Eco514-Game Theory Forward Induction Marciano siniscalchi January 10, 2000 Introduction One of the merits of the notion of sequential equilibrium is the emphasis on out-of- sets that should not be reached if a given equilibrium is plane( play)at information equilibrium beliefs-that is, on beliefs(about past and future pla The key insight of extensive-form analysis is that out-of-equilibrium beliefs deter- mine equilibrium behavior:. For instance, consider the simple two-stage entry deter- rence game in which a potential entrant decides 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 But things can become much more subtle-and interesting. In particular, se- quential 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 Outside Options and Burning Money nuch of the literature in this field, it's best to begin with a couple of ke examples 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 subgame-perfect equilibrium. In fact, you can convince yourself that it is sequential and trembling-hand perfect as well And yet, and yet... does it really make sense for Player 2 to expect Player 1 to follow In(a deviation) with B? Note that InB is strictly dominated, hence certainly irrational for Player 1: Out does strictly betterEco514—Game Theory Forward Induction Marciano Siniscalchi January 10, 2000 Introduction One of the merits of the notion of sequential equilibrium is the emphasis on out-of￾equilibrium beliefs—that is, on beliefs (about past and future play) at information sets that should not be reached if a given equilibrium is played. The key insight of extensive-form analysis is that out-of-equilibrium beliefs deter￾mine equilibrium behavior. For instance, consider the simple two-stage entry deter￾rence game in which a potential entrant decides 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. But things can become much more subtle—and interesting. In particular, se￾quential 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 Outside Options and Burning Money Following much of the literature in this field, it’s best to begin with a couple of key examples. 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 subgame-perfect equilibrium. In fact, you can convince yourself that it is sequential and trembling-hand perfect as well. And yet, and yet... does it really make sense for Player 2 to expect Player 1 to follow In (a deviation) with B? Note that InB is strictly dominated, hence certainly irrational for Player 1: Out does strictly better. 1