Learning and Transfer: An Experimental Investigation into Ultimatum Bargaining

Transcription

1 SCHOOL OF ECONOMICS HONOURS THESIS AUSTRALIAN SCHOOL OF BUSINESS Learning and Transfer: An Experimental Investigation into Ultimatum Bargaining Author: Justin CHEONG Supervisor: Dr. Ben GREINER Bachelor of Laws / Bachelor of Commerce Honours in Business Strategy and Economic Management 24 th October, 2011

2 DECLARATION I hereby declare that this thesis submission is my own work and that any contributions or material by other authors used herein have been appropriately acknowledged. This thesis has not been submitted to any other university or institution as part of the requirements for a degree or other award. Justin Cheong 24th October,

3 ACKNOWLEDGEMENTS I would like to first and foremost thank my talented supervisor Dr. Ben Greiner, whose patience and guidance made this thesis possible and helped me to learn a little more about the world. I gratefully thank the donors of the Australian School of Business Honours Scholarship and the Sir William Tyree Award in Commerce, whose financial support greatly assisted with my studies. Further thanks to the ASBLab Small Projects Fund for generously financing my experiments this year. Many thanks to Dr. Stanley Cho and Dr. Valentyn Panchenko for guiding us through the Honours year. Special thanks to A/Prof. Paul Pezanis-Christou for his comments, expertise and his kind charity in shouting fellow tutors to Yum Cha. Sincere thanks to my fellow Honours students of The diverse range of characters we had in this cohort made the Honours experience interesting and unforgettable. Special thanks to Chia, Jade, Hoon, Cencil, Ashley, Jankree, Jess, Patrick, Marcus, Riri, Shinvana and Silverbolt. No thanks to Frank Ciarliero, who refused to stop riddling my laptop with pointless software updates. Just kidding, thank you Frank, I know you meant the best. Special thanks to my close friends, especially the ones who were unfailingly there to listen to my rants, play Scrabble and visit Sydney Fish Markets. Here are the words you wanted to see in my thesis: Justingan, Baptême du feu, Like a boss. Finally, countless thanks to mum and dad for keeping me well-fed throughout the year and for making the best soup in the world. 3

4 ABSTRACT We experimentally examine individual learning behaviour and learning transfer across three variations of the ultimatum game. We implement repeated versions of a twoplayer ultimatum game, a proposer competition variation with two proposers and one responder and a responder competition variation with one proposer and two responders. Using a within-subjects design, we observe the effects of learning and transfer as these games are played in different orders. All six permutations of transfer are captured in a systematic experimental design. We further elicit responder preferences through minimum acceptance thresholds. We find evidence of forward-looking deliberation, reinforcement and directional learning as well as concept learning in subject behaviour. Our results show that prior interaction in responder competition leads to a significant decrease in the frequency of fair outcomes in the ultimatum game, but also that prior interaction in both proposer competition and responder competition leads to overall fair outcomes in the ultimatum game. We find various other history effects, suggesting that experience in prior interactions can significantly influence subjects decisions and learning depending on the order in which games are played. 4

5 TABLE OF CONTENTS 1 INTRODUCTION 7 2 LITERATURE REVIEW 10 3 THE EXPERIMENT Experimental Design Experimental Procedures Theoretical Predictions EXPERIMENTAL RESULTS Summary of Results in all treatments Analysis of Decisions within different permutations Analysis of Decisions under different histories Frequency of Fair Decisions under different histories OLS Regression Analysis DISCUSSION 55 6 CONCLUSION 61 7 REFERENCES 62 Appendix A Experimental Protocol 64 Appendix B Experimental Instructions 66 B.1 General Instructions B.2 Instructions for the Ultimatum Game B.3 Instructions for Proposer Competition

6 B.4 Instructions for Responder Competition Appendix C Post-experiment Questionnaire 72 Appendix D Supplementary Graphs 73 D.1 Distribution of Offers and Thresholds in UG D.2 Distribution of Offers and Thresholds in PC D.3 Distribution of Offers and Thresholds in 2R

7 1 INTRODUCTION In the behavioural sciences, learning is an evolutionary process by which an animal acquires and updates information about its surrounding environment. In the context of economics, learning plays an important role in explaining the dynamics of decisionmaking as agents gain experience. From a game theoretic perspective, this experience can be useful to players who learn about both the structure of a game as well as the preferences of other players. To this extent, learning theories can be crucial to our understanding of individual decision-making and can explain convergence to outcomes in strategic interactions (Fudenberg and Kreps, 1988). The need for learning models in the economics literature reflects the fundamental need to explain bounded rationality in human decision-making. While the underlying assumption of homo economicus and rationality in game theory implies that players are perfectly strategic, forward-looking deliberators, this assumption necessarily implies that decision-makers have, as described by Güth (2000), unlimited cognitive abilities. Evidence from experimental settings has consistently shown this assumption to be unrealistic, as humans demonstrate limited reasoning abilities when making decisions. Rather, individuals often require time, repetition and feedback to learn about their predicament before an optimal response can be observed and understood. For this reason, any complete theory used to explain individual behaviour must include consideration of backward-looking adaptation. However, as Güth (2000) further points out, this by no means implies that consideration of forward-looking behaviour should be abandoned. Instead, it is imperative to recognise an important difference between humans and other species in the animal kingdom in our ability to use skills such 7

