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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

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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)


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
  • Game of Incomplete Information
  • Belief Function Theory
  • Coq Proof Assistant
  • SSReflect Proof Language
  • MathComp Library


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  1. Reynald Affeldt and Cyril Cohen. Measure construction by extension in dependent type theory with application to integration, 2022. URL:
  2. Reynald Affeldt, Jacques Garrigue, and Takafumi Saikawa. A library for formalization of linear error-correcting codes. Journal of Automated Reasoning, 64(6):1123-1164, 2020. URL:
  3. Nahla Ben Amor, Hélène Fargier, Régis Sabbadin, and Meriem Trabelsi. Possibilistic Games with Incomplete Information. In Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence, pages 1544-1550., 2019. Google Scholar
  4. Alexander Bagnall, Samuel Merten, and Gordon Stewart. A Library for Algorithmic Game Theory in SSReflect/Coq. Journal of Formalized Reasoning, 10(1):67-95, 2017. Google Scholar
  5. Sylvie Boldo, François Clément, Florian Faissole, Vincent Martin, and Micaela Mayero. A Coq formalization of Lebesgue integration of nonnegative functions. Journal of Automated Reasoning, 66(2):175-213, 2022. URL:
  6. The Coq Development Team. The Coq Proof Assistant, 2022. URL:
  7. Arthur P. Dempster. Upper and Lower Probabilities Induced by a Multivalued Mapping. The Annals of Mathematical Statistics, 38:325-339, 1967. Google Scholar
  8. Christoph Dittmann. Positional determinacy of parity games. Available at browser_info/devel/AFP/Parity_Game/outline.pdf, 2016.
  9. Didier Dubois and Thierry Denoeux. Conditioning in Dempster-Shafer Theory: Prediction vs. Revision. In Belief Functions: Theory and Applications - Proceedings of the 2nd International Conference on Belief Functions, pages 385-392. Springer, 2012. Google Scholar
  10. Didier Dubois and Henri Prade. Possibility Theory: An Approach to Computerized Processing of Uncertainty. Plenum Press, 1988. Google Scholar
  11. Didier Dubois and Henri Prade. Focusing vs. belief revision: A fundamental distinction when dealing with generic knowledge. In Qualitative and quantitative practical reasoning, pages 96-107. Springer, 1997. Google Scholar
  12. Mnacho Echenim and Nicolas Peltier. The binomial pricing model in finance: A formalization in Isabelle. In CADE, volume 10395 of LNCS, pages 546-562. Springer, 2017. Google Scholar
  13. Hélène Fargier, Érik Martin-Dorel, and Pierre Pomeret-Coquot. Games of incomplete information: A framework based on belief functions. In Symbolic and Quantitative Approaches to Reasoning with Uncertainty - 16th European Conference, volume 12897 of LNCS, pages 328-341. Springer, 2021. URL:
  14. Ruobin Gong and Xiao-Li Meng. Judicious judgment meets unsettling updating: Dilation, sure loss and simpson’s paradox. Statistical Science, 36(2):169-190, 2021. Google Scholar
  15. John C Harsanyi. Games with Incomplete Information Played by “Bayesian” Players, I-III. Part I. The Basic Model. Management Science, 14(3):159-182, 1967. Google Scholar
  16. Johannes Hölzl and Armin Heller. Three chapters of measure theory in Isabelle/HOL. In Interactive Theorem Proving, pages 135-151, Berlin, Heidelberg, 2011. Springer Berlin Heidelberg. Google Scholar
  17. Joseph T Howson Jr and Robert W Rosenthal. Bayesian Equilibria of Finite Two-Person Games with Incomplete Information. Management Science, 21(3):313-315, 1974. Google Scholar
  18. Jean-Yves Jaffray. Linear Utility Theory for Belief Functions. Operations Research Letters, 8(2):107-112, 1989. Google Scholar
  19. Jean-Yves Jaffray. Linear Utility Theory and Belief Functions: a Discussion. In Progress in decision, utility and risk theory, pages 221-229. Springer, 1991. Google Scholar
