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Measure-Theoretic Analysis of Stochastic Competence Sets and Dynamic Shapley Values in Banach Spaces

Author

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  • Jih-Jeng Huang

    (Department of Computer Science & Information Management, Soochow University, No. 56, Section 1, Kueiyang Street, Chungcheng District, Taipei City 100, Taiwan)

  • Chin-Yi Chen

    (Department of Business Administration, Chung Yuan Christian University, No. 200, Zhongbei Rd, Zhongli District, Taoyuan City 320, Taiwan)

Abstract

We develop a measure-theoretic framework for dynamic Shapley values in Banach spaces, extending classical cooperative game theory to continuous-time, infinite-dimensional settings. We prove the existence and uniqueness of strong solutions to stochastic differential equations modeling competence evolution in Banach spaces, establishing sample path continuity and moment estimates. The dynamic Shapley value is rigorously defined as a càdlàg stochastic process with an axiomatic characterization. We derive a martingale representation for this process and establish its asymptotic properties, including a strong law of large numbers and a functional central limit theorem under α-mixing conditions. This framework provides a rigorous basis for analyzing dynamic value attribution in abstract spaces, with potential applications to economic and game-theoretic models.

Suggested Citation

  • Jih-Jeng Huang & Chin-Yi Chen, 2024. "Measure-Theoretic Analysis of Stochastic Competence Sets and Dynamic Shapley Values in Banach Spaces," Mathematics, MDPI, vol. 12(19), pages 1-18, October.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:19:p:3085-:d:1490807
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    References listed on IDEAS

    as
    1. Yu, Po L. & Zhang, Dazhi, 1990. "A foundation for competence set analysis," Mathematical Social Sciences, Elsevier, vol. 20(3), pages 251-299, December.
    2. Pierre Cardaliaguet & Catherine Rainer & Dinah Rosenberg & Nicolas Vieille, 2016. "Markov Games with Frequent Actions and Incomplete Information—The Limit Case," Mathematics of Operations Research, INFORMS, vol. 41(1), pages 49-71, February.
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