Proper equilibrium

Given a normal form game and a parameter $\backslash epsilon\; >\; 0$, a totally mixed strategy profile $\backslash sigma$ is defined to be $\backslash epsilon$-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

is at most $\backslash epsilon$ 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 $\backslash epsilon$ approaches 0, of a sequence of $\backslash epsilon$-proper strategy profiles.

The game to the right is a variant of Matching Pennies.

class="wikitable" align=right style="margin: 1em 1em 1em 1em; background: #f9f9f9; font-size: 95%;" |+ align=bottom |Matching Pennies with a twist
! Guess heads up ! Guess tails up ! Grab penny
Hide heads up |align=center|−1, 1 |align=center|0, 0 |align=center|−1, 1
Hide tails up |align=center|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. 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.

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.

• Roger B. Myerson. [https://www.kellogg.northwestern.edu/research/math/papers/295.pdf Refinements of the Nash equilibrium concept]. {{interlanguage link|International Journal of Game Theory|de}}, 15:133-154, 1978.
• Eric van Damme. "[https://link.springer.com/article/10.1007/BF01769861 A relationship between perfect equilibria in extensive form games and proper equilibria in normal form games]." {{interlanguage link|International Journal of Game Theory|de}} 13:1--13, 1984.

{{Game theory}}

Category:Game theory equilibrium concepts

Category:Non-cooperative games