The Annals of Mathematical Statistics

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  and . The derivation in  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  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 , 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  and that of Sobel  pertain to Section 2, and that of Buehler  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 .

Article information

Source
Ann. Math. Statist., Volume 38, Number 1 (1967), 242-250.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177699076

Digital Object Identifier
doi:10.1214/aoms/1177699076

Mathematical Reviews number (MathSciNet)
MR233596

Zentralblatt MATH identifier
0166.15803

JSTOR