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Learning in unprofitable games

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  • Gaunersdorfer, Andrea
  • Hofbauer, Josef

Abstract

A game is unprofitable if equilibrium payoffs do not exceed the maximin payoff for each player. In an unprofitable game, Nash equilibrium play has been notoriously difficult to justify. For a class of 3×3 games we analyze whether evolutionary and learning processes lead to Nash play. We find that neither the pure Nash equilibrium nor the pure maximin strategy are stable rest points under the studied dynamics whereas the mixed Nash equilibrium and the quantal response equilibrium may be attractors, repellors or surrounded by periodic orbits.

Suggested Citation

  • Gaunersdorfer, Andrea & Hofbauer, Josef, 2025. "Learning in unprofitable games," Games and Economic Behavior, Elsevier, vol. 151(C), pages 108-126.
  • Handle: RePEc:eee:gamebe:v:151:y:2025:i:c:p:108-126
    DOI: 10.1016/j.geb.2025.03.001
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    References listed on IDEAS

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    1. Charles Noussair & Marc Willinger, 2011. "Mixed strategies in an unprofitable game: an experiment," Working Papers 11-19, LAMETA, Universtiy of Montpellier, revised Nov 2011.
    2. Shachat, Jason & Swarthout, J. Todd & Wei, Lijia, 2015. "A Hidden Markov Model For The Detection Of Pure And Mixed Strategy Play In Games," Econometric Theory, Cambridge University Press, vol. 31(4), pages 729-752, August.
    3. George J. Mailath, 1998. "Do People Play Nash Equilibrium? Lessons from Evolutionary Game Theory," Journal of Economic Literature, American Economic Association, vol. 36(3), pages 1347-1374, September.
    4. Shachat, Jason & Swarthout, J. Todd, 2012. "Learning about learning in games through experimental control of strategic interdependence," Journal of Economic Dynamics and Control, Elsevier, vol. 36(3), pages 383-402.
    5. Friedman, Daniel & Zhao, Shuchen, 2021. "When are mixed equilibria relevant?," Journal of Economic Behavior & Organization, Elsevier, vol. 191(C), pages 51-65.
    6. McKelvey, Richard D. & Palfrey, Thomas R. & Weber, Roberto A., 2000. "The effects of payoff magnitude and heterogeneity on behavior in 2 x 2 games with unique mixed strategy equilibria," Journal of Economic Behavior & Organization, Elsevier, vol. 42(4), pages 523-548, August.
    7. R. J. Aumann & M. Maschler, 1972. "Some Thoughts on the Minimax Principle," Management Science, INFORMS, vol. 18(5-Part-2), pages 54-63, January.
    8. Larry Samuelson, 2002. "Evolution and Game Theory," Journal of Economic Perspectives, American Economic Association, vol. 16(2), pages 47-66, Spring.
    9. Hommes, Cars H. & Ochea, Marius I., 2012. "Multiple equilibria and limit cycles in evolutionary games with Logit Dynamics," Games and Economic Behavior, Elsevier, vol. 74(1), pages 434-441.
    10. Gaunersdorfer Andrea & Hofbauer Josef, 1995. "Fictitious Play, Shapley Polygons, and the Replicator Equation," Games and Economic Behavior, Elsevier, vol. 11(2), pages 279-303, November.
    11. McKelvey Richard D. & Palfrey Thomas R., 1995. "Quantal Response Equilibria for Normal Form Games," Games and Economic Behavior, Elsevier, vol. 10(1), pages 6-38, July.
    12. Borgers, Tilman & Sarin, Rajiv, 1997. "Learning Through Reinforcement and Replicator Dynamics," Journal of Economic Theory, Elsevier, vol. 77(1), pages 1-14, November.
    13. Jason Shachat & J. Todd Swarthout & Lijia Wei, 2011. "Man versus Nash An experiment on the self-enforcing nature of mixed strategy equilibrium," Working Papers 1101, Xiamen Unversity, The Wang Yanan Institute for Studies in Economics, Finance and Economics Experimental Laboratory, revised 21 Feb 2011.
    14. Dhami, Sanjit, 2016. "The Foundations of Behavioral Economic Analysis," OUP Catalogue, Oxford University Press, number 9780198715535.
    15. William A. Brock & Cars H. Hommes, 1997. "A Rational Route to Randomness," Econometrica, Econometric Society, vol. 65(5), pages 1059-1096, September.
    16. Holler, Manfred J, 1993. "Fighting Pollution When Decisions Are Strategic," Public Choice, Springer, vol. 76(4), pages 347-356, August.
    17. Drew Fudenberg & David K. Levine, 1998. "The Theory of Learning in Games," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262061945, December.
    18. Holler, Manfred J., 1990. "The unprofitability of mixed-strategy equilibria in two-person games : A second folk-theorem," Economics Letters, Elsevier, vol. 32(4), pages 319-323, April.
    19. Luciano Andreozzi, 2002. "Oscilliations in the Enforcement of Law: an Evolutionary Analysis," Homo Oeconomicus, Institute of SocioEconomics, vol. 18, pages 403-428.
    20. Hopkins, Ed, 1999. "A Note on Best Response Dynamics," Games and Economic Behavior, Elsevier, vol. 29(1-2), pages 138-150, October.
    21. Michel Benaïm & Josef Hofbauer & Sylvain Sorin, 2012. "Perturbations of Set-Valued Dynamical Systems, with Applications to Game Theory," Dynamic Games and Applications, Springer, vol. 2(2), pages 195-205, June.
    22. Morgan, John & Sefton, Martin, 2002. "An Experimental Investigation of Unprofitable Games," Games and Economic Behavior, Elsevier, vol. 40(1), pages 123-146, July.
    23. Vitaly Pruzhansky, 2011. "Some interesting properties of maximin strategies," International Journal of Game Theory, Springer;Game Theory Society, vol. 40(2), pages 351-365, May.
    24. William A. Brock & Cars H. Hommes, 2001. "A Rational Route to Randomness," Chapters, in: W. D. Dechert (ed.), Growth Theory, Nonlinear Dynamics and Economic Modelling, chapter 16, pages 402-438, Edward Elgar Publishing.
    25. Colin F. Camerer & Teck-Hua Ho & Juin Kuan Chong, 2004. "Behavioural Game Theory: Thinking, Learning and Teaching," Palgrave Macmillan Books, in: Steffen Huck (ed.), Advances in Understanding Strategic Behaviour, chapter 8, pages 120-180, Palgrave Macmillan.
    26. Goeree, Jacob K. & Holt, Charles A. & Palfrey, Thomas R., 2003. "Risk averse behavior in generalized matching pennies games," Games and Economic Behavior, Elsevier, vol. 45(1), pages 97-113, October.
    27. David Avis & Gabriel Rosenberg & Rahul Savani & Bernhard Stengel, 2010. "Enumeration of Nash equilibria for two-player games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 42(1), pages 9-37, January.
    28. George J. Mailath, 1998. "Do People Play Nash Equilibrium? Lessons from Evolutionary Game Theory," Journal of Economic Literature, American Economic Association, vol. 36(3), pages 1347-1374, September.
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    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games

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