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Games of incomplete information and myopic equilibria

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Abstract

We consider a finitely defined game where the payoff for each player at each terminal point of the game is not a fixed quantity but varies according to probability distributions on the terminal points induced by the strategies chosen. We prove that if these payoffs have an upper-semicontinuous and convex valued structure then the game has an equilibrium. For this purpose the concept of a myopic equilibrium is introduced, a concept that generalizes that of a Nash equilibrium and applies to the games we consider. We answer in the affirmative a question posed by A. Neyman: if the payoffs of an infinitely repeated game of incomplete information on one side are a convex combination of the undiscounted payoffs and payoffs from a finite number of initial stages, does the game have an equilibrium?

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References

  1. F. D. Ancel, The role of countable dimensionality in the theory of cell-like relations, Transactions of the American Mathematical Society 287 (1985), 1–40.

    Article  MathSciNet  Google Scholar 

  2. R. Aumann and M. Maschler, Repeated Games with Incomplete Information, MIT Press, Cambridge, MA, 1995.

    MATH  Google Scholar 

  3. C. Biasi and T. Monis, Weak local Nash equilibrium, Topological Methods of Nonlinear Analysis 41 (2013), 409–419.

    MathSciNet  MATH  Google Scholar 

  4. D. Blackwell, An analogue of the minimax theorem for vector payoffs, Pacific Journal of Mathematics 6 (1965), 1–8.

    Article  Google Scholar 

  5. W. Bolt and F. Tieman, On myopic equilibria in dynamic games with endogenous discounting, International Monetary Fund Working Papers, Paper no. 06/302, 2006.

  6. S. Govindan and R. Wilson, Structure theorems for game trees, Proceedings of the National Academy of Sciences of the United States of America 99 (2002), 9077–9080.

    Article  MathSciNet  Google Scholar 

  7. S. Govindan and R. Wilson, Essential equilibria, Proceedings of the National Academy of Sciences of the United States of America 102 (2005), 15706–15711.

    Article  MathSciNet  Google Scholar 

  8. S. Hart, Non-zero sum two-person repeated games with incomplete information, Mathematics of Operations Research 10 (1985), 117–153.

    Article  MathSciNet  Google Scholar 

  9. E. Kohlberg and J.-F. Mertens, On the strategic stability of equilibria, Econometrica 54 (1986), 1003–1037.

    Article  MathSciNet  Google Scholar 

  10. D. Kreps and R. Wilson, Sequential equilibria, Econometrica 50 (1982), 863–894.

    Article  MathSciNet  Google Scholar 

  11. H. Kuhn, Extensive games and the problem of information, in Contributions to the Theory of Games. I, Annals of Mathematics Studies, Vol. 28, Princeton University Press, Princeton, NJ, 1953, pp. 193–216.

    Google Scholar 

  12. J.-F. Mertens, Stable equilibria: A reformulation. Part I. Definition and basic properties, Mathematics of Operations Research 14 (1989), 575–625.

    Article  MathSciNet  Google Scholar 

  13. J.-F. Mertens, Stable equilibria: A reformulation. Part II., Discussion of the definition and further results, Mathematics of Operations Research 16 (1991), 694–753.

    Article  MathSciNet  Google Scholar 

  14. J. Nash, Equilibrium points in n-person games, Proceedings of the National Academy of Sciences of the United States of America 36 (1950), 48–49.

    Article  MathSciNet  Google Scholar 

  15. A. Neyman, private communication, 2016.

  16. W. H. Sandholm, Population Games and Evolutionary Dynamics, Economic Learning and Social Evolution, MIT Press, Cambridge, MA, 2010.

    Google Scholar 

  17. R. Selten, A reexamination of the perfectness concept for equilibrium points in extensive games, International Journal of Game Theory 4 (1975), 25–55.

    Article  MathSciNet  Google Scholar 

  18. R. S. Simon, S. Spież and H. Toruńczyk, The existence of equilibria in certain games, separation for families of convex functions and a theorem of Borsuk-Ulam type, Israel Journal of Mathematics 92 (1995), 1–21.

    Article  MathSciNet  Google Scholar 

  19. R. S. Simon, S. Spież and H. Toruńczyk, Equilibrium existence and topology in games of incomplete information on one side, Transactions of the American Mathematical Society 354 (2002), 5005–5026.

    Article  MathSciNet  Google Scholar 

  20. S. Sorin and S. Zamir, A two-person game with lack of information on one and one-half sides, Mathematics of Operations Research 10 (1985), 17–23.

    Article  MathSciNet  Google Scholar 

  21. C. Stapenhorst, Noisy Signalling in the Principal-Agent Problem, M. Sc. Dissertation, London School of Economics, 2016.

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Authors and Affiliations

  1. Department of Mathematics, London School of Economics, Houghton Street, London, WC2A 2AE, UK

    Robert Simon

  2. Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656, Warszawa, Poland

    Stanisław Spież

  3. Institute of Mathematics, University of Warsaw, Banacha 2, 02-097, Warszawa, Poland

    Henryk Toruńczyk

Authors
  1. Robert Simon
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  2. Stanisław Spież
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  3. Henryk Toruńczyk
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Correspondence to Robert Simon.

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Dedicated to the memory of Andrzej Granas (1929–2019)

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Simon, R., Spież, S. & Toruńczyk, H. Games of incomplete information and myopic equilibria. Isr. J. Math. 241, 721–748 (2021). https://doi.org/10.1007/s11856-021-2111-7

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  • DOI: https://doi.org/10.1007/s11856-021-2111-7

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