Rationalizability
| Rationalizability | |
|---|---|
| A solution concept in game theory | |
| Relationship | |
| Superset of | Nash equilibrium |
| Significance | |
| Proposed by | D. Bernheim and D. Pearce |
| Example | Matching pennies |
In game theory, rationalizability is a solution concept. The general idea is to provide the weakest constraints on players while still requiring that players are rational and this rationality is common knowledge among the players. It is more permissive than Nash equilibrium. Both require that players respond optimally to some belief about their opponents' actions, but Nash equilibrium requires that these beliefs be correct while rationalizability does not. Rationalizability was first defined, independently, by Bernheim (1984) and Pearce (1984).
Given a normal-form game, the rationalizable set of actions can be computed as follows: Start with the full action set for each player. Next, remove all actions which are never a best reply to any belief about the opponents' actions -- the motivation for this step is that no rational player could choose such actions. Next, remove all actions which are never a best reply to any belief about the opponents' remaining actions -- this second step is justified because each player knows that the other players are rational. Continue the process until no further actions are eliminated. In a game with finitely many actions, this process always terminates and leaves a non-empty set of actions for each player. These are the rationalizable actions.
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 pair of actions. A similar process can be repeated for (b, B).
As an example where not all strategies are 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. For this reason, c is not rationalizable.
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