All passable games are realizable as monotone set coloring games

TL;DR

This paper proves that all passable combinatorial games can be represented as monotone set coloring games, establishing an equivalence between these two classes of games.

Contribution

It demonstrates that every passable game is, up to equivalence, realizable as a monotone set coloring game, bridging the gap between these game classes.

Findings

All passable games are realizable as monotone set coloring games.

Equivalence between passable games and monotone set coloring games.

Provides a new characterization of passable games.

Abstract

The class of passable games was recently introduced by Selinger as a class of combinatorial games that are suitable for modelling monotone set coloring games such as Hex. In a monotone set coloring game, the players alternately color the cells of a board with their respective color, and the winner is determined by a monotone function of the final position. It is easy to see that every monotone set coloring game is a passable combinatorial game. Here we prove the converse: every passable game is realizable, up to equivalence, as a monotone set coloring game.