The Dynamics of Responder Behavior in Ultimatum Games: A Meta-study 8/28/2011

Transcription

1 The Dynamics of Responder Behavior in Ultimatum Games: A Meta-study 8/28/2011 David J. Cooper Department of Economics Florida State University Tallahassee, FL E. Glenn Dutcher Department of Economics Florida State University Tallahassee, FL Acknowledgements: We received data from Lisa Anderson, James Andreoni, Marco Castillo, John Duffy, Ernst Fehr, Nick Feltovich, Urs Fischbacher, Cristina Fong, Masahiro Okuno- Fujiwara, Ragan Petrie, Vesna Prasnikar, Yana Rodgers, Roger Rodriguez, Al Roth, Robert Slonim, Shmuel Zamir and Rami Zwick. We are very grateful to them, since this paper would not have been possible without their generosity. We would also like to thank John Kagel, Jacob Goeree, two anonymous referees, and seminar participants at Florida State for their insightful feedback.

2 The Dynamics of Responder Behavior in Ultimatum Games: A Meta-study Abstract: Using data aggregated from seven papers that study repeated play in standard ultimatum games with either stranger or absolute stranger matching, we show that the behavior of responders changes with experience. High offers are more likely to be accepted with experience and low offers are more likely to be rejected. At the individual level, there is a negative relationship between the likelihood that a given offer is accepted and the size of the preceding offer. We compare the results with predictions generated by static models of distributional preferences, implicitly dynamic models of preferences with reciprocity, and explicitly dynamic models of adaptive learning. The data is most consistent with models of preferences with reciprocity. Key Words: Ultimatum Games, Other-regarding Preferences, Learning JEL: C91, C78 1. Introduction: The ultimatum game (Güth, Schmittberger, and Schwarze, 1982) is quite likely the most influential and most studied game in the history of experimental economics. 1 Two players, the proposer and the responder, must decide how to divide a sum of money. The proposer starts the game by offering a non-negative portion of the money to the responder. The responder can either accept or reject this offer. If it is accepted then the money is divided as proposed, but if the responder rejects the offer then both players get nothing. In either case the game ends after the responder s decision. Assuming individuals only care about their own payoffs, subgame perfection predicts that the proposer offers the responder virtually nothing and all positive offers are accepted. 2 The data tells a very different story as proposers typically offer between 40-50% of the pie and offers below this range are often rejected. This pattern is quite robust, occurring in a wide variety of cultures (e.g. Roth, Prasnikar, Okuno-Fujiwara, and Zamir, 1991; Henrich, Boyd, Bowles, Camerer, Fehr, Gintis, McElreath, 2001; Oosterbeek, Sloof, and Van de Kuilen, 2004) 3 and holding up under large increases in stakes (Slonim and Roth, 1998; Cameron, 1999). Attempts to understand why responders walk away from positive offers played 1 For a recent survey of this literature see Chaudhuri, To be more technical, it is also necessary that payoffs of the game are common knowledge. In other words, players are self-regarding, know that the other player is self-regarding, know that the other player knows they are self-regarding, etc. 3 This is not to say that behavior is identical across locations and cultures. As ably documented by Oosterbeek et al s (2004) meta-study, behavior varies across locations and cultures. Our point is that the basic pattern of the data is generally similar mean offers are almost always at least a third of the pie and offers that are too low are rejected. The precise definition of too low does vary a fair amount across locations and cultures. 1

3 a central role in the emergence of the extensive literature on other-regarding behavior and preferences. A feature of behavior in ultimatum games that has received relatively little attention is how responders behavior changes with experience. This is an important issue for two reasons. First, there is a large literature studying whether anomalous behavior in the ultimatum game is robust (Roth, 1995). Persistence is an important element of robustness. If acceptance rates in the ultimatum game rise steadily, even at a slow rate, this suggests that the use of relatively large positive offers and frequent rejection of relatively small offers need not persist over time. In other words, long run play may converge to the subgame perfect outcome. To better understand how this could occur, consider a scenario with a fixed group of responders and a changing group of proposers where experienced proposers are constantly being replaced by new, inexperienced proposers. In the standard case where both populations are fixed, proposers learn that low offers will be rejected and the proportion of low offers therefore steadily declines. But if inexperienced proposers are constantly entering the population there instead will be a steady supply of low offers. This is precisely the type of situation where the adaptive learning model of Roth and Erev (1995) predicts movement toward the subgame perfect equilibrium as responders are slowly worn down by the constant need to sacrifice monetary payoffs by rejecting low offers. If responders eventually learn to accept low offers, it becomes attractive for proposers to make such offers and play can converge to the subgame perfect equilibrium. A second reason to study the dynamics of responder behavior is to better understand why responders reject positive offers. Speaking broadly, there are three major theoretical approaches to explaining the anomalous behavior observed in ultimatum games: static models of distributional preferences (e.g. Bolton and Ockenfels, 2000; Fehr and Schmidt, 1999; Kirchsteiger, 1994), implicitly dynamic models of preferences with reciprocity (e.g. Charness and Rabin, 2002; Dufwenberg and Kirchsteiger, 2004; Falk and Fischbacher, 2006; and Cox, Friedman, and Gjerstad, 2007), and explicitly dynamic models of adaptive learning (e.g. Roth and Erev, 1995; Gale, Binmore, and Samuelson, 1995). These models make different predictions about how the behavior of responders should change with experience. Thus, understanding the dynamics of responder behavior can help us sort out the relative importance of these alternative explanations. 2

4 Initial examinations found little evidence of responders behavior changing over time. Roth, Prasnikar, Okuno-Fujiwara, and Zamir (1991) find that acceptance rates change little with experience, but, as they note, this is only an informal examination of the issue with no attempt to control for changes in the distribution of offers. 4 Slonim and Roth (1998) offer a formal treatment of the issue and find no evidence for changes in responders behavior. More recent papers do find evidence of changes in acceptance rates. Duffy and Feltovich (1999) study ultimatum games with and without observation of another pair s outcomes. They find that acceptance rates fall, but only with observation of another pair s outcome. List and Cherry (2000) run a variant of Slonim and Roth s high-stakes experiment where proposers earn the right to propose rather than having it determined exogenously. With a resulting increase in the proportion of low offers, Cherry and List find that acceptance rates rise with experience. Most of this increase occurs in the last few periods, suggesting that it may be an endgame effect. Cooper, Feltovich, Roth, and Zwick (2003) manipulate the experience received by responders by doubling the number of proposers relative to responders. They use this treatment to generate indirect evidence that acceptance rates rise with experience. They also find that a history of receiving low offers makes subjects more likely to accept low offers. 5 While none of the preceding studies is definitive, taken together they indicate that the behavior of responders may not be static. However, even basic features of the data like the direction of change have not been established beyond dispute and more subtle features that make it possible to distinguish between models, like how acceptance rates change as a function of the offer and the adjustment process driving changes in responders choices, are largely unexamined. An important reason these questions have been little studied is the lack of power in existing datasets. Nobody disputes that changes in the behavior of responders, in aggregate, are subtle and hence difficult to detect in the relatively small datasets generated by experimenters. Low offers, which are particularly useful for establishing persistence and distinguishing between the different approaches to modeling ultimatum game behavior, are rare. To solve the power problem, we have created a dataset aggregating responder behavior from seven different studies 4 Roth et al is primarily concerned with differences between countries, and changes over time in responder behavior are a secondary issue that is only discussed briefly. 5 See also Winter and Zamir (1997), Abbink, Bolton, Sadrieh, and Tang (2001), Armantier (2006), and Andreoni and Blanchard (2006). 3