8 as reasoning and logical deduction. This key difference in cognitive ability suggests that humans are capable of developing more complex causal models to solve problems rather than simply relying on adaptive dynamics. As a result, it is necessary in any descriptive theory of human behaviour to recognise humans as sophisticated learners and decision-makers. This thesis explores the above ideas about learning by experimentally testing behaviour in the bargaining context. In particular, we examine decision-making in the ultimatum game developed by Güth et al (1982) and its competitive variations. In the traditional ultimatum game, a proposer suggests the division of a pie to a responder, who must then choose to either accept or reject the proposal. Should the responder accept the proposal, the pie is divided accordingly, but should she reject, both players receive nothing. The ultimatum game has been a subject of particular interest in the experimental economic literature as evidence shows that most proposers offer responders 40 to 50 per cent of the pie (Henrich et al., 2004), despite game theory predicting that the proposer keeps almost all of the pie in equilibrium. In the literature, this robust experimental result has been attributed to fairness, inequality aversion and strategic considerations (see e.g, Bahry and Wilson, 2005). However, experimental evidence also shows that subjects can converge to unfair outcomes when the ultimatum game is varied in certain ways, for instance, by introducing competing proposers or responders into the game. 1 In this thesis, we experimentally examine learning behaviour as subjects play three variations of the traditional ultimatum game. These variations comprise the stan- 1 E.g., the market game (Roth et al., 1991) or the responder competition game (Güth et al., 1997; Grosskopf, 2003). These are discussed in detail in Chapter 2. 8

9 dard ultimatum game, a proposer competition version with two competing proposers and a responder competition version with two competing responders. In our experiment, we search in particular for evidence of learning transfer, i.e., when subjects learning in one game affects their decisions in a different game. As the literature suggests that these three similar games lead to clearly different outcomes, we are able to systematically examine learning transfer as the games are played in different orders. In our results, we firstly find that players do converge to different outcomes in these three games without any prior experience. Outcomes in the ultimatum game are approximately fair, while outcomes in proposer competition favour the responder and outcomes in responder competition favour the proposer. Moreover, by comparing treatments with different histories, we further find evidence of forward-looking deliberation, reinforcement learning and concept learning in subject behaviour. We find that prior experience in responder competition leads to a significant decrease in the frequency of fair outcomes in the ultimatum game, suggesting that a breakdown of fairness considerations can persist across games. However, prior experience in both responder and proposer competition leads to approximately fair outcomes in the ultimatum game, suggesting that joint experience in situations of both bargaining advantage and disadvantage has a restoration effect on fairness considerations. Furthermore, we find that prior experience in the ultimatum game leads to a quicker increase in offers made by proposers in proposer competition. However, we also find that prior experience in responder competition slows down learning in proposer competition. Together, these results suggest that learning transfer is a real but complex phenomenon that occurs differently depending on the concepts learned in each game and the order in which games are played. 9

10 2 LITERATURE REVIEW In the economic literature, the dynamics of learning have been modelled extensively by different authors. One of the earlier working models by Fudenberg and Kreps (1988) describes learning as a process in which players use experimentation as a means to acquire information, which is the basis for which an equilibrium outcome is reached. Specifically, the model describes experimentation as a means to acquire information about out-of-equilibrium play, with such strategy profiles becoming unstable over time as experimentation reveals their suboptimal nature. Essentially, this model has lent support to equilibrium analysis by describing learning as the process used to converge to the equilibrium state in repeated interactions. In the past few decades, the experimental literature has seen the development of a number of prominent learning models, which have shared the similar goal of explaining how players reach an equilibrium state. The reinforcement learning model by Roth and Erev (1995) captures the basic law of effect, which is the idea that over time, more successful strategies tend to be reinforced over less successful strategies. The Experience Weighted Attraction model by Camerer and Ho (1999) follows a similar line of thinking by describing attractions to different strategies which become updated by experience, the EWA model suggests that choice probabilities ultimately depend on these levels of attraction. Meanwhile, authors such as Selten and Stoecker (1986) have modelled learning as a process in which players explore their strategy space in directions, and directionally adjust their strategies in response to the feedback they receive. The rule learning model by Stahl (1996) further suggests that players do not simply 10

11 choose between in-game strategies, but rather choose between more complex cognitive strategies from a rich space of behavioural rules. While these learning theories have achieved varying levels of success in predicting within-game learning, they share a common shortfall in their inability to answer a more difficult question: How do people learn across different games? While the simple reinforcement and directional learning models describe an initial propensity to explain the behaviour of agents in the first period of a given game, they are not able to explain where this initial propensity comes from or how it might change with prior experience in a different interaction. In addition, these learning theories do not explain how learning in one game might speed up, slow down, or even change the learning process in a subsequent game. 2 Thus, the fundamental question arises about how humans can take their learning from one situation and apply it to another. In the psychology literature, learning transfer is described as the process and the extent to which past learning affects behaviour in a new situation (see Ellis, 1965; Woodworth, 1938). Helfenstein and Saariluoma (2005) describe transfer as the case where mastery of one situation makes the subsequent handling of another easier or more difficult. While psychologists have shown learning transfer to be a cognitively demanding process, they recognise that transfer can occur under specific conditions. For instance, Johnson-Laird (1999) explains transfer to be the result of humans developing mental models in their learning, which can be evoked in a subsequent situation if the two situations share similar properties, characteristics or concepts. 2 Camerer, Ho and Chong (2000) describe this as learning about learning. See also Stahl (1996). 11

