Individual characteristics and behavior in repeated games: an experimental study

Abstract

Using a laboratory experiment, we investigate whether a variety of behaviors in repeated games are related to an array of individual characteristics that are popular in economics: risk attitude, time preference, trust, trustworthiness, altruism, strategic skills in one-shot matrix games, compliance with first-order stochastic dominance, ability to plan ahead, and gender. We do find some systematic relationships. A subject’s compliance with first-order stochastic dominance as well as, possibly, patience, gender, and altruism have some systematic effects on her behavior in repeated games. At the level of a pair of subjects who are playing a repeated game, each subject’s gender as well as, possibly, patience and ability to choose an available dominant strategy in a one-shot matrix game systematically affect the frequency of the cooperate–cooperate outcome. However, overall, the number of systematic relationships is surprisingly small.

1 Introduction

Laboratory experiments on repeated games have shown that there is substantial heterogeneity across subjects: some cooperate a lot while others hardly do so (e.g., see Dal Bó and Fréchette (2011a, b) and Davis et al. (2010)).¹ This raises the question of whether the behavior of subjects in repeated games is related to other individual characteristics. We address this question by focusing on an array of individual characteristics that are popular in economics and may plausibly be related to behavior in repeated games: risk attitude, time preference, trust, trustworthiness, altruism, strategic skills in one-shot matrix games, compliance with first-order stochastic dominance, ability to plan ahead, and gender.

Research on the relationship between individual characteristics and behavior in repeated games could be useful in at least three ways. First, it could provide insights into what motivates the different kinds of behavior in repeated games. Second, such research could tell us whether (i) at the individual level, a player’s individual characteristics help predict her behavior in repeated games and (ii) at the pair level, the individual characteristics of a pair of players help predict to what extent cooperation would emerge if these players are matched to play a repeated game. Third, such research could be used to guide theoretical developments on repeated games.

To better understand the sources of heterogeneous play, we conducted a laboratory experiment in which each subject attended two sessions. In session 1, subjects played for several matches a two-player repeated game based on a “Mini-Bertrand” stage game, a three-price version of the regular Bertrand game.² Following the repeated Mini-Bertrand (RMB) games, subjects played for several matches a repeated Prisoner’s Dilemma (RPD). In session 2, subjects performed an array of tasks meant to measure the individual characteristics of interest.

We do find some systematic relationships between individual characteristics and behavior in repeated games. At the individual level, a subject’s compliance with first-order stochastic dominance as well as, possibly, patience, gender, and altruism have some systematic effects on her behavior in repeated games. At the pair level, each subject’s gender as well as, possibly, patience and ability to choose an available dominant strategy in a one-shot matrix game systematically affect the frequency of the cooperate-cooperate outcome. None of the remaining individual characteristics systematically affect behavior at either the individual or the pair level. Overall, the number of systematic relationships we find is surprisingly small.

Our paper adds to a small, but growing, experimental literature exploring the connection between behavior in indefinitely repeated games and individual characteristics that are popular in economics.³ Dreber et al. (2011) find that, when cooperation is an equilibrium with selfish preferences, behavior in the RPD is largely unrelated to giving in the dictator game, answers to survey questions about prosocial behavior outside the lab, and individual characteristics (such as age, belief in God, and risk attitudes); it also appears that, in line with our study, men are more cooperative than women.⁴ Reuben and Suetens (2012) and Cabral et al. (2012) provide evidence that behavior in indefinitely repeated games is mostly driven by strategic motives rather than by other-regarding preferences or non-strategic reciprocity.⁵

In contrast with our findings, Sabater-Grande and Georgantzis (2002) find that cooperative behavior in the RPD is negatively correlated with risk aversion. The discrepancy may be due to the fact that Sabater-Grande and Georgantzis used a version of the RPD that is somewhat different from the version used in our study. In their study, the first 15 rounds were played with shrinking payoffs of the stage game and a continuation probability of 1; in the remaining rounds, the stage game payoffs were unchanged and there was a positive probability of the game ending after each round.⁶

Our study contributes beyond previous studies in that we consider a wide array of additional individual characteristics that are popular in economics. In particular, to the best of our knowledge, no previous study has looked at the connection between, on the one hand, behavior in indefinitely repeated games, and, on the other hand, patience, trust, trustworthiness, strategic skills in one-shot matrix games, compliance with first-order stochastic dominance, and ability to plan ahead.

The remainder of the paper is organized as follows. In Sect. 2, we explain the experimental design. Section 3 explains how we quantify subjects’ behavior in repeated games. Section 4 examines the possible connections between the individual characteristics we consider and behavior in repeated games. Section 5 contains the data analysis. Section 6 concludes.

2 Experimental design

In our experiment, each subject attended two sessions that were one week apart. Subjects were grouped into cohorts with all subjects in a cohort attending sessions 1 and 2 together. There were 92 participants who attended session 1 and were grouped in three cohorts of roughly equal size; 87 of these participants returned for the second session.⁷

Subjects were undergraduate students at Virginia Commonwealth University (VCU). The experiment was conducted at the Experimental Laboratory for Economics and Business Research at VCU. Each session lasted around 2 hours. The experiment was programmed and conducted with the software z-Tree (Fischbacher 2007).

