Here(did2, D) is not a SPE, but it is a Nash equilibrium. It fails the test of perfection because, upon being reached (contrary to the equilibrium prescription! Player 2"should" really expect Player 1 to continue with a2, not d2, because "Player 2 is certain that Player 1 is rational. But this argument is even more ambiguous Theorists have fiercely argued over these and similar examples. I cannot do justice to either side of the debate without a full-blown model of interactive beliefs specifically developed to account for dynamically evolving beliefs. However, for the time being, let me e a few informal but rigorous observation First, in a dynamic game, statements such as "Player 2 is certain that Player l is rational are meaningless. One must specify when, in the course of the game, that statement is assumed to be true-when, for example, Player 2 is certain that his opponent is rational, or Player 2 is certain that Player 1 expects Player 2 to choose D The reason is that, in a dynamic game(and especially in a game with observable actions! players acquire new information as the play unfolds. This information may lead them to update, i. e. "refine"their beliefs, or it may lead them to completely revise them For example, Player 2 might be certain, at the initial history, that Player 1 chooses di at the initial history. However, if Player 1 chooses al instead, Player 2 observes this, so it would be impossible for her to continue to be certain after the history(a,) that she chooses d1. Player 2 has to revise her beliefs Reasoning along these lines, the argument against backward induction in the Centipede game can be reformulated more convincingly. Suppose that, at the beginning of the game both players' beliefs are consistent with the unique SPE. Moreover, suppose that, at the beginning of the game, both players are certain that their opponent is rational (you can assume that there is common certainty of this: it is not going to matter Could it be the case that Player 1 chooses a1 at the initial history, contrary to the SPe prediction? The answer is, surprisingly, yes! For instance, this will be the unique best response of Player 1 if she is certain at the initial node that (1) Player 2 is certain, at the initial node, that Player I will play dyd2, and that Player I is rational(this much we had already assumed) (2) However, Player 2, upon observing al, will revise her beliefs and become certain at (an) that Player 1 is actually irrational and will choose a2 after(a1, A) (3)Player 2 is rational(again, this mud Ir assumptions Note that(2)+ 3)imply that Player 2, if he has to choose, will pick A and not D. But hen, of course, Player 1 will rationally choose an herself The key point is that, even if players'beliefs are concentrated on the equilibrium at the beginning of the game, it can still be the case that the way they revise their beliefs leads them to act differently, both off and on the equilibrium path To put it differently, in order to justify SPE one has to make assumptions about how players revise their beliefs. Characterizing backward induction in perfect-information gamesHere (d1d2, D) is not a SPE, but it is a Nash equilibrium. It fails the test of perfection because, upon being reached (contrary to the equilibrium prescription!) Player 2 “should” really expect Player 1 to continue with a2, not d2, because “Player 2 is certain that Player 1 is rational.” But this argument is even more ambiguous! Theorists have fiercely argued over these and similar examples. I cannot do justice to either side of the debate without a full-blown model of interactive beliefs specifically developed to account for dynamically evolving beliefs. However, for the time being, let me propose a few informal but rigorous observation. First, in a dynamic game, statements such as “Player 2 is certain that Player 1 is rational” are meaningless. One must specify when, in the course of the game, that statement is assumed to be true—when, for example, Player 2 is certain that his opponent is rational, or Player 2 is certain that Player 1 expects Player 2 to choose D. The reason is that, in a dynamic game (and especially in a game with observable actions!), players acquire new information as the play unfolds. This information may lead them to update, i.e. “refine” their beliefs, or it may lead them to completely revise them. For example, Player 2 might be certain, at the initial history, that Player 1 chooses d1 at the initial history. However, if Player 1 chooses a1 instead, Player 2 observes this, so it would be impossible for her to continue to be certain after the history (a1) that she chooses d1. Player 2 has to revise her beliefs. Reasoning along these lines, the argument against backward induction in the Centipede game can be reformulated more convincingly. Suppose that, at the beginning of the game, both players’ beliefs are consistent with the unique SPE. Moreover, suppose that, at the beginning of the game, both players are certain that their opponent is rational (you can assume that there is common certainty of this: it is not going to matter). Could it be the case that Player 1 chooses a1 at the initial history, contrary to the SPE prediction? The answer is, surprisingly, yes! For instance, this will be the unique best response of Player 1 if she is certain at the initial node that: (1) Player 2 is certain, at the initial node, that Player 1 will play d1d2, and that Player 1 is rational (this much we had already assumed). (2) However, Player 2, upon observing a1, will revise her beliefs and become certain at (a1) that Player 1 is actually irrational and will choose a2 after (a1, A). (3) Player 2 is rational (again, this much was already part of our assumptions). Note that (2)+(3) imply that Player 2, if he has to choose, will pick A and not D. But then, of course, Player 1 will rationally choose a1 herself! The key point is that, even if players’ beliefs are concentrated on the equilibrium at the beginning of the game, it can still be the case that the way they revise their beliefs leads them to act differently, both off and on the equilibrium path. To put it differently, in order to justify SPE one has to make assumptions about how players revise their beliefs. Characterizing backward induction in perfect-information games 5