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Regular Quantal Response Equilibrium

Published online by Cambridge University Press:  14 March 2025

Jacob K. Goeree ,
Charles A. Holt  and
Thomas R. Palfrey
Jacob K. Goeree
Affiliation:
Division of the Humanities and Social Sciences, California Institute of Technology, Mail code 228-77, Pasadena, CA 91125, USA
Charles A. Holt
Affiliation:
Department of Economics, University of Virginia, Charlottesville, VA 22904-4182, USA
Thomas R. Palfrey
Affiliation:
Department of Economics, Princeton University, 112 Fisher Hall, Princeton, NJ 08540, USA
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Abstract

The structural Quantal Response Equilibrium (QRE) generalizes the Nash equilibrium by augmenting payoffs with random elements that are not removed in some limit. This approach has been widely used both as a theoretical framework to study comparative statics of games and as an econometric framework to analyze experimental and field data. The framework of structural QRE is flexible: it can be applied to arbitrary finite games and incorporate very general error structures. Restrictions on the error structure are needed, however, to place testable restrictions on the data (Haile et al., 2004). This paper proposes a reduced-form approach, based on quantal response functions that replace the best-response functions underlying the Nash equilibrium. We define a regular QRE as a fixed point of quantal response functions that satisfies four axioms: continuity, interiority, responsiveness, and monotonicity. We show that these conditions are not vacuous and demonstrate with an example that they imply economically sensible restrictions on data consistent with laboratory observations. The reduced-form approach allows for a richer set of regular quantal response functions, which has proven useful for estimation purposes.

Keywords

quantal response equilibriumdiscrete choice modelsreduced-form approach

JEL classification

D62: Externalities C73: Stochastic and Dynamic Games • Evolutionary Games • Repeated Games

Information

Type
Research Article
Information
Experimental Economics , Volume 8 , Issue 4: Exploring the Error in Experimental Economics , December 2005 , pp. 347 - 367
Copyright
Copyright © 2005 Economic Science Association