2.1 Session 1

Table 1 Stage games

In session 1, we used an RMB game and an RPD game. The stage games are shown in the two panels of Table 1. The Mini-Bertrand stage game is an especially designed version of a regular Bertrand game in which players are allowed to post only three prices–a high price (H), medium price (M), and low price (L).⁸,⁹ In both the RMB and RPD, the continuation probability was $\delta =0.93$. That is, in each round there was a 0.07 probability that the game will end after that round.

In both the RMB and RPD, the most cooperative action (i.e., H/C in the RMB/RPD) can be sustained as part of a subgame-perfect Nash equilibrium.¹⁰ Nevertheless, the stage game in the RMB and the stage game in the RPD are very different strategically. In the latter, D is a dominant strategy for each player. In the former, there is a unique Nash equilibrium in which both players choose L, but there is no dominant strategy. This suggests, but does not guarantee, that the RMB and RPD games are in some sense also quite different from each other.

Subjects played in pairs one practice RMB and thirteen RMB games for cash and, after that, one practice RPD and thirteen RPD games for cash.¹¹ At the end of each match, subjects were randomly and anonymously rematched into new pairs.¹² In any round during a repeated game, except in round 1, each subject could see the choices she and the other subject had made in all previous rounds of the repeated game.

A subject’s earnings from a given match equaled the sum of the experimental currency units (ECU) earned in all rounds of that match. A subject’s total earnings for the session equaled a $6 show-up fee plus the accumulated earnings from all matches, converted at an exchange rate of 500 ECU/$1. Average earnings for the session (including the show-up fee) equaled $24.29.¹³,¹⁴

At the start of the session, the experimenter read the instructions aloud as subjects read along, seated at their computer terminals.¹⁵ After clarifying questions, subjects completed a short understanding test. Experimenters walked around checking subjects’ quizzes, answering questions, and explaining mistakes. In case a subject made a mistake, extra care was taken to make sure she understood the task.

2.2 Session 2

In session 2, subjects performed an array of tasks meant to measure individual characteristics that are popular in economics and may plausibly be related to behavior in repeated games: risk attitude, time preference, trust, trustworthiness, altruism, strategic skills in one-shot matrix games, compliance with first-order stochastic dominance, and ability to plan ahead. The order of the tasks was different across cohorts. We also recorded each participant’s gender through a dummy variable, $Male$, that equals 1 for a male participant and 0 for a female one.

Subjects were paid for one randomly chosen task.¹⁶ Average earnings, including the $6 show-up fee, equaled $24.07.

Before each task, the experimenter read the instructions for that task aloud as subjects read along, followed by clarifying questions.¹⁷ Below, we describe each task in more detail. After that, we address the possibility that these tasks may measure subjects’ “true” individual characteristics with error.

2.2.1 Risk attitude

Fig. 1

Screenshot from risk attitude task

Subjects made ten Holt and Laury (2002) style choices between a safe lottery (Option A) and a risky lottery (Option B). In each decision, the safe lottery offered a $20 prize with probability $p$ and a $16 prize with probability $1-p$. The risky lottery offered a $38.5 prize with probability $p$ and a $1 prize with probability $1-p$. In decision 1, $p$ equalled 0.1 and increased in increments of 0.1 for each subsequent decision, so that the risky lottery became progressively more appealing relative to the safe lottery. In decision 10, both lotteries paid the high prize with certainty, so that the risky lottery was clearly superior.¹⁸ A screenshot from this task is given in Fig. 1. Subjects were told to imagine that the computer rolls a ten-sided die to determine the outcome of a lottery. The vertical line in the description of a given lottery in the screenshot indicates that if the random number is to the left/right of this bar, the left/right prize is realized.

Subjects were told that the payment for this task, subject to this task being chosen for payment purposes, would be determined as follows. The computer randomly selects one of the ten decisions. Each subject is paid according to the lottery she chose in that decision, with the computer rolling an imaginary die to determine the lottery outcome.

Our measure of a subject’s behavior in this task is $N_{risky}$, the number of decisions in which she chose the risk lottery.¹⁹ A higher value of $N_{risky}$ indicates more risk loving. Two subjects always chose the safe lottery despite the fact that it is clearly inferior in decision 10. For these subjects, the low value of $N_{risky}$ (namely, $N_{risky}=0$) may indicate a kind of misunderstanding of the task that has nothing to do with risk attitude. Therefore, we create a dummy variable $A_{risk}$ that equals 1 for these two subjects and equals 0 for everyone else. Including this dummy in our regressions will allow them to treat these subjects differently, so that these subjects’ behavior will not affect the estimated coefficient on $N_{risky}$. (The estimated coefficient on $A_{risk}$ will be of no interest.)

