Dynamic Games with Almost Perfect Information

He, Wei and Sun, Yeneng (2015): Dynamic Games with Almost Perfect Information.

Abstract

This paper aims to solve two fundamental problems on finite or infinite horizon dynamic games with perfect or almost perfect information. Under some mild conditions, we prove (1) the existence of subgame-perfect equilibria in general dynamic games with almost perfect information, and (2) the existence of pure-strategy subgame-perfect equilibria in perfect-information dynamic games with uncertainty. Our results go beyond previous works on continuous dynamic games in the sense that public randomization and the continuity requirement on the state variables are not needed. As an illustrative application, a dynamic stochastic oligopoly market with intertemporally dependent payoffs is considered.

• Item Type: MPRA Paper
• Language: English
• Keywords: Dynamic games; almost perfect information; perfect information; subgame-perfect equilibirum
• Subjects: C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C62 - Existence and Stability Conditions of Equilibrium; C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C73 - Stochastic and Dynamic Games ; Evolutionary Games ; Repeated Games; D - Microeconomics > D0 - General

References

D. Abreu, D. Pearce and E. Stacchetti, Toward a Theory of Discounted Repeated Games with Imperfect Monitoring, Econometrica 58 (1990), 1041-1063.

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis: A Hitchhiker's Guide, Springer, Berlin, 2006.

G. S. Becker and K. M. Murphy, A Theory of Rational Addiction, Journal of Political Economy 96 (1988), 675-700.

V. I. Bogachev, Measure Theory, Volume I, II, Springer-Verlag, Berlin Heidelberg, 2007.

T. Borgers, Perfect Equilibrium Histories of Finite and Infinite Horizon Games, Journal of Economic Theory 47 (1989), 218-227.

T. Borgers, Upper Hemicontinuity of The Correspondence of Subgame-Perfect Equilibrium Outcomes, Journal of Mathematical Economics 20 (1991), 89-106.

L. Brown and B. M. Schreiber, Approximation and Extension of Random Functions, Monatshefte fur Mathematik 107 (1989), 111-123.

C. Castaing, P. R. De Fitte and M. Valadier, Young Measures on Topological Spaces: with Applications in Control Theory and Probability Theory, Vol. 571, Springer Science & Business Media, 2004.

C. Fershtman and M. I. Kamien, Dynamic Duopolistic Competition with Sticky Prices, Econometrica 55 (1987), 1151-1164.

R. Fierro, C. Martinez, and C. H. Morales, Caratheodory Selections for Multivalued Mappings, Nonlinear Analysis Theory, Methods and Applications 64 (2006), 1229-1235.

D. Fudenberg and D. Levine, Subgame-Perfect Equilibria of Finite and Infinite-Horizon Games, Journal of Economic Theory 31 (1983), 251-268.

D. Fudenberg and J. Tirole, Game Theory, MIT Press, Cambridge, 1991.

S. Goldman, Consistent Plans, Review of Economic Studies 47 (1980), 533-537.

E. J. Green and R. H. Porter, Noncooperative Collusion under Imperfect Price Information, Econometrica 52 (1984), 87-100.

C. Harris, Existence and Characterization of Perfect Equilibrium in Games of Perfect Information, Econometrica 53 (1985), 613-628.

C. Harris, The Existence of Subgame-Perfect Equilibrium with and without Markov Strategies: A Case for Extensive-Form Correlation, Discussion Paper No. 53, Nuffield College, Oxford.

C. Harris, P. J. Reny and A. Robson, The Existence of Subgame-Perfect Equilibrium in Continuous Games with Almost Perfect Information: A Case for Public Randomization, Econometrica 63 (1995), 507-544.

M. Hellwig and W. Leininger, On the Existence of Subgame-Perfect Equilibrium in Infinite-Action Games of Perfect Information, Journal of Economic Theory 43 (1987), 55-75.

M. Hellwig, W. Leininger, P. J. Reny and A. J. Robson, Subgame-Perfect Equilibrium in Continuous Games of Perfect Information: An Elementary Approach to Existence and Approximation by Discrete Games, Journal of Economic Theory 52 (1990), 406-422.

C. Hess, Set-Valued Integration and Set-Valued Probability Theory: An Overview, Handbook of Measure Theory, Chapter 14, North Holland, 2002, 617-673.

W. Hildenbrand, Core and Equilibria of a Large Economy, Princeton University Press, Princeton, NJ, 1974.

E. G. J. Luttmer and T. Mariotti, The Existence of Subgame-Perfect Equilibrium in Continuous Games with Almost Perfect Information: A Comment, Econometrica 71 (2003), 1909-1911.

T. Mariotti, Subgame-Perfect Equilibrium Outcomes in Continuous Games of Almost Perfect Information, Journal of Mathematical Economics 34 (2000), 99-128.

J.-F. Mertens, A Measurable `Measurable Choice' Theorem, in Proceedings of the NATO Advanced Study Institute on Stochastic Games and Applications, ed. by A. Neyman and S. Sorin. NATO Science Series. Dordrecht: Kluwer Academic, 2003, 107-130.

J.-F. Mertens and T. Parthasarathy, Equilibria for Discounted Stochastic Games, Research Paper 8750, CORE, 1987.

B. Peleg and M. Yaari, On the Existence of a Consistent Course of Action When Tastes Are Changing, Review of Economic Studies 40 (1973), 391-401.

E. Phelps and R. Pollak, On Second Best National Savings and Game Equilibrium Growth, Review of Economic Studies 35 (1968), 185-199.

P. J. Reny and A. J. Robson, Existence of Subgame-Perfect Equilibrium with Public Randomization: A Short Proof, Economics Bulletin 24 (2002), 1-8.

J. J. Rotemberg and G. Saloner, A Supergame-Theoretic Model of Price Wars during Booms, American Economic Review 76 (1986), 390-407.

H. E. Ryder and G. M. Heal, Optimal Growth with Intertemporally Dependent Preferences, Review of Economic Studies 40 (1973), 1-31.

R. Selten, Spieltheoretische Behandlung Eines Oligopolmodells Mit Nachfragetraheit, Z. Gesamte Staatwissenschaft 12 (1965), 301-324.

L. K. Simon and W. R. Zame, Discontinuous Games and Endogenous Sharing Rules, Econometrica 58 (1990), 861-872.

Y. N. Sun, Integration of Correspondences on Loeb Spaces, Transactions of the American Mathematical Society 349 (1997), 129-153.