Kybernetika 57 no. 2, 312-331, 2021

Incomplete information and risk sensitive analysis of sequential games without a predetermined order of turns

Rubén Becerril-Borja and Raúl Montes-de-OcaDOI: 10.14736/kyb-2021-2-0312

Abstract:

The authors introduce risk sensitivity to a model of sequential games where players don't know beforehand which of them will make a choice at each stage of the game. It is shown that every sequential game without a predetermined order of turns with risk sensitivity has a Nash equilibrium, as well as in the case in which players have types that are chosen for them before the game starts and that are kept from the other players. There are also a couple of examples that show how the equilibria might change if the players are risk prone or risk adverse.

Keywords:

incomplete information, sequential game, risk sensitive, turn selection process

Classification:

91A10, 91A18, 91A25

paper.pdf

References:

  1. C. D. Aliprantis and K. C. Border: Infinite Dimensional Analysis. Springer, Berlin Heidelberg 2006.   CrossRef
  2. K. J. Arrow: Aspects of the theory of risk-bearing. In: Essays in the Theory of Risk Bearing, Markham Publ. Co., Chicago 1971, pp. 90-109.   CrossRef
  3. A. Basu and M. K. Ghosh: Nonzero-sum risk-sensitive stochastic games on a countable state space. Math. Oper. Res. 43 (2018), 516-532.   DOI:10.1287/moor.2017.0870
  4. N. Bäuerle and U. Rieder: Zero-sum risk-sensitive stochastic games. Stoch. Processes Appl. 12 (2017), 2, 622-642.   DOI:10.1016/j.spa.2016.06.020
  5. R. Becerril-Borja and R. Montes-de-Oca: A family of models for finite sequential games without a predetermined order of turns. In: Operations Research and Enterprise Systems (B. Vitoriano, G. H. Parlier, eds.), Springer International Publishing, Cham 2017, 35-51.   DOI:10.1007/978-3-319-53982-9\_3
  6. K. C. Border: Fixed Point Theorems with Applications to Economics and Game Theory. Cambridge University Press, Cambridge 1985.   CrossRef
  7. L. Eeckhoudt, C. Gollier and H. Schlesinger: Economic and Financial Decisions under Risk. Princeton University Press, Princeton 2005.   DOI:10.1515/9781400829217
  8. W. H. Fleming and W. M. McEneaney: Risk sensitive optimal control and differential games. In: Stochastic Theory and Adaptive Control. Lecture Notes in Control and Information Sciences (T. E. Duncan and B. Pasik-Duncan, eds.), Springer, Berlin Heidelberg 1992, pp. 185-197.   CrossRef
  9. R. A. Howard and J. E. Matheson: Risk sensitive Markov decision processes. Management Sci. 18 (1972), 356-369.   DOI:10.1287/mnsc.18.7.356
  10. M. R. James, J. Baras and R. J. Elliott: Risk-sensitive control and dynamic games for partially observed discrete-time nonlinear systems. IEEE Trans. Automat. Control 39 (1994), 780-792.   DOI:10.1109/9.286253
  11. S. Kakutani: A generalization of Brouwer's fixed point theorem. Duke Math. J. 8 (1942), 457-459.   DOI:10.1215/S0012-7094-41-00838-4
  12. M. B. Klompstra: Nash equilibria in risk-sensitive dynamic games. IEEE Trans. Automat. Control 45 (2000), 1397-1401.   DOI:10.1109/9.867067
  13. A. S. Nowak: Notes on risk-sensitive Nash equilibria. In: Advances in Dynamic Games: Applications to Economics, Finance, Optimization and Stochastic Control (A. S. Nowak and K. Szajowski, eds.), Birkhäuser, Boston 2005, pp. 95-109.   CrossRef
  14. J. W. Pratt: Risk aversion in the small and in the large. Econometrica 32 (1964), 122-136.   DOI:10.2307/1913738
  15. Y. Shoham and K. Leyton-Brown: Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations. Cambridge University Press, Cambridge 2008.   CrossRef
  16. K. Sladký: Risk sensitive average optimality in Markov decision processes. Kybernetika 54 (2018), 1218-1230.   DOI:10.14736/kyb-2018-6-1218
  17. S. Tadelis: Game Theory: An Introduction. Princeton University Press, Princeton 2013.   CrossRef