Abstract
We experimentally study how mutual payoff information affects strategic play. Subjects play the Prisoner’s Dilemma or Stag Hunt game against randomly re-matched opponents under two information treatments. In our partial-information treatment, subjects are shown only their own payoff structure, while in our full-information treatment they are shown both their own and their opponent’s payoff structure. In both treatments, they receive feedback on their opponent’s action after each round. We find that mutual payoff information initially facilitates reaching the socially optimal outcome in both games. Play in the Prisoner’s Dilemma converges toward the unique Nash equilibrium of the game under both information treatments, while in the Stag Hunt mutual payoff information has a substantial impact on play and equilibrium selection in all rounds of the game. Belief-learning model estimations and simulations suggest these effects are driven by both initial play and the way subjects learn.
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Notes
By playing multiple rounds, subjects could learn about the game, but not their opponents.
For consistency with prior literature, we use the words “cooperate" and “defect" to describe the action choices in the PD with acknowledgement that they are only well-defined from the player’s perspective in the presence of mutual payoff information.
Strictly speaking, following Harsanyi and Selten (1988)’s canonical definition of risk dominance an equilibrium cannot be “risk-dominant” for a player who does not have access to full payoff information. For the sake of clarity and consistency, we will refer to the “risk-dominant” and “payoff-dominant” actions for both treatments in the SH game.
The basin of attraction for pure strategies X and Y thus remains constant across treatments in the SH. According to Embrey et al. (2017), the basin of attraction size for a strategy is positively correlated with its selection frequency.
Ghidoni et al. (2019) find that cooperation rates in a PD game with ten rounds are very similar when subjects are randomly re-matched in groups of 6 or with a new opponent each round.
Five of the 196 subjects initially answered some quiz questions incorrectly, but passed on the second attempt after receiving feedback and a new quiz version with different matrix entries.
We do not use session fixed effects in our main specification, since the estimate \(\hat{\beta }\) would only exploit the within-subjects data.
Results robust to using logit are found in Appendix Table D5.
The exact number of equilibria reached each round may partly depend on the random matching of pairs.
Our SH game also has a mixed strategy Nash equilibrium where subjects play X two-thirds and Y one-third of the time. Appendix Table D6 shows the share of subjects whose mix of actions are within 10pp of \(p_X = 0.667\).
The efficiency ratio compares the total payoffs of both subjects in a round to the total payoffs of the efficient outcome. Random re-matching of subjects introduces variation in this ratio.
Attractions are constrained to the range of possible payoffs of an action, for consistency and easier interpretation. For a given expected probability of one’s counterpart playing X in the first round, \(P^X(1)\), there is a one-to-one correspondence between the expected value of X and the expected value of Y. For example, if a player assesses in our SH game that \(\mathop {\mathrm {\mathbb {E}}}\limits (X) = 8\) in the first round, this implies \(P^X(1) = 0.7\) (note that \(0.7*11 + (1-0.7)*1 = 8)\), which in turn implies that \(\mathop {\mathrm {\mathbb {E}}}\limits (Y) = 0.7*9+(1-0.7)*5 = 7.8\). We allow both initial attractions to range independently of one another.
We allow for the possibility that symmetric outcomes (i.e., (X, X) or (Y, Y)) may be perceived as more likely.
\(\phi =1\) would indicate that all observed actions are weighted equally. Camerer and Ho (1999) comment that values of \(\phi\) are “presumably between zero and one.” While we do not constrain the value of \(\phi\) in our maximum-likelihood estimations, we note that estimated values of \(\phi\) all lie within this range.
The treatment effect is also (weakly) significant for parameter \(\phi\), but only for the SH. We thank an anonymous referee for pointing out that learning model parameter estimates can be heavily biased if subjects within a cell have parameter heterogeneity (Wilcox, 2006) We cannot test for such heterogeneity in our data.
In Alshaikhmubarak et al. (2021), a previous and more expansive version of this paper, we discuss how the interplay of social preferences and strategic uncertainty may play a role in shaping the information treatment effect.
References
Agresti, A., & Coull, B. A. (1998). Approximate is better than “exact" for interval estimation of binomial proportions. The American Statistician, 52(2), 119–126.
Alshaikhmubarak, H., Hales, D., Kogelnik, M., Schwarz, M., & Strauss K. (2021). Knowing me, knowing you: An experiment on mutual payoff information and strategic uncertainty. Available at SSRN 3915018.
