The modified stochastic game

Original Paper
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Abstract

We present a new tool for the study of multiplayer stochastic games, namely the modified game, which is a normal-form game that depends on the discount factor, the initial state, and for every player a partition of the set of states and a vector that assigns a real number to each element of the partition. We study properties of the modified game, like its equilibria, min–max value, and max–min value. We then show how this tool can be used to prove the existence of a uniform equilibrium in a certain class of multiplayer stochastic games.

Keywords

Stochastic games Modified game Uniform equilibrium 

Mathematics Subject Classfication

91A15 

JEL Classification

C72 C73 

References

  1. Altman E (1999) Constrained Markov decision processes. Chapman and Hall/CRC, Boca RatonGoogle Scholar
  2. Bewley T, Kohlberg E (1976) The asymptotic theory of stochastic games. Math Oper Res 1:197–208CrossRefGoogle Scholar
  3. Billingsley P (1995) Probability and measure. Wiley, HobokenGoogle Scholar
  4. Blackwell D, Ferguson TS (1968) The big match. Ann Math Stat 39:159–163CrossRefGoogle Scholar
  5. Buhuvsky L, Solan E, Solan ON (2018) Monovex sets, Fundamenta Mathematicae, ForthcomingGoogle Scholar
  6. Eilenberg S, Montgomery D (1946) Fixed point theorems for multi-valued transformations. Am J Math 68:214–222CrossRefGoogle Scholar
  7. Fink AM (1964) Equilibrium in a stochastic \(n\)-person game. J Sci Hiroshima Univ Ser A-I Math 28:89–93Google Scholar
  8. Flesch J, Schoenmakers G, Vrieze K (2008) Stochastic games on a product state space. Math Oper Res 33:403–420CrossRefGoogle Scholar
  9. Flesch J, Schoenmakers G, Vrieze K (2009) Stochastic games on a product state space: the periodic case. Int J Game Theory 38:263–289CrossRefGoogle Scholar
  10. Flesch J, Thuijsman F, Vrieze OJ (1997) Stochastic games with additive transitions. Eur J Oper Res 179:483–497CrossRefGoogle Scholar
  11. Flesch J, Thuijsman F, Vrieze OJ (1997) Cyclic Markov equilibria in stochastic games. Int J Game Theory 26:303–314CrossRefGoogle Scholar
  12. Flesch J, Thuijsman F, Vrieze OJ (2007) Stochastic games with additive transitions. Eur J Oper Res 179:483–497CrossRefGoogle Scholar
  13. Gillette D (1957) Stochastic games with zero stop probabilities. Contributions to the theory of games, vol 3. Princeton University Press, Princeton, pp 179–187Google Scholar
  14. Hörner J, Sugaya T, Takahashi S, Vieille N (2011) Recursive methods in discounted stochastic games: an algorithm for \(\delta \rightarrow 1\) and a Folk theorem. Econometrica 79:1277–1318CrossRefGoogle Scholar
  15. Kuhn HW (1957) Extensive games and the problem of information. In: Kuhn H, Tucker AW (eds) Contributions to the Theory of Games, vol 2. Annals of Mathematical Studies 2. Princeton University Press, Princeton, pp 193–216Google Scholar
  16. Mertens JF, Neyman A (1981) Stochastic games. Int J Game Theory 10:53–66CrossRefGoogle Scholar
  17. Mertens JF, Parthasarathy T (1987) Equilibria for discounted stochastic games, CORE Discussion Paper No. 8750. Appeared in Stochastic games and applications. In: Neyman A and Sorin S (eds), Kluwer Academic Publishers, pp 131–172Google Scholar
  18. Mertens JF, Sorin S, Zamir S (2015) Repeated Games. Cambridge University Press, CambridgeGoogle Scholar
  19. Neyman A (2003) Real algebraic tools in stochastic games. In: Neyman A, Sorin S (eds) Stochastic games and applications. Kluwer Academic Publishers, Dordrecht, pp 57–75CrossRefGoogle Scholar
  20. Shapley LS (1953) Stochastic games. Proc Nat Acad Sci U.S.A. 39:1095–1100Google Scholar
  21. Simon RS (2007) The structure of non-zero-sum stochastic games. Adv Appl Math 38:1–26CrossRefGoogle Scholar
  22. Simon RS (2012) A topological approach to quitting games. Math Oper Res 37:180–195CrossRefGoogle Scholar
  23. Solan E (1999) Three-player absorbing games. Math Oper Res 24:669–698CrossRefGoogle Scholar
  24. Solan E (2000) Absorbing team games. Games Econ Behav 31:245–261CrossRefGoogle Scholar
  25. Solan E (2017) The modified stochastic game. arXiv:1703.04026
  26. Solan E (2018) Acceptable strategy profiles in stochastic games. Games and Economic Behavior, ForthcomingGoogle Scholar
  27. Solan E, Vieille N (2001) Quitting games. Math Oper Res 26:265–285CrossRefGoogle Scholar
  28. Solan E, Vieille N (2002) Correlated equilibrium in stochastic games. Games Econ Behav 38:362–399CrossRefGoogle Scholar
  29. Solan E, Vohra R (2002) Correlated equilibrium payoffs and public signalling in absorbing games. Int J Game Theory 31:91–121CrossRefGoogle Scholar
  30. Takahashi M (1964) Equilibrium points of stochastic non-cooperative \(n\)-person games. J Sci Hiroshima Univ Ser A-I Math 28:95–99Google Scholar
  31. Thuijsman F (2003) The Big Match and the Paris Match. In: Neyman A, Sorin S (eds) Stochastic games and applications, vol 570. NATO Science Series (Series C: Mathematical and Physical Sciences). Springer, Dordrecht, pp 195–204CrossRefGoogle Scholar
  32. Vieille N (2000a) Two-player stochastic games I: a reduction. Isr J Math 119:55–91CrossRefGoogle Scholar
  33. Vieille N (2000b) Two-player stochastic games II: the case of recursive games. Isr J Math 119:93–126CrossRefGoogle Scholar
  34. Vrieze OJ, Thuijsman F (1989) On equilibria in repeated games with absorbing states. Int J Game Theory 18:293–310CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.The School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael

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