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New Solution based on Hodge Decomposition for Abstract Games

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  • Yihao Luo
  • Jinhui Pang
  • Weibin Han
  • Huafei Sun

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.

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  • Yihao Luo & Jinhui Pang & Weibin Han & Huafei Sun, 2021. "New Solution based on Hodge Decomposition for Abstract Games," Papers 2109.14539, arXiv.org, revised Jul 2024.
    • Handle: RePEc:arx:papers:2109.14539
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    References listed on IDEAS

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    1. Georges Bordes, 1976. "Consistency, Rationality and Collective Choice," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 43(3), pages 451-457.
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    3. Jörg Stoye, 2011. "Statistical decisions under ambiguity," Theory and Decision, Springer, vol. 70(2), pages 129-148, February.
    4. Brandt, Felix, 2011. "Minimal stable sets in tournaments," Journal of Economic Theory, Elsevier, vol. 146(4), pages 1481-1499, July.
    5. Kenneth J. Arrow & Herve Raynaud, 1986. "Social Choice and Multicriterion Decision-Making," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262511754, April.
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