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'.
The strategy profile of the game is said to be a proper equilibrium if it is a limit point, as approaches 0, of a sequence of -proper strategy profiles.
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.
Proper equilibria of extensive gamesedit
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.
Referencesedit
Roger B. Myerson. Refinements of the Nash equilibrium concept. International Journal of Game Theory, 15:133-154, 1978.
Eric van Damme. "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.
April 11, 2024
proper, equilibrium, refinement, nash, equilibrium, roger, myerson, further, refines, reinhard, selten, notion, trembling, hand, perfect, equilibrium, assuming, that, more, costly, trembles, made, with, significantly, smaller, probability, than, less, costly, . Proper equilibrium is a refinement of Nash Equilibrium by 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 Proper equilibriumA solution concept in game theoryRelationshipSubset ofTrembling hand perfect equilibriumSignificanceProposed byRoger B Myerson Contents 1 Definition 2 Example 3 Proper equilibria of extensive games 4 ReferencesDefinition editGiven a normal form game and a parameter ϵ gt 0 displaystyle epsilon gt 0 nbsp a totally mixed strategy profile s displaystyle sigma nbsp is defined to be ϵ displaystyle epsilon nbsp 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 s i lt u s s i displaystyle u s sigma i lt u s sigma i nbsp then the probability assigned to s is at most ϵ displaystyle epsilon nbsp 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 ϵ displaystyle epsilon nbsp approaches 0 of a sequence of ϵ displaystyle epsilon nbsp proper strategy profiles Example editThe game to the right is a variant of Matching Pennies Matching Pennies with a twist Guess heads up Guess tails up Grab pennyHide heads up 1 1 0 0 1 1Hide tails up 0 0 1 1 1 1Player 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 Proper equilibria of extensive games editOne 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 References editRoger B Myerson Refinements of the Nash equilibrium concept International Journal of Game Theory 15 133 154 1978 Eric van Damme 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 Retrieved from https en wikipedia org w index php title Proper equilibrium amp oldid 1143560623, wikipedia, wiki, book, books, library,