Babichenko, Y. (2010). Uncoupled automata and pure Nash equilibria. International Journal of Game Theory, 39(3), 483–502.
Battalio, R., Samuelson, L., & Van Huyck, J. (2001). Optimization incentives and coordination failure in laboratory Stag Hunt games. Econometrica, 69(3), 749–764.
Brown, L. D., Cai, T., & DasGupta, A. (2001). Interval estimation for a binomial proportion. Institute of Mathematical Statistics, 16(2), 101–117.
Camerer, C., & Ho, T.-H. (1999). Experience-weighted attraction learning in normal form games. Econometrica, 67(4), 827–874.
Embrey, M., Fréchette, G. R., & Yuksel, S. (2017). Cooperation in the finitely repeated Prisoner’s Dilemma. The Quarterly Journal of Economics, 133(1), 509–551.
Feltovich, N., & Oda, S. H. (2014). Effect of matching mechanism on learning in games played under limited information. Pacific Economic Review, 3, 260–277.
Fischbacher, U. (2007). z-Tree: Zurich toolbox for ready-made economic experiments. Experimental Economics, 10, 171–178.
Foster, D., & Young, H. (2006). Regret testing leads to Nash Equilibrium. Theoretical Economics, 1, 341–367.
Fudenberg, D., & Levine, D. K. (2009). Self-confirming equilibrium and the Lucas critique. Journal of Economic Theory, 144(6), 2354–2371.
Ghidoni, R., Cleave, B. L., & Suetens, S. (2019). Perfect and imperfect strangers in social dilemmas. European Economic Review, 116, 148–159.
Greiner, B. (2015). Subject pool recruitment procedures: Organizing experiments with ORSEE. Journal of the Economic Science Association, 1, 114–125.
Harsanyi, J., & Selten, R. (1988). A General Theory of Equilibrium Selection in Games. MIT Press.
Hart, S., & Mas-Colell, A. (2006). Stochastic uncoupled dynamics and Nash Equilibrium. Games and Economic Behavior, 57(2), 286–303.
Kneeland, T. (2015). Identifying higher-order rationality. Econometrica, 83(5), 2065–2079.
Knoepfle, D. T., Wang, J.T.-Y., & Camerer, C. F. (2009). Studying learning in games using eye-tracking. Journal of the European Economic Association, 7(2–3), 388–398.
Nagel, R. (1995). Unraveling in guessing games: An experimental study. American Economic Review, 85(5), 1313–1326.
Nash, J. (1950). Non-cooperative games. Ph.D. Dissertation, Princeton University.
Polonio, L., & Coricelli, G. (2019). Testing the level of consistency between choices and beliefs in games using eye-tracking. Games and Economic Behavior, 113, 566–586.
Stahl, D., & Wilson, P. (1994). Experimental evidence on players? Models of other players. Journal of Economic Behavior & Organization, 25(3), 309–327.
Wilcox, N. T. (2006). Theories of learning in games and heterogeneity bias. Econometrica, 74(5), 1271–1292.
Young, H. P. (2009). Learning by trial and error. Games and Economic Behavior, 65(2), 626–643.
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Molly Schwarz: The opinions expressed in this article are those of the author and do not necessarily represent the views of the Federal Communications Commission or the United States Government.
A previous version of this paper was circulated as “Knowing Me, Knowing You: An Experiment on Mutual Payoff Information and Strategic Uncertainty.” We thank Ryan Oprea, Emanuel Vespa, and Sevgi Yuksel for their guidance and encouragement. We are grateful to participants of the UCSB Experimental Economics Seminar, Terri Kneeland, and Yi Zheng for helpful comments and suggestions. Emanuel Vespa inspired and instigated this work by noting that partial-information games are understudied. Funding from the UCSB Department of Economics is gratefully acknowledged. This study obtained IRB approval at UCSB. The replication material for the study is available at http://doi.org/10.17605/OSF.IO/Y2JS7.
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Alshaikhmubarak, H., Hales, D., Kogelnik, M. et al. Knowing me, knowing you: an experiment on mutual payoff information in the stag hunt and Prisoner’s Dilemma. J Econ Sci Assoc (2024). https://doi.org/10.1007/s40881-024-00167-5
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DOI: https://doi.org/10.1007/s40881-024-00167-5