On the combinatorial value of Hex positions

TL;DR

This paper introduces a new combinatorial game theory framework tailored for Hex and similar games, analyzing properties like monotonicity and passability to better understand game positions.

Contribution

It develops a novel theoretical approach to analyze Hex positions using monotone and passable game concepts, establishing their equivalence and applicability.

Findings

Monotone games are always passable.

Passability and monotonicity are equivalent under certain conditions.

The theory provides new insights into Hex position analysis.

Abstract

We develop a theory of combinatorial games that is appropriate for describing positions in Hex and other monotone set coloring games. We consider two natural conditions on such games: a game is monotone if all moves available to both players are good, and passable if in each position, at least one player has at least one good move available. The latter condition is equivalent to saying that if passing were permitted, no player would benefit from passing. Clearly every monotone game is passable, and we prove that the converse holds up to equivalence of games. We give some examples of how this theory can be applied to the analysis of Hex positions.