Chapter 8 Ratio n a liz a b ility The definition of a gam e (N,S, u1,...,un) imp lic it ly a ssu mes that 1. the set of players is N, the set of a vailable strategies to a pla yer i is Si,and the player i tries to maximize the expected va lu e of ui : S R accordingtosome → belief, and that 2. eac h player knows 1, and that 3. eac h player knows 2, and that ... n each play er kno ws n − 1 ... ad infinitum. Tha t is, it is imp licitly assumed that it is common knowledge among the play ers that the game is (N, S,u1,...,un) and that play ers are rational (i.e. they are expected utility max im ize rs). As a solution concept, Rationalizab ility yields the strategies that are consistent with th ese assu m ption s, captu ring w ha t is imp lied by the m odel (i.e. th e game ). Other solution concepts impose further assumptions, usually on play er s’ beliefs, to obtain sharper predictio ns. In this lecture, I will form ally introduce rationalizab ility and present some of its applications. The outline is as follow s. I will first illustrate the idea on a simple example. I will then presen t the formal theory. I will fina lly apply rationalizability to Cournot and Bertrand competitions. 7374 CHAPTER 8. RATIONALIZABILITY 8.1 Examp le Consid er the following game. 1\2L R T 2,0 -1,1 0,10 0,0 -1,-6 2,0 (8.1) M B A pla yer is said to be rational if he plays a best response to a belief about the other pla yers’ strategies. Wha t 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 payoffsreveals that hisbest play dependson whathethinksthe other pla yer does. Let’s then write p for the proba bility he assigns to L (as Player 2’s play ), representing his belief about Player 2’s strategy. His expected pay offsfrom playing T, M, and B are UT =2p − (1 − p)=3p − 1, UM =0, UB = −p +2(1− p)=2− 3p, respectiv ely. These values as a function of p are plotted in the follo w ing graph: U 0 1p UM UB UT 2 0 -175 8.1. EXAMPLE As it is clear from the graph, UT is the largest when p>1/2,and UB is the largest when p<1/2.At p =1/2, UT = UB > 0. Hence, if pla yer 1 is rational, then he will play B when p<1/2,D when p>1/2,and B orD if p =1/2. Notice that, if Player 1 is rationa l, then he will never play M– no matter what he believ es about the Play er 2’s play. Therefore, if we assum e that Player 1 is rational (and that the game is as it is described abo ve), then we can conclude that Player 1 will not play M. Th is is because M is a strictly dominate d str ategy. In particular, the mixed strategy that puts probability 1/2 on T and probab ility 1/2 on B yields a higher expected payoff than strategy M no m a tter what (pu re) strategy Player 2 p lays. A consequence of this is that M is never a weak best response to a belief p, a general fact that will be established mom entarily. We no w w an t to understand the implications of the assumption that play ers know that the other players are also rational. Now, rationality of pla yer 1 requires that he does not pla y M. For Play er 2, her both actions can be a best reply. If she thinks that Player 1isnot likely to play M, then shemustplayR,and if shethinksthatitisvery likely that Player 1 will play M, then she must play L. Hence, ration a lity of pla yer 2 does not put any restriction on her behavior. But, what if she thinks that it is very lik ely that play e r 1 is ratio na l (and that his pay off are as in (8.1))? In that case, since a ration al player 1 does not play M, she must assign very small probability for pla yer 1 playing M. In fact, if she knows that player 1 is rational, then she must be sure that he will not play M. In that case, being rationa l, she must pla y R. In summ ary, if player 2 is rational and she knows that player 1 is rational, then she must play R. Notice that we first eliminated all of the strategies that are strictly dominated (namely M ), then taking the resulting gam e, w e eliminated again a ll of the strate-gies that are strictly dom inated (namely L). This is called twice iterated elim in a tion of strictly dominated strategies. The resulting strategies are the strategies that are consis-tent with the assumption that pla yers are rational and they know that the other players are rational. As w e impose further assumptio ns about rationality, we k eep iterativ e ly eliminating all strictly dom ina ted strategies (if there remains any). Recall that rationality of player 1 requires him to play T or B, and kno wledge of the fact that player 2 is also rational does not put any restriction on his beha vior– as rationality itself does not restrict Pla yer76 CHAPTER 8. RATIONALIZABILITY 2’s behavior. Now , assume that Player 1 also kno w s (i) that Player 2 is rational and (ii) that Player 2 knows that Pla yer 1 is rational (and that the game is as in (8.1)). Then, as the above analysis sno w s, Pla yer 1 m u st kno w that Play er 2 will pla y R. In that case, being rational he m u st play B. Therefore, comm on knowledge of rationalit y imp lies that Player 1 plays B and Player 2 plays R. In the next section, I will apply these ideas more generally. 8.2 Theory Fix a game (N, S,u1,...,un). To be concrete, define the concepts of belief, best response, and rationalit y as follows. Definition 26 For any player i,a (correlated) belief of i ab out the other players’ strate-gies is a probability distribution μ on S−i = Qj=i Sj.−i 6The essential part of this definition is that the belief μ−i of pla y er i allo w s correlation between the other pla yers’ strategies. For example, in a game of three players in which each pla yer is to choose bet ween Left and Righ t , Player 1 may believe that with proba-bilit y 1/2 both of the other pla yers will play Left and with proba bility 1/2 both pla yers will pla y Righ t. Hence, viewed as mixed strategies, it may appear as though Pla yers 2 and 3 use a common random ization device, contradicting the fact that Players 2 and 3 make their decisions independently. One may then find such a correlated belief unrea-sonable. …