5 of repeated play (with stranger or absolute stranger matching) in standard ultimatum games. We have at least ten rounds of play for 387 responders. Analyzing this large dataset allows us to definitively establish that responders behavior changes with experience. These changes are broad-based (i.e. consistent across papers and not due to the data of a single paper), are not due to endgame effects, 6 and are inconsistent with models of reputation building or strategic teaching. The direction of change is a function of the offer size. For large offers (20% of the pie or greater) the acceptance rate rises over time while for small offers (less than 20% of the pie) the acceptance rate falls. Looking at the process underlying changes in responder behavior, there is a strong negative relationship between the previous offer received and the likelihood that the current offer is accepted, holding the current offer fixed. In other words, receiving a higher (lower) offer makes a responder less (more) likely to accept the next offer, all else being equal. Our findings suggest that play will not converge to the subgame perfect outcome even under ideal conditions. Once again consider a situation where a fixed group of responders face a steady stream of new proposers and hence a steady stream of low offers. The prediction that play will move toward the subgame perfect outcome relies on acceptance rates for all offers rising over time. We observe decreasing acceptance rates over time for low offers (less than 20% of the pie) and therefore expect, even in the very long run, to observe substantial positive offers with small positive offers getting rejected. Our findings are more consistent with models of preferences with reciprocity than either static models of distributional preferences or adaptive learning models. Intuitively, suppose that responders are more likely to accept an offer that is perceived as kind than an offer that is seen as unkind. If the perception of whether an offer is kind or unkind depends on responders beliefs about what offers are typical, responders behavior should change as they see more offers and change their beliefs about the distribution of offers. For example, suppose a responder is initially pessimistic about the distribution of offers. As he learns that offers are higher than expected, low offers that initially seem acceptable become unkind and hence are rejected. Likewise, a responder who is initially too optimistic will learn to accept more high offers over 6 Endgame effects refer to changes in behavior for the last game or two of experiments using repeated trials and a stranger matching. These aren t repeated game effects in the typical sense, since subjects aren t playing repeated games against a fixed opponent. While models of reputation building, particularly with stereotyping (Healy, 2007), can predict some endgame effects, pure irrationality almost certainly is playing a role as well. 4

6 time. The predicted pattern of changes is exactly what we observe in the data: acceptance rates rise for high offers, fall for low offers, and observing a relatively high (low) offer makes future offers, ceteris paribus, seem less (more) kind and hence less (more) likely to be accepted. On a general level our results illustrate that the dynamics of other-regarding behavior, a topic which has received little attention, are deserving of greater attention. The experimental literature already contains cases where aggregate changes in other regarding behavior are quite large (e.g. Cooper and Stockman, 2002; Brosig, Reichmann, and Weimann, 2009) and we have now shown that behavior changes over time in what is probably the canonical example of other-regarding behavior, responder behavior in the ultimatum game. While these changes are subtle in aggregate (hence the need for a large dataset), they are quite substantial at the individual level. More work is needed to understand why and how these changes are occurring. Our results have several methodological implications. First, a number of papers in the literature on other-regarding behavior rely on data from one-shot games or repeated games without feedback to estimate parameters for various models or to classify subjects into types (e.g. Andreoni and Miller, 2002; Engelmann and Stoebel, 2004). There are obvious advantages to this approach: subjects are forced to focus on each decision knowing they won t be able to learn from experience, repeated game effects are eliminated, and clean within subject comparisons are possible. However, our results show that other-regarding behavior changes with experience. As such, these estimation exercises must be treated with caution. Second, our results suggest that unobserved beliefs play an important role in other-regarding behavior. To the extent that beliefs change across treatments, changes in behavior that are attributed to changes in preferences may actually reflect changes in beliefs. Careful identification of the causes underlying treatment effects therefore requires some attempt to measure beliefs. The primary focus of our work is studying the dynamics of responder behavior, but the large dataset we have constructed also makes it possible to revisit the question of how proposers behavior changes over time. A number of previous papers have found that the distribution of offers changes in response to the incentives generated by responders behavior (e.g. Roth, Prasnikar, Okuno-Fujiwara, and Zamir, 1991; Slonim and Roth, 1998; Cooper et al, 2003). We confirm and extend these findings. Holding the lagged offer fixed, proposers make lower offers if the previous offer was accepted than if it was rejected. Due to this adjustment pattern, the average amount earned by proposers increases with experience and the probability of the payoff 5

7 maximizing category of offers rises. If proposers preferences incorporate any of the standard versions of distributional preferences (e.g. inequality aversion, social welfare, maxmin), the long run distribution of offers is predicted to be shifted above the payoff maximizing offer. We find no evidence for this, indicating that, in the long run, positive offers in the ultimatum game are driven more by expected payoff maximization than distributional preferences. This is consistent with earlier findings in the literature. 7 The remainder of this paper is structured as follows: Section 2 lays out theoretical predictions for changes in behavior for the ultimatum game. Section 3 describes how the dataset was assembled. Section 4 presents the results. Section 5 concludes the paper with a discussion of the results. 2. Theory and Changes in Responder Behavior: Theoretical explanations of behavior in ultimatum games, particularly the willingness of responders to reject positive amounts of money, fall into three broad categories. These approaches have very different implications for the nature of other-regarding behavior and make distinctly different predictions about how the behavior of responders changes with experience. Rather than developing specific predictions for each of the myriad papers in this literature, our goal in this section is to describe the defining characteristics of each category of theory and to provide an intuitive explanation of what each category of model predicts. 8 (1) Distributional Preferences: Individuals are assumed to have stable preferences that are a function of their own payoff and the payoffs of others. Bolton and Ockenfels (2000) and Fehr and Schmidt (1999) provide the best known examples of this approach. In these models, responders reject positive offers because the disutility of receiving a lower payoff than the proposer is greater than the utility from receiving a positive payoff. Because these models define utility over monetary payoffs and because responders face a simple binary choice with no payoff or strategic uncertainty, responders behavior, controlling for the amount offered, is not predicted to change with experience. 9 7 For examples see Roth et al and Prasnikar and Roth, To quote Roth (1995, p. 287),... the [proposers] seem to be adapting to the experience of the prior rounds in a manner roughly consistent with simple income maximization. 8 For a recent survey of the enormous theory literature on other-regarding preferences, see Cooper and Kagel (2009). 9 See Kirchsteiger (1994) for a model of distributional preferences where rejections are driven by envy rather than inequality aversion. 6

8 (2) Preferences with Reciprocity: Ample experimental evidence exists that responders do not just care about the offer they receive but also about the process that led to the offer. 10 This leads naturally to the idea that rejections reflect an attempt to punish unkind actions. In other words, rejections are driven by negative reciprocity. Well known models formalizing this intuition include Charness and Rabin (2002), Dufwenberg and Kirchsteiger (2004), Falk and Fischbacher (2006), and Cox, Friedman, and Gjerstad (2007). 11 Unlike models that rely solely on distributional preferences to explain rejections of positive offers, models that incorporate reciprocity are implicitly dynamic. Defining an action as kind or unkind is a matter of beliefs. If responders expect to be offered 30% of the pie, an offer of 40% seems kind. If they expect an even split, the same 40% offer becomes unkind. If responders beliefs respond to the stream of offers they observe, the behavior of responders should change as they learn the distribution of offers. To get some intuition for how acceptance rates will change over time, consider the following variation on the reduced form model of Charness and Rabin. This is a simple model, but the intuition extends to more complex cases. Normalize the pie size to 1. Let Ω i t be the offer received by responder i in round t and let H i t be the history of offers received by responder i prior to round t: H i t = (Ω i 1, Ω i 2,..., Ω i t-1). Assume that responder i believes that Ω i t, is drawn from a distribution over the range [0,1] with mean ω i (H i t). We do not assume a common prior. Instead, some responders are initially more optimistic than others about the offers they will receive. Let π P and π R be the proposer and responder s respective monetary payoffs. Suppose that responder i s utility in round t is given by the following: i( ) i,, i, i, i i i u H v i H * q *. t P R t t P R t t P The first term corresponds to some variety of static distributional preferences. The function is an indicator function that equals one if the offer is less than the expected offer subject to the observed history of offers and equals zero otherwise. The variable q is a positive constant that determines the weight on disutility received from accepting an unkind offer. 10 Blount (1995) provides a striking example of this. She compares the behavior of responders in the standard ultimatum game and in an ultimatum game where a computer randomly chooses offers for the proposers. Low offers are for more likely to be accepted when the computer is responsible. 11 A related approach is taken by Levine (1998) who models individuals as wanting to be kinder to others who they believe to be altruistic. This approach turns the ultimatum game into a signaling game, where high (low) offers convey the information that the sender is (not) altruistic. Relevant beliefs are over the distribution of types rather than the distribution of actions per se. 7

