博弈论 前向归纳法 forward induction

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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. 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 Sexes.
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 more, including deviations from the equilibrium path, is intentional and purposeful. Rationalization: Players recognize this, and therefore attempt to rationalize, i.e. explain, deviations (or, more generally, unexpected moves) by guessing their objective.
Consider the profile (OutB, R) in the game in Figure 1.This is clearly a Nash equilibrium; moreover, since B is a best reply to R and conversely, (OutB, R) is actually a subgameperfect equilibrium. In fact, you can convince yourself that it is sequential and tremblinghand perfect as well. And yet, and yet…dose it really make sense for Player 2 to expect Player 1 to following In (a deviation) with B? Note that InB is strictly dominated, hence certainly irrational for Player 1:Out dose strictly better.
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!
The key insight of extensive-form analysis is that out-of-equilibrium beliefs determine equilibrium behavior. For instance, consider the simple two-stage entry deterrence game in which a potential entrant whether to enter a market or stay out, and the incumbent decides whether to fight or acquiesce after the entrant’s move. The Nash equilibrium in which the entrant stays out is supported by the belief that the incumbent will fight; but if we think that fighting is not a credible threat, this equilibrium collapses.
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.
But things can be become much more subtle—and interesting. In particular, sequential equilibrium takes care of the issue of non-credible threats, but it sometimes still relies on “implausible inferences” out of equilibrium. This lecture addresses this point, focusing on the notion of forward induction.