Proper equilibrium
Proper equilibrium | |
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A Roger B. Myerson |
Proper equilibrium is a refinement of
Definition
Given a normal form game and a parameter , a
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.
Example
The game to the right is a variant of
Guess heads up | Guess tails up | Grab penny | |
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Hide heads up | -1, 1 | 0, 0 | -1, 1 |
Hide tails up | 0, 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
Proper equilibria of extensive games
One may apply the properness notion to
References
- 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.