12 Few learning models in the economic literature have been able to broadly capture learning transfer, although authors such as Fudenberg and Kreps (1988) have always considered it to be an issue. In a signalling game experiment, Cooper and Kagel (2009) discuss transfer as the result of learning about concepts. They explain that by understanding why strategic play works in a prior game and recognising that similar concepts apply in a subsequent game, agents can use their learning to more effectively adjust to new situations, resulting in a positive, or successful, transfer of learning. 3 Critically, Cooper and Kagel (2008) distinguish this type of sophisticated learning from simple reinforcement learning. They comment that learning transfer must reflect an ability to use some underlying concept, and not merely continuing use of a previously successful strategy. This commentary resonates with the discussion by Fudenberg and Kreps (1988), where players are described as forming mental models and applying these mental models to infer what will happen in future interactions. In an experimental study, Grosskopf (2003) examines learning and transfer in the context of an ultimatum bargaining interaction. The ultimatum game, as developed by Güth et al. (1982), involves two players a proposer and a responder who must agree on the division of a pie. In the typical ultimatum game, the proposer must make a suggestion of how to divide a sum of money between himself and a responder. Upon observing this proposal, the responder then chooses either to accept this division, or to reject it, leaving both players with nothing. In the experimental literature, the ultimatum game has captured the interest of many economists as the Subgame perfect equilibrium is rarely observed, despite the clear game-theoretic prediction. Whereas the equilibrium prediction involves the responder accepting any positive amount and by backwards in- 3 In their paper, the authors show that use of context can also enhance learning transfer, which is explained as a way for subjects to draw analogies and associations between different situations. 12

13 duction, the proposer offering to give the responder only the least amount possible, experimental results show that most proposers offer between 40 to 50 per cent of the pie to responders, while responders on average do not accept offers of less than 20 per cent (Henrich et al., 2004). This particular result has consistently been used as evidence against the joint rationality and selfishness assumption of homo economicus, showing that fairness considerations are a robust phenomenon in bargaining interactions. In the cross-game experimental study by Grosskopf (2003), subjects interacted in both a standard ultimatum game and a modified version of the ultimatum game with multiple responders. In the version with multiple responders, a single proposer makes an offer which could be accepted or rejected by each of three responders. If more than one responder accepts, a random draw decides which accepting responder would receive the offer made by the proposer. Originating in a paper by Güth et al. (1997), it has been shown that fairness considerations can decrease dramatically in this version of the game, with proposers offering much lower amounts to responders over time, while competing responders reject unfair offers much less often. In Grosskopf s experiment, the behaviour of subjects was examined as they played the two games in different orders. In one treatment, subjects played six rounds of the traditional ultimatum game followed by six rounds of the modified ultimatum game, while in another treatment subjects would do the reverse. The results of the experiment raised ambiguity about how transfer of learning occurs in this type of bargaining interaction. On the one hand, proposers tended to drastically increase their offers when switching from the modified responder game to the ultimatum game, as if they recognised a change in fairness considerations. On the other hand, proposers switching from the traditional ultimatum game to the modified ultimatum game did not adjust their of- 13

14 fers quickly when entering the modified game. These results suggest that learning about underlying concepts in games can be a complex process affected by the order in which games are played. In this thesis, we take Grosskopf as the primary motivation for conducting a cross-game experiment in order to examine learning and transfer across the ultimatum game and the responder competition variation. We further utilise a modification of the market game (Roth et al., 1991), or the proposer competition game for our experiment, which is a variation of the ultimatum game in which multiple proposers make competing offers to a single responder. In addition, we use the minimum acceptance threshold as a type of strategy method to elicit the preferences of responders. 4 This thesis contributes to the existing literature by exploring how different learning behaviours can manifest in the context of a bargaining experiment. In using our experiment, we are able to examine backward-looking and forward-looking behaviour as described by Güth (2000), while also exploring the properties of concept learning and learning transfer as discussed by Cooper and Kagel (2009). Through our implementation of multiple permutations as inspired by Grosskopf s (2003) experiment, we aim to systematically discover how learning can be transferred from one type of bargaining situation to another. 4 While the literature shows that responder behaviour can vary with different elicitation methods (Güth et al., 2001; Osterbeek et al., 2004; Güth and Tietz, 1990), this is not of concern as we use the same method across all treatments, and any differences are explained as due to subject heterogeneity. 14

