Proper equilibrium
Proper equilibrium
Proper equilibrium in game theory is a refinement of nash-equilibrium by 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 <math>\epsilon > 0</math>, a totally mixed strategy profile <math>\sigma</math> is defined to be <math>\epsilon</math>-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 <math> u(s,\sigma_{-i})<u(s',\sigma_{-i})</math>), then the probability assigned to s is at most <math>\epsilon</math> times the probability assigned to s'.
The strategy profile of the game is said to be a proper equilibrium if it is a limit point, as <math>\epsilon</math> approaches 0, of a sequence of <math>\epsilon</math>-proper 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. 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.