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.
| H | T | |
|---|---|---|
| h | 1, -1 | -1, 1 |
| t | -1, 1 | 1, -1 |
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.
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“It is the sin of omission, the second kind of sin,
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—Ogden Nash (1902–1971)