# Proper equilibrium

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, 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', then the probability assigned to s
is at most 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 approaches 0, of a sequence of -proper strategy profiles.

## Example

The game to the right is a variant of Matching Pennies.
Player 1 hides a
penny and if Player 2 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 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.
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.