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
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Anonymous. Program evaluation research task. Summary report Phase 1 and 2, U.S. Government Printing Office, Washington, D.C. (1958)
Berthot, Y., Castéran, P.: Interactive Theorem Proving and Program Development Coq’Art: The Calculus of Inductive Constructions. Springer, Heidelberg (2004)
Blackwell, D.: An analog of the minimax theorem for vector payoffs. Pacific Journal of Mathematics 6, 1–8 (1956)
Blanqui, F., Coupet-Grimal, S., Delobel, W., Hinderer, S., Koprowski, A.: CoLoR, a Coq Library on rewriting and termination. In: Workshop on Termination (2006)
Kahn, A.B.: Topological sorting of large networks. Commun. ACM 5(11), 558–562 (1962)
Knuth, D.E.: The Art of Computer Programming, 2nd edn., vol. 1. Addison Wesley, Reading (1973)
Kreps, D.M.: Notes on the Theory of Choice. Westview Press, Inc., Boulder (1988)
Krieger, T.: On pareto equilibria in vector-valued extensive form games. Mathematical Methods of Operations Research 58, 449–458 (2003)
Kuhn, H.W.: Extensive games and the problem of information. Contributions to the Theory of Games II (1953)
Lasser, D.J.: Topological ordering of a list of randomly-numbered elements of a network. Commun. ACM 4(4), 167–168 (1961)
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)
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)
Le Roux, S.: Generalisation and formalisation in game theory. Ph.d. thesis, Ecole Normale Supérieure de Lyon (January 2008)
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)
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)
Osborne, M.J., Rubinstein, A.: A Course in Game Theory. The MIT Press, Cambridge (1994)
Pratt, V.: Origins of the calculus of binary relations. In: Logic in Computer Science (1992)
Selten, R.: Spieltheoretische Behandlung eines Oligopolmodells mit Nachfrageträgheit. Zeitschrift für die desamte Staatswissenschaft 121 (1965)
Simon, H.A.: A behavioral model of rational choice. The Quarterly Journal of Economics 69(1), 99–118 (1955)
Szpilrajn, E.: Sur l’extension de l’ordre partiel. Fund. Math. 16 (1930)
Vestergaard, R.: A constructive approach to sequential Nash equilibria. Information Processing Letter 97, 46–51 (2006)
Zermelo, E.: Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels. In: Proceedings of the Fifth International Congress of Mathematicians, vol. 2 (1912)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Le Roux, S. (2009). Acyclic Preferences and Existence of Sequential Nash Equilibria: A Formal and Constructive Equivalence. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds) Theorem Proving in Higher Order Logics. TPHOLs 2009. Lecture Notes in Computer Science, vol 5674. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03359-9_21
Download citation
DOI: https://doi.org/10.1007/978-3-642-03359-9_21
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03358-2
Online ISBN: 978-3-642-03359-9
eBook Packages: Computer ScienceComputer Science (R0)