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On Repeated Zero-Sum Games with Incomplete Information and Asymptotically Bounded Values

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Abstract

We consider repeated zero-sum games with incomplete information on the side of Player 2 with the total payoff given by the non-normalized sum of stage gains. In the classical examples the value \(V_N\) of such an N-stage game is of the order of N or \(\sqrt{N}\) as \(N\rightarrow \infty \). Our aim is to find what is causing another type of asymptotic behavior of the value \(V_N\) observed for the discrete version of the financial market model introduced by De Meyer and Saley. For this game Domansky and independently De Meyer with Marino found that \(V_N\) remains bounded as \(N\rightarrow \infty \) and converges to the limit value. This game is almost-fair, i.e., if Player 1 forgets his private information the value becomes zero. We describe a class of almost-fair games having bounded values in terms of an easy-checkable property of the auxiliary non-revealing game. We call this property the piecewise property, and it says that there exists an optimal strategy of Player 2 that is piecewise constant as a function of a prior distribution p. Discrete market models have the piecewise property. We show that for non-piecewise almost-fair games with an additional non-degeneracy condition the value \(V_N\) is of the order of \(\sqrt{N}\).

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Notes

  1. Usually one considers the expected average total payoff, i.e., the expected sum of stage gains divided by N, to ensure that the sequence of values \(V_N\) is bounded as \(N\rightarrow \infty \). However, we do not follow this convention as \(V_N\) remains bounded in the games we are interested in without any normalization.

  2. For almost-fair games with infinite \(I\), \(J\), and \(K\) the value can also grow as \(N^\alpha \) with \(\alpha \in (0.5,1)\), see Sandomirskiy [30].

  3. I am grateful to Eilon Solan for telling me about the paper of Mannor and Perchet [22] which allows to use the second approach to prove boundedness of \(V_N\), see Remark 8 for details.

  4. Sometimes this metric is called the Wasserstein distance (named after Leonid Vaserstein). Note that Kantorovich introduced this metric to study optimal transportation problems 27 years before Vaserstein used it in dynamical system context (see the discussion by Vershik [32]).

References

  1. Aumann R, Maschler M (1995) Repeated games with incomplete information. MIT Press, Cambridge

    MATH  Google Scholar 

  2. De Meyer B (1996) Repeated games and partial differential equations. Math Oper Res 21(1):209–236

    Article  MathSciNet  MATH  Google Scholar 

  3. De Meyer B (1996) Repeated games, duality and the central limit theorem. Math Oper Res 21(1):237–251

    Article  MathSciNet  MATH  Google Scholar 

  4. De Meyer B (1998) The maximal variation of a bounded martingale and the central limit theorem. Ann. de l’IHP Probabilités et statistiques 34(1):49–59

    Article  MathSciNet  MATH  Google Scholar 

  5. De Meyer B (1999) From repeated games to Brownian games. Ann. de l’IHP Probabilités et statistiques 35(1):1–48

    Article  MathSciNet  MATH  Google Scholar 

  6. De Meyer B, Saley HM (2003) On the strategic origin of Brownian motion in finance. Int J Game Theory 31(2):285–319

    Article  MathSciNet  MATH  Google Scholar 

  7. De Meyer B, Marino A (2004) Repeated market games with lack of information on both sides. Cahier de la MSE 2004/66, Université Paris 1

  8. De Meyer B, Marino A (2005) Continuous versus discrete Market games. Cowles Foundation discussion paper 1535. Yale University

  9. De Meyer B (2010) Price dynamics on a stock market with asymmetric information. Games Econ Behav 69:42–71

    Article  MathSciNet  MATH  Google Scholar 

  10. De Meyer B, Fournier G (2015) Price dynamics on a risk averse market with asymmetric information. Documents de travail du Centre d’Economie de la Sorbonne 2015.54. ISSN: 1955-611X \(<\)halshs-01169563\(>\)

  11. Domansky V (2007) Repeated games with asymmetric information and random price fluctuations at finance markets. Int J Game Theory 36(2):241–257

    Article  MathSciNet  MATH  Google Scholar 

  12. Domansky V, Kreps V (1994) “Eventually revealing” repeated games with incomplete information. Int J Game Theory 23(2):89–99

    Article  MathSciNet  MATH  Google Scholar 

  13. Domansky V, Kreps V (1995) Repeated games and multinomial distributions. Math Meth Oper Res 42(3):275–293

    Article  MathSciNet  MATH  Google Scholar 

  14. Domansky V, Kreps V (1999) Repeated games with incomplete information and transportation problems. Math Meth Oper Res 49(2):283–298