15 3 THE EXPERIMENT 3.1 Experimental Design To explore learning transfer, we implement in our experiment three variations of the ultimatum game that utilises the minimum acceptance threshold as a strategy method for responder decisions (Selten, 1967). We name these three variations UG (ultimatum game), PC (proposer competition) and 2R (responder competition). Each of these games is repeated for 10 rounds to allow for sufficient learning and convergence to equilibria. 5 All three games involve two types of players, proposers and responders. In the ultimatum game, a proposer must decide the division of 100 points (AUD$4.00) between himself and a responder by making an offer to the responder. Meanwhile, the responder must simultaneously decide an acceptance threshold, which is the minimum amount of points she must receive in order for her to accept an offer. If the proposer s offer satisfies the threshold of the responder, then that offer is accepted and 100 points is divided accordingly. Otherwise, the offer is rejected and both participants receive nothing. The PC game is identical to the ultimatum game, but with two proposers making offers instead of one. The payoff is determined as follows. If only one of the two offers satisfies the acceptance threshold of the responder, then the proposer who made that offer receives the division of the points, while the other proposer receives nothing. If neither of the offers satisfies the acceptance threshold of the responder, then all players receive nothing. If both of the offers satisfy the responder s threshold, then the responder is shown the two offers and is given the ability to choose the offer she prefers, 5 Grosskopf (2003) uses six rounds in her experiment; Roth et al (1991), and Güth et al (1997) have ten rounds. These studies suggest that ten rounds are sufficient to observe convergence of behaviour. 15

16 which results in the proposer who made the other offer receiving nothing. This is a modification of the market game by Roth et al. (1991) in that we do not define competition to the players exogenously. By providing responders with the choice of which offer to accept, we allow responders to express inequality aversion by selecting the lower offer. 6 When given the choice between offers, responders are not shown the identities of the proposers making those offers, so as to reduce any reputation effects across rounds. The 2R game is identical to the UG game, but with two responders setting thresholds instead of one. The payoff is determined as follows. If the proposer s offer satisfies only one of the two responders thresholds, the points are divided with the responder who set that threshold, while the other responder receives nothing. If the proposer s offer satisfies neither of the two responders thresholds, all players receive nothing. If the proposer s offer satisfies both of the two responders thresholds, the proposer can select one of the two responders to divide the pie with, which results in the other responder receiving nothing. For the proposer s decision in the last case, no information is provided to the proposer about either responder s threshold. In this sense, the selection decision is equivalent to the random respondership case in the experiment by Güth et al. (1997), where responder selection is imposed exogenously and randomly. To encourage learning, feedback is provided to participants at the conclusion of each round. This information includes the earnings of all players in the given game, the offer(s) made in that round and whether or not the offer(s) were accepted. Other responders thresholds are not made known to the players. In the PC game, feedback is 6 Experimentally, our results show that when faced between two different offers, responders choose the higher offer 91 per cent of the time. 16

17 also provided on how many offers met the single responder s threshold. In the 2R game, further feedback is given on how many thresholds the single proposer s offer met. We limit offers and thresholds to values between 2 and 98 in order to reduce the predicted equilibria in each game, while the values of offers and thresholds are limited to increments of 1. 7 Each experimental session consisted of three phases, where in each phase subjects would play one of the three games. The order in which subjects played the three games depended on which of the six treatments they were allocated to. All permutations of transfer are studied in this design, as shown in Table 1. Subjects were allocated to these treatments using the matching schedule shown in Table 2. Table 1. Six permutations of subject play in experimental design 1. UG PC 2R 4. UG 2R PC 2. PC 2R UG 5. PC UG 2R 3. 2R UG PC 6. 2R PC UG 7 With these values, there exists a set of Subgame perfect equilibrium in the PC game where both proposers offer 98. For consistency, we impose these upper and lower limits on offers and thresholds in all three games. 17

18 Table 2. Matching schedule of players in experimental design. P denotes proposer i and R denotes responder j. 1. P P P P 4. P P P P R R R R R R R R 2. P P P P 5. P P P P R R R R R R R R 3. P P P P 6. P P P P R R R R R R R R Within this design, proposers P to P and responders R to R would play three different games in the experiment, but retain their roles as either proposer or responder throughout. Meanwhile, proposers P and P played only the PC game throughout all three phases of the experiment, while responders R and R played only the 2R game in all phases of the experiment. 8 While subjects were informed in advance that the experiment would consist of three phases, they were not informed in advance of what games they would play. Through this design, we were able to examine the progression of offers and acceptance thresholds under each permutation in order to evaluate learning transfer. 8 The inclusion of P, R, P and R was necessary for the design of the experiment. The results show that the average behaviour of these subjects remained consistent in each phase of the experiment, making them suitable controls for the experiment. 18

