New Solution based on Hodge Decomposition for Abstract Games

TL;DR

This paper introduces Hodge Potential Choice (HPC), a novel solution for abstract games using geometric tools, which improves upon traditional methods and is validated through extensive digital experiments.

Contribution

The paper develops HPC, a new geometric solution for abstract games that extends existing methods and demonstrates superior statistical performance.

Findings

HPC coincides with Copeland Choice in complete games.

HPC overcomes weaknesses of conventional solutions.

Experimental results show HPC's statistical advantage.

Abstract

This paper proposes Hodge Potential Choice (HPC), a new solution for abstract games with irreflexive dominance relations. This solution is formulated by involving geometric tools like differential forms and Hodge decomposition onto abstract games. We provide a workable algorithm for the proposed solution with a new data structure of abstract games. From the view of gaming, HPC overcomes several weaknesses of conventional solutions. HPC coincides with Copeland Choice in complete cases and can be extended to slove games with marginal strengths. It will be proven that the Hodge potential choice possesses three prevalent axiomatic properties: neutrality, strong monotonicity, dominance cycle s reversing independence, and sensitivity to mutual dominance. To compare the HPC with Copeland Choice in large samples of games, we design digital experiments with randomly generated abstract games with different sizes and completeness. The experimental results present the advantage of HPC in the statistical sense.