9 The first term of the utility function does not change with experience, but the second term changes as beliefs change. For simplicity, assume offers are drawn from a stable distribution. With any reasonable learning process, ω i (H i t) will (on average) converge toward the true mean over time. Individuals who are initially too optimistic will learn to expect lower offers. As such, they will come to realize that offers that initially seemed unkind are in fact perfectly normal and hence acceptable. Likewise, individuals who are too pessimistic initially should learn to expect higher offers. Offers that previously seemed generous may no longer seem so kind, and hence no longer be accepted. Acceptance rates are predicted to change with experience but in a way that varies by the offer size relatively high offers should be accepted more and relatively low offers should be accepted less. (3) Reinforcement Learning: As pointed out by Roth and Erev (1995) and Gale, Binmore, and Samuelson (1995), anomalous behavior in the ultimatum game can be explained via reinforcement learning models without resorting to distributional preferences. Suppose responders have standard preferences (i. e., utility is defined solely in terms of monetary payoffs) but are not payoff maximizers. Instead, responders have an exogenously determined initial distribution over their available strategies, usually a discrete set of cutoff strategies. 12 After a strategy has been randomly selected and played, the probability of it being used in the next period is increased. Critically, the amount of this increase is proportional to the realized payoff. Thus, strategies that realize higher payoffs tend to be played more frequently over time. 13 With reinforcement learning, reaching the subgame perfect equilibrium is a two step process: responders learn to stop rejecting low offers and then proposers learn that low offers will be accepted and hence are highly profitable. The problem is that proposers make little money from low offers due to the high rejection rate, and therefore stop making them. If proposers learn to stop making low offers faster than responders learn to accept them, play fails to converge to the subgame perfect equilibrium. 14 Reinforcement learning models are explicitly dynamic. Unlike models of preferences with reciprocity, acceptance rates are predicted to rise for all offers. To give some intuition for this 12 Cutoff strategies are strategies that specify a minimal acceptable offer (MAO). Any offer greater than or equal to the MAO is accepted and lower offers are rejected. 13 See Camerer (2003, ch. 6) for a survey of the large experimental literature exploring what type of model best characterizes learning in games. For our purposes, the specific form of the learning model is unimportant. What matters is that strategies that earn higher payoffs are played more frequently over time. 14 Gale et al use an evolutionary model, but the intuition is much the same. 8

10 prediction, consider a simple version of the Roth-Erev model. Once again normalize the size of the pie to equal 1. For simplicity assume that only a discrete set of offers 1 ω 1 > ω 2 > ω N 0 is available. The set of available strategies for responders is the corresponding set of cutoff strategies. With each cutoff strategy we associate a weight w i j (t) > 0 where i indexes individuals, j indexes cutoff strategies, and t indexes rounds. Following Roth and Erev, we refer to these weights as propensities. The probability of responder i choosing cutoff j in round t, p i j (t), is given by the following: i p () t j i w () t N k 1 j i w () t The model is made dynamic by adding a transition rule. If cutoff j is chosen by responder i in round t and earns payoff π i (t), then w i j (t+1) = w i j (t) + π i (t). The propensities for all other strategies are unchanged. Take any specific offer. To prove that the probability of acceptance for ω must increase, on average, over time, it is sufficient to show that the probability of responder i accepting ω must increase, on average, in round t + 1 following i s receipt of any possible offer ω i (t) in round t. Given the structure of cutoff strategies, this is trivial if ω i (t) ω since only cutoff strategies which would accept ω can receive positive reinforcement. The more difficult case is when ω i (t) > ω, since it is possible for responder i to use a cutoff strategy which accepts this offer but rejects ω. In this case, the realized probability of ω being accepted falls. However, note that the expected value of p i j (t+1) is a weighted average of p i j (t) and the following probability where π(ω j ω i (t)) is the profit earned by play of cutoff j subject to individual i receiving an offer of ω i (t): i p () t j N i i p ( t)* ( ( t)) k 1 j k i i p ( t)* ( ( t)) Since π(ω j ω i (t)) = ω i (t) for all ω j ω(t) and π(ω j ω i i i (t)) = 0 otherwise, p ( t) p ( t) for any j such that ω j ω. The result follows directly. Although we have developed this prediction using the simplest version of the Roth-Erev learning model, the main intuition holds for any learning model that has a component of k j k j j 9

11 reinforcement learning. 15 Accepting always leads to a higher payoff than rejecting an offer. Since reinforcement models move in the direction of higher payoffs as long as all strategies are played with positive probability, the probability of an acceptance must increase over time. Critically, this statement is true for all offers, not just high offers. Summary of Predictions: Models based on stable preferences over the distribution of payoffs predict no changes in responder behavior over time. Models that also incorporate reciprocity predict rising acceptance rates for high offers and falling acceptance rates for low offers. Reinforcement learning models predict that acceptance rates should increase for all offers. 3. The Data: Table 1 summarizes the sources for the data used in this paper. For a more detailed summary of the data sources, see Appendix A. (Insert Table 1 Here) In choosing papers to include in this meta-study we applied a number of criteria beyond the obvious one that the authors were willing to share the data. Most of the papers include multiple treatments, but we only used the subset of the data that met these criteria. (1) All data was from standard ultimatum games. Standard in this sense refers to games where there was one proposer and one responder, play was anonymous, 16 proposers and responders were rematched in each round, proposers and responders had the same amount of experience with the ultimatum game, and each bargainer only learned the result of their own negotiation. Four of the papers use stranger matching and three use absolute stranger matching. 17 In practice, no experiment allows proposers to choose from a continuous set of offers, but we restricted ourselves to experiments where proposers could offer amounts in at least 1/10 increments of the 15 So, for example, this prediction will hold for EWA (Camerer and Holt, 1999), since this learning model nests reinforcement learning. It will not hold for pure belief based models such as fictitious play, since these models assume individuals know payoffs and optimize (possibly with noise). Unless we exogenously force the noise to decrease over time, such models predict no change in responder behavior over time. 16 We refer to anonymity in the weak sense that no identifiers were used that would allow proposers and responders to identify each other either during the experiment or after the fact. This limits the possibility of uncontrolled supergame effects. These experiments were not anonymous in the sense that the experimenter could identify specific actions with specific subjects (see Hoffman, McCabe, Shachat, and Smith, 1994; Bolton and Zwick, 1995). 17 With stranger matching, responders are matched with a new proposer in each round. Absolute stranger matching adds the restriction that a responder will never be matched with a proposer more than once. Two of the papers with stranger matching have some restrictions on rematching: Cooper et al did not allow a proposer and responder to be matched in two consecutive rounds and Andreoni et al did not allow a proposer and responder to be matched more than twice in a session. 10

12 entire pie (eliminating cardinal ultimatum games where proposers only have a few choices). We did not use studies that employ the strategy method. Responders observed the offer and could accept or reject with the typical consequences tied to their decisions. We restricted ourselves to standard ultimatum games to limit the sources of heterogeneity between datasets and to reduce the possibility that any observed changes in responder behavior were due to an idiosyncratic feature of the experimental design. (2) The included experiments all involved at least 10 rounds of play, with a maximum of 60 rounds. The goal was to have long enough runs that changes in responder behavior had a realistic chance to emerge. (3) Proposers and responders stayed in the same role throughout the entire experiment. This avoids the issue of whether experience is calculated in terms of the total number of games played or in terms of games as a responder and also allows us to avoid questions about how responders changing behavior is due to their experiences as proposers. (4) One of the papers, Anderson, Rodgers and Rodriguez (2000), used a within subject design where the standard ultimatum game was played in the same session as another treatment. We only used data from sessions where the standard ultimatum game was played first to avoid any spillovers. Subjects did not know beforehand that another game followed the first. We utilized Google Scholar to search through hundreds of studies in the summer of We repeated the Google search in Spring of 2011 and also ed the ESA s discussion board. For more detail on how the Google search was conducted, see Appendix A. We asked for data from eight studies that met all of our criteria and obtained data for seven of them. 18 It is quite possible that we missed other studies that would have met our criteria given the sheer volume of work on the ultimatum game, but the studies we found contain 7,188 observations from 387 subjects in 44 sessions, sufficient for us to detect subtle changes in responder behavior. 4. Results: The main result of our analysis can be seen in Figure 1. This figure only shows data from the first ten rounds of play, giving the same number of observations from all responders and eliminating any distortions due to sessions dropping out. See Appendix B for an equivalent 18 There were several common reasons why studies did not meet our criteria. Many of the ultimatum game studies did not last at least our minimum required length of 10 periods and many added nuances to the standard game in order to study a different research question. To give some examples of why studies that almost fit our criteria were not included, Sutter, Kocher, and Strauss (2003) had responders face a mixture of human and simulated proposers, Armantier (2006) had responders submit a minimum acceptable offer with a 50% chance of getting an opportunity for revision after seeing the offer, and Buchan, Croson, and Johnson (2004) used the strategy method. 11