2.2.2 Time preference

Fig. 2

Screenshot from time preference task

Subjects made ten choices between receiving $18 today (Option A) and receiving $$x$ in one week (Option B), where $x=18.09$ in decision 1 and $x$ increases in subsequent decisions. A screenshot from this task is given in Fig. 2.

Subjects were told that the payment for this task, subject to this task being chosen for payment purposes, would be operationalized as follows. The computer randomly selects one of the ten decisions and each subject is paid according to her preferred option in that decision. In particular, if the subject chose the “today” option, she receives a check that can be cashed immediately; if she chose the “in-one-week” option, she receives a check with a date on it that is one week in the future, i.e., the subject has to wait for one week before being able to cash the check.

Our measure of a subject’s behavior in this task is $N_{patient}$, the number of decisions in which she was patient and chose the “in-one-week” option.²⁰

2.2.3 Trust and trustworthiness

Subjects played a binary-choice trust game in which a first mover decides whether to keep $16 or pass them to the second mover. If the first mover decides to pass, the money is tripled, so that the second mover has $48 at her disposal. The second mover then decides whether to keep all $48 or keep only $24 and pass back $24 to the first mover.

Subjects were anonymously paired to play the game. Each subject had to decide how she would behave both as a first mover and as a second mover. Subjects were told that the payment for this task, subject to this task being chosen for payment purposes, would be determined as follows. After each subject in a pair has made her decisions in the role of a first mover and a second mover, the computer randomly assigns the first mover role to one of the subjects and the second mover role to the other subject. Each subject’s payment is determined based on the choice she made for her assigned role and the choice the other subject made for the other role.

Our measures of a subject’s behavior in this task are: (i) $Trust$, a dummy variable that equals 1 if the subject’s decision as a first mover was to pass the $16 to the other subject and equals 0 otherwise and (ii) $Trustworthy$, a dummy variable that equals 1 if the subject’s decision as a second mover was to pass back $24 and equals 0 otherwise.

2.2.4 Altruism

Each subject was endowed with $20 and had to decide how much of this amount to keep for herself (any integer between 0 and $20 was allowed) and how much to donate to the charity “Feed the Children,” about which subjects were given some basic information. Subjects were told that, at the end of the experiment, the experimenter would add up all donations and write a check to the charity for the total amount. Subjects were also told that the check would be mailed immediately after the session, and were encouraged to accompany the experimenter to the closest mailbox (which was across the street). Our measure of a subject’s behavior in this task is $Alt$, the fraction of the endowment contributed to the charity.

2.2.5 Strategic skills in one-shot matrix games

Table 2 One-shot matrix game

Subjects were anonymously matched into pairs and each pair played two matrix games without feedback. One of the subjects in a pair played as the row player while the other subject played as the column player. The first matrix game is presented in Table 2. Notice that CENTER is a dominant strategy for the column player. The second matrix game is a transposed version of the first one, in which, to disguise the relationship with the first game, we scrambled the order of the rows and columns and subtracted $1 from each payoff. Thus, in effect, each subject played the game in Table 2 from both sides.²¹ Subjects were told that, subject to this task being chosen for payment purposes, the computer would randomly choose one of the two matrix games and subjects would be paid based on that game.

Our measures of a subject’s behavior in this task are: (i) $Dom$, a dummy that equals 1 if the subject plays a dominant strategy when one is available and equals 0 otherwise and (ii) $Str$, a dummy that equals 1 if a subject chooses a best-response to the opponent’s dominant strategy (whenever the opponent has a dominant strategy) and equals 0 otherwise. $Dom$ is meant to capture very basic rationality in a game; $Str$ is meant capture a kind of strategic thinking that goes one level deeper, namely whether a subject thinks about her opponent’s incentives for choosing different strategies.

2.2.6 Compliance with first-order stochastic dominance and ability to plan ahead

Fig. 3

Decision tree task

Subjects performed a task developed by Bone et al. (2009) to test whether people plan ahead. As a by-product, the task also measures whether people respect first-order stochastic dominance. The task is based on the decision tree presented in Fig. 3. In this decision tree, the subject moves first by choosing UP or DOWN. Next, the computer moves by choosing UP or DOWN with 50-50 chance. Then, the subject moves again UP or DOWN, followed by another 50-50 UP or DOWN move by the computer. The sequence of moves by the subject and the computer leads to a payoff which the subject receives if this task is selected for payment purposes.

The decision tree has the following two key features. First, whenever a subject has to make her second move, one of her two choices first-order stochastically dominates the other one. Second, if a subject plans ahead and anticipates this when making her first move, DOWN first-order stochastically dominates UP at the initial node. In particular, fixing a subject’s choices at all four second-move nodes in a way that respects first-order stochastic dominance, a choice of UP at the initial node leads to a uniform-distribution lottery over $6, $8, $16, $20 while a choice of DOWN leads to a uniform-distribution lottery over $8, $15, $17, $20.

Our measures of a subject’s behavior in this task are: (i) $FOSD$, a dummy that equals 1 if the subject respects first-order stochastic dominance at her second move and equals 0 otherwise and (ii) $PA$, a dummy that equals 1 if a subject chooses DOWN at her first move and equals 0 otherwise. $FOSD$ is meant to capture very basic rationality in choice under risk; $PA$ is meant to capture whether a subject plans ahead to the future in deciding what is currently optimal.

