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.
-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
The strategy profile of the game is said to be a proper equilibrium if it is a limit point, as
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.
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).
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.
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.
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.