13 Acceptance Rate figure using data from all rounds the same qualitative patterns are seen as in the restricted dataset. Offers have been broken into categories by tenths of the pie, with all offers greater than or equal to 50% of the pie combined into a single category. For each offer category we separately display the acceptance rates for Rounds 1 5 and Rounds The numbers above the bars are the number of observations in the offer category for the relevant time period Figure 1: Acceptance Rate as a Function of Experience Note: The numbers above the bars give the number of observations for that bar Offer < Offer < Offer < Offer < Offer < 50 Offer 50 % Offered Rounds 1-5 Rounds 6-10 The figure shows a clear pattern in how acceptance rates change over time. For the two lowest categories of offers, acceptance rates are lower for Rounds 6 10 than in Rounds 1 5. For all the remaining categories the acceptance rates are higher in Rounds Figure 1 also makes it clear why a large dataset is needed. The increase in acceptance rates for high offers is small and would be hard to identify with fewer observations. The magnitude of the decrease for small offers is larger, but given the infrequency of low offers we would once again struggle to identify the effect without a sizable dataset. Because the effects shown in Figure 1 are subtle, it is particularly important to establish statistical significance. The regression analysis shown in Table 2 does so. These regressions use 12

14 all observations in the dataset (rather than only using Rounds 1 10). The dependent variable is a dummy for whether the offer was accepted. Since this is a binary variable we use a probit model. It is critical that we control for session effects. Only 15 of the 44 sessions in our dataset (4 of 7 papers) last more than ten rounds. Without controls for session effects, changes due to experience are confounded with any idiosyncratic features of the 15 long sessions. All of the regressions therefore include session dummies. Parameter estimates for these dummies are not reported in Table 2 as they are of little economic importance and take up a lot of space. Full copies of the regression output are available upon request. Standard errors are corrected for clustering at the individual responder level. (Insert Table 2 Here) The primary independent variables are a dummy for late rounds and the (truncated) percentage of the pie offered to the responder. Late rounds are defined as Round 6 and later. More than 2/3 of the subjects are in sessions that have only ten rounds, so a late round corresponds to the second half of a modal session. 19 The amount offered is given as a percentage of the pie truncated at 50% to reflect the obvious kink in acceptance rates at the offer. 20 Table 2 reports parameter estimates rather than marginal effects. The numbers in parentheses are standard errors, and one, two, and three stars indicate statistical significance at the 10%, 5%, and 1% levels respectively. Model 1 controls for the effect of experience in a naïve fashion, only including the dummy for late rounds. The associated parameter estimate is positive but small, failing to achieve statistical significance at even the 10% level. As Figure 1 suggests, it is difficult to identify changes in responder behavior without allowing for differing effects at large and small offers since the decreasing acceptance rate for low offers partially offsets the increasing acceptance rate for high offers. Model 2 therefore adds an interaction term between the dummy for late rounds and the truncated amount offered. This model picks up a strong effect from experience as the parameter estimate for late rounds is large and negative while the parameter estimate of the interaction term is positive and large. Both estimates are statistically significant at the 1% level, 19 We have experimented with moving the breakpoint for late rounds to different time points. As long as the shift is small (i.e. a couple of rounds) there is little qualitative effect. 20 The qualitative results look the same without truncation, but the fit is significantly worse. 13

15 and the respective marginal effects are -17.2% and 0.6%. The estimated marginal effect for the interaction term looks small in absolute terms compared to the large marginal effect for the late rounds dummy, but note that this is the effect of a small (1%) change in the amount offered. Taken together, these parameter estimates imply a significant increase in the acceptance rate for large offers and a significant decrease for small offers. Recall that the changing behavior of responders in List and Cherry was consistent with endgame effects. The same issue could be present here, so Model 3 adds a dummy for the final round of a session. 21 The resulting estimate is positive and significant at the 1% level, but has little effect on the estimated effects of experience. Model 4 adds a dummy for observations from the 11 th Round or later as well as an interaction between this dummy and the truncated amount offered. 22 Adding these variables allows us to examine whether the marginal effect of experience is decreasing. The results indicate that experience beyond the first ten rounds has almost no additional impact, as both parameter estimates are small and fail to approach statistical significance even at the 10% level. To check whether our results are sensitive to the method used to control for session and individual effects, we have rerun Model 2 using a probit model with random effects at the individual level (rather than clustering to correct for individual effects), a logit model with nested random effects at the session and individual levels, and a logit model with individual fixed effects. The results of these specification checks are reported in Table 3. Model 1 on Table 3 is the same as Model 2 from Table 2, and is included to ease comparisons across models. Parameter effects are reported, not marginal effects, so comparisons of parameter magnitudes across models is not appropriate. (Insert Table 3 Here) The qualitative results are largely unaffected by how we control for session and individual effects. In all four models the parameter estimate for late rounds is large, negative, and statistically significant at the 1% level while the parameter estimate of the interaction term is 21 Adding an additional dummy for the next to last round did not have a statistically significant effect. 22 These two parameters measure differences from Rounds 6 10, not Rounds 1 5. Specifically, the Round 11 parameter captures the difference between the constant in Rounds 6 10 and the constant in later rounds. Likewise, the parameter for (Round 11) * Truncated Amount Offered captures the difference between the slope for Rounds 6 10 and the slope in later rounds. 14

16 large, positive, and statistically significant at the 1% level. We prefer the probit models that correct for individual effects by clustering rather than random effects because this approach is more conservative in terms of finding significant effects, requires fewer assumptions about functional form, and corrects the standard errors for autocorrelation. To generate a large data set, we ve aggregated data from seven papers. The papers were selected to be similar along many dimensions, but there remains variation in the details of how the sessions were run, the number of periods used, and the make-up of the subject pools. It is therefore possible that our results might reflect a single anomalous paper. To check this issue, we have rerun Model 2 from Table 2 with seven restricted datasets, each of which drops all observations from one of the papers. The results are reported in the top panel of Table 4. (Insert Table 4 Here) The top panel of Table 4 demonstrates that no one paper is responsible for our main results. The parameter values vary as different papers are dropped, but the sign of the estimates remain unchanged, the magnitude of the parameter estimates is similar across all seven regressions, and the two parameters of interest, the dummy for late observations and the interaction between this dummy and the truncated amount offered are always statistically significant at the 1% level. We are identifying a broad effect, not one that is occurring only in a single paper. Along similar lines, we have also checked to see whether some of the papers show a dramatically different pattern than the dataset as a whole. Specifically, we ran Model 2 from Table 2 separately for the dataset from each of the seven papers. The results of this exercise are reported as the bottom panel of Table 4. For three of the seven papers (Anderson et al, Slonim and Roth, and Cooper et al), neither parameter estimate (late periods or late periods interacted with truncated amount demanded) is statistically significant at even the 10% level, in one paper only the interaction term is significant (Duffy and Feltovich), and for three papers both parameters are significant (Roth et al, Andreoni et al, and Fischbacher et al). All of the statistically significant parameter estimates have the same sign as those reported for the entire dataset, negative for late periods and positive for the interaction term between late period and the truncated amount demanded. For two of the papers (Anderson et al, Slonim and Roth) the signs of these estimates are reversed, but these are the two papers where the parameter estimates are 15

