Game Value Distributions I

Abstract

This paper is concerned with the distribution of the value of a game with random payoffs. Two types of games are considered: matrix games with iid matrix elements, and games of perfect information with iid terminal payoffs. Let $\| x_{ij}\|, i:1, 2, \cdots, m; j:1, 2, \cdots, n$, be the matrix of player I's payoffs in a zero-sum two-person game, and let $v(\|x_{ij}\|)$ be its (possibly mixed) value. Consider the random value $V_{m,n}(f) \equiv v(\| X_{ij}\|)$, where the $X_{ij}$ are $mn$ iid random variables, each distributed according to the density $f$. It is pointed out in Section 2 that the conditional distribution of $V_{m,n}$, given that it is pure, is that of the $n$th largest of $m + n - 1$ iid random variables, each distributed according to $f$. For $f$ uniform on (0, 1) (i.e., $f = u$), a method is given for determining the conditional distribution of $V_{2,n}(u)$, given that it is mixed. This leads to an elementary expression for the distribution of $V_{2,2}(u)$ and the asymptotic distribution of $V_{2,n}(u)$. Consider as well two players alternately choosing one of two alternative moves, with $n$ choices to be made in all by each. Corresponding to each of the $4^n$ possible sequences of moves, there are $4^n$ payoffs $x(i_1, i_2, \cdots, i_{2n})$ for player I, $i_k = 1$ or 2, where the odd and even locations indicate, respectively, the successive alternatives chosen by players I and II. The (pure) value $v(\{x(i_1, \cdots, i_{2n})\})$ of such a game is $\max_{i_1} \min_{i_2} \max_{i_3} \min_{i_4} \cdots \max_{i_{2n - 1}} \min_{i_{2n}} x(i_1, \cdots, i_{2n}).$ Now replace the $4^n$ numbers $x(i_1, \cdots, i_n)$ by independent uniformly distributed random variables $X(i_1, \cdots, i_{2n})$. The asymptotic behavior of the random value $V_n \equiv v(\{X(i_1, \cdots, i_{2n})\})$ is investigated in Section 3; it is shown that the asymptotic distribution $L$ of $V_n$ is everywhere continuous and monotone-increasing, and satisfies a certain functional equation; it is also shown that the moments of the normed $V_n$ converge to those of $L$. It is planned, in a subsequent paper, to explore games of perfect information in greater depth. After this paper was submitted, Thomas M. Cover drew our attention to [3] and [9]. The derivation in [9] of the expected value of a $2 \times n$ game, conditionally on there being a $2 \times 2$ kernel, is based on essentially the geometric considerations leading to our distribution (5); however, since the argument in [9] is not aimed at obtaining distributions, and is thus rather different in detail, a sketch of our derivation of (5) has not been deleted. In [3], the probability is computed, in the case of payoff distributions symmetric about zero, that an $m \times n$ game has positive value. Also, the work of Efron [4] and that of Sobel [8] pertain to Section 2, and that of Buehler [1] to Section 3. Finally, closely related to this paper, and indeed the source of our original interest in this area, is the work of Chernoff and Teicher [2].