  20. Cezary Kaliszyk and Julian Parsert. Formal microeconomic foundations and the first welfare theorem. In Proceedings of the 7th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2018, pages 91-101. ACM, 2018. Google Scholar
  21. Christoph Lange, Colin Rowat, and Manfred Kerber. The formare project - formal mathematical reasoning in economics. In MKM/Calculemus/DML, volume 7961 of LNCS, pages 330-334. Springer, 2013. Google Scholar
  22. Rida Laraki, Jérôme Renault, and Sylvain Sorin. Mathematical foundations of game theory. Springer, 2019. Google Scholar
  23. Stéphane Le Roux, Érik Martin-Dorel, and Jan-Georg Smaus. An Existence Theorem of Nash Equilibrium in Coq and Isabelle. In Proceedings Eighth International Symposium on Games, Automata, Logics and Formal Verification, volume 256 of Electronic Proceedings in Theoretical Computer Science, pages 46-60, 2017. URL:
  24. Stéphane Le Roux, Érik Martin-Dorel, and Jan-Georg Smaus. Existence of Nash equilibria in preference priority games proven in Isabelle. In Kurt Gödel Day and Czech Gathering of Logicians, 2021. Google Scholar
  25. Pierre Lescanne and Matthieu Perrinel. "backward" coinduction, Nash equilibrium and the rationality of escalation. Acta Informatica, 49(3):117-137, 2012. Google Scholar
  26. Assia Mahboubi and Enrico Tassi. Mathematical Components, 2022. URL:
  27. Érik Martin-Dorel and Sergei Soloviev. A Formal Study of Boolean Games with Random Formulas as Payoff Functions. In 22nd International Conference on Types for Proofs and Programs, TYPES 2016, volume 97 of Leibniz International Proceedings in Informatics, pages 14:1-14:22, 2016. Google Scholar
  28. Érik Martin-Dorel and Enrico Tassi. SSReflect in Coq 8.10. In The Coq Workshop 2019, Portland State University, OR, USA, 2019. URL:
  29. Oskar Morgenstern and John Von Neumann. Theory of Games and Economic Behavior. Princeton University Press, 1953. Google Scholar
  30. Roger B Myerson. Game Theory. Harvard university press, 2013. Google Scholar
  31. John Nash. Non-Cooperative Games. Annals of Mathematics, pages 286-295, 1951. Google Scholar
  32. Christos H. Papadimitriou and Tim Roughgarden. Computing Correlated Equilibria in Multi-Player Games. Journal of the Association for Computing Machinery, 55(3):1-29, 2008. Google Scholar
  33. Julian Parsert and Cezary Kaliszyk. Towards formal foundations for game theory. In Interactive Theorem Proving - 9th International Conference ITP 2018, volume 10895 of LNCS, pages 495-503. Springer, 2018. Google Scholar
  34. Bernard Planchet. Credibility and Conditioning. Journal of Theoretical Probability, 2(3):289-299, 1989. Google Scholar
  35. Pierre Pomeret-Coquot, Hélène Fargier, and Érik Martin-Dorel. Games of incomplete information: A framework based on belief functions. International Journal of Approximate Reasoning, 151:182-204, 2022. URL:
  36. Stéphane Le Roux. Acyclic preferences and existence of sequential Nash equilibria: A formal and constructive equivalence. In Proc. Theorem Proving in Higher Order Logics, 22nd International Conference, volume 5674 of LNCS, pages 293-309. Springer, 2009. URL:
  37. Glenn Shafer. A Mathematical Theory of Evidence. Princeton University Press, 1976. Google Scholar
  38. Philippe Smets. Jeffrey’s Rule of Conditioning Generalized to Belief Functions. In Uncertainty in artificial intelligence, pages 500-505. Elsevier, 1993. Google Scholar
  39. Philippe Smets and Robert Kennes. The Transferable Belief Model. Artificial Intelligence, 66(2):191-234, 1994. Google Scholar
  40. René Vestergaard. A constructive approach to sequential Nash equilibria. Inormation. Processing Letters, 97(2):46-51, 2006. URL:
  41. Peter Walley. Statistical Reasoning with Imprecise Probabilities. Chapman & Hall, 1991. Google Scholar
  42. Elena Yanovskaya. Equilibrium Points in Polymatrix Games. Lithuanian Mathematical Journal, 8:381-384, 1968. Google Scholar
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