Proper equilibrium

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Proper equilibrium
A solution concept in game theory
Subset of Trembling hand perfect equilibrium
Proposed by Roger B. Myerson

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.

Reinhard Selten German economist

Reinhard Justus Reginald Selten was a German economist, who won the 1994 Nobel Memorial Prize in Economic Sciences. He is also well known for his work in bounded rationality and can be considered as one of the founding fathers of experimental economics.

In game theory, trembling hand perfect equilibrium is a refinement of Nash equilibrium due to Reinhard Selten. A trembling hand perfect equilibrium is an equilibrium that takes the possibility of off-the-equilibrium play into account by assuming that the players, through a "slip of the hand" or tremble, may choose unintended strategies, albeit with negligible probability.



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 ), then the probability assigned to s is at most times the probability assigned to s'.

In abstract rewriting, an object is in normal form if it cannot be rewritten any further. Depending on the rewriting system and the object, several normal forms may exist, or none at all.

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.


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

Matching Pennies with a twist
Guess heads upGuess tails upGrab penny
Hide heads up-1, 10, 0-1, 1
Hide tails up0, 0-1, 1-1, 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.

In game theory, the Nash equilibrium, named after the mathematician John Forbes Nash Jr., is a proposed solution of a non-cooperative game involving two or more players in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy.

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.

Quasi-perfect equilibrium is a refinement of Nash Equilibrium for extensive form games due to Eric van Damme.

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Further reading

Eric van Damme Dutch economist

Eric Eleterius Coralie van Damme is a Dutch economist and Professor of Economics at the Tilburg University, known for his contributions to game theory.