User:Brqano/sandbox

In game theory, rationalizability or rationalizable equilibria is a solution concept which generalizes Nash equilibrium. The general idea is to provide the weakest constraints on players while still requiring rational players. It was first discovered independently by Bernheim (1984) and Pearce (1984).

Constraints on beliefs
Consider a simple coordination game (the payoff matrix is to the right). The row player can play a if she can reasonably believe that the column player could play A, since a is a best response to A. She can reasonably believe that the column player can play A if it is reasonable for the column player to believe that the row player could play a. He can believe that she will play a if it is reasonable for him to believe that she could play a, etc.

This provides an infinite chain of consistent beliefs that result in the players playing (a, A). This makes (a, A) a rationalizable equilibrium. A similar process can be repeated for (b, B).

Not every strategy in every game is rationalizable. Consider a prisoner's dilemma pictured to the left. Row player would never play c, since c is not a best response to any strategy by the column player. This is an example of a more general fact, that a strategy which is strictly dominated cannot be part of a rationalizable equilibrium.

Conversely, for two-player games, the set of all rationalizable strategies can be found by iterated elimination of strictly dominated strategies. For this method to hold however, one also needs to consider strict domination by mixed strategies. Consider the game on the right with payoffs of the column player omitted for simplicity. Notice that "b" is not strictly dominated by either "t" or "m" in the pure strategy sense, but it is still dominated by a strategy that would mix "t" and "m" with probability of each equal to 1/2. This is due to the fact that given any belief about the action of the column player, the mixed strategy will always yield higher expected payoff. This implies that "b" is not rationalizable.

Moreover, "b" is not a best response to either "L" or "R" or any mix of the two. This is because an action that is not rationalizable can never be a best response to any opponent's strategy (pure or mixed). This would imply another version of the previous method of finding rationalizable strategies as those that survive the iterated elimination of strategies that are never a best response (in pure or mixed sense).

In games with more than two players, however, there may be strategies that are not strictly dominated, but which can never be the best response. By the iterated elimination of all such strategies one can find the rationalizable strategies for a multiplayer game.

Rationalizability and Nash equilibria
It can be easily proved that every Nash equilibrium is a rationalizable equilibrium; however, the converse is not true. Some rationalizable equilibria are not Nash equilibria. This makes the rationalizability concept a generalization of Nash equilibrium concept.

As an example, consider the game matching pennies pictured to the right. In this game the only Nash equilibrium is row playing h and t with equal probability and column playing H and T with equal probability. However, all the pure strategies in this game are rationalizable.

Consider the following reasoning: row can play h if it is reasonable for her to believe that column will play H. Column can play H if its reasonable for him to believe that row will play t. Row can play t if it is reasonable for her to believe that column will play T. Column can play T if it is reasonable for him to believe that row will play h (beginning the cycle again). This provides an infinite set of consistent beliefs that results in row playing h. A similar argument can be given for row playing t, and for column playing either H or T.