2.2.7 Measurement error

Later, we will use the individual characteristics variables defined above as independent variables in regressions in an attempt to see if they predict behavior in repeated games. However, it is possible that some of these variables capture subjects’ “true” individual characteristics with error. Such measurement error would lead to inconsistent coefficient estimates in our regressions.

There are two possible sources of measurement error. First, some subjects could make a mistake and not choose in accordance with their true preferences/abilities. Although some such mistakes invariably occur, the danger that they are large and/or widespread is limited by the fact that we use standard tasks that are well established in the literature.

Second, $Dom, Str, FOSD$, and $PA$ may involve measurement error even if subjects are not making mistakes. $Dom$ measures basic rationality in a game with error because a subject who does not recognize the dominant strategy may nevertheless choose it by fluke. Similarly, $FOSD$ measures basic rationality in choice under risk with error because a subject who does not recognize first-order stochastic dominance may nevertheless make the right choice at her second move in the decision tree by fluke.

$Str$ may measure the kind of strategic thinking we are after with error because (i) even a subject who does not think about the opponent’s incentives may still choose the best-response to the opponent’s dominant strategy by fluke and (ii) even a subject who thinks about the opponent’s incentives may still have serious doubts about the opponent’s rationality and, hence, may not choose the best-response to the opponent’s dominant strategy. The possibility for measurement error due to (i) is diminished by evidence that naive players choose the strategy with the highest average payoff, which in our games is different from the best-response to the opponent’s dominant strategy (see Costa-Gomes et al. 2001). (For example, in Table 2 DOWN is a best-response for the row player to the column player’s dominant strategy while UP has the higher average payoff.) The possibility for measurement error due to (ii) is diminished by the fact that, in our games, a subject who thinks about the opponent’s incentives need not be certain that the opponent will play her dominant strategy: in Table 2, it suffices that the row player assigns a probability greater than $\frac{15}{22}$ to the opponent playing her dominant strategy for DOWN to be optimal (assuming linear utility).

$PA$ may measure the kind of planning ahead we are after with error because even a subject who does not plan ahead may still choose DOWN at her first move by fluke. This possibility for measurement error is diminished under the plausible assumption that subjects who do not plan ahead simply compare the top eight payoffs with the bottom eight payoffs: the top eight payoffs simply look better (for example, a lottery that assigns $1/8{\mathrm{th}}$ chance to each of the top eight payoffs first-order stochastically dominates a lottery that assigns $1/8{\mathrm{th}}$ chance to each of the bottom eight payoffs).

When we run regressions with subjects’ individual characteristics as independent variables, what will be the direction of inconsistency of the estimated regression coefficients as a result of measurement error? In general, it is not possible to give a definitive answer, largely because there are multiple independent variables.²² However, as shown in the appendix, under certain assumptions one can say that the inconsistency will be toward zero, i.e., that there will be so-called attenuation bias. The key assumptions are that (i) the individual characteristic variables are uncorrelated and (ii) our non-dummy variables ($N_{risky}, N_{patient}$, and $Alt$) satisfy the classical errors-in-variables assumption. Although these assumptions do not hold exactly in our experiment, they probably do provide a reasonable approximation. (See the appendix for details.) Thus, our best guess is that, if anything, measurement error would lead to attenuation bias.

Perhaps the more important question concerns the size of the inconsistency, which, of course, depends on the size of the measurement errors. Although the elicitation tasks probably do involve some measurement error ($Dom$ and $FOSD$ almost surely do), for the reasons given above any measurement error should not be so large as to completely dilute away (or, in the unlikely case of inconsistency away from zero, to magnify greatly) any genuine effects of individual characteristics on behavior in repeated games. Having said that, one should bear in mind the possibility of inconsistency (most probably toward zero) when interpreting our regression results in Sect. 5.

3 Aspects of behavior in repeated games

To relate behavior in a repeated game to subjects’ individual characteristics, we need to be able to somehow formally quantify behavior in the repeated game. The theoretical notion that does this is that of a strategy: a strategy describes how a player behaves at each possible history in the repeated game. The problem is that strategies are notoriously difficult to estimate.²³

Instead of estimating players’ full-blown strategies, we use six summary measures of each player’s behavior, called aspects, that were developed in Davis et al. (2014). Before we present these aspects, let us introduce some notation. Consider an RPD game with two players, $i$ and $j$. Let C stand for “cooperate” and D for “defect”. Let (UV, XY,...) denote a history in which i played U and j played V in round 1, i played X and j played Y in round 2, etc., where U,V,X,Y $\in \{\hbox {C,D}\}$. For each subject $i$, we consider the following aspects.

• Round-1 cooperation ($C1$): the probability with which $i$ plays C in round 1. Thus, $C1$ is about how a player starts off the game.

• Lenience ($Len$): the probability with which $i$ plays C at history (CD). Thus, $Len$ is about not immediately retaliating after a unilateral defection by the other player.