17 the weakest both in terms of magnitude and statistical significance. It probably is not a coincidence that these are also the two papers with the lowest percentage of low offers (offers less than 20% of the pie) in Rounds Once again the evidence indicates that we have identified a broadly occurring effect. The stranger and absolute stranger matching protocols used by the six papers in our dataset are intended to eliminate repeated game effects. If subjects nonetheless treat the session as a single repeated game rather than a series of one-shot games, changing responder behavior could be due to reputation building or strategic teaching. When responders are frequently matched with the same proposers, there is an incentive to reject low offers to maintain a reputation for toughness, hopefully deterring future low offers. 24 As the experiment reaches its end, the benefits of a reputation for toughness decrease and acceptance rates should rise. The endgame effects we observe (see Model 3 in Table 2) suggest that some reputation building is present. However, for a number of reasons we do not believe our results can be explained by reputation building: (1) The decreasing acceptance rate for low offers is contrary to the predictions of a model of reputation building. (2) The rate of change in responders behavior flattens out over time. This would not be predicted by a model of reputation building. (3) Reputation effects should be weaker in the three papers that only have ten rounds and use an absolute stranger matching (Roth et al, Slonim and Roth, and Anderson et al). Looking at the bottom panel of Table 4, two of the absolute stranger papers (Slonim and Roth, and Anderson et al) show no significant changes in responder behavior with experience, but the other (Roth et al) shows a very strong effect. The effect of experience on responder behavior is not consistently different when an absolute stranger matching is used. To make this point more formally, we have rerun Model 2 from Table 2 using only the data from the three papers that use an absolute stranger matching. The same pattern emerges as for the full dataset over time low offers are rejected more frequently and high offers are rejected less frequently. The effect is weaker than in the full dataset, but remains statistically significant For rounds 1-5, the proportion of low offers is 2.0% for Slonim and Roth and 2.7% for Anderson et al. For the remaining papers, the proportion of low offers in rounds 1-5 is 5.7% for Roth et al, 11.7% for Andreoni et al, 5.7% for Cooper et al, 7.5% for Duffy and Feltovich, and 7.8% for Fischbacher et al. 24 Equivalently, this can be thought of as an attempt to teach proposers to make higher offers in the future. 25 The parameters for Round 6 and (Round 6) *Truncated Amount Offered are (s.e. =.308; p =.06) and.018 (s.e. =.008; p =.02) respectively. Jointly, the two parameters are significant at the 1% level. The parameter estimate for Truncated Amount Offered is.089 with a standard error of.006. As an alternative we have modified Model 2 from Table 2 to include interaction terms between a dummy for data from papers using an 16

18 Our dataset pools sessions from seven different countries. The session dummies absorb any differences in levels of acceptance rates across countries, but as an additional check that our main result is not driven by uncontrolled country effects we have rerun Model 2 from Table 2 using only the data from the United States. The restricted dataset still includes 4,720 observations, 66% of the full data set with five of the seven papers being represented. The same pattern emerges as for the full dataset and remains strongly statistically significant. 26 Conclusion 1: With experience, acceptance rates rise for relatively large offers and fall for relatively small offers. This result is robust to controls for end game effects, is not driven by a small subset of the data, and cannot be explained via models of reputation building or strategic teaching. Having established the effect of experience on acceptance rates, we now turn to the adjustment process underlying this effect. Figure 2 illustrates how the probability of acceptance responds to the difference between the previous and current round s offers. To control for the size of the current offer, the data has been broken into offer categories by tenths of the pie. For each category, the probability of acceptance is displayed as a function of whether the current offer is less than, the same as, or greater than the previous offer received by the responder. Since lagged data is being used, data from the first round is dropped. Otherwise, Figure 2 includes data from all rounds. Figure 2 does not include data for offers less than 20 or greater than 50. For all of these categories one of the three cells was virtually empty. The numbers above the bars are the number of observations in the cell. For all three offer categories the pattern is the same. The probability of acceptance is an increasing function of the change in offers. This cannot be attributed to the current offer being high or low in absolute terms since we are (roughly) controlling for the size of the offer. This is precisely the pattern we should see if changes in acceptance rates are driven by changes in beliefs about what constitutes a kind or unkind offer. If a relatively low offer is observed, beliefs about distribution of offers should also shift lower making all offers seem more kind and hence be more likely to be accepted. Likewise, a relatively high offer shifts beliefs upward and leads to fewer acceptances. absolute strangers matching and the three independent variables. None of the three estimates for the interaction variables is statistically significant nor are the three new variables jointly significant (d.f. = 3; χ 2 = 3.46; p =.33). 26 The parameters for Round 6 and (Round 6) *Truncated Amount Offered are (s.e. =.400; p <.01) and.035 (s.e. =.010; p <.01) respectively. Jointly, the two parameters are significant at the 1% level. The parameter estimate for Truncated Amount Offered is.083 with a standard error of

19 Acceptance Rate 1 Figure 2: Acceptance Rate as a Function of Previous Offer Category Note: The numbers above the bars give the number of observations for that bar Offer < Offer < Offer < 50 % Offered Lower than Previous Same as Previous Higher than Previous The regressions shown in Table 5 put the preceding observations on a firmer statistical basis. The base model is the same as Model 2 from Table 2. Recall that this model controls for the current amount offered (unlike Figure 2 which only controls for the offer category). The dataset is smaller than in Table 2 since the use of lagged variables necessitates dropping observations from the first round. Model 1 replicates Model 2 from Table 2 with this reduced dataset. Model 2 adds the change in the truncated amount offered as an independent variable. 27 The effect on the probability of acceptance of receiving a higher offer than the previous round is positive and statistically significant at the 1% level. This supports our observations based on Figure 2. Model 3 adds an interaction term between the change in offer and the dummy for late rounds (Round 6). This interaction term is negative but not statistically significant. The effect of the previous round s offer decreases with experience as would be expected in any sort of learning model, although the effect is weak. Model 4 breaks down changes in the offers by whether there is an increase or decrease. The effect of increases on the current probability of acceptance is 27 This is the difference between the current truncated offer and the truncated amount offered to the same responder in the previous round. 18

20 Average Current Offer larger than the effect of decreases, but the difference is not statistically significant at even the 10% level. 28 (Insert Table 5 Here) Conclusion 2: Holding the current offer fixed, the acceptance rate is an increasing function of the change from the previous offer received. 60 Figure 3: Current Offers as a Function of Lagged Outcome Note: The numbers above the bars give the number of observations for that bar Offer < Offer < Offer < Offer < Offer < 50 Offer 50 Lagged Percent Offered Lagged Offer Rejected Lagged Offer Accepted The primary focus of this paper is the dynamics of responders behavior, but the large dataset we have gathered also provides an opportunity to examine how proposers behavior changes with experience. Figure 3 shows the relationship between a proposer s current offer and the 28 For simplicity, we have restricted ourselves to the immediate lagged offer. However, the models discussed in Section 2 imply that longer lags should matter as well. To check that our results are robust if we allow for more lagged offers to affect the current acceptance rate, we have replicated the results on Table 4 using the difference with the moving average of the previous three rounds rather than just the previous round. Qualitatively, this has no effect on our conclusions. For example, in Model 2 the estimate of the key parameter, Change in Amount Offered, is.0163 with a standard error of This estimate has the same size as the estimate in the original model, somewhat larger magnitude, and once again is statistically significant at the 1% level. 19

21 response (reject or accept) to his lagged offer. Lagged offers are broken into categories defined, as in Figure 1, by tenths of the pie, with all offers greater than or equal to 50% of the pie combined into a single category. Offers from the first round are excluded since there is no lagged offer or response. Otherwise, Figure 3 includes data from all rounds. For all categories of lagged offers, the average current offer is lower if the lagged offer was rejected than if it was accepted. In other words, proposers who have had their previous offer accepted are systematically more aggressive than those who have had their previous offer rejected. 29 This pattern is consistent with reinforcement learning, directional learning, or just about any other sensible learning model. To document the existence of this adjustment pattern in a more systematic fashion, we ran a tobit regression with the current offer as the dependent variable. The independent variables were session dummies, the lagged offer, a dummy for whether the lagged offer was accepted, and a dummy for late rounds (Round 6). Standard errors are corrected for clustering at the individual proposer level. The parameter estimate for the lagged response is with a standard error of meaning, holding the lagged offer fixed, the current offer is 4% lower if the lagged offer was accepted. The parameter estimate is statistically significant at the 1% level. We have also run a variant of this regression that checks whether the size of the adjustment decreases with experience. In this regression, the lagged offer and lagged response are interacted with dummies for early rounds (Round 5) and late rounds (Round 6). The parameter estimate for the lagged response in the early rounds (-6.740) is greater than the estimate for the late rounds (-3.402). Both estimates are significant at the 1% level and the difference between the two estimates is also significant at the 1% level. 30 Our findings are consistent with the commonplace observation that learning curves become flatter with experience (Blackburn, 1936) since any one piece of information about responders behavior matters less for the proposers as they gain experience. As a consequence of this adjustment process, proposers offers tend to converge to the payoff maximizing offer. To confirm this claim, we divided offers into categories by tenths of the pie, as in Figures 1 and 3, and calculated the offer category that yields the highest expected payoff for the proposers in each session. The horizontal axis in Figure 4 shows the proportion of offers in 29 It is not true that offers tend to go up following a rejection and down following an acceptance. There is strong regression to the mean in the data, so proposers who have made a low (high) offer previously tend to offer more (less) regardless of whether the previous offer was accepted or not. The lagged response (accept or reject) affects the strength of this adjustment. 30 The respective standard errors are.789 and

