14.123 Rationalizability Lecture Summary

# 14.123 Rationalizability Lecture Summary - Chapter 8...

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Chapter 8 Rationalizability The de fi nition of a game ( N, S, u 1 , . . . , u n ) implicitly assumes that 1. the set of players is N , the set of available strategies to a player i is S i , and the player i tries to maximize the expected value of u i : S R according to some belief, and that 2. each player knows 1, and that 3. each player knows 2, and that . . . n each player knows n 1 . . . ad in fi nitum. That is, it is implicitly assumed that it is common knowledge among the players that the game is ( N, S, u 1 , . . . , u n ) and that players are rational (i.e. they are expected utility maximizers). As a solution concept, Rationalizability yields the strategies that are consistent with these assumptions, capturing what is implied by the model (i.e. the game). Other solution concepts impose further assumptions, usually on players’ beliefs, to obtain sharper predictions. In this lecture, I will formally introduce rationalizability and present some of its applications. The outline is as follows. I will fi rst illustrate the idea on a simple example. I will then present the formal theory. I will fi nally apply rationalizability to Cournot and Bertrand competitions. 73

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74 CHAPTER 8. RATIONALIZABILITY 8.1 Example Consider the following game. 1 \ 2 L R T 2,0 -1,1 0,10 0,0 -1,-6 2,0 (8.1) M B A player is said to be rational if he plays a best response to a belief about the other players’ strategies. What does rationality imply for this game? Consider Player 1. He is contemplating about whether to play T, or M, or B. A quick inspection of his payo ff s reveals that his best play depends on what he thinks the other player does. Let’s then write p for the probability he assigns to L (as Player 2’s play), representing his belief about Player 2’s strategy. His expected payo ff s from playing T, M, and B are U T = 2 p (1 p ) = 3 p 1 , U M = 0 , U B = p + 2(1 p ) = 2 3 p, respectively. These values as a function of p are plotted in the following graph: U 0 1 p U M U B U T 2 0 -1
75 8.1. EXAMPLE As it is clear from the graph, U T is the largest when p > 1 / 2 , and U B is the largest when p < 1 / 2 . At p = 1 / 2 , U T = U B > 0 . Hence, if player 1 is rational, then he will play B when p < 1 / 2 , D when p > 1 / 2 , and B or D if p = 1 / 2 . Notice that, if Player 1 is rational, then he will never play M–no matter what he believes about the Player 2’s play. Therefore, if we assume that Player 1 is rational (and that the game is as it is described above), then we can conclude that Player 1 will not play M. This is because M is a strictly dominated strategy . In particular, the mixed strategy that puts probability 1/2 on T and probability 1/2 on B yields a higher expected payo ff than strategy M no matter what (pure) strategy Player 2 plays. A consequence of this is that M is never a weak best response to a belief p , a general fact that will be established momentarily.

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