Skip to main content
SpringerLink
Log in

Nonzero-sum Risk-Sensitive Average Stochastic Games: The Case of Unbounded Costs

  • Published:
Dynamic Games and Applications Aims and scope Submit manuscript

Abstract

In this paper, we study discrete-time nonzero-sum stochastic games under the risk-sensitive average cost criterion. The state space is a denumerable set, the action spaces of players are Borel spaces, and the cost functions are unbounded. Under suitable conditions, we first introduce the risk-sensitive first passage payoff functions and obtain their properties. Then, we establish the existence of a solution to the risk-sensitive average cost optimality equation of each player for the case of unbounded cost functions and show the existence of a randomized stationary Nash equilibrium in the class of randomized history-dependent strategies. Finally, we use a controlled population system to illustrate the main results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aliprantis C, Border K (2006) Infinite dimensional analysis. Springer, New York

    MATH  Google Scholar 

  2. Altman E, Hordijk A, Spieksma FM (1997) Contraction conditions for average and \(\alpha \)-discount optimality in countable state Markov games with unbounded rewards. Math Oper Res 22:588–618

    Article  MathSciNet  Google Scholar 

  3. Asienkiewicz H, Balbus Ł (2019) Existence of Nash equilibria in stochastic games of resource extraction with risk-sensitive players. TOP 27:502–518

    Article  MathSciNet  Google Scholar 

  4. Balaji S, Meyn SP (2000) Multiplicative ergodicity and large deviations for an irreducible Markov chain. Stoch Process Appl 90:123–144

    Article  MathSciNet  Google Scholar 

  5. Başar T (1999) Nash equilibria of risk-sensitive nonlinear stochastic differential games. J Optim Theory Appl 100:479–498

    Article  MathSciNet  Google Scholar 

  6. Basu A, Ghosh MK (2014) Zero-sum risk-sensitive stochastic games on a countable state space. Stoch Process Appl 124:961–983

    Article  MathSciNet  Google Scholar 

  7. Basu A, Ghosh MK (2018) Nonzero-sum risk-sensitive stochastic games on a countable state space. Math Oper Res 43:516–532

    Article  MathSciNet  Google Scholar 

  8. Bäuerle N, Rieder U (2017) Zero-sum risk-sensitive stochastic games. Stoch Process Appl 127:622–642

    Article  MathSciNet  Google Scholar 

  9. Bertsekas DP, Shreve SE (1996) Stochastic optimal control: the discrete-time case. Athena Scientific, Belmont

    MATH  Google Scholar 

  10. Borkar VS, Meyn SP (2002) Risk-sensitive optimal control for Markov decision processes with monotone cost. Math Oper Res 27:192–209

    Article  MathSciNet  Google Scholar 

  11. Cavazos-Cadena R (2010) Optimality equations and inequalities in a class of risk-sensitive average cost Markov decision chains. Math Methods Oper Res 71:47–84

    Article  MathSciNet  Google Scholar 

  12. Cavazos-Cadena R, Hernández-Hernández D (2019) The vanishing discount approach in a class of zero-sum finite games with risk-sensitive average criterion. SIAM J Control Optim 57:219–240

    Article  MathSciNet  Google Scholar 

  13. Filar J, Vrieze K (1997) Competitive Markov decision processes. Springer, New York

    MATH  Google Scholar 

  14. Fink AM (1964) Equilibrium in a stochastic n-person game. J Sci Hiroshima Univ Ser A-I Math 28:89–93

    MathSciNet  MATH  Google Scholar 

  15. Hernández-Lerma O, Lasserre JB (1996) Discrete-time Markov control processes: basic optimality criteria. Springer, New York

    Book  Google Scholar 

  16. Hernández-Lerma O, Lasserre JB (1999) Further topics on discrete-time Markov control processes. Springer, New York

    Book  Google Scholar 

  17. Jaśkiewicz A, Nowak AS (2006) Zero-sum ergodic stochastic games with Feller transition probabilities. SIAM J Control Optim 45:773–789

    Article  MathSciNet  Google Scholar 

  18. Jaśkiewicz A, Nowak AS (2011) Stochastic games with unbounded payoffs: applications to robust control in economics. Dyn Games Appl 1:253–279

    Article  MathSciNet  Google Scholar 

  19. Jaśkiewicz A, Nowak AS (2014) Stationary Markov perfect equilibria in risk sensitive stochastic overlapping generations models. J Econ Theory 151:411–447

    Article  MathSciNet  Google Scholar 

  20. Jaśkiewicz A, Nowak AS (2016) Stationary almost Markov perfect equilibria in discounted stochastic games. Math Oper Res 41:430–441