22 % of Offers the expected payoff maximizing category as well as the two categories above (labeled +1 and +2 ) and below (labeled -1 and -2 ). For example, if the expected payoff maximizing category is 30 Offer < 40, the category labeled -1 is 20 Offer < 30 and the category labeled +1 is 40 Offer < 50. Data is only shown from the first ten rounds of play, giving the same number of observations from all proposers and eliminating any distortions due to sessions dropping out. The data is divided between Rounds 1 5 and Rounds Figure 4: Distribution of Offers Note: The numbers above the bars give the number of observations for that bar Categories Away from Expected Payoff Maximizing Category Rounds 1-5 Rounds 6-10 The expected payoff maximizing category is played more frequently with experience, rising from 33% in the Rounds 1 5 to 39% in Rounds All of the other categories become less frequent with experience with the exception of the offer category just below the expected payoff maximizing category. The trend is more extreme if we consider all rounds rather than just the first ten. After the first 10 rounds, 55% of offers fall in the expected payoff maximizing category. See Appendix B for a version of Figure 4 that includes data from all rounds. Standard models of distributional preferences (e.g. Bolton and Ockenfels, 2000; Fehr and Schmidt, 1999; Charness and Rabin, 2002) predict that the expected utility maximizing offer will 21

23 be greater than the expected payoff maximizing offer. 31 There is no reason to expect this shading toward offers above the payoff maximizing offer to be immediately apparent, since proposers need to learn about responders behavior which is also changing over time. However, if subjects learn to maximize their utility and have a preference for avoiding inequality, we predict the distribution of offers to be shaded above the expected payoff maximizing offer in the long run. As seen in Figure 4, the data provides no evidence of the predicted effect. The only category other than the payoff maximizing category that gains weight is the one that involves offering slightly less to the responder! 32 Conclusion 3: Responses to rejections and acceptance cause offers to converge toward the expected payoff maximizing category of offers. In the long run positive offers appear to be largely driven by expected payoff maximization rather than any standard variety of otherregarding preferences. 5. Conclusion: The results of our meta-study show that responders behavior changes with experience: high offers are more likely to be accepted and low offers are less likely to be accepted. Underlying this general result is an adjustment process where receiving a high (low) offer makes it more likely that the next offer, ceteris paribus, will be rejected (accepted). The point of our paper is not that the aggregate change in responders behavior is enormous. If it was, we would not have needed to gather such an extensive dataset. Instead, our results make two main points: (1) There is little reason to believe behavior in the ultimatum game will converge to the subgame perfect equilibrium in the long run. Responders are not being worn down by experience, but instead are learning that low offers are unkind and hence becoming even less likely to accept them. It follows that low offers are becoming less, not more, attractive for proposers over time. Reflecting this, the proportion of low offers also falls with experience. 33 Even if a steady supply of new proposers entered the population so that low offers didn t die out, these new proposers would also face incentives that move them away from low offers. (2) 31 For models of inequality aversion, a proposer who increases his offer decreases advantageous inequality if the offer is accepted. Since the marginal effect of increasing the offer on expected payoffs must be zero at the expected payoff maximizing offer, the derivative of the utility with respect to the offer must be positive due to this decrease. Similar arguments apply to other common varieties of distributional preferences such as a desire to increase social welfare or maxmin preferences (Engelmann and Strobel, 2004). 32 Our conclusions drawn from Figure 4 are robust to dividing the data into smaller offer categories (bin width of 5 rather than 10) and allowing for the fact that the expected payoff maximizing category can change over time. 33 In rounds 1 5, 5% of offers are for less than 20% of the pie. This falls to 2% for round 6 and later. 22

24 Models based on stable preferences over the distribution of payoffs predict no changes in responder behavior and reinforcement learning models predict that acceptance rates should rise for all offers. The data is not consistent with either of these predictions. Models that allow for reciprocity predict rising acceptance rates for high offers and falling acceptance rates for low offers. This is the pattern we observe in the data. 34 It is worth noting that our results are consistent with existing papers that find changes in responders behavior. Manipulations designed to yield more low offers (Winter and Zamir, 2005; Cherry and List, 2000; Cooper et al, 2003) find evidence of rising acceptance rates. Duffy and Feltovich observe decreasing acceptance rates, but only in their treatment with observation. This treatment has the effect of significantly increasing offers. Assuming subjects don t anticipate these treatment effects, having lower (higher) than expected offers should lead all offers being perceived as more (less) kind over time, and hence rising (falling) acceptance rates. 35 Though not the emphasis of our paper, we show that proposer behavior changes with experience. Specifically, over time proposers behavior moves in the direction of expected profit maximization. Consistent with existing results (e.g. Roth et al. 1991; Prasnikar and Roth, 1992; and Roth, 1995), there is little evidence that the long run behavior of proposers is heavily influenced by distributional concerns. In general, our results illustrate the need for more study of the dynamic nature of otherregarding behavior. First, there is a need to develop and test models of these dynamics. While we have shown that the changes observed for responders are broadly consistent with models allowing for reciprocity, we know neither the exact form such a model should take nor whether this is the best possible dynamic model of responder behavior. For example, the hybrid model of Cooper and Stockman (2002) is also broadly consistent with the observed dynamics. Second, methodological work is needed to establish how to best estimate preferences and distributions 34 This is not to say that the entirety of responder behavior can be explained by models of reciprocity alone. For example, the odd results of Yamagishi, Horita, Takagishi, Shinada, Tanida and Cook (2009), showing that positive offers are rejected in impunity games even though there are no repercussions for proposers, do not fit neatly with models of reciprocity. 35 It is worth noting that our conclusion that play will not converge to the subgame perfect equilibrium in the long run assumes a fairly standard distribution of offers. If some feature of the environment (i.e. a more extreme version of the Winter and Zamir design) leads to a steady stream of extremely low offers, we predict convergence to the subgame perfect equilibrium as proposers come to view very low offers as normal and hence not unkind. We find it hard to imagine a situation in which this will naturally occur. 23

25 over preference types in a dynamic environment. Our results also indicate the need for more work on belief elicitation, as changes in beliefs may underlie many observed treatment effects. 24

26 References Abbink, K., Bolton, G., Sadrieh, A. and Tang, F. (2001). Adaptive Learning versus Punishment in Ultimatum Bargaining. Games and Economic Behavior, Volume 37, Number 1, pp Anderson, L., Rodgers, Y. and Rodriguez R. (2000). Cultural differences in attitudes toward bargaining. Economics Letters, Volume 69, Number 1, pp Andreoni, J. and Blanchard, E. (2006). Testing subgame perfection apart from fairness in ultimatum games. Experimental Economics, Volume 9, Number 4, pp Andreoni,J. Castillo,M. and Petrie,R. (2004). Revealing preferences for fairness in ultimatum bargaining. Working Paper Andreoni, J. and Miller, J. (2002). Giving According to GARP: An Experimental Test of the Consistency of Preferences for Altruism. Econometrica, Volume 70, Number 2, pp Armantier, O. (2006). Do Wealth Differences Affect Fairness Considerations? International Economic Review, Volume 47, Number 2, pp Blackburn, J. (1936). The acquisition of skill: an analysis of learning curves. London, H. M. Stationery off. Blount, S. (1995). When Social Outcomes Aren t Fair: The Effect of Causal Attributions on Preferences. Organizational Behavior and Human Decision Processes, Volume 63, Number 2, pp Bolton, G. and Ockenfels, A. (2000). ERC: A theory of equity, reciprocity, and competition. American Economic Review, Volume 90, Number 1, pp Bolton, G. and Zwick, R. (1995). Anonymity versus Punishment in Ultimatum Bargaining. Games and Economic Behavior, Volume 10, Number 1, pp Brosig, J., Reichmann, T. and Weimann, J. (2009). Selfish in the end? An investigation of consistency and stability of individual behavior. Working paper, University of Duisberg-Essen. 25

