Acyclic Preferences and Existence of Sequential Nash Equilibria: A Formal and Constructive Equivalence

  • Stéphane Le Roux
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5674)

Abstract

In a game from game theory, a Nash equilibrium (NE) is a combination of one strategy per agent such that no agent can increase its payoff by unilaterally changing its strategy. Kuhn proved that all (tree-like) sequential games have NE. Osborne and Rubinstein abstracted over these games and Kuhn’s result: they proved a sufficient condition on agents’ preferences for all games to have NE. This paper proves a necessary and sufficient condition, thus accounting for the game-theoretic frameworks that were left aside. The proof is formalised using Coq, and contrary to usual game theory it adopts an inductive approach to trees for definitions and proofs. By rephrasing a few game-theoretic concepts, by ignoring useless ones, and by characterising the proof-theoretic strength of Kuhn’s/Osborne and Rubinstein’s development, this paper also clarifies sequential game theory. The introduction sketches these clarifications, while the rest of the paper details the formalisation.

Keywords

Coq induction sequential game theory abstraction effective generalisation 

References

  1. 1.
    Anonymous. Program evaluation research task. Summary report Phase 1 and 2, U.S. Government Printing Office, Washington, D.C. (1958)Google Scholar
  2. 2.
    Berthot, Y., Castéran, P.: Interactive Theorem Proving and Program Development Coq’Art: The Calculus of Inductive Constructions. Springer, Heidelberg (2004)CrossRefMATHGoogle Scholar
  3. 3.
    Blackwell, D.: An analog of the minimax theorem for vector payoffs. Pacific Journal of Mathematics 6, 1–8 (1956)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Blanqui, F., Coupet-Grimal, S., Delobel, W., Hinderer, S., Koprowski, A.: CoLoR, a Coq Library on rewriting and termination. In: Workshop on Termination (2006)Google Scholar
  5. 5.
    Kahn, A.B.: Topological sorting of large networks. Commun. ACM 5(11), 558–562 (1962)CrossRefMATHGoogle Scholar
  6. 6.
    Knuth, D.E.: The Art of Computer Programming, 2nd edn., vol. 1. Addison Wesley, Reading (1973)MATHGoogle Scholar
  7. 7.
    Kreps, D.M.: Notes on the Theory of Choice. Westview Press, Inc., Boulder (1988)Google Scholar
  8. 8.
    Krieger, T.: On pareto equilibria in vector-valued extensive form games. Mathematical Methods of Operations Research 58, 449–458 (2003)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Kuhn, H.W.: Extensive games and the problem of information. Contributions to the Theory of Games II (1953)Google Scholar
  10. 10.
    Lasser, D.J.: Topological ordering of a list of randomly-numbered elements of a network. Commun. ACM 4(4), 167–168 (1961)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Le Roux, S.: Non-determinism and Nash equilibria for sequential game over partial order. In: Computational Logic and Applications, CLA 2005. Discrete Mathematics & Theoretical Computer Science (2006)Google Scholar
  12. 12.
    Le Roux, S.: Acyclicity and finite linear extendability: a formal and constructive equivalence. In: Schneider, K., Brandt, J. (eds.) Theorem Proving in Higher Order Logics: Emerging Trends Proceedings, September 2007, pp. 154–169. Department of Computer Science, University of Kaiserslautern (2007)Google Scholar
  13. 13.
    Le Roux, S.: Generalisation and formalisation in game theory. Ph.d. thesis, Ecole Normale Supérieure de Lyon (January 2008)Google Scholar
  14. 14.
    Le Roux, S., Lescanne, P., Vestergaard, R.: A discrete Nash theorem with quadratic complexity and dynamic equilibria. Research report IS-RR-2006-006, JAIST (2006)Google Scholar
  15. 15.
    Jarnagin, M.P.: Automatic machine methods of testing pert networks for consistency. Technical Memorandum K-24/60, U. S. Naval Weapons Laboratory, Dahlgren, Va (1960)Google Scholar
  16. 16.
    Osborne, M.J., Rubinstein, A.: A Course in Game Theory. The MIT Press, Cambridge (1994)MATHGoogle Scholar
  17. 17.
    Pratt, V.: Origins of the calculus of binary relations. In: Logic in Computer Science (1992)Google Scholar
  18. 18.
    Selten, R.: Spieltheoretische Behandlung eines Oligopolmodells mit Nachfrageträgheit. Zeitschrift für die desamte Staatswissenschaft 121 (1965)Google Scholar
  19. 19.
    Simon, H.A.: A behavioral model of rational choice. The Quarterly Journal of Economics 69(1), 99–118 (1955)CrossRefGoogle Scholar
  20. 20.
    Szpilrajn, E.: Sur l’extension de l’ordre partiel. Fund. Math. 16 (1930)Google Scholar
  21. 21.
    Vestergaard, R.: A constructive approach to sequential Nash equilibria. Information Processing Letter 97, 46–51 (2006)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Zermelo, E.: Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels. In: Proceedings of the Fifth International Congress of Mathematicians, vol. 2 (1912)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Stéphane Le Roux
    • 1
  1. 1.LIX, École Polytechnique, CEA, CNRS, INRIAFrance

Personalised recommendations