Economic Theory

, Volume 63, Issue 2, pp 407–430 | Cite as

Characterizing existence of equilibrium for large extensive form games: a necessity result

Research Article

Abstract

What is the minimal structure that is needed to perform equilibrium analysis in large extensive form games? To answer this question, this paper provides conditions that are simultaneously necessary and sufficient for the existence of a subgame perfect equilibrium in any well-behaved perfect information game defined on a large game tree. In particular, the set of plays needs to be endowed with a topology satisfying two conditions. (a) Nodes are closed as sets of plays; and (b) the immediate predecessor function is an open map.

Keywords

Backwards induction Subgame perfection Equilibrium existence Large extensive form games Perfect information 

JEL Classification

C72 C62 

References

  1. Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis: A Hitchhiker’s Guide. Springer, Berlin (1999)CrossRefGoogle Scholar
  2. Alós-Ferrer, C., Ritzberger, K.: Equilibrium existence for large perfect information games. J. Math. Econ. 62, 5–18 (2016)Google Scholar
  3. Alós-Ferrer, C., Kern, J., Ritzberger, K.: Comment on ‘Trees and extensive forms’. J. Econ. Theory 146(5), 2165–2168 (2011)CrossRefGoogle Scholar
  4. Alós-Ferrer, C., Ritzberger, K.: Trees and decisions. Econ. Theory 25(4), 763–798 (2005)CrossRefGoogle Scholar
  5. Alós-Ferrer, C., Ritzberger, K.: Trees and extensive forms. J. Econ. Theory 43(1), 216–250 (2008)CrossRefGoogle Scholar
  6. Alós-Ferrer, C., Ritzberger, K.: Large extensive form games. Econ. Theory 52(1), 75–102 (2013)CrossRefGoogle Scholar
  7. Bertrand, J.: Théorie Mathématique de la Richesse Sociale. Journal des Savants 67, 499–508 (1883)Google Scholar
  8. Cournot, A.A.: Recherches Sur les Principes Mathématiques de la Théorie des Richesses. Hachette, Paris (1838)Google Scholar
  9. Fedorchuck, V.: Fully closed mappings and the compatibility of some theorems in general topology with the axioms of set theory. Mat. Sb. 99, 3–33 (1976)Google Scholar
  10. Flesch, J., Kuipers, J., Mashiah-Yaakovi, A., Schoenmakers, G., Solan, E., Vrieze, K.: Perfect-information games with lower-semicontinuous payoffs. Math. Oper. Res. 35(4), 742–755 (2010)CrossRefGoogle Scholar
  11. Fudenberg, D., Levine, D.K.: Subgame-perfect equilibria of finite and infinite horizon games. J. Econ. Theory 31(2), 251–268 (1983)CrossRefGoogle Scholar
  12. Gödel, K.: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I. Monatsh. Math. Phys. 38, 173–198 (1931)CrossRefGoogle Scholar
  13. Harris, C.: A characterization of the perfect equilibria of infinite horizon games. J. Econ. Theory 33, 461–481 (1985a)Google Scholar
  14. Harris, C.: Existence and characterization of perfect equilibrium in games of perfect information. Econometrica 53, 613–628 (1985b)CrossRefGoogle Scholar
  15. Hellwig, M., Leininger, W.: On the existence of subgame-perfect equilibrium in infinite-action games of perfect information. J. Econ. Theory 43, 55–75 (1987)CrossRefGoogle Scholar
  16. Kuhn, H.: Extensive games and the problem of information. In: Kuhn, H., Tucker, A. (eds.) Contributions to the Theory of Games, vol. II. Princeton University Press, Princeton (1953)Google Scholar
  17. Luttmer, E.G.J., Mariotti, T.: The existence of subgame-perfect equilibrium in continuous games with almost perfect information: a comment. Econometrica 71, 1909–1911 (2003)CrossRefGoogle Scholar
  18. Osborne, M.J., Rubinstein, A.: A Course in Game Theory. MIT Press, Cambridge (1994)Google Scholar
  19. Ostaszewski, A.J.: On countably compact, perfectly normal spaces. J. Lond. Math. Soc. 2, 505–516 (1976)CrossRefGoogle Scholar
  20. Purves, R.A., Sudderth, W.D.: Perfect-information games with upper-semicontinuous payoffs. Math. Oper. Res. 36(3), 468–473 (2011)CrossRefGoogle Scholar
  21. Ritzberger, K.: Foundations of Non-cooperative Game Theory. Oxford University Press, Oxford (2001)Google Scholar
  22. Rubinstein, A.: Perfect equilibrium in a bargaining model. Econometrica 50, 97–109 (1982)CrossRefGoogle Scholar
  23. Selten, R.: Spieltheoretische Behandlung eines Oligopolmodells mit Nachfrageträgheit. Z. Gesamte Staatswiss. 121, 301–324, 667–689 (1965)Google Scholar
  24. Shapley, L.: Stochastic games. Proc. Natl. Acad. Sci. USA 39, 1095–1100 (1953)CrossRefGoogle Scholar
  25. Solan, E., Vieille, N.: Deterministic multi-player Dynkin games. J. Math. Econ. 1097, 1–19 (2003)Google Scholar
  26. Steen, L.A., Seebach Jr, J.A.: Counterexamples in Topology, 2nd edn. Springer, Berlin (1978)CrossRefGoogle Scholar
  27. von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton (1944)Google Scholar
  28. von Stackelberg, H.: Marktform und Gleichgewicht. Springer, Heidelberg (1934)Google Scholar
  29. Weiss, W.: Countably compact spaces and Martin’s axiom. Can. J. Math. 30, 243–249 (1978)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of EconomicsUniversity of CologneCologneGermany
  2. 2.Vienna Graduate School of Finance and Institute for Advanced StudiesViennaAustria

Personalised recommendations