Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-10T12:35:01.691Z Has data issue: false hasContentIssue false
  • English
  • Français

Two-person red-and-black games with bet-dependent win probability functions

Part of: Game theory

Published online by Cambridge University Press:  14 July 2016

May-Ru Chen  and
Shoou-Ren Hsiau
May-Ru Chen*
Affiliation:
National Changhua University of Education
Shoou-Ren Hsiau*
Affiliation:
National Changhua University of Education
*
∗Postal address: Department of Mathematics, National Changhua University of Education, No. 1 Jin-De Rd., Changhua 500, Taiwan, R. O. C.
∗Postal address: Department of Mathematics, National Changhua University of Education, No. 1 Jin-De Rd., Changhua 500, Taiwan, R. O. C.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper a two-person red-and-black game is investigated. We suppose that, at every stage of the game, player I's win probability, f, is a function of the ratio of his bet to the sum of both players' bets. Two results are given: (i) if f is convex then a bold strategy is optimal for player I when player II plays timidly; and (ii) if f satisfies f(s)f(t) ≤ f(st) then a timid strategy is optimal for player II when player I plays boldly. These two results extend two formulations of red-and-black games proposed by Pontiggia (2005), and also provide a sufficient condition to ensure that the profile (bold, timid) is the unique Nash equilibrium for players I and II. Finally, we give a counterexample to Pontiggia's conjecture about a proportional N-person red-and-black game.

Keywords

Red-and-black gamebold strategytimid strategyconvex functionNash equilibrium

MSC classification

Primary: 91A15: Stochastic games
Secondary: 91A05: 2-person games 91A06: $n$-person games, $n>2$
Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 4 , December 2006 , pp. 905 - 915
Copyright
© Applied Probability Trust 2006 

References

Dubins, L. E. and Savage, L. J. (1976). Inequalities for Stochastic Processes: How to Gamble if You Must, 2nd edn. Dover, New York.Google Scholar
Maitra, A. P. and Sudderth, W. D. (1996). Discrete Gambling and Stochastic Games. Springer, New York.Google Scholar
Pontiggia, L. (2005). Two-person red-and-black with bet-dependent win probability. Adv. Appl. Prob. 37, 75–89.Google Scholar
Ross, S. M. (1974). Dynamic programming and gambling models. Adv. Appl. Prob. 6, 598–606.Google Scholar
Secchi, P. (1997). Two-person red-and-black stochastic games. J. Appl. Prob. 34, 107–126.Google Scholar