Quantal response equilibrium

Quantal response equilibrium
	A solution concept in game theory
Relationship
Superset of	Nash equilibrium, Logit equilibrium
Significance
Proposed by	Richard McKelvey and Thomas Palfrey
Used for	Non-cooperative games
Example	Traveler's dilemma

Quantal response equilibrium (QRE) is a solution concept in game theory. First introduced by Richard McKelvey and Thomas Palfrey,^[1]^[2] it provides an equilibrium notion with bounded rationality. QRE is not an equilibrium refinement, and it can give significantly different results from Nash equilibrium. QRE is only defined for games with discrete strategies, although there are continuous-strategy analogues.

In a quantal response equilibrium, players are assumed to make errors in choosing which pure strategy to play. The probability of any particular strategy being chosen is positively related to the payoff from that strategy. In other words, very costly errors are unlikely.

The equilibrium arises from the realization of beliefs. A player's payoffs are computed based on beliefs about other players' probability distribution over strategies. In equilibrium, a player's beliefs are correct.

Application to data

When analyzing data from the play of actual games, particularly from laboratory experiments, particularly from experiments with the matching pennies game, Nash equilibrium can be unforgiving. Any non-equilibrium move can appear equally "wrong", but realistically should not be used to reject a theory. QRE allows every strategy to be played with non-zero probability, and so any data is possible (though not necessarily reasonable).

Logit equilibrium

The most common specification for QRE is logit equilibrium (LQRE). In a logit equilibrium, player's strategies are chosen according to the probability distribution:

$P_{{ij}}={\frac {\exp(\lambda EU_{{ij}}(P_{{-i}}))}{\sum _{k}{\exp(\lambda EU_{{ik}}(P_{{-i}}))}}}$

$P_{ij}$ is the probability of player $i$ choosing strategy $j$ . $EU_{{ij}}(P_{{-i}}))$ is the expected utility to player $i$ of choosing strategy $j$ under the belief that other players are playing according to the probability distribution $P_{{-i}}$ .

Of particular interest in the logit model is the non-negative parameter λ (sometimes written as 1/μ). λ can be thought of as the rationality parameter. As λ→0, players become "completely non-rational", and play each strategy with equal probability. As λ→∞, players become "perfectly rational", and play approaches a Nash equilibrium.

For dynamic games

For dynamic (extensive form) games, McKelvey and Palfrey defined agent quantal response equilibrium (AQRE). AQRE is somewhat analogous to subgame perfection. In an AQRE, each player plays with some error as in QRE. At a given decision node, the player determines the expected payoff of each action by treating their future self as an independent player with a known probability distribution over actions. As in QRE, in an AQRE every strategy is used with nonzero probability.

Critiques

Non-falsifiability

Work by Haile et al. has shown that QRE is not falsifiable in any normal form game, even with significant a priori restrictions on payoff perturbations.^[3] The authors argue that the LQRE concept can sometimes restrict the set of possible outcomes from a game, but may be insufficient to provide a powerful test of behavior without a priori restrictions on payoff perturbations.

References

↑ McKelvey, Richard; Palfrey, Thomas (1995). "Quantal Response Equilibria for Normal Form Games". Games and Economic Behavior. 10: 6–38. doi:10.1006/game.1995.1023.
↑ McKelvey, Richard; Palfrey, Thomas (1998). "Quantal Response Equilibria for Extensive Form Games". Experimental Economics. 1: 9–41. doi:10.1007/BF01426213.
↑ Haile, Philip A.; Hortaçsu, Ali; Kosenok, Grigory (2008). "On the Empirical Content of Quantal Response Equilibrium". American Economic Review. 98 (1): 180–200. doi:10.1257/aer.98.1.180.

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[MKP1-1] McKelvey, Richard; Palfrey, Thomas (1995). "Quantal Response Equilibria for Normal Form Games". Games and Economic Behavior. 10: 6–38. doi:10.1006/game.1995.1023.

[MKP2-2] McKelvey, Richard; Palfrey, Thomas (1998). "Quantal Response Equilibria for Extensive Form Games". Experimental Economics. 1: 9–41. doi:10.1007/BF01426213.

[Haile-3] Haile, Philip A.; Hortaçsu, Ali; Kosenok, Grigory (2008). "On the Empirical Content of Quantal Response Equilibrium". American Economic Review. 98 (1): 180–200. doi:10.1257/aer.98.1.180.

Topics in game theory
Definitions	Cooperative game Determinacy Escalation of commitment Extensive-form game First-player and second-player win Game complexity Graphical game Hierarchy of beliefs Information set Normal-form game Preference Sequential game Simultaneous game Simultaneous action selection Solved game Succinct game
Equilibrium concepts	Nash equilibrium Subgame perfection Mertens-stable equilibrium Bayesian Nash equilibrium Perfect Bayesian equilibrium Trembling hand Proper equilibrium Epsilon-equilibrium Correlated equilibrium Sequential equilibrium Quasi-perfect equilibrium Evolutionarily stable strategy Risk dominance Core Shapley value Pareto efficiency Gibbs equilibrium Quantal response equilibrium Self-confirming equilibrium Strong Nash equilibrium Markov perfect equilibrium
Strategies	Dominant strategies Pure strategy Mixed strategy Strategy-stealing argument Tit for tat Grim trigger Collusion Backward induction Forward induction Markov strategy
Classes of games	Symmetric game Perfect information Repeated game Signaling game Screening game Cheap talk Zero-sum game Mechanism design Bargaining problem Stochastic game n-player game Large Poisson game Nontransitive game Global game Strictly determined game Potential game
Games	Chess Infinite chess Checkers Tic-tac-toe Prisoner's dilemma Optional prisoner's dilemma Traveler's dilemma Coordination game Chicken Centipede game Volunteer's dilemma Dollar auction Battle of the sexes Stag hunt Matching pennies Ultimatum game Rock–paper–scissors Pirate game Dictator game Public goods game Blotto game War of attrition El Farol Bar problem Fair division Fair cake-cutting Cournot game Deadlock Diner's dilemma Guess 2/3 of the average Kuhn poker Nash bargaining game Prisoners and hats puzzle Trust game Princess and Monster game Rendezvous problem
Theorems	Arrow's impossibility theorem Aumann's agreement theorem Folk theorem Minimax theorem Nash's theorem Purification theorem Revelation principle Zermelo's theorem
Key figures	Albert W. Tucker Amos Tversky Ariel Rubinstein Claude Shannon Daniel Kahneman David K. Levine David M. Kreps Donald B. Gillies Drew Fudenberg Eric Maskin Harold W. Kuhn Herbert Simon Hervé Moulin Jean Tirole Jean-François Mertens Jennifer Tour Chayes John Harsanyi John Maynard Smith Antoine Augustin Cournot John Nash John von Neumann Kenneth Arrow Kenneth Binmore Leonid Hurwicz Lloyd Shapley Melvin Dresher Merrill M. Flood Olga Bondareva Oskar Morgenstern Paul Milgrom Peyton Young Reinhard Selten Robert Axelrod Robert Aumann Robert B. Wilson Roger Myerson Samuel Bowles Suzanne Scotchmer Thomas Schelling William Vickrey
See also	All-pay auction Alpha–beta pruning Bertrand paradox Bounded rationality Combinatorial game theory Confrontation analysis Coopetition First-move advantage in chess Game mechanics Glossary of game theory List of game theorists List of games in game theory No-win situation Solving chess Topological game Tragedy of the commons Tyranny of small decisions