R 1,10.0 B|1,0-1,2 Figure 1: Correct beliefs This game has a unique Nash equilibrium, (T, L). Suppose that we assume that:(1) Players are rational; (2. 1)Player 1 expects Player 2 to choose L; and(2.2)Player 2 expects Player 1 to choose R. Are these assumptions sufficient to conclude that Player 1 will choose T and Player 2 will choose L? The answer is no, because a rational Player 1 might equally well choose B if she thinks hat Player 2 will choose L. Thus, "correct beliefs "must imply something more than this We need to consider the following assumption (2.1)Player 1 expects Player 2 to play whatever action he actually chooses (2.2) Player 2 expects Player I to play whatever action she actually chooses This will be crystal-clear when we develop a model where assumptions about beliefs be formalized. However, the basic idea should still be easy to grasp: what is needed is a restriction that relates beliefs and actual behavior. The problem with this assumption is that of course, it does not go much beyond the definition of Nash equilibrium Yet, it delivers the required result, for although(B, L) is consistent with assumptions (1),(2.1)and(2.2), it fails(2.2): for Player 2 to choose L, it must be the case that he expects Player 1 to choose T with high enough probability, but in fact Player 1 chooses B with probability one. Again, these considerations will be clearer once we develop a model of interactive beliefs By way of comparison, recall that the assumptions we used to justify(correlated)ra tionalizability(following the eductive approach) were of the following form:(1)Players are rational;(2) Players believe that their opponents are rational; 3)Players believe that their opponents believe that their own opponents are rational; and so on. Note that assumptions (2),(3). do not involve specific actions(as our assumptions(2. 1)and(2. 2) above), and do not in any way refer to what players may actually do in a given "state of the world Thus, these assumptions are truly decision-theoretical in nature; those required to moti- ate Nash equilibrium are somewhat of a hybrid (indeed, fixpoint )nature One could correctly argue, however, that the above point is only valid for equilibria in which players have multiple best-replies to their equilibrium beliefs; that is, for non-strict equilibria in game-theoretic parlance. One could then observe that in "generic"games(i.e games with payoffs in general position) all pure-action equilibria are strictL R T 1,1 0,0 B 1,0 -1,2 Figure 1: Correct beliefs This game has a unique Nash equilibrium, (T,L). Suppose that we assume that: (1) Players are rational; (2.1) Player 1 expects Player 2 to choose L; and (2.2) Player 2 expects Player 1 to choose R. Are these assumptions sufficient to conclude that Player 1 will choose T and Player 2 will choose L? The answer is no, because a rational Player 1 might equally well choose B if she thinks that Player 2 will choose L. Thus, “correct beliefs” must imply something more than this. We need to consider the following assumption: (2.1’) Player 1 expects Player 2 to play whatever action he actually chooses; (2.2’) Player 2 expects Player 1 to play whatever action she actually chooses. This will be crystal-clear when we develop a model where assumptions about beliefs can be formalized. However, the basic idea should still be easy to grasp: what is needed is a restriction that relates beliefs and actual behavior. The problem with this assumption is that, of course, it does not go much beyond the definition of Nash equilibrium! Yet, it delivers the required result, for although (B,L) is consistent with assumptions (1), (2.1) and (2.2), it fails (2.2’): for Player 2 to choose L, it must be the case that he expects Player 1 to choose T with high enough probability, but in fact Player 1 chooses B with probability one. Again, these considerations will be clearer once we develop a model of interactive beliefs. By way of comparison, recall that the assumptions we used to justify (correlated) ra￾tionalizability (following the eductive approach) were of the following form: (1) Players are rational; (2) Players believe that their opponents are rational; (3) Players believe that their opponents believe that their own opponents are rational; and so on. Note that assumptions (2), (3)... do not involve specific actions (as our assumptions (2.1) and (2.2) above), and do not in any way refer to what players may actually do in a given “state of the world.” Thus, these assumptions are truly decision-theoretical in nature; those required to moti￾vate Nash equilibrium are somewhat of a hybrid (indeed, fixpoint) nature. One could correctly argue, however, that the above point is only valid for equilibria in which players have multiple best-replies to their equilibrium beliefs; that is, for non-strict equilibria in game-theoretic parlance. One could then observe that in “generic” games (i.e. games with payoffs in general position) all pure-action equilibria are strict. 6