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Bel-Games: A Formal Theory of Games of Incomplete Information Based on Belief Functions in the Coq Proof Assistant

Authors Pierre Pomeret-Coquot , Hélène Fargier , Érik Martin-Dorel

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Pierre Pomeret-Coquot
  • IRIT, Université de Toulouse, CNRS, Toulouse INP, UT3, Toulouse, France
Hélène Fargier
  • IRIT, CNRS, Toulouse, France
Érik Martin-Dorel
  • IRIT, Université de Toulouse, CNRS, Toulouse INP, UT3, Toulouse, France

Funding

  • Pomeret-Coquot, Pierre: ANITI, funded by the French ”Investing for the Future – PIA3” program under the Grant agreement n°ANR-19-PI3A-0004.

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Pierre Pomeret-Coquot, Hélène Fargier, and Érik Martin-Dorel. Bel-Games: A Formal Theory of Games of Incomplete Information Based on Belief Functions in the Coq Proof Assistant. In 14th International Conference on Interactive Theorem Proving (ITP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 268, pp. 25:1-25:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ITP.2023.25

Abstract

Decision theory and game theory are both interdisciplinary domains that focus on modelling and {analyzing} decision-making processes. On the one hand, decision theory aims to account for the possible behaviors of an agent with respect to an uncertain situation. It thus provides several frameworks to describe the decision-making processes in this context, including that of belief functions. On the other hand, game theory focuses on multi-agent decisions, typically with probabilistic uncertainty (if any), hence the so-called class of Bayesian games. In this paper, we use the Coq/SSReflect proof assistant to formally prove the results we obtained in [Pierre Pomeret{-}Coquot et al., 2022]. First, we formalize a general theory of belief functions with finite support, and structures and solutions concepts from game theory. On top of that, we extend Bayesian games to the theory of belief functions, so that we obtain a more expressive class of games we refer to as Bel games; it makes it possible to better capture human behaviors with respect to lack of information. Next, we provide three different proofs of an extended version of the so-called Howson-Rosenthal’s theorem, showing that Bel games can be casted into games of complete information, i.e., without any uncertainty. We thus embed this class of games into classical game theory, enabling the use of existing algorithms.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic and verification
  • Theory of computation → Type theory
  • Theory of computation → Higher order logic
  • Theory of computation → Algorithmic game theory
  • Theory of computation → Solution concepts in game theory
  • Theory of computation → Representations of games and their complexity
Keywords
  • Game of Incomplete Information
  • Belief Function Theory
  • Coq Proof Assistant
  • SSReflect Proof Language
  • MathComp Library

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