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Equilibrium selection in similar repeated games: experimental evidence on the role of precedents

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Abstract

We report on an experiment examining behavior and equilibrium selection in two similar, infinitely repeated games, Stag Hunt and Prisoner’s Dilemma under anonymous random matching. We are interested in the role that historical precedents may play for equilibrium selection between these two repeated games. We find that a precedent for efficient play in the repeated Stag Hunt game does not carry over to the repeated Prisoner’s Dilemma game despite the possibility that efficient play can be sustained as an equilibrium of the indefinitely repeated game. Similarly, a precedent for inefficient play in the repeated Prisoner’s Dilemma game does not extend to the repeated Stag Hunt game. We conclude that equilibrium selection between similar repeated games may have less to do with historical precedents and might instead depend more on strategic considerations associated with the different payoffs of these similar repeated games.

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Notes

  1. We use the terminology “indefinitely repeated” rather than “infinitely repeated” to refer to the type of repeated games we can play in the laboratory. Infinitely repeated games cannot be implemented in the laboratory but indefinitely repeated games involving a constant probability that the game continues from one round to the next, can be implemented in the laboratory. One can interpret the continuation probability as the discount factor of the infinitely repeated game.

  2. Other well-known equilibrium selection mechanisms include risk dominance, payoff dominance, or embedding the coordination game in some kind of incomplete information setting e.g., as in the global game approach of Carlson and van Damme (1993).

  3. There is also a related literature that investigates how changes in institutional details can affect equilibrium play in the same game. For example, Brandts and Cooper (2006) explored the salience of equilibrium payoffs, Ahn et al. (2001), Bohnet and Huck (2004) or Duffy and Ochs (2009) studied reputation building using different matching protocols and Knez (1998) or Weber (2006) explored a change in group size. In contrast to this literature, we keep the institutional details fixed and consider changes in off-equilibrium payoffs so as to focus on behavior in consecutively played similar games.

  4. Indefinitely repeated games have been studied in the laboratory e.g., by Dal Bò (2005) Duffy and Ochs (2009), Camera and Casari (2009), Dal Bò and Fréchette (2011), Blonski et al. (2011) and Fudenberg et al. (2012) among others. These papers all consider equilibrium selection within the same class of indefinitely repeated Prisoner’s dilemma games, and do not consider, as we do in this paper, whether the equilibrium selected in an indefinitely repeated Prisoner’s dilemma game carries over to a similar but different class of indefinitely repeated game, in our case, the Stag Hunt game.

  5. An alternative notion of “structural similarity” for normal-form games is given by Germano (2006) who defines similarity between two games using the geometry of the best response correspondences. According to that criterion, the prisoner’s dilemma and stag hunt stage games that we study are not structurally similar. However, as we are exploring indefinitely repeated versions of these normal form stage games and the parameterizations that we study for the indefinitely repeated prisoner’s dilemma game allow for play of both of the pure strategy equilibria in the repeated stag hunt game, we believe that payoff similarity, rather than structural similarity, is the more relevant similarity concept for our purposes. See for example, Dufwenberg et al. (2010) or Mengel and Sciubba (2014) for experimental evidence of learning spillovers in “structurally similar” normal-form games.

  6. Allowing subjects to roll a die provides the most credible means of establishing the indefiniteness of the repeated game.

  7. Thus in the very first round of each new supergame it was very evident to subjects whether or not the payoff parameter, T, of the stage game had changed.

  8. The experiment was computerized using z-Tree (Fischbacher 2007). Subjects were paid their game payoffs in cents (US$) from all rounds of all supergames played. Total earnings for subjects averaged about $17 (including a $5 show-up fee), and sessions typically lasted about 90 min. For more procedural details see Appendix B of the supplementary material. Instructions are found in Appendix C of the supplementary material.

  9. In the experimental instructions we refer to a “supergame” as an indefinite “sequence” of rounds. In the remainder of the paper we use the terms “supergame” and “sequence” interchangeably.