• Forgiveness ($Forg$): the probability with which $i$ plays C at history (CD,DC). Thus, $Forg$ is about returning to cooperation after punishing the opponent for a unilateral defection, which the opponent was quick to correct.

• Loyalty ($Loyal$): the probability with which $i$ is not weakly first to play D in a game that starts out with (CC).²⁴,²⁵ Thus, $Loyal$ is about not breaking a streak of mutual cooperation.

• Leadership ($Lead$): the probability with which $i$ is weakly first to play C in a game that starts out with (DD).²⁶ Thus, $Lead$ is about breaking a streak of mutual defections by being the first to cooperate.

• Following ($Foll$): the probability with which $i$ plays C at history (DC). Thus, $Foll$ is about switching from D to C in response to the opponent playing C.

In the RMB game, we define our aspects exactly as above except that we take “C” to stand for “H” and “D” to stand for “L” or “M.”²⁷

For each subject $i$ and each repeated game (i.e., the RMB or RPD), we estimate each aspect in that repeated game by computing the corresponding empirical frequencies based on the 13 matches in which the game was played; we denote these frequencies by $\widehat{C1}, \widehat{Len}, \widehat{Forg}, \widehat{Loyal}, \widehat{Lead}$, and $\widehat{Foll}$, respectively. For example, if in the RPD $i$ faced history (CD) 5 times during the 13 matches and played C at that history 3 times, $\widehat{Len}=0.6$.²⁸ Although Davis et al. (2014) argue that it is a very crude measure of behavior in repeated games, for completeness sake we also consider $\widehat{C}$, the aggregate frequency with which $i$ plays C.²⁹

Note that the empirical frequencies are obtained from $i$’s behavior over several matches. Thus, either one has to make the implicit assumption that the probabilities in the definitions of each aspect are constant across matches³⁰ or, alternatively, one could view these probabilities as average probabilities across matches.³¹

4 Possible connections between individual characteristics and behavior in repeated games

We choose the particular individual characteristics used in our study for three main reasons. First, they are standard characteristics that play a large role in economics. Second, in contrast with “soft” personality traits used in psychology, they have clear definitions in terms of behavior. Third, although there exists no formal theory linking these individual characteristics to behavior in indefinitely repeated games, there are plausible intuitive arguments for why most of these characteristics might affect behavior in repeated games. Below, we discuss for each individual characteristic whether/how it might affect behavior in repeated games.

Risk attitude: In practice, each player $i$ does not know how the other player $j$ would choose at each history either because $i$ does not know $j$’s strategy or because $j$’s strategy involves mixing. Thus, a player’s attitude to risk may affect her behavior in repeated games in potentially systematic ways. We had no prior expectation about the sign of the effect. For example, it is plausible that risk aversion might induce a subject to defect because this guarantees she will earn at least 20 ECU per round and she will never be the “sucker” who earns 0 ECU. On the other hand, risk aversion may also prompt a subject to cooperate out of fear of disrupting existing or potential mutual cooperation.

Time preference: Cooperating at any point in an indefinitely repeated game requires foregoing current benefits for the sake of expected future benefits. Thus, presumably, patient players would be more likely to cooperate at any point in the game. Thus, we expected $N_{patient}$ to be positively correlated with our aspects.³²

Trust: Cooperating requires some trust that the opponent will not defect (in the current round or in the near future or ever). Thus, we expected $Trust$ to be positively correlated with our aspects.

Trustworthiness: One would expect trustworthiness to be positively correlated with $\widehat{Loyal}$ ($\widehat{Loyal}$ is to a large extent about not betraying the other player) and, possibly, with $\widehat{Foll}$ ($\widehat{Foll}$ is to some extent about rewarding the other player for having displayed trust by choosing the most cooperative action in round 1).

Altruism: Altruism provides additional motives for cooperating. Thus, we expected $Alt$ to be positively correlated with our aspects, except possibly with $\widehat{Len}$ and $\widehat{Forg}$. Just because a subject is generous towards a charity (as captured by $Alt$) does not mean that she is unconditionally altruistic and would have tolerance for the other player’s defections (as captured by $\widehat{Len}$ and $\widehat{Forg}$).

Strategic skills in one-shot matrix games: Behavior in repeated games requires strategic thinking: “if I do this, the opponent might react like that...”. Thus, we thought that basic measures of strategic thinking, such as $Dom$ and $Str$, might be correlated with our aspects. We had no prior expectation about the signs of the correlations.

Compliance with first-order stochastic dominance: The measure $FOSD$ emerges as a by-product in the task used for eliciting subjects’ ability to plan ahead. $FOSD$ seems like a basic measure of rationality and we included it in our regressions. We had no prior expectation about how this measure might correlate with our aspects.

Ability to plan ahead: In a repeated game, a player needs to think about how her current actions affect the opponent’s future behavior. Thus, we thought that an independent measure of subjects’ ability to plan ahead might be correlated with our aspects. Moreover, because a lack of planning ahead would probably favor defections, we expected the correlations to be positive.