19 3.2 Experimental Procedures Our experiment was conducted over 10 sessions in the ASB Experimental Research Laboratory, at the University of New South Wales in August and September, Sixteen subjects were recruited for each session through ORSEE (Greiner, 2004), for a total of 160 recruitments from the ASBLab subject pool. This comprised a total of 89 males and 71 females. As described, each session was split into three phases. To discourage any endowment effects across the three phases, subjects were paid for only one phase of the experiment, which was decided by a public die roll. Subjects received general instructions on paper for the whole experiment and specific instructions for each phase of the experiment. Following each phase, instructions were collected from participants and new instructions for the following phase were distributed. Subjects entered their decisions anonymously into computer screens, with the experiment program created on ztree (Fischbacher, 2007). Each session lasted approximately 60 minutes. Participants received a show-up fee of AUD$5.00 plus an average performance payment of AUD$18.40, with a standard deviation of AUD$8.40. To minimise any context effects, we limited the use of language to neutral words in the experiment instructions. 9 At the conclusion of each session, subjects were asked to complete a questionnaire. This contained questions asking subjects to describe the experiment in their own words, explain how they reached their decisions and whether their experience in one part of the experiment affected subsequent decisions in any way See Appendix A for the protocol used by the experimenter. See Appendices B for the experiment instructions. 10 See Appendix C for the post-experiment questionnaire. 19

20 3.3 Theoretical Predictions Game-theoretic predictions We describe the game theoretic prediction in the UG game as the Perfect equilibrium refinement of Nash equilibrium described by Selten (1975). In the UG game, the responder s lowest possible acceptance threshold (in this case, =2) weakly dominates all other thresholds. Therefore, under Perfect equilibrium the responder chooses the lowest possible threshold, =2, and the proposer offers the responder the lowest possible amount, =2. In the PC game, sequentiality implies that the game theoretic solution is a set of Subgame perfect equilibria. In the last stage of the game, assuming both offers meet the responder s threshold, the rational responder observes two offers and chooses to accept the higher offer. By backwards induction, both proposers will therefore set the maximum possible offer. Thus, the solution to PC is the set of Subgame perfect equilibria where both proposers offer the maximum amount, =98, while the responder sets any threshold and chooses the higher offer in the second stage. 11 In the 2R game, the game theoretic prediction is the extensive form Perfect equilibrium where the proposer selects a responder to divide the points with in the second stage of the game, with equal probability that either responder is selected. In the first stage of the game, both responders set the lowest possible (weakly dominant) threshold amount, =2, while the proposer offers the lowest possible amount, =2. 11 Note again that the responder s lowest possible acceptance threshold =2 weakly dominates all other possible threshold choices. Therefore in an extensive form Perfect equilibrium, the responder sets the threshold =2. 20

21 Experimental and learning predictions Contrary to the game theoretic prediction, most experimental evidence shows that proposers offer an equal division (50-50) of the pie in the ultimatum game due to fairness and strategic considerations. 12 Consequently, we expect subjects in the experiment to converge to the fair outcome in the UG game. Contrastingly for the PC game, we expect subjects to converge to the equilibrium where the responder receives the maximum amount possible, while for the 2R game we expect convergence to the equilibrium where the proposer receives the maximum amount possible. We hypothesise that there are three types of subject behaviour. These are forward-looking deliberation (or strategic thinking ) and backward-looking reinforcement-type behaviour as described by Güth (2000), and concept learning as described by Cooper and Kagel (2009). 13 We use these types of agent behaviour to hypothesise possible observations in learning transfer. Strategic thinking or forward-looking deliberation suggests that there are no history effects and that players will make decisions based on complete understanding of a game and its payoff rules and equilibria at the outset. With the strategic thinking hypothesis, we expect subjects to play the predicted outcome from the first round of each game until the end. Backward-looking, reinforcement-type behaviour predicts that subjects will engage in reinforcement or directional types of learning within games. This suggests that proposers will learn through feedback to increase their offers in PC, while responders will learn to decrease their thresholds in 2R. In general, the reinforcement learning 12 This has also been shown for the repeated ultimatum game see Brenner and Vriend (2006). 13 For a discussion of these theories, refer back to Chapter 2. 21

22 prediction suggests that players learn to adapt their decisions within a game until the predicted equilibrium is reached. However, we make the further assumption that reinforcement types of behaviour can occur between games. Under this assumption, we hypothesise that subjects first round propensities in a new game are reinforcements of their end-game decisions in the previous game. In other words, under this hypothesis, subjects can simply reinforce their last decision in a previous game in the first round of a current game. Finally, concept learning predicts that players will apply their learning of concepts across games, based on prior experience in different games. Effectively this is a sophisticated learning hypothesis, which predicts that players decisions will be influenced by history effects. As an example, suppose that players could learn and acquire the concept of competition over 10 rounds in PC. Given this is the case, we expect prior experience in PC to speed up convergence to the predicted outcome in 2R, due to the similar concept of competition applying in both games. Similarly, when playing PC after experiencing 2R, we expect players to more quickly converge to the outcome that favours the responder. Effectively, through the concept learning hypothesis we expect that prior experiences can influence subjects level of forward-looking deliberation. 22

