee; thus, best responding to opponents’ strategic behavior has lead to trust collapsing in the finite game. There is only one strategy that we infer across all supergames for the finite game Trustees. This strategy (Always Return) unconditionally plays Return in every round that the Trustor plays Send. Inferring this strategy across all supergames indicates that trustworthiness does not decrease across rounds for at least some proportion of Trustees, even in the last round when there is no possible future interaction in the repeated game. This strategy suggests that some Trustees may have other regarding preferences (e.g., see Bolton and Ockenfels, 2000; Fehr and Schmidt, 1999). However, this strategy may also be inferred since there are not many times that the Trustors play Send in the late rounds of the repeated game. For instance, by the last round of the last supergame, no Trustors ever play Send. Further, to the extent that the Trustors do play Send in the late rounds, it is often conditional on Trustees playing Return in all previous rounds, indicating that there is selection bias regarding which Trustees make choices in the later rounds of the Supergames. Thus, inferring the Always Return strategy suggests that trustworthiness is robust across rounds, but there may be alternative explanations for this behavior. In the indefinite game, we infer only one strategy for the Trustors. This strategy, called the Grim Trigger, plays Send as long as the Trustor plays Return and permanently plays Don’t Send if the Trustor is ever untrustworthy.22 This strategy is always a best response for one of the inferred Trustee strategies. For the Trustees, we infer between two and four strategies during each interval, and one of these strategies is always a best response to the inferred Trustor strategy.23 If the inferred best response Trustor and Trustee strategies play each other, then trust will occur in all rounds of the supergame; thus, best responding to opponents’ strategic behavior does not lead to any collapse of trust in the indefinite game. In sum, strategies evolve in a best response manner across players and institutions. As subjects gain experience, the inferred strategies trust increasingly less often in the finite but not indefinite game. Despite analysis of actions that suggest that inexperienced subject behavior was similar across the institutions, we infer distinct strategies for inexperienced Trustors across the institutions. Finally, we find that the strategic behavior that results in trust in the indefinite game appears to be due to the Trustors playing the Grim Trigger strategy. 22 Renner and Tyran (2004) restrict the space of possible punishment strategies to a grim strategy, thus cleanly isolating the difference between the effects of price shocks that are and are not common knowledge in a customer market. Our results suggest that the restriction is empirically relevant. 23 We also infer a second Trustee strategy across all supergames. This four-period counter strategy is trustworthy for the first three rounds and untrustworthy thereafter. It is interesting that we infer this strategy since it is never a best response to the inferred Trustor strategy. Some Trustees may use this strategy if they suffer from the gambler’s fallacy, and increasingly anticipate that the repeated game will end soon after the fourth round (however, since we do not see evidence of the gambler’s fallacy in the Trustors’ behavior, we are hesitant to draw this conclusion).

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5. Conclusion We present an experiment that investigates concern for the future by examining a trust game and comparing an institution in which relationships have definite ending points with an institution in which the ending point is indefinite. We find that efficiency in these two institutions is remarkably similar when subjects are inexperienced, but diverges as subjects gain experience. The data show that after acquiring experience, but not before, Trustors in games with definite ending points are the most likely to break off relationships with histories of full trust and reciprocity. A look at inferred repeated-game strategies shows that there is substantial decision-making heterogeneity and that this heterogeneity was qualitatively different across the two institutions. However, best response dynamics were similar across player types and institutions, and best response dynamics can explain the evolution of behavior in both institutions. A particularly strong theoretical implication stemming from the strategy inference results is the existence of the grim trigger strategy in the indefinitely repeated game. While the Folk Theorem of Repeated Games generally admits an embarrassment of riches with respect to equilibria, we uniquely inferred the strategy with the maximum possible punishment (i.e., permanent punishment) across all periods of the game for the Trustors. Inference of this strategy not only confirms the theoretical prediction, but also reduces the number of possible solutions that theory predicts are possible. Further empirical evidence for the reduction of the repeated-game strategy space can be found in Engle-Warnick and Ruffle (2002). Our results verify that concern for the future of a relationship with repeated interaction is important for trust to persist. We find that subjects apply different strategies to the different institutions and best respond to these strategies. We advise cautious interpretation of the repeated game behavior; we examine behavior across two simple institutions, and even in these simple environments subjects require substantial experience before we notice differences in trust. The interpretation of behavior, particularly in the finite game, would be different had we not collected fifty supergames of observations since differences in trust behavior emerge after twenty repeated games are played. Also, the interpretation of how the two institutions affect trust would be different had we not inferred underlying repeated-game strategies; without the strategy inference the data analysis suggests the institutions have no effect on inexperienced subject behavior, but the strategy inference suggests that the players use distinct strategies initially and always. Finally, though similar levels of trust occur across the institutions when subjects are inexperienced, suggesting trust exists regardless of the institution, we find that without consideration for the future, trust disappears, but with consideration for the future, trust persists.

Acknowledgments We thank Iris Bohnet, James Cox, Rachel Croson and Simon G¨achter for many helpful comments on an earlier version of this paper, and seminar participants at the University of Edinburgh, University of Oxford Experimental Economics Seminar (2002), the ESRC

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Centre for Economic Learning and Social Evolution (2002) and the Trust and Institutions Conference (2003) at the Radcliffe Institute for Advanced Study at Harvard University. We are also grateful to the University of Pittsburgh Economics Department and the Weatherhead School of Management at Case Western Reserve University for use of the experimental laboratory and funding. Much of this work was completed while Engle-Warnick was at Nuffield College, University of Oxford.

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