Two-Person Red-and-Black Games with Bet-Dependent Win Probability Functions

Abstract

In this talk, a two-person red-and-black game is investigated. We suppose that, at every stage of the game, each player's win probability is a function of the ratio of his/her bet to the sum of both players' bets. Two simple criteria are given: the first ensure a bold strategy is optimal for player I when player II plays timidly and the second ensure a timid strategy is optimal for player II when player I plays boldly. These two criteria extend two formulations of red-and-black game proposed by Pontiggia, and also provide a sufficient condition for the profile (bold, timid) of players I and II to be the unique Nash equilibrium. At last, we give a counterexample to Pontiggia's conjecture about a proportional N-person red-and-black game.