Proper equilibrium

Proper equilibrium
A solution concept in game theory
Relationships
Subset of	Trembling hand perfect equilibrium
Significance
Proposed by	Roger B. Myerson

Proper equilibrium is a refinement of Nash Equilibrium due to Roger B. Myerson. Proper equilibrium further refines Reinhard Selten's notion of a trembling hand perfect equilibrium by assuming that more costly trembles are made with significantly smaller probability than less costly ones.

Definition

Given a normal form game and a parameter $\epsilon > 0$ , a totally mixed strategy profile $\sigma$ is defined to be $\epsilon$ -proper if, whenever a player has two pure strategies s and s' such that the expected payoff of playing s is smaller than the expected payoff of playing s' (that is $u(s,\sigma_{-i})<u(s',\sigma_{-i})$ ), then the probability assigned to s is at most $\epsilon$ times the probability assigned to s'.

A strategy profile of the game is then said to be a proper equilibrium if it is a limit point, as $\epsilon$ approaches 0, of a sequence of $\epsilon$ -proper strategy profiles.

Example

The game to the right is a variant of Matching Pennies.

*Matching Pennies with a twist*
	Guess heads up	Guess tails up	Grab penny
Hide heads up	-1, 1	0, 0	-1, 1
Hide tails up	0, 0	-1, 1	-1, 1

Player 1 (row player) hides a penny and if Player 2 (column player) guesses correctly whether it is heads up or tails up, he gets the penny. In this variant, Player 2 has a third option: Grabbing the penny without guessing. The Nash equilibria of the game are the strategy profiles where Player 2 grabs the penny with probability 1. Any mixed strategy of Player 1 is in (Nash) equilibrium with this pure strategy of Player 2. Any such pair is even trembling hand perfect. Intuitively, since Player 1 expects Player 2 to grab the penny, he is not concerned about leaving Player 2 uncertain about whether it is heads up or tails up. However, it can be seen that the unique proper equilibrium of this game is the one where Player 1 hides the penny heads up with probability 1/2 and tails up with probability 1/2 (and Player 2 grabs the penny). This unique proper equilibrium can be motivated intuitively as follows: Player 1 fully expects Player 2 to grab the penny. However, Player 1 still prepares for the unlikely event that Player 2 does not grab the penny and instead for some reason decides to make a guess. Player 1 prepares for this event by making sure that Player 2 has no information about whether the penny is heads up or tails up, exactly as in the original Matching Pennies game.

Proper equilibria of extensive games

One may apply the properness notion to extensive form games in two different ways, completely analogous to the two different ways trembling hand perfection is applied to extensive games. This leads to the notions of normal form proper equilibrium and extensive form proper equilibrium of an extensive form game. It was shown by van Damme that a normal form proper equilibrium of an extensive form game is behaviorally equivalent to a quasi-perfect equilibrium of that game.

Roger B. Myerson. Refinements of the Nash equilibrium concept. International Journal of Game Theory, 15:133-154, 1978.
Eric van Damme. "A relationship between perfect equilibria in extensive form games and proper equilibria in normal form games." International Journal of Game Theory 13:1--13, 1984.

Topics in game theory

Definitions	Normal-form game Extensive-form game Escalation of commitment Graphical game Cooperative game Succinct game Information set Hierarchy of beliefs Preference

Equilibrium concepts	Nash equilibrium Subgame perfection Mertens-stable equilibrium Bayesian-Nash Perfect Bayesian Trembling hand Proper equilibrium Epsilon-equilibrium Correlated equilibrium Sequential equilibrium Quasi-perfect equilibrium Evolutionarily stable strategy Risk dominance Core Shapley value Pareto efficiency Quantal response equilibrium Self-confirming equilibrium Strong Nash equilibrium Markov perfect equilibrium

Strategies	Dominant strategies Pure strategy Mixed strategy Tit for tat Grim trigger Collusion Backward induction Forward induction Markov strategy

Classes of games	Symmetric game Perfect information Simultaneous game Sequential game Repeated game Signaling game Cheap talk Zero-sum game Mechanism design Bargaining problem Stochastic game n-player game Large Poisson game Nontransitive game Global games Strictly determined game

Games	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 games 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 Screening game Prisoners and hats puzzle Trust game Princess and monster game Monty Hall problem Rendezvous problem

Theorems	Minimax theorem Nash's theorem Purification theorem Folk theorem Revelation principle Arrow's impossibility theorem

Key figures	Kenneth Arrow Robert Aumann Kenneth Binmore Samuel Bowles Melvin Dresher Merrill M. Flood Drew Fudenberg Donald B. Gillies John Harsanyi Leonid Hurwicz David K. Levine Daniel Kahneman David M. Kreps Harold W. Kuhn Eric Maskin Jean-François Mertens Paul Milgrom Oskar Morgenstern Hervé Moulin Roger Myerson John Nash John von Neumann Ariel Rubinstein Thomas Schelling Reinhard Selten Herbert Simon Lloyd Shapley John Maynard Smith Jean Tirole Albert W. Tucker Amos Tversky William Vickrey Robert B. Wilson Peyton Young

See also	Tragedy of the commons Tyranny of small decisions All-pay auction List of games in game theory Confrontation analysis List of game theorists Combinatorial game theory Alpha–beta pruning Bertrand paradox Bounded rationality No-win situation Coopetition

This article is issued from Wikipedia - version of the Friday, December 27, 2013. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.

Proper equilibrium

Definition

Example

Proper equilibria of extensive games

Further reading