  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.