23 4 EXPERIMENTAL RESULTS In this chapter, experimental results are divided into five sections. In Section 4.1, summary statistics on offers and thresholds are reported for all treatments. In Section 4.2, we analyse learning transfer within all six permutations of the experiment by comparing end-game and first round decisions between games within those permutations. In Section 4.3, we study how decisions vary in the UG, PC and 2R games with different player histories. In Section 4.4, we examine the frequency of fair decisions in each treatment by comparing the number of offers and thresholds between 45 and 55 with different treatment histories. Finally, in Section 4.5, we analyse the results of OLS regressions on proposer and responder decisions with controls for player histories and within-game learning. Statistical tests conducted in all of the results sections are two-tailed Summary of Results in all treatments Figure 1 shows the mean progression of offers and thresholds in all three games with no prior history. Summing data across 10 rounds, we observe mean offers (thresholds) of 48.4 (43.5) in the UG game, 60.6 (46.8) in the PC game and 38.3 (30.9) in the 2R game. Using the Kruskal-Wallis test, we find a statistically significant difference between first round offers in UG, PC and 2R (p=0.0291). Through pair-wise comparison, we find no significant difference in first round offers between UG and PC (Mann-Whitney U test, p=0.340), but we find that the average first round 2R offer of 38.9 is significantly dif- 14 For all results, we limit our data analysis to the 12 players who play all three versions of the game in the experiment. For players who play the same game across all three phases as described in Chapter 3, we include their first 10 rounds of decisions in our analysis only. 23

24 Figure 1. Mean progression of offers and thresholds in treatments UG, PC and 2R with no prior history Offer / Threshold Offers UG PC 2R Round Thresholds UG PC 2R Round ferent from the first round UG offer of 45.9 (p=0.0493), and from the first round PC offer of 47.8 (p=0.0143). For responders, we similarly find a significant difference in first round thresholds between the three games (Kruskal-Wallis test, p=0.0399). We find that the average first round 2R threshold of 32.7 is significantly different from the first round UG threshold of 40.7 (Mann-Whitney U test, p=0.087), and from the first round PC threshold of 44.9 (p=0.024). We do not find a statistically significant difference between first round UG and PC thresholds (p=0.3388). Together, these results show that both proposers and responders entering 2R make different decisions to those players entering UG or PC. We find a significant difference between the average end-game offers of 68.7 in PC, 47.9 in UG and 36.2 in 2R (Kruskal Wallis test, p<0.001; pair-wise Mann- Whitney U tests all return p<0.001). Similarly, we find that end-game thresholds are significantly different depending on the game (Kruskal Wallis test, p<0.001). We find 24

25 Table 3. Mean offers and acceptance thresholds across 10 rounds for all treatments for the six permutations of game order. Standard deviations are shown in brackets. First phase data for treatments UG, PC and 2R are pooled. 1. UG PC 2R 4. UG 2R PC Offer Thres. Offer Thres. Offer Thres. Offer Thres. Offer Thres. Offer Thres (7.1) (14.4) (23.2) (18.2) (15.5) (15.9) (7.1) (14.4) (8.3) (15.7) (13.2) (18.4) 2. PC 2R UG 5. PC UG 2R Offer Thres. Offer Thres. Offer Thres. Offer Thres. Offer Thres. Offer Thres (17.3) (19.8) (10.8) (15.7) (9.4) (12.7) (17.3) (19.8) (10.1) (12.7) (14.1) (15.5) 3. 2R UG PC 6. 2R PC UG Offer Thres. Offer Thres. Offer Thres. Offer Thres. Offer Thres. Offer Thres (10.3) (18) (10.5) (22.7) (13.0) (23.8) (10.3) (18) (15.6) (6.6) (6.5) (3.1) 25

26 that the average end-game 2R threshold of 28.8 is significantly different from the endgame UG threshold of 46.5 (Mann-Whitney U test, p<0.001), and from the end-game PC threshold of 48.5 (p<0.001). We do not find a statistically significant difference between end-game UG thresholds and PC thresholds (p=0.4987). These results show that without any prior history, players in these three games converge to different outcomes over 10 rounds. Table 3 summarises the mean offers and thresholds across 10 rounds for all treatments. The lowest overall UG average offer is 41.9 in treatment 3, while the highest average UG offer is 50.7 in treatment 2. The lowest average PC offer is 57.5 in treatment 6, while the highest average PC offer is 66.5 in treatment 1. The lowest average 2R offer is 34.2 in treatment 1, while the highest average 2R offer is 41.1 in treatment Analysis of Decisions within different permutations In this section we analyse learning transfer across all six permutations of the experiment. For each treatment, we conduct Wilcoxon signed-rank tests comparing the first round offer in a given game to the last round offer in the previous game. This effectively allows us to analyse how players respond to a change in the rules of a game. We further conduct Wilcoxon signed-rank tests comparing first round offers between games to examine whether players change their first round decisions when playing different games. Firstly, we examine learning transfer in the two treatments where UG is played first. Figure 2(1) shows the mean progression of offers and thresholds in treatments UG- PC-2R and UG-2R-PC, while Figure 2(2) shows the mean progression of thresh- 26

27 Figure 2. Mean progression of offers and thresholds in treatments with history UG. The phase in which the game is played is indicated in brackets. (1) Offers UG(1) PC(2) 2R(3) 2R(2) PC(3) Offer Round (2) Thresholds UG(1) PC(2) 2R(3) 2R(2) PC(3) Threshold Round Table 4. Comparison of first round decision in current game to last and first round decision in previous game. Each cell in the table represents the results of a Wilcoxon signed-rank test. Proposers Responders Current game Previous game Last round offer First round offer Last round threshold First round threshold PC(2) UG(1) ** * 2R(2) UG(1) ** 2R(3) PC(2) ** ** PC(3) 2R(2) ** ** ** *** P-values: * <0.10, ** <0.05, *** <0.01, Not statistically significant 27

