Winning Criteria for Open Games: A Game-Theoretic Approach to Prefix Codes

TL;DR

This paper explores the conditions under which a player has a winning strategy in infinite tree games with open winning sets, linking game theory with algebraic structures like prefix codes.

Contribution

It establishes a novel equivalence between winning sets and maximal prefix codes, providing algebraic criteria for winning strategies in open games on infinite trees.

Findings

Necessary conditions for first player's winning strategy

Equivalence between winning sets and maximal prefix codes

Use of algebraic conditions and coverings to analyze game outcomes

Abstract

We study two-player games with alternating moves played on infinite trees. Our main focus is on the case where the trees are full (regular) and the winning set is open (with respect to the product topology on the tree). Gale and Stewart showed that in this setting one of the players always has a winning strategy, though it is not known in advance which player. We present simple necessary conditions for the first player to have a winning strategy, and establish an equivalence between winning sets that guarantee a win for the first player and maximal prefix codes. Using this equivalence, we derive a necessary algebraic condition for winning, and exhibit a family of games for which this algebraic condition is in fact equivalent to winning. We introduce the concept of coverings, and show that by covering the tree of the game with an infinite labeled tree corresponding to the free group, we can use "game-theoretic tools" to derive a simple trait of maximal prefix codes.