The Annals of Mathematical Statistics

Game Value Distributions II

David R. Thomas

Abstract

An earlier paper  has been concerned with the distribution of the value of perfect information games with random payoffs of a certain very special type: two alternatives were assumed available to each player at every move, and the terminal payoffs were assumed to be iid and uniform. This paper considers a more general class of games, with $p$ and $q$ alternatives available, respectively, for players I and II at every move, and with the terminal payoffs arbitrarily distributed, though still iid. Specifically, consider a two-person zero-sum perfect information game, with player I and player II alternately choosing one of several alternative moves, with $n$ choices to be made in all by each. It is assumed that there are always $p$ and $q$ alternatives available respectively to players I and II. Corresponding to each of the $(pq)^n$ possible sequences of moves, there are $(pq)^n$ payoffs (to player $I$) $x(i_1, i_2, \cdots, i_{2n})$, where the indices $i_1, i_3, \cdots, i_{2n - 1}$, each with range $(1, 2, \cdots, p)$ indicate the successive alternatives chosen by player I, and the indices $(i_2, i_4, \cdots, i_{2n})$, each with range $(1, 2, \cdots, q)$, indicate the successive alternatives chosen by player II. The value $v(\{x(i_1, \cdots, i_{2n})\})$ of such a game is $\max_{i_1} \min_{i_2} \max_{i_3} \min_{i_4} \cdots \max_{i_{2n}} x(i_1, \cdots, i_{2n}).$ Now replace the $(pq)^n$ numbers $x(i_1, \cdots, i_{2n})$ by independent random variables $X(i_1, \cdots, i_{2n})$, each with cdf. $F$. This paper is concerned with the limiting behavior of the random values $V_n(F) \equiv v(\{X(i_1, \cdots, i_{2n})\})$. The limiting behavior of $V_n(F)$ is investigated in Section 2 for uniform $F (F = U)$. Analogous to the results for $p = q = 2$ obtained in , the limiting distribution for the sequence $\{V_n(U)\}$ is everywhere continuous and monotone increasing, and satisfies a certain functional equation. Limiting distributions arising from arbitrary $F$ are considered in Section 3. Section 4 is devoted to some results concerning norming sequences and domains of attraction. The final corollary of Section 4 establishes that all of the common cdf's lead to the same limiting distribution. This study bears a strong resemblance to Gnedenko's  study of extremes. Since, in Gnedenko's case, the limiting distributions for $Z_{(n)} = \max (Z_1, \cdots, Z_n)$, where $Z_n$ are independent identically distributed random variables, must in effect be limiting distributions for $Z_{(k^n)}$ for every positive integer $k$, Gnedenko's argument involves an infinite sequence of functional equations , p. 431, namely, one functional equation for every $k$. In the present treatment the limiting distributions for $V_n(F)$ must satisfy only the single functional equation (6). It may thus be worth noting that even a stronger resemblance would exist between this paper and a study of the limiting distributions for $Z_{(k^n)}, k$ fixed. I would like to acknowledge Professor H. T. David for his many helpful discussions concerning this research.

Article information

Source
Ann. Math. Statist., Volume 38, Number 1 (1967), 251-260.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177699077

Digital Object Identifier
doi:10.1214/aoms/1177699077

Mathematical Reviews number (MathSciNet)
MR233597

Zentralblatt MATH identifier
0183.24002

JSTOR