Proper equilibrium

Infobox equilibrium
name=Proper equilibrium
subsetof=Trembling hand perfect equilibrium
discoverer=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 lesscostly ones.

Definition

Given a normal form game and a parameter $epsilon\; >\; 0$, a totally mixed strategy profile for the game is defined to be $epsilon$-proper if the following holds: If, for two pure strategies s and s' for the same playerit is the case that the expected payoff of playing s is smaller than the expected payoff of playing s' (againstthe mixed strategies of the other players in the profile), then the probability assigned to sis at most $epsilon$ times the probability assigned to s'.

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

Example

The game to the right is a variant of Matching Pennies. 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. Inthis 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 pennywith probability 1. Any mixed strategy of Player 1 is in (Nash) equilibrium with this pure strategyof 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 aboutleaving Player 2 uncertain about whether it is heads up or tails up. However, it can be seenthat 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 thepenny and instead for some reason decides to make a guess. Player 1 prepares for this event bymaking 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 perfectionis applied to extensive games. This leads to the notions of normal form proper equilibriumand extensive form proper equilibrium of an extensive form game. It was shown by vanDamme that a normal form proper equilibrium of an extensive form game is behaviorally equivalent toa quasi-perfect equilibrium of that game.

References

* 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.