Gender: Many studies (including some on repeated games) have focused on the role of gender. Including this characteristic was easy and seemed natural.

5 Results

We start the data analysis by exploring the connection between the individual characteristics variables defined in Sect. 2.2 and the aspects defined in Sect. 3. After that, we investigate whether two subjects’ individual characteristics help predict the frequency of the CC outcome (i.e., the outcome in which both players choose C in the RPD or H in the RMB) if these two players are matched to play a repeated game. Then, we consider whether a player’s individual characteristics predict her earnings in the repeated games.³³,³⁴

In our data analysis, we will run a number of regressions with the individual characteristics variables as independent variables and variables that are based on behavior in either the RMB or RPD as dependent variables. Each regression will be run separately for the RMB and RPD. In our regressions, we will focus on coefficients that, for both the RMB and RPD, (i) have the same sign and (ii) are statistically significant at the 10-percent level.³⁵ In this way, we hope to capture some kind of systematic effects that are at work in different repeated games. In addition, if one estimates many coefficients (as we do), some are likely to turn out statistically significant by fluke (more on this below). However, the possibility that a coefficient comes out statistically significant by fluke for both the RMB and RPD is smaller. By way of language, whenever in a regression a coefficient satisfies conditions (i) and (ii), we say that the corresponding individual characteristic variable has a systematic (positive or negative) effect on the dependent variable.

5.1 Can individual characteristics explain aspects?

Table 3 Regressions of aspects in RMB and RPD on individual characteristics. Robust standard errors in parentheses

Table 3 reports the results from OLS regressions of each aspect, as computed separately from the RMB and RPD games, on the individual characteristics variables. Based on this table, we can now state our first result.

Result 1

$N_{patient}$ has a systematic positive effect on $\widehat{Len}$.³⁶ $Alt$ has a systematic negative effect on $Foll$. $FOSD$ has a systematic negative effect on $\widehat{Loyal}$. $Male$ has a systematic positive effect on $\widehat{C}$ and $\widehat{C1}$. No other individual characteristic variable (except for $A_{risk}$) has a systematic effect on any aspect. ³⁷

Perhaps, the most interesting and surprising feature of Result 1 is the overall paucity of relationships between individual characteristics and behavior in repeated games. $N_{risky}, Trust, Trustworthy, Dom, Str$, and $PA$ have no systematic effects on behavior. For each of the four variables that do have systematic effects, these effects are limited in the sense that they apply to only one or, at most, two aspects.³⁸

5.2 Can individual characteristics explain the frequency of the CC outcome?

Table 4 Regressions of $FreqCC$ in the RMB and RPD on individual characteristics

Consider two subjects, $i$ and $j$, who are matched together to play a repeated game. Let $FreqCC$ denote the frequency (on a 0–1 scale, i.e., not in percent) of the CC outcome in that repeated game and, for each individual characteristic variable, let a superscript $i/j$ indicate that we are referring to $i$’s/$j$’s individual characteristic. (For example, $N_{risky}^i$ refers to the value of $N_{risky}$ for $i$.)

We run the following OLS regression (separately for the RMB and RPD):

$$\begin{aligned} FreqCC&= &\beta _0 + \beta _1 (A_{risk}^i + A_{risk}^j) + \beta _2 (N_{risky}^i + N_{risky}^j) + \beta _3 (N_{patient}^i + N_{patient}^j) \\&+ \beta _4 (Trust^i + Trust^j) + \beta _5 (Trustworthy^i + Trustworthy^j) + \beta _6 (Alt^i + Alt^j) \\&+ \beta _7 (Dom^i + Dom^j) + \beta _{8} (Str^i + Str^j) + \beta _{9} (FOSD^i + FOSD^j) \\&+ \beta _{10} (PA^i + PA^j) + \beta _{11} (Male^i + Male^j) + \varepsilon , \end{aligned}$$

where $\varepsilon$ is the idiosyncratic error term.³⁹,⁴⁰

The first and second columns in Table 4 report the result for the RMB and RPD, respectively. Based on the table, we can state the following result.

Result 2

$N_{patient}^l, Male^l$, and $Dom^l\,(l \in \{i,j\})$ have a systematic positive effect on $FreqCC$. No other individual characteristic variable (except for $A_{risk}^l, l \in \{i,j\}$) has a systematic effect on $FreqCC$.

In the RMB/RPD, $FreqCC$ increases (i) by 0.2/0.1 if one replaces an impatient player for whom $N_{patient}=0$ with a patient player for whom $N_{patient}=10$, (ii) by 0.18/0.15 if one replaces a female player with a male one, and (iii) by 0.07/0.17 if one replaces a player who does not choose a dominant strategy in a one-shot matrix game with a player who does. These effects are not only statistically, but also economically significant.⁴¹ However, $N_{risky}, Trust, Trustworthy, Alt, Str, FOSD$ and $PA$ have no systematic effects on $FreqCC$.