27 Buchan, N., Croson, R., and Johnson, E., (2004) When Do Fair Beliefs Influence Bargaining Behavior? Experimental Bargaining in Japan and the United States. The Journal of Consumer Research, Volume 31, Number 1, pp Camerer, C. (2003). Behavioral Game Theory: Experiments in Strategic Interaction. Princeton, NJ: Princeton University Press. Cameron, L. (1999). "Raising the Stakes in the Ultimatum Game: Experimental Evidence from Indonesia." Economic Inquiry, Volume 37, Number 1, pp Charness, G. and Rabin, M. (2002). Understanding social preferences with simple tests. Quarterly Journal of Economics, Volume 117, Number 3, pp Chaudhuri, A. (2009). Experiments in Economics, Playing Fair with Money. London: Routledge. Cooper, D., Feltovich, N., Roth, A. and Zwick, R. (2003). Relative versus Absolute Speed of Adjustment in Strategic Environments: Responder Behavior in Ultimatum Games. Experimental Economics, Volume 6, Number 2, pp Cooper, D. and Kagel, J. (2009). Other Regarding Preferences: A Selective Survey of Experimental Results, in John H. Kagel and Alvin E. Roth, eds., The Handbook of Experimental Economics, Vol 2, forthcoming. Cooper, D. and Stockman, C. (2002). Fairness and learning: an experimental examination. Games and Economic Behavior, Volume 41, Number 1, pp Cox, J., Friedman, D. and Gjerstad, S. (2007). A tractable model of reciprocity and fairness. Games and Economic Behavior, Volume 59, Number 1, pp Duffy, J. and Feltovich, N. (1999). Does observation of others affect learning in strategic environments? An experimental study. International Journal of Game Theory, Volume 28, Number 1, pp Dufwenberg, M. and Kirchsteiger, G. (2004). A theory of sequential reciprocity. Games and Economic Behavior, Volume 47, Number 2, pp Engelmann, D and Strobel, M. (2004). Inequality Aversion, Efficiency, and Maximin Preferences in Simple Distribution Experiments. The American Economic Review, Volume 94, Number 4, pp Falk, A. and Fischbacher, U. (2006). A Theory of Reciprocity. Games and Economic Behavior Volume 54, Number 2, pp

28 Fehr, E. and Schmidt, K. (1999). Theory of fairness, competition, and cooperation. Quarterly Journal of Economics, Volume 114, Number 3, pp Fischbacher, U. Fong, C. and Fehr, E. (2009). Fairness, errors and the power of competition Journal of Economic Behavior and Organizaions. Volume 72, pp Gale, J., Binmore, K. and Samuelson, L. (1995). Learning to be imperfect: The ultimatum game. Games and Economic Behavior, Volume 8, Number 1, pp Gneezy, U. and List, J. (2006). Putting Behavioral Economics to Work: Testing for Gift Exchange in Labor Markets Using Field Experiments., Econometrica, Volume 74, Number 5, pp Guth, W., Schmittberger, R. and Schwartz, R. (1982). An Experimental Analysis of Ultimatum Bargaining. Journal of Economic Behavior and Organization, Volume 3, Number 4, pp Healy, P.J. (2007). Group Reputations, Stereotypes, and Cooperation in a Repeated Labor Market. The American Economic Review, Volume 97, Number 5, pp Henrich, J., Boyd, R., Bowles, S., Camerer, C., Fehr, E., Gintis, H. and McElreath, R. (2001). In Search of Homo Economicus: Behavioral Experiments in 15 Small-Scale Societies. The American Economic Review P&P, Volume 91, Number 2, pp Hoffman, E., McCabe, K., Shachat, K., and Smith, V. (1994). Preferences, Property Rights and Anonymity in Bargaining Games. Games and Economic Behavior, Volume 7, Number 3, pp Kirchsteiger, G. (1994) The Role of Envy in Ultimatum Games. Journal of Economic Behavior & Organization, Volume 25, pp Levine, D. (1998). Modeling Altruism and Spitefulness in Experiments. Review of Economic Dynamics, Volume 1, pp List, J. and Cherry, T. (2000). Learning to accept in ultimatum games: Evidence from an experimental design that generates low offers. Experimental Economics, Volume 3, Number 1, pp Oosterbek, H., Sloof, R., and Van de Kuilen, G. (2004). Cultural Differences in Ultimatum Game Experiments: Evidence from a Meta-Analysis. Experimental Economics, Volume 7, Number 2, pp

29 Roth, Alvin E. (1995) Bargaining Experiments, in John H. Kagel and Alvin E. Roth, eds., The Handbook of Experimental Economics. Princeton: Princeton University Press, pp Roth, A. and Erev, I. (1995). Learning in Extensive-Form Games: Experimental Data and Simple Dynamic Models in the Intermediate Term. Games and Economic Behavior, Volume 8, Number 1, pp Roth, A., Prasnikar, V., Okunofujiwara, M. and Zamir, S. (1991). Bargaining and market behavior in Jerusalem, Ljubljana, Pittsburgh, and Tokyo: An experimental study. American Economic Review, Volume 81, Number 5, pp Slonim, R., and Roth, A. (1998). Learning in high stakes ultimatum games: An experiment in the Slovak republic. Econometrica, Volume 66, Number 3, pp Sutter, M., Kocher, M., Strauß, S. (2003). Bargaining under time pressure in an experimental ultimatum game. Economics Letters Volume 81, Number 3, pp Yamagishi, T., Horita Y., Takagishi H., Shinada M., Tanida S., and Cook K. (2009) The Private Rejection of Unfair Offers and Emotional Commitment. PNAS Volume 106, Number 28, pp Winter, E. and Zamir, S. (2005). An Experiment with Ultimatum Bargaining in a Changing Environment. Japanese Economic Review, Volume 56, Number 3, pp

30 Table 1: Papers Included in Data Set Paper Roth, Prasnikar, Okuno- Fujiwara, and Zamir, 1991 Number of Sessions Number of Responders Number of Rounds Slonim and Roth, Matching Absolute Strangers Absolute Strangers Location(s) USA, Slovenia, Japan, Israel Slovakia Duffy and Feltovich, Strangers USA Anderson, Rodgers, Rodriguez, 2000 Cooper, Feltovich, Roth, and Zwick, 2003 Andreoni, Castillo, and Petrie, 2004 Fischbacher, Fong, and Fehr, (43 obs) 60 (13 obs) Absolute Strangers Strangers USA, Honduras USA Strangers USA Strangers Switzerland 29

31 Table 2 Regression Analysis: Effect of Experience on Acceptance Rates Variable Model 1 Model 2 Model 3 Model 4 Truncated Amount Offered.1021**.0866***.0866***.0868*** (.0053) (.0053) (.0053) (.0053) Late Rounds (Round 6).0667 (.0522) *** (.2655) *** (.2660) *** (.2562) Late Rounds * Truncated Amount Offered.0268*** (.0067).0267*** (.0067).0272*** (.0065) Last Round Played.2382***.2365** (.0882) (.0884) Round (.4227) (Round 11) * Truncated Amount Offered (.0111) Notes: All regressions included session dummies. Standard errors are corrected for clustering at the individual (responder) level. Three (***), two (**), and one (*) stars indicate statistical significance at the 1%, 5%, and 10% respectively. All regressions are based on 7188 observations from 387 subjects in 44 sessions. 30

32 Table 3 Regression Analysis: Alternative Controls for Individual and Session Effects Variable Model 1 Model 2 Model 3 Model 4 Probit/Logit Probit Probit Logit Logit Session Effects Fixed Effects Fixed Effects Random Effects N/A Individual Effects Clustering Random Effects Nested Random Fixed Effects Truncated Amount Offered.0866***.1252***.2388***.2517*** (.0053) (.0055) (.0112) (.0124) Late Rounds *** *** *** *** (Round 6) Late Rounds * Truncated Amount Offered (.2655).0268*** (.0067) (.2439).0328*** (.0062) (.4729).0602*** (.0120) (.5331).0586*** (.0133) Notes: Model 1 is the same as Model 2 on Table 2 in the text. Three (***), two (**), and one (*) stars indicate statistical significance at the 1%, 5%, and 10% respectively. All regressions are based on 7188 observations from 387 subjects in 44 sessions. 31

