T3,10,0 B0.01,3 Figure 1: The Battle of the Sexes with an Outside Option On the other hand, while In constitutes a deviation from equilibrium play, it is not necessarily an irrational choice per se: if Player 1 expects Player 2 to choose L with sufficiently high probability, then In followed by T is actually optimal Wait a moment, you will say: how can Player 1 expect Player 2 to choose L when the equilibrium profile says he should play R? Well, that's a good question. However, note that the same equilibrium profile also says that Player 1 should not deviate to In; so, in some sense, its prescriptions should be taken with a grain of salt! More rigorously, faced with a deviation to In, Player 2 may either think that this was merely the result of a mistake, a tremble, in which case he is entitled to continue to believe that Player 1 expects R and hence plays B; or, he can abandon his faith in this equilibrium, and attempt to come up with some alternative theory of Player 1 behavior The first route is the story implicit in sequential equilibrium, THPE, and backward induction in general. According to this story, (OutB, R)is a fine equilibrium The second route leads to forward induction. This notion is quite distinct from backward induction-and sometimes even at odds with it! But it makes a lot of sense, at least in the opinion of this writer. The two main tenets of forward induction are Intentionality: any move, including deviations from the equilibrium path, is intentional and purposeful Rationalization: players recognize this, and therefore attempt to ratio- nalize, i.e. explain, deviations (or, more generally, unexpected moves)by guessing their objective1 ￾ ￾ ￾ ￾ ￾￾ Out 2,2 In ❅ ❅ ❅ ❅ ❅❅ 2 L R 1 T B 3,1 0,0 0,0 1,3 Figure 1: The Battle of the Sexes with an Outside Option On the other hand, while In constitutes a deviation from equilibrium play, it is not necessarily an irrational choice per se: if Player 1 expects Player 2 to choose L with sufficiently high probability, then In followed by T is actually optimal! Wait a moment, you will say: how can Player 1 expect Player 2 to choose L when the equilibrium profile says he should play R? Well, that’s a good question. However, note that the same equilibrium profile also says that Player 1 should not deviate to In; so, in some sense, its prescriptions should be taken with a grain of salt! More rigorously, faced with a deviation to In, Player 2 may either think that this was merely the result of a mistake, a tremble, in which case he is entitled to continue to believe that Player 1 expects R and hence plays B; or, he can abandon his faith in this equilibrium, and attempt to come up with some alternative theory of Player 1’s behavior. The first route is the story implicit in sequential equilibrium, THPE, and backward induction in general. According to this story, (OutB, R) is a fine equilibrium. The second route leads to forward induction. This notion is quite distinct from backward induction—and sometimes even at odds with it! But, it makes a lot of sense, at least in the opinion of this writer... The two main tenets of forward induction are: Intentionality: any move, including deviations from the equilibrium path, is intentional and purposeful. Rationalization: players recognize this, and therefore attempt to ratio￾nalize, i.e. explain, deviations (or, more generally, unexpected moves) by guessing their objective. 2