To check the robustness of Result 2, we tried three alternative specifications of regression equation (1): the first replaces each sum $x^i+x^j$ with $max(x^i,x^j)$, the second replaces this sum with $min(x^i,x^j)$, and the third replaces it with the product, $x^i x^j$. $Male$ retains its systematic effect on $FreqCC$ under all three alternative specifications. $N_{patient}$ retains its systematic effect both under the min and the product specifications, but not under the max specification.⁴² $Dom$ loses its systematic effect under all three alternative specifications because, under each of them, its coefficient is not significant for the RPD. The bottom line from this robustness exercise is that (i) the systematic effects of gender and, largely, of patience on $FreqCC$ are confirmed, and (ii) the systematic effect of $Dom$ becomes more questionable.⁴³

5.3 Individual characteristics and individual profits

Table 5 Regressions of individual earnings (in USD) in RMB and RPD on individual characteristics

The first and second columns in Table 5 report the results from OLS regressions of $Earnings$–a subject’s earnings (in USD) in the RMB and RPD games, respectively–on the individual characteristics variables. Based on the table, we can state:

Result 3

$Male$ has a systematic positive effect on earnings. No other individual characteristic variable has a systematic effect on earnings.

5.4 Testing multiple hypotheses

For each coefficient estimated in Tables 3, 4 and 5, except for the constant and the coefficient on $A_{risk}$, we were interested in testing the hypothesis that it equals 0. This gives us 180 hypotheses, which means that some rejections are likely to be false positives. We try to guard against false positives by restricting attention to coefficients that are statistically significant in both the RMB and RPD. This approach also has the advantage that it allows us to focus on what are hopefully systematic effects operating in different repeated games.

An alternative approach is to use an existing formal procedure designed to deal with the problem of testing multiple hypotheses. Such a procedure would not exploit any information contained in the fact that the coefficient on a given individual characteristic comes out significant in both the RMB and RPD. Nevertheless, it makes sense to employ such a procedure as a robustness check on Results 1, 2, and 3.

We use the Holm–Bonferroni procedure (Holm 1979) that controls at some level, $\alpha$, the so-called familywise error rate, i.e., the probability that there are one or more false positives. Note that by trying to limit the risk of even a single false positive, this kind of procedure is inherently conservative.⁴⁴ The coefficients in Tables 3, 4 and 5 that remain significant after the Holm–Bonferroni procedure for 180 hypotheses and $\alpha =0.1$ are underlined in the tables. There are seven such coefficients.⁴⁵

The Holm–Bonferroni procedure confirms our finding in Result 1 that $FOSD$ has an effect on $Loyal$ for both the RMB and RPD. Our findings in Result 1 regarding the effects of $N_{patient}, Alt$, and $Male$ on individual behavior in repeated games as well as our finding in Result 3 regarding the effect of $Male$ on earnings are not confirmed. Although this does not necessarily negate these findings, it does cast some doubt on them.

Regarding Result 2, the Holm–Bonferroni procedure confirms that $Male$ has an effect on $FreqCC$ in both the RMB and RPD. This procedure also partially confirms the findings about the effect of $N_{patient}$ and $Dom$ on $FreqCC$–“partially” because the coefficient on each of $N_{patient}$ and $Dom$ remains significant only in one of the repeated games. The bottom line is that, the Holm–Bonferroni procedure leads to an even greater paucity of statistically significant relationships between individual characteristics and behavior in repeated games.⁴⁶,⁴⁷

6 Concluding remarks

In the current paper, we investigate the relationship between several individual characteristics that are popular in economics and behavior in repeated games. Although we do find some evidence of systematic relationships, their number is surprisingly small overall.

Although game theory provides no guidance regarding how individual characteristics might be related to behavior in indefinitely repeated games, our intuitive expectation was that, at a minimum, patience, trust, trustworthiness, altruism, and an ability to plan ahead would be strongly positively related to most aspects and, possibly, to the frequency of the CC outcome. Instead, we find that trust, trustworthiness, and an ability to plan ahead have no effect on behavior both at the individual and at the pair level; altruism has no effect at the pair level and has the opposite from the expected effect at the individual level. Patience affects only one aspect ($Len$) at the individual level (and the effect does not survive the Holm–Bonferroni procedure). Attitude to risk and strategic thinking in one-shot matrix games (as captured through $Str$) also seem to have no effect on behavior both at the individual and at the pair level. Overall, our data indicate that behavior in repeated games is driven by factors that are largely independent of many of the individual characteristics we consider.

In conclusion, we view our study as taking a step towards answering some interesting questions about the connections (or lack thereof) between individual characteristics and behavior in repeated games. It is definitely not the final word on the matter. To deal with the issue of testing multiple hypotheses, one needs to confirm any effects we do find in further studies. To reduce the possibility of measurement error in eliciting the individual characteristics, one might employ multiple tasks to elicit each characteristic (possibly focusing on fewer individual characteristics due to time constraints). Finally, one needs to investigate whether our results extend to other repeated games.