  10. Subjects expect less cooperation in PD30-SH10 than in SH10-PD30 (\(p<0.01\), two-sided t-test) as well as less cooperation in PD25-SH15 than in SH15-PD25 (\(p<0.01\), two-sided t-test).

  11. These findings are corroborated by subjects’ beliefs. On average, subjects expect cooperative play to be approximately the same in the PD, \(T=30\) game (0.56) as in the PD, \(T=25\) game (0.63) (\(p>0.31\), two-sided t-test). In the SH games, subjects’ expectations are on average closer to actual observed behavior. Indeed, they expect a bit more cooperation in the SH, \(T=10\) game (0.88) than in SH, \(T=15\) game (0.76). This difference in beliefs is marginally statistically significant (\(p=0.094\), two-sided t-test).

  12. See Feltovich (2003) for a discussion of the robust rank-order test.

  13. It is important to note that this result does not depend on the inclusion of the two groups with a low cooperation rates (group 9 and 11). As Table 3 shows, cooperation rates in the SH, \(T=15\) game are in all stages lower than for the SH, \(T=10\) game for each group.

  14. Recall that we elicited subjects’ beliefs about cooperative play in a matching group in the first round of a sequence except in the first session of PD30-SH10 and SH10-PD30.

  15. The idea behind Kandori’s contagious equilibrium is that cooperation can be sustained only if all players cooperate because defection by a single player would initiate a contagious spread of defection within the entire community (group) and this process cannot be stopped by re-igniting cooperation.

  16. Note that low best response rates in the SH, \(T=15\) are not due to the two groups who coordinate on the inefficient all-Y equilibrium. Rather, it is the case that best response rates are in general lower in the SH, \(T=15\) than in the SH, \(T=10\).

  17. Note that it is technically not possible to report equal probabilities, since subjects had to indicate how many of the other nine group members would choose cooperation (X). Thus, the squared deviation of 0.2 reflects the case where more weight −5 out of 9 others—is placed on the actual choice.

  18. A Wilcoxon signed-rank test comparing the mean squared deviations of all games having the same parameterization with the benchmark of 0.20 yields the following p values: \(p=0.046\) (PD, \(T=30\)), \(p=0.027\) (SH, \(T=10\)), \(p=0.58\) (PD, \(T=25\)) and \(p=0.34\) (SH, \(T=15\)).

  19. The correlation coefficient in PD25-SH15 is −0.08 (\(p=0.62\)), whereas coefficients range between −0.54 and −0.56 in PD30-SH10, SH10-PD30 and SH15-PD25 (p values \(< 0.015\)).

  20. The correlation coefficients are 0.14 (\(p=0.54\)) in PD30-SH10, 0.04 (\(p<0.87\)) in SH10-PD30, 0.31 (\(p=0.054\)) in PD25-SH15 and 0.35 (\(p=0.026\)) in SH15-PD25.

  21. For a discussion of such demand effects, see, e.g., Zizzo (2010) who defines experimenter demand effects as “changes in behavior by experimental subjects due to cues about what constitutes appropriate behavior.”

  22. Vespa and Wilson (2015) have studied Markov games in the laboratory. Dal Bò et al. (2017) report on an experiment where subjects endogenously choose, via a majority rule vote, whether to participate in a Prisoner’s dilemma game or an alternative (anti-PD game) where cooperation is a dominant strategy; they find that a slight majority (53.6%) vote for the PD game!

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Authors and Affiliations

  1. University of California, Irvine, CA, USA

    John Duffy

  2. WZB, Berlin, Germany

    Dietmar Fehr

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  1. John Duffy
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  2. Dietmar Fehr
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Correspondence to John Duffy.

Additional information

For helpful comments and suggestions we thank an anonymous referee and participants at the Workshop in Memory of John Van Huyck held at SMU in October 2015. Dietmar Fehr gratefully acknowledges financial support from the Deutsche Forschungsgemeinschaft (DFG) through CRC 1029.

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Duffy, J., Fehr, D. Equilibrium selection in similar repeated games: experimental evidence on the role of precedents. Exp Econ 21, 573–600 (2018). https://doi.org/10.1007/s10683-017-9531-6

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