A Complete Classification of Ideal Chomp Games on Low-Rank Algebras

TL;DR

This paper fully classifies winning strategies in the Ideal Chomp Game on low-rank a0a0-algebras, identifying all cases where the first player has a winning strategy for ranks up to 6.

Contribution

It provides a complete classification of winning strategies for the Ideal Chomp Game on a0a0-algebras of rank at most 6, combining game theory, algebraic structure, and computational methods.

Findings

Player A has a winning strategy on all a0a0-algebras up to rank 6 except five specific cases.

The classification uses a combination of theoretical analysis and computational verification.

The paper discusses open questions for higher-dimensional a0a0-algebras.

Abstract

We completely classify winning strategies in the Ideal Chomp Game played on $\bar{K}$-algebras R of rank at most 6. In this two-player combinatorial game, players alternately add generators to build an ideal inside a given ring R, with the player who builds an ideal equal to the entire ring losing. We prove that player A has a winning strategy on all $\bar{K}$-algebras R up to rank 6 except for five specific cases: $\bar{K}$ itself, $\bar{K}[x, y]/(x, y)^2$, and three other local algebras. Our methods combine game-theoretic analysis with the structure theory of Artinian rings and computational verification. We also discuss a classical result of Henson on winning strategies in the Ideal Chomp Game, as well as ideas and open questions about the Ideal Chomp Game on higher-dimensional $\bar{K}$-algebras.