28 olds. The first phase data for the UG game is pooled. Table 4 represents the results of Wilcoxon signed-rank tests conducted to examine difference between the first round decision of a current game and the last round decision of a previous game. As proposers enter PC with history UG, they do not make an offer that is significantly different to their last round offer in UG. Similarly, as proposers leave UG to enter 2R, there is no statistically significant difference between proposers first round 2R offer and their last round UG offer. However, as proposers then switch from PC to 2R, there is a clear decrease from an end-game PC offer of 70.1 to a first-round 2R offer of 44.5 (significant at the 5% level). Similarly, as proposers switch from 2R to PC there is an increase from an average offer of 39.8 to 49.5 (significant at the 5% level). In examining responder behaviour, we notably observe that within-game 2R thresholds decrease from 31.6 to 24.6 over 10 rounds when history is UG, but decreases from 33.0 to 9.1 when history is UGPC. On average, responders decrease their last round UG threshold of 48.5 to a first round 2R threshold of 31.6 (significant at the 5% level). We do not find a significant difference when responders adjust from PC to 2R. Meanwhile, the statistically significant differences we find between first-round decisions in Table 4 suggest that players do make different first-round decisions when entering a different game within these permutations. Next, we examine learning transfer in the two treatments where PC is played first. Figure 3(1) shows the mean progression of offers in these two treatments, PC-UG- 2R and PC-2R-UG, while Figure 3(2) shows the mean progression of thresholds. Table 5 represents the results of Wilcoxon signed-rank tests comparing end-game and first round decisions. Proposers average last-round offer of 68.7 in PC falls to an average 28

29 Figure 3. Mean progression of offers and thresholds in treatments with history PC. The phase in which the game is played is indicated in brackets. (1) Offers PC(1) UG(2) 2R(3) 2R(2) UG(3) Offer Round (2) Thresholds PC(1) UG(2) 2R(3) 2R(2) UG(3) Threshold Round Table 5. Comparison of first round decision in current game to last and first round decision in previous game. Each cell in the table represents the results of a Wilcoxon signed-rank test. Proposers Responders Current game Previous game Last round offer First round offer Last round threshold First round threshold UG(2) PC(1) *** * 2R(2) PC(1) *** *** 2R(3) UG(2) * * ** UG(3) 2R(2) *** ** *** P-values: * <0.10, ** <0.05, *** <0.01, Not statistically significant 29

30 first-round offer of 41.1 in UG (significant at the 1% level), and an average first-round offer of 32.3 in 2R (significant at the 1% level). We further compare the PC-to-2R adjustment with the PC-to-2R adjustment in offers and do not reject the null hypothesis that these adjustments are the same (Mann-Whitney U test, p=0.5703). Therefore, proposers entering UG and proposers entering 2R make similar adjustments in reducing their last-round offers from PC. In the third phase, proposers entering 2R from UG lower their end-game UG offer of 51.6 to a first-round 2R offer of 43.4 (significant level at the 10% level). Meanwhile, proposers entering third phase UG from second phase 2R increase their offer from 34.0 to 47.8 (significant at the 1% level). Examining responder behaviour, we observe that responders entering UG or 2R after playing PC do not adjust their thresholds at any statistically significant level. However in the third phase, we observe that responders entering 2R from UG do reduce their threshold from 41.7 to 30.4 (significant at the 10% level), while responders entering UG from 2R increase their threshold from 29.5 to 42.8 (significant at the 1% level). However, we can observe in Table 5 that there are no significant differences between first-round thresholds of responders in all three games of permutation PC-2R-UG. This result shows that under the permutation PC-2R-UG, responders first-round decision is similar in every game. Finally, we examine learning transfer in the two treatments where 2R is played first. Figure 4(1) shows the mean progression of offers in treatments 2R-UG-PC and 2R-PC-UG, while Figure 4(2) shows the mean progression of thresholds. Table 6 represents the results of Wilcoxon signed-rank tests comparing end-game and first round decisions in these two treatments. When entering UG from 2R, proposers do not 30

31 Figure 4. Mean progression of offers and thresholds in treatments with history PC. The phase in which the game is played is indicated in brackets. (1) Offers 2R(1) UG(2) PC(3) PC(2) UG(3) Offer Round (2) Thresholds 2R(1) UG(2) PC(3) PC(2) UG(3) Threshold Round Table 6. Comparison of first round decision in current game to last and first round decision in previous game. Each cell in the table represents the results of a Wilcoxon signed-rank test. Proposers Responders Current game Previous game Last round offer 31 First round offer Last round threshold First round threshold UG(2) 2R(1) PC(2) 2R(1) * *** ** PC(3) UG(2) ** ** UG(3) PC(2) *** P-values: * <0.10, ** <0.05, *** <0.01, Not statistically significant

