Random perfect information games

TL;DR

This paper introduces a measure space for zero-sum perfect information games with probabilistic elements, characterizes the distribution of game values, and analyzes optimal strategies and plays.

Contribution

It defines a natural measure space for these games, characterizes the value distribution via fixed points, and explores probabilistic properties of strategies and plays.

Findings

Characterization of the value distribution using fixed points.

Necessary and sufficient conditions for the value to exceed a threshold.

Analysis of probabilistic properties of optimal strategies.

Abstract

The paper proposes a natural measure space of zero-sum perfect information games with upper semicontinuous payoffs. Each game is specified by the game tree, and by the assignment of the active player and of the capacity to each node of the tree. The payoff in a game is defined as the infimum of the capacity over the nodes that have been visited during the play. The active player, the number of children, and the capacity are drawn from a given joint distribution independently across the nodes. We characterize the cumulative distribution function of the value $v$ using the fixed points of the so-called value generating function. The characterization leads to a necessary and sufficient condition for the event $v \geq k$ to occur with positive probability. We also study probabilistic properties of the set of Player I's $k$-optimal strategies and the corresponding plays.