Game Values of Strong Placement Games

TL;DR

This paper explores the range of game values in strong placement games like Domineering by analyzing their legal complexes, revealing that all numbers and nimbers can occur as game values.

Contribution

It introduces a novel approach using legal complexes to study game values in strong placement games and characterizes the possible values.

Findings

All numbers and nimbers can be realized as game values.

Legal complex structures influence which game values are achievable.

The study links combinatorial structures to game-theoretic outcomes.

Abstract

The legal positions of a strong placement game, such as Domineering, form a simplicial complex called the legal complex. In this paper, we use the legal complex to study the game values taken on by the class of strong placement games using the legal complex. We show that many interesting values are possible, including all numbers and nimbers. We further consider how structures of the legal complex influence which values are possible.