Proper equilibrium explained
Proper equilibrium 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 lesscostly ones.
Definition
Given a normal form game and a parameter
, a totally mixed strategy profile
is defined to be
-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 sis at most
times the probability assigned to s'.
The strategy profile of the game is said to be a proper equilibriumif it is a limit point, as
approaches 0, of a sequence of
-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 | align=center | -1, 1 | 0, 0 | align=center | -1, 1 |
|---|
| Hide tails up | 0, 0 | align=center | -1, 1 | align=center | -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. 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
