Abstract
We study the ability of subjects to transfer principles between related coordination games. Subjects play a class of order statistic coordination games closely related to the well-known minimum (or weak-link) and median games (Van Huyck et al. in Am Econ Rev 80:234–248, 1990, Q J Econ 106(3):885–910, 1991). When subjects play a random sequence of games with differing order statistics, play is less sensitive to the order statistic than when a fixed order statistic is used throughout. This is consistent with the prediction of a simple learning model with transfer. If subjects play a series of similar stag hunt games, play converges to the payoff dominant equilibrium when a convention emerges, replicating the main result of Rankin et al. (Games Econ Behav 32:315–337, 2000). When these subjects subsequently play a random sequence of order statistic games, play is shifted towards the payoff dominant equilibrium relative to subjects without previous experience. The data is consistent with subjects absorbing a general principle, play of the payoff dominant equilibrium, and applying it in a new related setting.
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Notes
See Ochs (1995) for a summary of evidence at the time.
This point is a simple matter of arithmetic. Let p be the probability that a subject plays the payoff dominant equilibrium. In choosing whether I should play the payoff dominant equilibrium in a stag hunt game, the relevant number is my belief about p. In an eight person minimum game, what matters is my belief about p7 which is a much smaller number. For example, suppose p = 0.8. The probability the one person plays the payoff dominant equilibrium is 80%, but the probability that a group of seven people all play the payoff dominant equilibrium (assuming independence) is only 21%.
See Fudenberg and Levine (2009) for a recent survey.
A related issue is the effect of spillover, the effect on play of participating in related games simultaneously (Bednar et al. 2012; Cason et al. 2012; Savikhin and Sheremeta 2013; Cason and Gangadharan 2013). Because play is simultaneous, effects in one game cannot rely on feedback about play or outcomes in the other. Hence, learning does not play a role in spillover effects.
A major difficulty in the literature on transfer is the absence of any clear metric for similarity between games. Our definition of structural similarity is a little bit like Stewart’s famous definition of pornography: I know it when I see it. The basic idea is that there exist elements of a game’s construction that make the similarity of two games transparent independent of any deeper principles.
The games differ because your actions do not affect the minimum you are trying to match. You therefore cannot not guarantee matching the minimum by choosing 1.
For additional related papers see Rick and Weber (2012) and Grimm and Mengel (2012). More broadly, there is a large psychology literature on learning and transfer with generally discouraging findings. See Gick and Holyoak (1983) for a classic article on the topic and Alfieri et al. (2013) for a recent meta-study.
See Duffy and Fehr (2015) for a case where equilibrium selection principles fail to transfer.
Use of an exclusive order statistic has several advantages. Most important, a player’s choice does not influence the value of \({\text{EOS}}\left( {{\text{X}}_{{ - {\text{i}}}} ,{\text{OS}}} \right)\) Play of the maximum game, for example, would be trivial if the player’s own choice is included in the calculation of \({\text{EOS}}\left( {{\text{X}}_{{ - {\text{i}}}} ,{\text{OS}}} \right)\) Using the order statistic also makes it possible to use the same payoff table for all seven games. Otherwise the minimum and maximum games require different payoff tables.
This was truncated in the final block as there were only five rather than seven rounds.
For the Fixed treatment, there were seven sessions. Each session in this treatment included cohorts with different values of the order statistic (OS), with no two cohorts from the same session having the same order statistic. This was done to avoid confounding any session effects with effects of the order statistic. Table 4 shows three sessions for each value of OS. This reflects the cohorts for each value of OS being spread across three sessions.
One subject had an exam and one was worried about being late to his job. Although both had been told in advance about the time commitment needed for the experiment, both were allowed to leave and paid for the portion of the experiment they had completed as per IRB regulations.
The average earnings reported above do not include the two problematic sessions.
These are simplifying assumptions but not critical for our conclusions.
In terms of the model, we can devise similarity functions that yield H4. The similarity function determines how payoffs from past actions are weighted when determining weights over currently available actions. For example, suppose payoffs from Action A in the stag hunt games reinforce only Action 3 in the order statistic games since both are maximin choices. Likewise suppose payoffs from Action B in the stag hunt games only reinforce Action 7 in the order statistic games since both are played in the payoff dominant equilibrium. If play in the stag hunt games converges to the payoff dominant equilibrium, it follows that play of the payoff dominant equilibrium is reinforced in the order statistic games and prior experience with the stag hunt games shifts play toward the payoff dominant equilibrium in the ROS games. The preceding yields H4 as a prediction but relies entirely on the assumed form of the similarity function, begging the main empirical question: Do subjects use general principals such as payoff dominance to identify choices in different games (i.e. stag hunt and order statistic games) as being similar?
Both regressions are one observation short. In the ROS treatment session that had to be cut short, there was one subject who never entered her final choice.
The number of clusters in the dataset for these regressions is low, only 12, biasing us in favor of finding statistical significance (a common rule of thumb is that you should have at least 20 clusters). We have experimented with the “wild bootstrap method” developed by Cameron et al. (2008) to generate correct standard errors with a low number of clusters. This isn’t perfect, since we have to use a linear probability model, but gives a sense of whether our result is driven by the low number of clusters. The statistical significance of the parameter estimate for “ % Payoff Dominant Equilibrium (Group)” is lower (p = 0.06) as expected, but our qualitative conclusions are unchanged. (Oddly, the change in p values is more due to use of a linear probability model than the method used to generate standard errors.) Play consistent with the payoff dominant equilibrium in the stag hunt games has significant explanatory power for play in the ROS games, and adding the two first Round 1 variables has a negligible effect.
Model 1 pools data from the ROS and SHROS treatments. If the model estimates separate responses to the lagged order statistic, the parameter estimates are virtually identical for the two treatments (0.071 and 0.072 respectively).
Running a Wilcoxon matched pairs signed rank test, the difference between the first and last blocks in the Fixed treatments is significant at the 5% level (z = 2.42; p = 0.02). Each observation is the average over a cohort for the seven round block. Oddly, the decrease in the ROS treatment is also significant (z = −2.31; p = 0.02).
Peysakhovich and Rand (2016) provide an example of cross-game transfer between repeated prisoners’ dilemma games and various one shot games. They show that the transfer is driven by a norm of pro-social behavior rather any notion of equilibrium selection, as transfer occurs for non-strategic settings such as the dictator game.
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Acknowledgements
Funding was provided by the NSF (SES-0214310). I would like to thank Phil Brookins, Laura Magee, and Joe Stinn for their fine work as research assistants. This paper would not have been possible without the extraordinary help of Catherine Eckel who recovered a large number of documents from John Van Huyck’s computer and file cabinets. We received helpful comments from seminar participants at FSU and the Workshop in Honor of John Van Huyck, Ed Hopkins, Roberto Weber, Yan Chen, and two anonymous referees.
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Cooper, D.J., Van Huyck, J. Coordination and transfer. Exp Econ 21, 487–512 (2018). https://doi.org/10.1007/s10683-017-9521-8
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DOI: https://doi.org/10.1007/s10683-017-9521-8