Abstract
History-dependent strategies are often used to support cooperation in repeated game models. Using the indefinitely repeated common-pool resource assignment game and a perfect stranger experimental design, this paper reports novel evidence that players who have successfully used an efficiency-enhancing turn taking strategy will teach other players in subsequent supergames to adopt this strategy. We find that subjects engage in turn taking frequently in both the Low Conflict and the High Conflict treatments. Prior experience with turn taking significantly increases turn taking in both treatments. Moreover, successful turn taking often involves fast learning, and individuals with turn taking experience are more likely to be teachers than inexperienced individuals. The comparative statics results show that teaching in such an environment also responds to incentives, since teaching is empirically more frequent in the Low Conflict treatment with higher benefits and lower costs.
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Notes
Duersch et al. (2010) study how subjects learn to play against computers that are programmed to follow one of a number of standard learning algorithms. They find that teaching occurs frequently and that all learning algorithms are subject to exploitation with the notable exception of imitation.
Unlike the contributions discussed above but like our paper, Camerer et al. (2002) consider teaching in repeated game strategies. Their main concern, however, is on teaching by a player who faces a sequence of different players in a finitely repeated trust game.
Fudenberg et al. (2012) identify the repeated game strategies most commonly used by players in an indefinitely repeated PD when intended actions are implemented with a noise. Engle-Warnick and Slonim (2006b) employ a statistical approach to identify repeated game strategies in finitely and indefinitely repeated trust games.
Note that for any probability \(p\) with which player 2 may play Tough, player 1 gets a higher payoff by playing Tough instead of Soft, and the difference in payoff between using these two different responses is given by \(\frac{\lambda }{1+\theta }\left( {\theta -0.5} \right)-0.5p\lambda \), which is increasing in \(\theta \). Hence, other things being equal, an increase in the degree of conflict increases the gain from playing Tough instead of Soft.
Thus, a change from the Low Conflict game to the High Conflict game can be thought as representing a change in the physical environment, where the total amount of fish available in the community remains unchanged, but some fish had migrated to the good spot. The laboratory allows us to test comparative statics results in a controlled environment where clean ceteris paribus counterfactual changes in the environment faced by the players may be hard to observe in the field.
More generally, suppose that a subject in the High Conflict treatment has a belief \(p_H\) that her opponent will play Tough, and a subject in the Low Conflict treatment has a belief \(p_L\). Then the difference in the cost of teaching for such two subjects will be \((110-70p_{H})-(77-70p_{L})\). By design, this difference in teaching costs is constant at 33 if the two subjects hold the same belief \(p_{H} =p_{L}\) in both treatments. While differences in the treatments’ degree of conflict might lead to differences in beliefs across treatments, so long as \(p_H -p_L <33/70\), the subject in the High Conflict treatment will face a higher cost of teaching.
This discussion assumes risk neutrality, but allowing for risk aversion will not change the implication that the differences in cost and benefit imply that teaching is more likely in the Low Conflict treatment.
The exchange rate was \(7.8\,\text{ HK}{\$}\, \approx 1\,\text{ US}{\$}\) when the experiment was conducted.
The mean match length was 10.1 periods, with a median of 7 periods and an interquartile range of 4–13 periods. The maximum match length was 50 periods. The 12 sessions each had 7 matches, and the mean total periods per session was 69.9 with a median of 65.5 and an interquartile range of 51–82.5 periods.
If the pair members who start with Tough–Soft–Tough pattern were actually teachers, they might just as well start with Soft–Tough–Soft as Tough–Soft–Tough, which would result in many “ties” where both pair members play Soft–Tough–Soft or both pair members play Tough–Soft–Tough simultaneously. But this is not commonly observed in the data; the vast majority of “ties” are cases in which one pair member plays Soft–Tough–Soft and the other plays Tough–Soft–Tough simultaneously. This suggests that those who play Tough–Soft–Tough are much more typically “fast learners” rather than teachers.
