Uncertainty Aversion and Equilibrium in Extensive Games

Jörn Rothe

Contributions to game theory and management, Tome 4 (2011), pp. 389-406.

Résumé

This paper formulates a rationality concept for extensive games in which deviations from rational play are interpreted as evidence of irrationality. Instead of confirming some prior belief about the nature of non-rational play, we assume that such a deviation leads to genuine uncertainty. Assuming complete ignorance about the nature of non-rational play and extreme uncertainty aversion of the rational players, we formulate an equilibrium concept on the basis of Choquet expected utility theory. Equilibrium reasoning is thus only applied on the equilibrium path, maximin reasoning applies off the equilibrium path. The equilibrium path itself is endogenously determined. In general this leads to strategy profiles differ qualitatively from sequential equilibria, but still satisfy equilibrium and perfection requirements. In the centipede game and the finitely repeated prisoners' dilemma this approach can also resolve the backward induction paradox.

Détail

Keywords: rationality, extensive game, uncertainty aversion, perfect equilibrium, backward induction, maximin, Choquet expected utility theory.

Bibliographie

[1] Anscombe F. J., Aumann R. J., “A definition of subjective probability”, Annals of Mathematical Statistics, 34 (1963), 199–205 | DOI | MR | Zbl

[2] Arrow K. J., Hurwicz L., “An optimality criterion for decision-making under ignorance”, Carter Ford, 1972, 11 pp.

[3] Aumann R. J., “Backward induction and common knowledge of rationality”, Games and Economic Behavior, 8 (1995), 6–19 | DOI | MR | Zbl

[4] Aumann R. J., “Reply to Binmore”, Games and Economic Behavior, 17 (1996), 138–46 | DOI

[5] Aumann R. J., “On the centipede game”, Games and Economic Behavior, 23 (1998), 97–105 | DOI | MR | Zbl

[6] Binmore K. G., “Modelling rational players. I; II”, Economics and Philosophy, 3 (1987), 179–214 ; 4 (1988), 9–55 | DOI | DOI

[7] Binmore K. G., “A note on backward induction”, Games and Economic Behavior, 17 (1996), 135–137 | DOI

[8] Carter C. F., Ford J. L. (eds.), Uncertainty and Expectations in Economics, Basil Blackwell, Oxford, 1972

[9] Choquet G., “Theory of capacities”, Annales de l'Institut Fourier (Grenoble), 5 (1953), 131–295 | DOI | MR

[10] Cohen M., Gilboa I., Jaffray J.-Y., Schmeidler D., An experimental study of updating ambiguous beliefs, mimeo, 1999

[11] Dempster A. P., “Upper and lower probabilities induced by a multivalued mapping”, Annals of Mathematical Statistics, 38 (1967), 325–339 | DOI | MR | Zbl

[12] Dow J., Werlang S. R. d. C., “Nash equilibrium under Knightian uncertainty: Breaking down backward induction”, Journal of Economic Theory, 64 (1994), 305–324 | DOI | MR | Zbl

[13] Eichberger J., Kelsey D., Non-additive beliefs and game theory, Discussion Paper 9410, Center for Economic Research, 1994

[14] Eichberger J., Kelsey D., Signalling games with uncertainty, mimeo, 1995

[15] Eichberger J., Kelsey D., “Uncertainty aversion and preference for randomisation”, Journal of Economic Theory, 71 (1996), 31–43 | DOI | Zbl

[16] Ellsberg D., “Risk, ambiguity and the savage axioms”, Quarterly Journal of Economics, 75 (1961), 643–669 | DOI

[17] Epstein L. G., “Preference, rationalizability and equilibrium”, Journal of Economic Theory, 73 (1997), 1–29 | DOI | MR | Zbl

[18] Epstein L. G., Uncertainty aversion, mimeo, 1997

[19] Epstein L. G., Breton M. L., “Dynamically consistent beliefs must be bayesian”, Journal of Economic Theory, 61 (1993), 1–22 | DOI | MR | Zbl

[20] Fagin R., Halpern J. Y., A new approach to updating beliefs, mimeo, 1990

[21] Fudenberg D., Tirole J., Game Theory, MIT press, Cambridge, MA, 1991 | MR

[22] Ghirardato P., Marinacci M., Ambiguity made precise: A comparative foundation and some implications, mimeo, University of Toronto, 1997