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References

Anderson, S.P., Jacob, K.G., and Charles, A.H. (1998). “Rent Seeking with Bounded Rationality: An Analysis of the All-Pay Auction.” Journal of Political Economy. 106(4), 828–853.10.1086/250031CrossRefGoogle Scholar
Anderson, S.P., Jacob, K.G., and Charles, A.H. (2001). “Minimum-Effort Coordination Games: Stochastic Potential and Logit Equilibrium.” Games and Economic Behavior. 34(2), 177–199.10.1006/game.2000.0800CrossRefGoogle Scholar
Anderson, S.P., Palma, Andre de, and Thisse, J.-F. (1992). Discrete Choice Theory of Product Differentiation. Cambridge, MA: MIT Press.10.7551/mitpress/2450.001.0001CrossRefGoogle Scholar
Camerer, C.F.H. T.-H., and Chong, J.-K. (2002). “Sophisticated Learning and Strategic Teaching.” Journal of Economic Theory. 104, 137–188.10.1006/jeth.2002.2927CrossRefGoogle Scholar
Capra, C.M., Jacob, K.G., Rosario, G., and Charles, A.H. (1999). “Anomalous Behavior in a Traveler's Dilemma.” American Economic Review. 89(3): 678–690.10.1257/aer.89.3.678CrossRefGoogle Scholar
Chong, J.-K., Colin, F.C., and H., T.-H. (2005). “A Learning-based Model of Repeated Games with Incomplete Information.” Games and Economic Behavior. forthcoming.Google Scholar
Dvoretsky, A., Abraham, W., and Jacob, W. (1951). “Elimination of Randomization in Certain Statistical Decision Problems and Zero-Sum Two-person Games.” Annals of Mathematical Statistics. 22, 1–21.10.1214/aoms/1177729689CrossRefGoogle Scholar
Goeree, J.K. and Charles, A.H. (2001). “Ten Little Treasures of Game Theory and Ten Intuitive Contradictions.” American Economic Review. 91(5), 1402–1422.CrossRefGoogle Scholar
Goeree, J.K. and Charles, A.H. (forthcoming) “An Explanation of Anomalous Behavior in Models of Political Participation.” American Political Science Review.Google Scholar
Goeree, J.K., Charles, A.H., and Thomas, R.P. (2002). “Quantal Response Equilibrium and Overbidding in First- Price Auctions.” Journal of Economic Theory. 104, 247–272.10.1006/jeth.2001.2914CrossRefGoogle Scholar
Goeree, J.K., Charles, A.H., and Thomas, R.P. (2003). “Risk Averse Behavior in Generalized Matching Pennies Games.” Games and Economic Behavior. 45, 97–113.10.1016/S0899-8256(03)00052-6CrossRefGoogle Scholar
Goeree, J.K., Thomas, R.P., and Brian, W.R. (2005). “Rank Dependent Choice in Games.” Working paper, California Institute of Technology.Google Scholar
Haile, P.A., Ali, H., and Grigory, K. (2004). “On the Empirical Content of Quantal Response Models.” Working paper, University of Wisconsin.Google Scholar
Harless, D.W. and Colin, F.C. (1994). “The Predictive Utility of Generalized Expected Utility Theories.” Econometrica. 62:6 (November), 1251–1289.CrossRefGoogle Scholar
Harsanyi, J.A. (1973). “Games with Randomly Distributed Payoffs: A New Rationale for Mixed-Strategy Equilibrium Points.” International Journal of Game Theory. 2, 1–23.10.1007/BF01737554CrossRefGoogle Scholar
Hey, J.D. and Chris, O. (1994). “Investigating Generalization of Expected Utility Theory Using Experimental Data.” Econometrica. 62:6 (November), 1291–1326.10.2307/2951750CrossRefGoogle Scholar
Hey, J.D. (2005). “Why We Should Not Be Silent About Noise.” Working paper, Universities of York and Bari.10.1007/s10683-005-5373-8CrossRefGoogle Scholar
Holt, C.A. (2004). “Markets, Games, and Strategic Behavior.” unpublished manuscript: http://www.people.virginia.edu/~cah2k/expbooknsf.pdf.Google Scholar
Karlin, S. (1966). A First Course in Stochastic Processes. Academic Press: New York.Google Scholar
Laury, S.K. and Charles, A.H. (2002). “Risk Aversion and Incentive Effects.” American Economic Review. December, 93(5).Google Scholar
Loomes, G. (2005). “Modelling the Stochastic Component of Behaviour: Some Implications and Problems for Testing Models.” Working paper, Unievrsity of East Anglia.Google Scholar
Luce, D. (1959). Individual Choice Behavior. New York: Wiley.Google Scholar
McFadden, D. (1981). “Econometric Models of Probabilistic Choice.” in Structural Analysis of Discrete Data with Econometric Applications. Manski, C. and McFadden, D., eds., MIT Press, Cambridge, 198–272.Google Scholar
McKelvey, R.D. and Thomas, R.P. (1992). “An Experimental Study of the Centipede Game.” Econometrica. 60(4), 803–836.10.2307/2951567CrossRefGoogle Scholar
McKelvey, R.D. and Thomas, R.P. (1994). “Quantal Response Equilibrium for Normal Form Games.” Social Science Working Paper # 883. California Institute of Technology, March.Google Scholar
McKelvey, R.D. and Thomas, R.P. (1995). “Quantal Response Equilibria for Normal Form Games.” Games and Economic Behavior. 10(1), 6–38.10.1006/game.1995.1023CrossRefGoogle Scholar
McKelvey, R.D. and Thomas, R.P. (1996). “A Statistical Theory of Equilibrium in Games.” Japanese Economic Review. 47(2): pp. 186–209.CrossRefGoogle Scholar
McKelvey, R.D. and Thomas, R.P. (1998). “Quantal Response Equilibria for Extensive Form Games.” Experimental Economics. 1(1), 9–41.10.1023/A:1009905800005CrossRefGoogle Scholar
McKelvey, R.D., Thomas, R.P., and Roberto, W. (2000). “The Effects of Payoff Magnitude and Heterogeneity on Behavior in 2 × 2 Games with a Unique Mixed-Strategy Equilibrium.” Journal of Economic Behavior and Organization. 42, 523–548.10.1016/S0167-2681(00)00102-5CrossRefGoogle Scholar
Myerson, R.B. (1978). “Refinements of the Nash Equilibrium Concept.” International Journal of Game Theory. 7, 73–80.CrossRefGoogle Scholar
Ochs, J. (1995). “Games with a Unique Mixed-Strategy Equilibrium: An Experimental Study.” Games and Economic Behavior. 10, 202–217.10.1006/game.1995.1030CrossRefGoogle Scholar
Rosenthal, R.W. (1989). “A Bounded Rationality Approach to the Study of Non-Cooperative Games.” International Journal of Game Theory. 2, 65–67.10.1007/BF01737559CrossRefGoogle Scholar
Selten, R. (1965). “Spieltheoretische Behandlung eines Oligopolmodells mit Nachfrage-tragheit.” Zeitschrift für die Gesamte Staatswissenschaft. 121, 301–324, 667-689.Google Scholar
Selten, R. (1975). “Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games.” International Journal of Game Theory. 4, 25–55.10.1007/BF01766400CrossRefGoogle Scholar
von Neumann, J. (1928). “Zur Theorie der Gesellschaftsspiele.” Mathematische Annalen. 100, 295–320.10.1007/BF01448847CrossRefGoogle Scholar
von Neumann, J. and Oskar, M. (1944). Theory of Games and Economic Behavior. Princeton: Princeton University Press.Google Scholar
Wald, A. (1945). “Statistical Decision Fucntions Which Minimize the Maximum Risk.” Annals of Mathematics. 46, 265–80.10.2307/1969022CrossRefGoogle Scholar