Pair members who may attempt to teach others to take turns by alternating Soft–Tough–Soft sometimes encounter subjects who simultaneously play Soft in early periods. These periods of miscoordination in which both pair members choose Soft occur less than one percent of the time (65 out of 6,712 period-pair observations), however. When miscoordination occurs, pairs nevertheless usually reach the turn taking path, and 52 of the 65 periods of miscoordination occur during the first three periods of a match—nearly always in the first or second period. When both pair members have turn taking experience, the miscoordination rate increases to 2.6 % in all periods of matches, and it is over 16 % in the first period of matches.
Lau and Mui (2012) study how players may use the Turn Taking with Independent Randomization (TTIR) strategy to support turn taking as equilibrium in infinitely repeated \(2\times 2\) games such as the assignment game. The TTIR strategy specifies that players randomize independently between Tough and Soft in the initial phase of the repeated assignment game and then engage in turn taking once the asymmetric outcome of either \((T,\,S)\) or \((S,\,T)\) is reached, with any defection from turn taking punished by the play of the Nash equilibrium \((T,\,T)\) forever. The TTIR equilibrium is designed to generate predictions about how changes in payoff parameters affect the incidence of successful turn taking for players who have not played the repeated assignment game before, and by design does not take into account how experience may affect behavior. The TTIR equilibrium correctly predicts that the incidence of turn taking is higher in the Low Conflict treatment than in the High Conflict treatment. The data, however, show that for both treatments, the TTIR equilibrium overpredicts the incidence of turn taking for the early matches in each session, but under-predicts the actual incidence of turn taking when both subjects have turn taking experience in later matches.
We also do not find evidence that “playing dumb” is empirically profitable. The average earnings per period for subjects who never played Soft, compared to those earned by subjects who play Soft in matches after they experience turn taking, are significantly less in the Low Conflict game (56.39 vs. 62.59), and an equivalent amount in the High Conflict game (63.05 vs. 63.03).
The same interaction term is not significant in a similar specification for column (a) based on successful turn taking matches (\(p\) value = 0.20). Note that male subjects and those with a high grade point average (GPA) are more likely to choose Soft in match pairings that do not result in turn taking and that students majoring in Economics and Finance are less likely to be identified as teachers in turn taking matches. These gender and major results are consistent with other studies concluding that women are less willing to incur risks than men (Croson and Gneezy 2009) and that economics majors tend to be less cooperative than non-economics majors (e.g., Faravelli 2007).
Alternatively, we could use a more stringent definition to classify a teacher as a subject who alternates between Soft and Tough. For example, we considered the definition of teaching for the unsuccessful turn taking matches to be at least one pattern of Soft–Tough–Soft by the teacher. We only observed 74 matches that could be classified as unsuccessful teaching by this definition, however. Conclusions regarding the relative costs of teaching in the two games are qualitatively similar, so to conserve space we only report the version based on the first choice of Soft.
Since the mean payoff of successful teachers is 65.3 and that of unsuccessful teachers is 48.3, the minimum success rate of 4.2 % is obtained by solving \(q(65.3)+(1-q)(48.3)=49\).
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We are grateful to the Research Grants Council of Hong Kong (project HKU7223/04H) for financial support. We also thank Kyle Hyndman, two anonymous referees, and an associate editor, as well as conference and seminar participants at Monash, Purdue, Université Paris 1, the Australia New Zealand Workshop on Experimental Economics and the Economic Science Association for valuable comments. We retain responsibility for any errors. Ishita Chatterjee and Julian Chan provided valuable research assistance. The experiment was programmed and conducted with the software z-Tree (Fischbacher 2007)
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Cason, T.N., Lau, SH.P. & Mui, VL. Learning, teaching, and turn taking in the repeated assignment game. Econ Theory 54, 335–357 (2013). https://doi.org/10.1007/s00199-012-0718-y
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DOI: https://doi.org/10.1007/s00199-012-0718-y