[23] Gilboa I., “Expected utility with purely subjective non-additive probabilities”, Journal of Mathematical Economics, 16 (1987), 65–88 | DOI | MR | Zbl

[24] Gilboa I., Schmeidler D., “Maxmin expected utility with non-unique prior”, Journal of Mathematical Economics, 18 (1989), 141–153 | DOI | MR | Zbl

[25] Gilboa I., Schmeidler D., “Updating ambiguous beliefs”, Journal of Economic Theory, 59 (1993), 33–49 | DOI | MR | Zbl

[26] Hendon E., Jacobsen H. J., Sloth B., Tranaes T., Nash equilibrium in lower probabilities, mimeo, 1995

[27] Klibanoff P., Uncertainty, decision and normal norm games, mimeo, 1993

[28] Kreps D. M., Wilson R. B., “Reputation and imperfect information”, Journal of Economic Theory, 27 (1982), 253–79 | DOI | MR

[29] Kreps D. M., Wilson R. B., “Sequential equilibria”, Econometrica, 50 (1982), 863–894 | DOI | MR | Zbl

[30] Kreps D. M., Milgrom P., Roberts J., Wilson R. B., “Rational cooperation in the finitely repeated prisoners' dilemma”, Journal of Economic Theory, 27 (1982), 245–252 | DOI | MR | Zbl

[31] Lo K. C., Nash equilibrium without mutual knowledge of rationality, mimeo, University of Toronto, 1995

[32] Lo K. C., Extensive form games with uncertainty averse players, mimeo, University of Toronto, 1995

[33] Marinacci M., Equilibrium in ambiguous games, mimeo, 1994

[34] Milgrom P. R., Roberts J., “Limit pricing and entry under incomplete information: An equilibrium analysis”, Econometrica, 50 (1982), 443–460 | DOI

[35] Milgrom P. R., Roberts J., “Predation, reputation and entry deterrence”, Journal of Economic Theory, 27 (1982), 280–312 | DOI | MR | Zbl

[36] Mukerji S., A theory of play for games in strategic form when rationality is not common knowledge, mimeo, 1994

[37] Reny P. J., “Common belief and the theory of games with perfect information”, Journal of Economic Theory, 59 (1993), 257–74 | DOI | MR

[38] Rosenthal R. W., “Games of perfect information, predatory pricing and the chain store paradox”, Journal of Economic Theory, 25 (1981), 92–100 | DOI | MR | Zbl

[39] Rothe J., Uncertainty aversion and equilibrium, mimeo, 1996

[40] Rothe J., “Uncertainty aversion and equilibrium”, Contributions to game theory and management, II, eds. L. A. Petrosjan, N. A. Zenkevich, 2009, 363–382 | MR

[41] Rothe J., “Uncertainty aversion and equilibrium in normal form games”, Contributions to game theory and management, III, eds. L. A. Petrosjan, N. A. Zenkevich, 2010, 342–367 | MR

[42] Ryan M., A refinement of Dempster–Shafer equilibrium, mimeo, University of Auckland, 1997

[43] Sarin R., Wakker P. P., “A simple axiomatization of nonadditive expected utility”, Econometrica, 60 (1992), 1255–1272 | DOI | MR | Zbl

[44] Savage L. Y., The Foundations of Statistics, Wiley, New York, 1954 ; 2nd edn., Dover, 1972 | MR | Zbl | Zbl

[45] Schmeidler D., “Integral representation without additivity”, Proceedings of the American Mathematical Society, 97 (1986), 255–261 | DOI | MR | Zbl

[46] Schmeidler D., “Subjective probability and expected utility without additivity”, Econometrica, 57 (1989), 571–587 | DOI | MR | Zbl

[47] Selten R., “Re-examination of the perfectness concept for equilibrium points in extensive games”, International Journal of Game Theory, 4 (1975), 25–55 | DOI | MR | Zbl

[48] Shafer G., A Mathematical Theory of Evidence, Princeton University Press, Princeton, NJ, 1976 | MR | Zbl

[49] Shapley L. S., “Cores of convex games”, International Journal of Game Theory, 1 (1971), 11–26 | DOI | MR | Zbl

[50] Smithson M. J., Human judgment and imprecise probabilities, mimeo, Imprecise Probability Project, 1997

[51] Wakker P. P., Additive Representations of Preferences, Kluwer Academic Publishers, Dordrecht, 1989 | MR | Zbl