Multi-round card game with arbitration and random amount of bet

摘要

We consider zero-sum game which is usually called poker. Each of two players (I and II) independently draws a number (x and y, respectively) according to uniform distribution in [0, 1]. Also they jointly draw a number z according to the distribution with cdf F(z), O < z < ∞. After observing this number and the "center card" z each player chooses either to accept (A) or to reject (R) his number. If both players choose A, showdown is made and I wins (loses) if x > (<) y, getting reward z(-z). If both players choose R, then the numbers are rejected and the game proceeds to the next round with the newly drawn numbers x', y', and z'. If players choose different choices, then arbitration comes in, and forces to take the same choices as I's (II's) with probability p (p-bar). Arbitration is fair (unfair) if p = (≠) 1/2. The game is played in n rounds. Player I (II) aims to maximize (minimize) the expected reward to I. Explicit solution is derived to this sequential game, and examples are shown for the cases where z obeys discrete and continuous uniform distributions.