A Stochastic Maximin Fixed-Point Equation Related to Game Tree Evaluation

Abstract

After suitable normalization the asymptotic root valueWof a minimax game tree of orderb≥ 2 with independent and identically distributed input values having a continuous, strictly increasing distribution function on a subinterval ofRappears to be a particular solution of the stochastic maximin fixed-point equationWξ max1≤i≤bmin1≤j≤bWi,j, whereWi,jare independent copies ofWanddenotes equality in law. Moreover, ξ=g'(α) > 1, whereg(x) := (1 − (1 −x)b)band α denotes the unique fixed point ofgin (0, 1). This equation, which takes the formF(t) =g(F(t/ξ)) in terms of the distribution functionFofW, is studied in the present paper for a reasonably extended class of functionsgso as to encompass more general stochastic maximin equations as well. A complete description of the set of solutionsFis provided followed by a discussion of additional properties such as continuity, differentiability, or existence of moments. Based on these results, it is further shown that the particular solution mentioned above stands out among all other ones in that its distribution function is the restriction of an entire function to the real line. This extends recent work of Ali Khan, Devroye and Neininger (2005). A connection with another class of stochastic fixed-point equations for weighted minima and maxima is also discussed.