Advertisement

Dynamic Games and Applications

, Volume 8, Issue 1, pp 180–198 | Cite as

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

  • Fedor SandomirskiyEmail author
Article
  • 164 Downloads

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 \(\sqrt{N}\).

Keywords

Repeated games with incomplete information Error term Bidding games Piecewise games Asymptotics of the value Transportation problems Kantorovich metric 

Notes

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.

References

  1. 1.
    Aumann R, Maschler M (1995) Repeated games with incomplete information. MIT Press, CambridgezbMATHGoogle Scholar
  2. 2.
    De Meyer B (1996) Repeated games and partial differential equations. Math Oper Res 21(1):209–236MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    De Meyer B (1996) Repeated games, duality and the central limit theorem. Math Oper Res 21(1):237–251MathSciNetCrossRefzbMATHGoogle Scholar
  4. 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):49–59MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    De Meyer B (1999) From repeated games to Brownian games. Ann. de l’IHP Probabilités et statistiques 35(1):1–48MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    De Meyer B, Saley HM (2003) On the strategic origin of Brownian motion in finance. Int J Game Theory 31(2):285–319MathSciNetCrossRefzbMATHGoogle Scholar
  7. 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 1Google Scholar
  8. 8.
    De Meyer B, Marino A (2005) Continuous versus discrete Market games. Cowles Foundation discussion paper 1535. Yale UniversityGoogle Scholar
  9. 9.
    De Meyer B (2010) Price dynamics on a stock market with asymmetric information. Games Econ Behav 69:42–71MathSciNetCrossRefzbMATHGoogle Scholar
  10. 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\(>\) Google Scholar
  11. 11.
    Domansky V (2007) Repeated games with asymmetric information and random price fluctuations at finance markets. Int J Game Theory 36(2):241–257MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Domansky V, Kreps V (1994) “Eventually revealing” repeated games with incomplete information. Int J Game Theory 23(2):89–99MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Domansky V, Kreps V (1995) Repeated games and multinomial distributions. Math Meth Oper Res 42(3):275–293MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Domansky V, Kreps V (1999) Repeated games with incomplete information and transportation problems. Math Meth Oper Res 49(2):283–298MathSciNetzbMATHGoogle Scholar
  15. 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\(>\) Google Scholar
  16. 16.
    Domansky V, Kreps V (2013) Repeated games with asymmetric information modeling financial markets with two risky assets. RAIRO Oper Res 47(3):251–272MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Domansky V, Kreps V (2016) Bidding games with several risky assets. Autom Remote Control 77(4):722–733MathSciNetCrossRefGoogle Scholar
  18. 18.
    Gensbittel F (2015) Extensions of the Cav \((u)\) theorem for repeated games with one-sided information. Math Oper Res 40(1):80–104MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Gensbittel F (2013) Covariance control problems of martingales arising from game theory. SIAM J Control Optim 51(2):1152–1185MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Heuer M (1991) Optimal strategies for the uninformed player. Int J Game Theory 20(1):33–51MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Karlin S (1959) Mathematical methods and theory in games, programming, and economics. Addison-Wesley Publishing Company, ReadingzbMATHGoogle Scholar
  22. 22.
    Mannor S, Perchet V (2013) Approachability, fast and slow. In: Proceedings of COLT 2013. JMLR Workshop Conf Proc 30:474–488Google Scholar
  23. 23.
    Mertens J-F, Zamir S (1976) The normal distribution and repeated games. Int J Game Theory 4:187–197MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Mertens J-F, Zamir S (1977) The maximal variation of a bounded martingale. Israel J Math 27(3–4):252–276MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Mertens J-F, Zamir S (1995) Incomplete information games and the normal distribution. CORE DP 9520Google Scholar
  26. 26.
    Mertens JF, Sorin S, Zamir S (2015) Repeated games. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  27. 27.
    Neyman A (2013) The maximal variation of martingales of probabilities and repeated games with incomplete information. J Theor Probab 26(2):557–567MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Revuz D, Yor M (1999) Continuous martingales and Brownian motion. Springer, BerlinCrossRefzbMATHGoogle Scholar
  29. 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. 30.
    Sandomirskiy F (2014) Repeated games of incomplete information with large sets of states. Int J Game Theory 43(4):767–789MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Sorin S (2002) A first course on zero-sum repeated games. Mathematiques & Applications Springer, BerlinzbMATHGoogle Scholar
  32. 32.
    Vershik A (2013) Long history of the Monge–Kantorovich transportation problem. Intell Math. doi: 10.1007/s00283-013-9380-x
  33. 33.
    Villani C (2008) Optimal transport: old and new. Science & Business Media Springer, BerlinzbMATHGoogle Scholar
  34. 34.
    Zamir S (1971) On the relation between finitely and infinitely repeated games with incomplete information. Int J Game Theory 1:179–198MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Zamir S (1992) Repeated games of incomplete information: zero-sum. Handb Game Theory 1:109–154MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.National Research University Higher School of EconomicsSt. PetersburgRussian Federation
  2. 2.St. Petersburg Institute for Economics and Mathematics of Russian Academy of SciencesSt. PetersburgRussian Federation

Personalised recommendations