    MathSciNet  MATH  Google Scholar 

  15. Domansky V, Kreps V (2009) Repeated games with asymmetric information and random price fluctuations at finance markets: the case of countable state space. Documents de travail du Centre d’Economie de la Sorbonne 2009.40. ISSN: 1955-611X \(<\)halshs-00390701\(>\)

  16. Domansky V, Kreps V (2013) Repeated games with asymmetric information modeling financial markets with two risky assets. RAIRO Oper Res 47(3):251–272

    Article  MathSciNet  MATH  Google Scholar 

  17. Domansky V, Kreps V (2016) Bidding games with several risky assets. Autom Remote Control 77(4):722–733

    Article  MathSciNet  Google Scholar 

  18. Gensbittel F (2015) Extensions of the Cav \((u)\) theorem for repeated games with one-sided information. Math Oper Res 40(1):80–104

    Article  MathSciNet  MATH  Google Scholar 

  19. Gensbittel F (2013) Covariance control problems of martingales arising from game theory. SIAM J Control Optim 51(2):1152–1185

    Article  MathSciNet  MATH  Google Scholar 

  20. Heuer M (1991) Optimal strategies for the uninformed player. Int J Game Theory 20(1):33–51

    Article  MathSciNet  MATH  Google Scholar 

  21. Karlin S (1959) Mathematical methods and theory in games, programming, and economics. Addison-Wesley Publishing Company, Reading

    MATH  Google Scholar 

  22. Mannor S, Perchet V (2013) Approachability, fast and slow. In: Proceedings of COLT 2013. JMLR Workshop Conf Proc 30:474–488

  23. Mertens J-F, Zamir S (1976) The normal distribution and repeated games. Int J Game Theory 4:187–197

    Article  MathSciNet  MATH  Google Scholar 

  24. Mertens J-F, Zamir S (1977) The maximal variation of a bounded martingale. Israel J Math 27(3–4):252–276

    Article  MathSciNet  MATH  Google Scholar 

  25. Mertens J-F, Zamir S (1995) Incomplete information games and the normal distribution. CORE DP 9520

  26. Mertens JF, Sorin S, Zamir S (2015) Repeated games. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  27. Neyman A (2013) The maximal variation of martingales of probabilities and repeated games with incomplete information. J Theor Probab 26(2):557–567

    Article  MathSciNet  MATH  Google Scholar 

  28. Revuz D, Yor M (1999) Continuous martingales and Brownian motion. Springer, Berlin

    Book  MATH  Google Scholar 

  29. Sandomirskaia M (2016) Repeated bidding games with incomplete information and bounded values: on the exponential speed of convergence. Int Game Theory Rev. doi:10.1142/S0219198916500171

  30. Sandomirskiy F (2014) Repeated games of incomplete information with large sets of states. Int J Game Theory 43(4):767–789

    Article  MathSciNet  MATH  Google Scholar 

  31. Sorin S (2002) A first course on zero-sum repeated games. Mathematiques & Applications Springer, Berlin

    MATH  Google Scholar 

  32. Vershik A (2013) Long history of the Monge–Kantorovich transportation problem. Intell Math. doi:10.1007/s00283-013-9380-x

  33. Villani C (2008) Optimal transport: old and new. Science & Business Media Springer, Berlin

    MATH  Google Scholar 

  34. Zamir S (1971) On the relation between finitely and infinitely repeated games with incomplete information. Int J Game Theory 1:179–198

    Article  MathSciNet  MATH  Google Scholar 

  35. Zamir S (1992) Repeated games of incomplete information: zero-sum. Handb Game Theory 1:109–154

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

I am thankful to Vita Kreps for many inspiring discussions and for her care. I thank Misha Gavrilovich and two anonymous referees for suggestions that significantly improved presentation of the results.

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Authors and Affiliations

  1. National Research University Higher School of Economics, Soyza Pechatnikov str., 16, St. Petersburg, Russian Federation

    Fedor Sandomirskiy

  2. St. Petersburg Institute for Economics and Mathematics of Russian Academy of Sciences, Serpuhovskaya str., 38, St. Petersburg, Russian Federation

    Fedor Sandomirskiy

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  1. Fedor Sandomirskiy
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Correspondence to Fedor Sandomirskiy.

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To the memory of Victor Domansky.

Support from the Basic Research Program of the National Research University Higher School of Economics and from Grants 13-01-00462, 13-01-00784, and 16-01-00269 of the Russian Foundation for Basic Research is gratefully acknowledged.

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Sandomirskiy, F. On Repeated Zero-Sum Games with Incomplete Information and Asymptotically Bounded Values. Dyn Games Appl 8, 180–198 (2018). https://doi.org/10.1007/s13235-017-0217-7

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