Fixed Point for Negamaxing Probability Distributions on Regular Trees

Consider a rooted tree with branching factor b ? 1 having b h leafs, each at distance h from the root. Suppose the leafs are assigned real values chosen i.i.d. from some probability density, and the interior nodes of the tree are then also assigned values recursively, according to the negamax rule: The value of a node is the maximum of the negated values of its children. The root will then be a random variable, with some probability distribution \Phi h (x) depending on b, h, and the distribution \Phi 0 (x) of leaf values, and indeed obeying certain recursive relations. We find a closed form for \Phi h (x). It then turns out that (usually) the behavior of the distribution at the root, when h becomes sufficiently large, is, except for scaling, asymptotically independent of the nature of \Phi 0 (x) and h and depends only on b. The function merely scales self-similarly as h becomes larger. This may be thought of as a new kind of central limit theorem. The limiting form \Psi b (x), is ap..