Two-person Red-and-black Games with Bet-dependent Win Probability Functions

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.