    Article  MathSciNet  Google Scholar 

  21. Jaśkiewicz A, Nowak AS (2018) Zero-sum stochastic games. In: Basar T, Zaccour G (eds) Handbook of dynamic game theory, vol I. Springer, Cham, pp 215–279

    Chapter  Google Scholar 

  22. Jaśkiewicz A, Nowak AS (2018) Non-zero-sum stochastic games. In: Basar T, Zaccour G (eds) Handbook of dynamic game theory, vol I. Springer, Cham, pp 281–344

    Chapter  Google Scholar 

  23. Klompstra MB (2000) Nash equilibria in risk-sensitive dynamic games. IEEE Trans Autom Control 45:1397–1401

    Article  MathSciNet  Google Scholar 

  24. Maitra A, Sudderth W (2003) Stochastic games with limsup payoff. In: Neyman A, Sorins S (eds) Stochastic games and applications. Kluwer, Dordrecht, pp 357–366

    Chapter  Google Scholar 

  25. Maitra A, Sudderth WD (2007) Subgame-perfect equilibria for stochastic games. Math Oper Res 32:711–722

    Article  MathSciNet  Google Scholar 

  26. Mertens JF (2002) Stochastic games. In: Aumann RJ, Hart S (eds) Handbook of game theory with economic applications, vol 3. North Holland, Amsterdam, pp 1809–1832

    Google Scholar 

  27. Mertens JF, Neyman A (1981) Stochastic games. Int J Game Theory 10:53–66

    Article  MathSciNet  Google Scholar 

  28. Mertens JF, Parthasarathy T (2003) Equilibria for discounted stochastic games. In: Neyman A, Sorin S (eds) Stochastic games and applications. Kluwer, Dordrecht, pp 131–172

    Chapter  Google Scholar 

  29. Meyn SP (1999) Algorithms for optimization and stabilization of controlled Markov chains. Sādhanā 24:339–367

    Article  MathSciNet  Google Scholar 

  30. Meyn SP, Tweedie RL (2009) Markov chains and stochastic stability. Cambridge University Press, New York

    Book  Google Scholar 

  31. Neyman A, Sorin S (2003) Stochastic games and applications. Kluwer, Dordrecht

    Book  Google Scholar 

  32. Nowak AS, Altman E (2002) \(\varepsilon \)-Equilibria for stochastic games with uncountable state space and unbounded costs. SIAM J Control Optim 40:1821–1839

    Article  MathSciNet  Google Scholar 

  33. Nowak AS, Raghavan TES (1992) Existence of stationary correlated equilibria with symmetric information for discounted stochastic games. Math Oper Res 17:519–526

    Article  MathSciNet  Google Scholar 

  34. Parthasarathy T, Sinha S (1989) Existence of stationary equilibrium strategies in nonzero-sum discounted stochastic games with uncountable state space and state-independent transitions. Int J Game Theory 18:189–194

    Article  Google Scholar 

  35. Shapley LS (1953) Stochastic games. Proc Nat Acad Sci USA 39:1095–1100

    Article  MathSciNet  Google Scholar 

  36. Takahashi M (1962) Stochastic games with infinitely many strategies. J Sci Hiroshima Univ Ser A-I Math 26:123–134

    MathSciNet  MATH  Google Scholar 

  37. Vega-Amaya O (2003) Zero-sum average semi-Markov games: fixed-point solutions of the Shapley equation. SIAM J Control Optim 42:1876–1894

    Article  MathSciNet  Google Scholar 

  38. Wei QD, Chen X (2019) Risk-sensitive average equilibria for discrete-time stochastic games. Dyn Games Appl 9:521–549

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are greatly indebted to the associate editor and the referees for the valuable comments and suggestions which have greatly improved the presentation. The research of the first author was supported by National Natural Science Foundation of China (Grant No. 11601166). The research of the second author was supported by National Natural Science Foundation of China (Grant No. 11701483).

Author information

Authors and Affiliations

  1. School of Economics and Finance, Huaqiao University, Quanzhou, 362021, People’s Republic of China

    Qingda Wei

  2. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, People’s Republic of China

    Xian Chen

Authors
  1. Qingda Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xian Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xian Chen.

Ethics declarations

Conflict of interest

We declare that no conflict of interest exists in this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wei, Q., Chen, X. Nonzero-sum Risk-Sensitive Average Stochastic Games: The Case of Unbounded Costs. Dyn Games Appl 11, 835–862 (2021). https://doi.org/10.1007/s13235-021-00380-5

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13235-021-00380-5

Keywords

Mathematics Subject Classification

Search

Navigation