Proper equilibrium is a refinement of

Nash Equilibrium In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equili ...

due to Roger B. Myerson. Proper equilibrium further refines

Reinhard Selten Reinhard Justus Reginald Selten (; 5 October 1930 – 23 August 2016) was a German economist, who won the 1994 Nobel Memorial Prize in Economic Sciences (shared with John Harsanyi and John Nash). He is also well known for his work in bound ...

's notion of a

trembling hand perfect equilibrium A tremor is an involuntary, somewhat rhythmic, muscle contraction and relaxation involving oscillations or twitching movements of one or more body parts. It is the most common of all involuntary movements and can affect the hands, arms, eyes, fa ...

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

\epsilon > 0

, a totally mixed strategy profile

\sigma

is defined to be

\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

u(s,\sigma_)), then the probability assigned to s
is at most \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 \epsilon approaches 0, of a sequence of \epsilon -proper strategy profiles.

Example

The game to the right is a variant of

Matching Pennies Matching pennies is the name for a simple game used in game theory. It is played between two players, Even and Odd. Each player has a penny and must secretly turn the penny to heads or tails. The players then reveal their choices simultaneously ...

. 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 In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equili ...

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

game.

Proper equilibria of extensive games

One may apply the properness notion to

extensive form game An extensive-form game is a specification of a game in game theory, allowing (as the name suggests) for the explicit representation of a number of key aspects, like the sequencing of players' possible moves, their choices at every decision point, th ...

s 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 Quasi-perfect equilibrium is a refinement of Nash Equilibrium for extensive form games due to Eric van Damme. Informally, a player playing by a strategy from a quasi-perfect equilibrium takes observed as well as potential future mistakes of hi ...

of that game.

References

* Roger B. Myerson
Refinements of the Nash equilibrium concept
''International Journal of Game Theory'', 15:133-154, 1978. *

Eric van Damme Eric Eleterius Coralie van Damme (born 27 July 1956CURRICULUM VITAE E ...

.
A relationship between perfect equilibria in extensive form games and proper equilibria in normal form games
" ''International Journal of Game Theory'' 13:1--13, 1984. {{Game theory Game theory equilibrium concepts Non-cooperative games