Appendices

Appendix 1: Subjects’ forecasts

As mentioned in footnote 13, in each round of a given repeated game subjects also had to forecast the current-round choice of the other player.⁴⁸ Our primary concern in this paper is to study the possible effects of individual characteristics on behavior in repeated games. However, it may be insightful to further break down these effects by considering the following three questions:

• (1) Do individual characteristics affect subjects’ forecasts?

• (2) Are subjects’ forecasts related to their behavior?

• (3) How do our results in Table 3 change if we control for subjects’ forecasts in the regressions?

To address these questions, one needs to somehow quantify each subject’s forecasts. We do so in a way that parallels how we quantified subjects’ behavior through our aspects. In particular, we define the following summary measures of the forecasts of a given subject $i$ who is matched with another subject $j$ to play an RPD.

• $\widehat{C1}_F$: the frequency with which $i$ forecasts (that $j$ will play) C in round 1. Thus, $\widehat{C1}_F$ captures $i$’s forecasts when she is deciding how to start off the game.

• $\widehat{Len}_F$: the frequency with which $i$ forecasts C at history (CD). Thus, $\widehat{Len}_F$ captures $i$’s forecasts when she is deciding whether to be lenient.

• $\widehat{Forg}_F$: the frequency with which $i$ forecasts C at history (CD,DC). Thus, $\widehat{Forg}_F$ captures $i$’s forecasts when she is deciding whether to be forgiving.

• $\widehat{Loyal}_F$: the frequency with which $i$ is not weakly first to forecast D in a game that starts out with (CC).⁴⁹,⁵⁰ Thus, $\widehat{Loyal}_F$ is about a subject’s propensity to be the first one to suspect that the other will defect in a game in which $i$ needs to decide whether to be loyal.

• $\widehat{Lead}_F$: the frequency with which $i$ is weakly first to forecast C in a game that starts out with (DD).⁵¹ Thus, $\widehat{Lead}_F$ is about a subject’s propensity to be the first one to expect that the other will cooperate in a game in which $i$ needs to decide whether to lead.

• $\widehat{Foll}_F$: the frequency with which $i$ plays C at history (DC). Thus, $\widehat{Foll}_F$ captures $i$’s forecasts when she is deciding whether to follow $j$ in cooperating.

• $\widehat{C}_F$: the overall frequency with which $i$ forecasts that $j$ will play C.

In the RMB game, we define these measures exactly as above except that we take “C” to stand for “H” and “D” to stand for “L” or “M.”

We are now ready to address the three questions above. To address the first question, we run the analogues of the regressions presented in Table 3 with each independent variable replaced by the corresponding variable defined above. The results are presented in Table 6. Based on the Table, we see that only $Male$ has a systematic positive effect on $\widehat{C}_F$. Thus, individual characteristics are even less useful for predicting subjects’ forecasts than they are for predicting subjects’ behavior in repeated games.

Table 6 Regressions of variables summarizing subjects’ forecasts in RMB and RPD on individual characteristics

Turning to the second question, we look at the correlations between each aspect and the corresponding variable summarizing forecasts.⁵² These correlations are presented in Table 7. Based on the Table, we see that, with minor exceptions, these correlations are quite large and statistically significant. However, except for the correlation between $\widehat{C}$ and $\widehat{C}_F$ in both the RMB and RPD, they are far from perfect.⁵³

Table 7 Correlations between each aspect and the corresponding variable summarizing forecasts

To address the third question, we rerun the regressions from Table 3 by adding the relevant measure of forecasts as a control variable. E.g., in the regressions with $\widehat{C}$ as left-hand side variable, we add $\widehat{C}_F$ as a control. By way of notation, in a regression with $\widehat{C}/\widehat{C1}/\widehat{Len}/\widehat{Forg}/\widehat{Loyal}/\widehat{Lead}/\widehat{Foll}$ as left-hand-side variable, let $Control$ denote $\widehat{C}_F/\widehat{C1}_F/\widehat{Len}_F/\widehat{Forg}_F/\widehat{Loyal}_F/\widehat{Lead}_F/\widehat{Foll}_F$. The results are presented in Table 8. Note that, based on the Table, Result 1 does not change much when we control for forecasts. The only differences in Table 8 from Table 3 are the following: (i) $N_{patient}$ now also has a systematic positive effect on $\widehat{C}$, (ii) $Dom$ has a systematic positive effect on $\widehat{C1}$, (iii) $Male$ loses its systematic positive effect on $\widehat{C}$. Note that (i) and (ii) are not due to differences in the estimated coefficients. Rather their standard errors change just enough in one of the repeated games for the effect to qualify as systematic according to our criterion.⁵⁴

Table 8 Regressions of aspects in RMB and RPD on individual characteristics, controlling for forecasts

Appendix 2: Summary statistics for individual characteristics and aspects

Table 9 Individual characteristics averages across subjects

Table 10 Correlations between individual characteristics

Table 11 Aspect averages across subjects for matches 1–7, 8–13, and 1–13 (with sample standard errors in parentheses) as well as aspect correlations between matches 1–7 and 8–13 (*/**/*** indicates statistical significance at the 10/5/1 percent level)