Game Value Distributions II

Abstract

An earlier paper [4] 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 [4], 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 [1] 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 [1], 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.