33 Table 4: Regression Analysis Robustness Across Papers Regressions with One Paper Dropped Paper Dropped Roth et al, Slonim and Duffy and Anderson et Cooper et al, Andreoni et Fischbacher 1991 Roth, 1998 Feltovich, 1999 al, al, 2004 et al, 2009 Number of Responders Number of Observations Truncated Amount Offered.0901***.0839***.0893***.0848***.0847***.0859***.0881*** (.0067) (.0057) (.0055) (.0057) (.0055) (.0054) (.0055) Late Rounds (Round 6) *** (.3239) *** (.2850) *** (.2681) *** (.2847) *** (.2657) *** (.2817) *** (.2846) Late Rounds * Truncated Amount Offered.0228*** (.0083).0302*** (.0073).0228*** (.0068).0307*** (.0072).0259*** (.0066).0271*** (.0072).0264*** (.0072) Regression with Only One Paper Paper Roth et al, Slonim and Duffy and Anderson et Cooper et al, Andreoni et Fischbacher 1991 Roth, 1998 Feltovich, 1999 al, al, 2004 et al, 2009 Number of Responders Number of Observations Truncated Amount Offered.0816***.1053***.0629***.0979***.0964***.0917***.0706*** (.0088) (.0127) (.0127) (.0118) (.0156) (.0221) (.0160) Late Rounds (Round 6) *** (.3991).3597 (.6149) (.7504).0629 (.5871) (.6310) *** (1.1734) ** (.4991) Late Rounds * Truncated Amount Offered.0336*** (.0097) (.0155).0346* (.0203) (.0146).0211 (.0165).0873*** (.0282).0318*** (.0126) Notes: All regressions included session dummies. Standard errors are corrected for clustering at the individual (responder) level. Three (***), two (**), and one (*) stars indicate statistical significance at the 1%, 5%, and 10% respectively. 32

34 Table 5 Regression Analysis: Effect of Previous Offer on Acceptance Rates Variable Model 1 Model 2 Model 3 Model 4 Truncated Amount Offered.0895***.0797***.0781***.0815*** (.0059) (.0062) (.0068) (.0064) Late Rounds (Round 6) *** (.2819) *** (.2840) *** (.3193) *** (.2809) Late Rounds * Truncated Amount Offered.0253*** (.0071).0259*** (.0072).0285*** (.0079).0256*** (.0071) Change in Amount Offered.0099***.0121*** (.0027) (.0042) Late Rounds * Change in Amount Offered (.0053) Change in Amount Offered * Positive Change.0133*** (.0042) Change in Amount Offered * Negative Change.0064 (.0044) Notes: All regressions included session dummies. Standard errors are corrected for clustering at the individual (responder) level. Three (***), two (**), and one (*) stars indicate statistical significance at the 1%, 5%, and 10% respectively. All regressions are based on 6801 observations from 387 subjects in 44 sessions. 33

35 Appendix A: Methods of Search and Summary of Papers Methods of Online Search: The search for papers was carried out as follows. Key phrases were entered into Google Scholar and the number of results was recorded. Since it would be impossible to read every UG paper published, we restricted our search to what we felt were reasonable limits. Thus, for each search item, the first 50 papers generated were examined. We cut the search off after the first 50 papers because the fit of papers to our criteria declined significantly after the initial papers. We went well beyond this point to make certain we did not miss any relevant papers, but searching through thousands of irrelevant papers would have served no useful purpose. In examining the papers, the first thing analyzed was the title. If from the title the paper was obviously not well suited for our study, it was skipped. An example of a title that was skipped is Irrational economic decision-making after ventromedial prefrontal damage: evidence from the Ultimatum Game. If from the title it could not be determined if the study may include some form of our standard game, the abstract was then read. If it became evident after reading the abstract that the paper did not contain any session that met our criteria, the paper was skipped. If a paper met the above two criteria, the experimental procedures section was examined. The following lists the key phrase searches used and the number of results for each. The following is filtering for papers published before learning in ultimatum games 13,900 results ultimatum games 13,700 results repeated ultimatum games 15,800 results ultimatum games 10 periods 9,020 results repeated ultimatum games 10 periods 6,610 results standard ultimatum games 15,600 results Using the above search criteria returned no additional papers which meet our criteria. When the filter is used to look for those papers published more recently than 2008, the following results are obtained. For the more recent papers, only the first 20 papers in each search category were examined (once again due to reaching obviously irrelevant papers). learning in ultimatum games 3,320 results ultimatum games 3,550 results repeated ultimatum games 3,750 results ultimatum games 10 periods 1,670 results 34

36 repeated ultimatum games 10 periods 1,220 results standard ultimatum games 3,570 results Summary of Included Papers: Roth et al (1991) presents a cross-cultural study of ultimatum and market games. The games were conducted in Israel, Japan, Yugoslavia and the United States. All sessions lasted for 10 rounds in which one was randomly chosen for payment. The players in this game bargained over 1,000 tokens. These tokens were equivalent to $10 (or its equivalent in countries outside the U.S.) in all treatments except one in which they were worth $30. From this paper we were able to use data from 127 responders. In all four countries play varied substantially from the predictions of subgame perfection. More importantly, significant differences were observed in bargaining behavior between the four countries, differences which can be attributed to cultural differences. Slonim and Roth (1998) focused on the effects of high stakes on behavior in ultimatum games. Sessions were conducted in the Slovak Republic because of the benefit of being able to pay subjects a higher amount in relative terms. Again, subjects were bargaining over 1,000 tokens. These tokens were translated to 60, 300, and 1500 Slovak Crowns (SC). Average monthly wage rate was 5500 Slovak Crowns, thus the subjects were bargaining over anywhere from 2-3 hours pay up to a little over a week s worth of pay. There were three sessions worth 60 and 1500 SC s and four worth 300 SC s. Each session consisted of 10 rounds and one was randomly chosen for payment. We were able to use data from 82 responders from this study. In the higher stakes treatments, they observed fewer rejections. With experience, offers are lower as well with high stakes. Andreoni, Castillo and Petrie (2004) were interested in seeing if they could predict behavior in standard ultimatum games using preferences fit from data in a convex ultimatum game. This experiment lasted 20 rounds in which the subjects were paid for all rounds of play. Subjects bargained over how to split ten quarters. From this paper, we were able to use observations from 24 responders. The standard ultimatum game data was unexceptional and the prediction exercise worked quite well. Anderson, Rodgers and Rodriguez (2000) focused on the perceived bargaining aversion in the U.S. and the lack thereof in Honduras. They ran experiments in both countries. Subjects bargain over 10 tokens with token value set to equilibrate purchasing power in the two countries. 35

37 The experiment lasted 10 rounds and a random round was picked for payment. All subjects played a block of ultimatum games and a block of dictator games, with the order systematically varied between sessions. We only use the data from the sessions in which the ultimatum game was run first. Note, the subjects did not know that another game was following the first. We were able to use the data from 60 responders. The ultimatum game results show a difference between the two countries as about one token more is proposed in Honduras than the U.S. and the rejection rate is also higher in the Honduras than the U.S. Duffy and Feltovich (1999) studied whether observing the actions of others is important to learning in bargaining games. To do this, they ran the best shot and ultimatum games where the decisions of a different pair was or was not observed. We only used the data from ultimatum games where the subjects did not observe the decisions of another pair. They ran 40 rounds of the ultimatum game where one round was chosen at random for payment. The subjects in this experiment were bargaining over $10. From this, we were able to use the responses of 16 responders. Duffy and Feltovich find that observation moves play away from the subgame perfect equilibrium, with higher offers and lower acceptance rates. Cooper et al (2003) were interested in seeing if learning by responders takes place in ultimatum games. There were two treatments, one in which proposers only played every other round (there were twice as many proposers as responders) and one which was standard (i.e. proposers play every round). The games ranged from rounds in which subjects were bargaining over $10 in increments of $1. A random round was chosen to pay the subjects. We only used the standard results for our current analysis which amounted to 56 responders. Cooper et al observe a lower rejection rate in the 2x1 treatment than in the 1x1 (standard) treatment, consistent with learning by responders. Fischbacher et al (2009) were interested in understating how bargaining behavior would change once competition was added to the basic ultimatum game framework. In their experiments, they conducted a control treatment where subjects participated in a standard ultimatum game for 20 rounds. In this treatment, there were 46 subjects who bargained over 100 tokens and earned an average of CHF. In addition to this, they also had treatments where there were 2 or 5 responders or 2 proposers. Their conclusion is that a purely self-interested, rational model does a poor job of predicting behavior in this setting. 36

38 Appendix B: Versions of Figure 1 and 4 Using All Rounds 37