Ised in the literature, but neither is"standard. " I only hope they are suggestive een Note: I am making these terms up as I write these notes. The second has b Us Going back to Figure 1, the point is that, if In is interpreted as an intentional move, then Player I must be planning to follow it with T: there is no other way to rationalize In, because the alternative possibility(Player 1 plans to follow In with B)is irrational. Hence, Player 2 must expect T in the subgame, and hence play L But of course this upsets our equilibrium:(OutB, R)is not stable with respect to forward-induction reasoning On the other hand, the equilibrium(InT, L) is consistent with forward induction (FI henceforth). Thus, we conclude that, if we believe in FI, this should be our edict Note what happens in this game: the addition of an outside option for Player 1 makes her stronger, and enables her to"force " her preferred equilibrium in the Battle of the Se There is an even more striking example of this fact. Suppose that, prior to playin the Battle of the Sexes, Player 1 has an option to burn S T, where I E(1, 2). The game is depicted in Figure 2 0 B0,01,3 Figure 2: Burning Money In this game, the equilibrium(OBB, RR) is sequential; the notation means"Player 1 chooses not to burn, and then plays B if a and B if 0; Player 2 plays R if r and R However, note that z BB and BR are strictly dominated for 1 (by OBB or OTB hence, if Player 2 observes he should expect l to continue with T, not B. Hence Player 2 should choose L, and the(T, L)equilibrium should prevail in the LHS game But then, if Player 1 anticipates this, she will never follow 0 with B: she gets at most 1 by doing so, whereas in the LHS subgame she can secure a payoff of 3-x>1Note: I am making these terms up as I write these notes. The second has been used in the literature, but neither is “standard.” I only hope they are suggestive. Going back to Figure 1, the point is that, if In is interpreted as an intentional move, then Player 1 must be planning to follow it with T: there is no other way to rationalize In, because the alternative possibility (Player 1 plans to follow In with B) is irrational. Hence, Player 2 must expect T in the subgame, and hence play L. But of course this upsets our equilibrium: (OutB, R) is not stable with respect to forward-induction reasoning. On the other hand, the equilibrium (InT,L) is consistent with forward induction (FI henceforth). Thus, we conclude that, if we believe in FI, this should be our prediction. Note what happens in this game: the addition of an outside option for Player 1 makes her stronger, and enables her to “force” her preferred equilibrium in the Battle of the Sexes. There is an even more striking example of this fact. Suppose that, prior to playing the Battle of the Sexes, Player 1 has an option to burn $x, where x ∈ (1, 2). The game is depicted in Figure 2. 1 ￾ ￾ ￾ ￾ ￾￾ x 2 L R 1 T B 3 − x,1 −x,0 −x,0 1 − x,3 0 ❅ ❅ ❅ ❅ ❅❅ 2 L R 1 T B 3,1 0,0 0,0 1,3 Figure 2: Burning Money In this game, the equilibrium (0BB,RR) is sequential; the notation means “Player 1 chooses not to burn, and then plays B if x and B if 0; Player 2 plays R if x and R if 0.” However, note that xBB and xBR are strictly dominated for 1 (by 0BB or 0TB); hence, if Player 2 observes x, he should expect 1 to continue with T, not B. Hence, Player 2 should choose L, and the (T,L) equilibrium should prevail in the LHS game. But then, if Player 1 anticipates this, she will never follow 0 with B: she gets at most 1 by doing so, whereas in the LHS subgame she can secure a payoff of 3−x > 1. 3