32 adjust their offer at any statistically significant level. When entering PC from 2R, proposers increase their last-round 2R offer of 36.2 to a first-round PC offer of 41.2 (significant at the 10% level). Meanwhile, responders switching from 2R to PC increase their threshold from 28.8 to 49.4 (significant at the 1% level), but responders switching from 2R to UG do not adjust their threshold at any significant level. In the third phase, when entering UG from PC, proposers reduce their offer from 73.4 to 41.5 (significant at the 1% level), while proposers entering PC from UG increase their offer from 38.6 to 49.0 (significant at the 5% level). From the results shown in Table 6, we do not reject the null hypothesis that first round offers in all three games 2R, PC and UG are the same when they are played by proposers in that order. Furthermore, we do not reject the null hypothesis that first round thresholds in all three games are the same when played by responders in order 2R-UG-PC. For responders in particular, we do not reject the null hypothesis that the lastround threshold of 2R is the same as the first-round threshold of UG. In addition, we do not reject the null hypothesis that those responders set the same threshold when leaving UG to enter PC. In other words, it appears that responder decisions remain fairly constant going from 2R to UG to the start of PC. Examining Figure 4(2), we can see that responders UG and PC thresholds are clearly different depending on the order that they are played. We will explore these differences in detail in the next section. 32

33 4.3 Analysis of Decisions under different histories The following section examines between-treatment differences in the distribution of offers and thresholds in the three games UG, PC and 2R. For each of these three games, Kruskal-Wallis tests are conducted to check for differences in offers and thresholds between these games with different histories. Furthermore, pair-wise Mann-Whitney U tests are conducted to examine where these differences originate. These tests were all conducted on individual median offers/thresholds across (a) all 10 rounds, (b) the first 5 rounds and (c) the final 5 rounds. The mean progression of offers and thresholds in UG treatments are shown graphically in Figure 5, while the results of pair-wise Mann-Whitney U tests are shown in Table 7. In comparing offers, we reject the null hypothesis that the offers come from the same distributions across all 10 rounds (Kruskal-Wallis test, p=0.003), for the first 5 rounds (p=0.072) or for the final 5 rounds (p=0.034). More closely examining pair-wise comparisons between treatments, we find that most of the statistical difference is between the treatment with history 2R and all of the other treatments. The only exception is that there is no statistically significant difference between the last 5 rounds of UG offers with history 2R and the last 5 rounds of UG offers with history PC2R. We further find a statistical difference between UG offers with history PC and no history (significant at the 5% level), and between offers with history PC and history PC2R (significant at the 10% level). Meanwhile for thresholds, we do not reject the null hypothesis that UG thresholds are the same for all 10 rounds (Kruskal-Wallis, p=0.1941) or for the last five rounds (p=0.5268). However, we do reject the null hypothesis that UG thresholds are the same for the first 5 rounds (p=0.0273). Pair-wise analysis shows statistical differ- 33

34 Figure 5. Mean progression of offers and thresholds in UG treatments with different histories Offer Threshold No history History PC History 2R History PC2R History 2RPC Offer / Threshold Round Table 7. Comparison of median UG offers (thresholds) between treatments with different histories. Each cell (p p p ) represents the results of three Mann-Whitney U tests between median offers (thresholds) for all 10 rounds (p ), for the first 5 rounds (p ) and for the last 5 rounds (p ). Comparison of UG offers Comparison of UG thresholds History History History PC 2R PC2R 2RPC History PC 2R PC2R 2RPC No history ** ** *** *** ** No history ** *** PC *** ** ** * PC * * 2R * ** ** * * 2R * PC2R PC2R P-values: * <0.10, ** <0.05, *** <0.01, Not statistically significant 34

35 ence in the first 5 rounds of UG thresholds between the history PC and no history (significant at the 5% level), between history 2RPC and no history (significant at the 1% level), between history PC and history 2R (significant at the 10% level) and between history 2R and history 2RPC (significant at the 10% level). These results show that UG offers are clearly lower throughout all 10 rounds when played with history 2R. Meanwhile, for UG thresholds, the above results suggest that responders initially make different decisions depending on history, but converge to a similar outcome over 10 rounds. Next, we examine behaviour in the PC game under different histories. The mean progression of offers and thresholds in PC treatments are shown graphically in Figure 6. We reject the null hypothesis that PC offers are the same between treatments for all 10 rounds (Kruskal-Wallis test, p=0.03) or the first 5 rounds (p=0.004), but we do not reject the null hypothesis that PC offers are the same for the final 5 rounds (p=0.337). Examining the pair-wise comparisons in Table 8, we observe a statistically significant difference in PC offers for the first 5 rounds between the treatment with history 2R and all other treatments. Specifically, the first 5 rounds of PC offers in the treatment with history 2R appear to be lower than in the treatments with different histories. Meanwhile, the first 5 rounds of PC offers with history UG appear to be higher than those in treatments with no history, history 2R and history UG2R. For responders, we do not to reject the null hypothesis that PC thresholds are the same between treatments for all 10 rounds (Kruskal-Wallis test, p=0.737), for the first 5 rounds (p=0.597) or the last 5 rounds (p=0.6042). Pair-wise comparisons also reveal no significant differences between PC thresholds in treatments with different histories. 35