Minimax functions on Galton-Watson trees

TL;DR

This paper analyzes the behavior of minimax recursions on Galton-Watson trees, revealing diverse limiting distributions and examining how much of the tree's information is needed for near-optimal play, depending on offspring distribution.

Contribution

It extends previous work by characterizing the limiting behavior of minimax values on Galton-Watson trees with general offspring distributions and explores endogeny and information requirements.

Findings

Unrescaled game values can converge to constants, discrete, or continuous distributions.

Distributional limits depend on the offspring distribution.

The amount of tree information needed for near-optimal play varies with offspring distribution.

Abstract

We consider the behaviour of minimax recursions defined on random trees. Such recursions give the value of a general class of two-player combinatorial games. We examine in particular the case where the tree is given by a Galton-Watson branching process, truncated at some depth $2n$, and the terminal values of the level-$2n$ nodes are drawn independently from some common distribution. The case of a regular tree was previously considered by Pearl, who showed that as $n\to\infty$ the value of the game converges to a constant, and by Ali Khan, Devroye and Neininger, who obtained a distributional limit under a suitable rescaling. For a general offspring distribution, there is a surprisingly rich variety of behaviour: the (unrescaled) value of the game may converge to a constant, or to a discrete limit with several atoms, or to a continuous distribution. We also give distributional limits under suitable rescalings in various cases. We also address questions of endogeny. Suppose the game is played on a tree with many levels, so that the terminal values are far from the root. To be confident of playing a good first move, do we need to see the whole tree and its terminal values, or can we play close to optimally by inspecting just the first few levels of the tree? The answers again depend in an interesting way on the offspring distribution. We also mention several open questions.