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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.
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Abstract

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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 games91A06: $n$-person games, $n>2$

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 4 , December 2006, pp. 905 - 915
Copyright
© Applied Probability Trust 2006 

References

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