The Probability that a Random Game is Unfair

Abstract

To every matrix $A$ of real valued payoffs there corresponds a unique quantity $V(A)$, the value of a game with payoff matrix $A$, defined by $V(A) = \min_y \max_x x^t Ay = \max_x \min_y x^t Ay,$ where the maximum and minimum are taken over all $x$ and $y$ having nonnegative components summing to one. We wish to show that, for a relatively wide class of probability distributions on the class of all $m \times n$ game matrices, the probability $P(m, n)$ that the value of a random game is greater than zero is given by the cumulative binomial distribution $P(m, n) = (\frac{1}{2})^{m + n - 1} \sum^{m - 1}_{k = 0} \binom{m + n -1}{k}.$ The observation that, for any $\epsilon > 0$, $\lim_{m \rightarrow \infty} P(m, (1 + \epsilon)m) = 0, P(m, m) = \frac{1}{2},\quad \text{for all m}, \lim_{m \rightarrow \infty} P(m, (1 - \epsilon)m) = 1,$ suggests that large rectangular games tend to be strongly biased in favor of the player having the greater number of alternatives. Thus we characterize the extent to which the nonsquareness of a game is reflected in a bias for one of the players. Our method involves an application of a theorem in combinatorial geometry due to Schlafli [5]. A discussion of the consequences of Schlafli's theorem in geometrical probability is given in [2], [3], [9].