Equilibrium Refer to OR for definitions The basic idea is as follows The first k*-1 entrants will stay Out, where k*= mink: b-k+1>6. Hence, nothing will be learned about the incumbents type in equilibrium during the first k*-1 rounds: t(h)< h* implies u(h(T) If, however, one of the first k*-1 entrants does enter, the Regular incumbent Fights Note that this implies that, even in this case, nothing will be learned about the in- cumbent's type; hence, the continuation play following such a deviation resembles equilibrium play Entrant k" goes In if E=u(h)(r)<b-k+, and randomizes between In and Out in case of equality. In either case, if there is entry, the Incumbent reacts by randomizing between C and F. If there is no entry, beliefs are unchanged, so the next entrant tainly enters Beliefs are thus updated via Bayes'rule. Thus, as soon as the Incumbent plays C, the following Entrants all choose In. Also, whenever an entrant goes In and the incumbent Fights. the next entrant is indifferent between In and Out. and hence randomizes e In the last round. the Incumbent Concedes You should convince yourself that the equilibrium described in OR achieves precisely this. In particular, you should see that, whenever u(h)(T)=b-k+, the probability of a Fight in the following round is exactly b (so the next entrant is indeed willing to randomize Also, the posterior probability of T given F is bk-k, as per the equilibrium beliefs. Finally as claimed in the previous section, it is easy to see that, following entry at or after Stage k*, the continuation payoff to the incumbent if he plays F is indeed 0, as it must be to justify randomization Plausible beliefs The preceding arguments go a long way(but not all the way) towards proving that reputation can be maintained in a PBE. However, the original result due to Kreps and wilson is stronger first, they use the notion of Sequential Equilibrium; second, they show that any SE which satisfies a certain restriction on beliefs induces the reputation outcome. The restriction on beliefs has the following form: consider two histories h and h' such that h' differs from h only in that, at some subhistories following entry, the incumbentEquilibrium [Refer to OR for definitions] The basic idea is as follows. • The first k ∗ − 1 entrants will stay Out, where k ∗ = min{k : b K − k + 1 ≥ }. Hence, nothing will be learned about the incumbent’s type in equilibrium during the first k ∗ − 1 rounds: t(h) < k∗ implies µ(h)(T) = . • If, however, one of the first k ∗ − 1 entrants does enter, the Regular incumbent Fights. Note that this implies that, even in this case, nothing will be learned about the in￾cumbent’s type; hence, the continuation play following such a deviation resembles equilibrium play. • Entrant k ∗ goes In if  = µ(h)(T) < bK−k ∗+1, and randomizes between In and Out in case of equality. In either case, if there is entry, the Incumbent reacts by randomizing between C and F. If there is no entry, beliefs are unchanged, so the next entrant certainly enters. • Beliefs are thus updated via Bayes’ rule. Thus, as soon as the Incumbent plays C, the following Entrants all choose In. Also, whenever an entrant goes In and the incumbent Fights, the next entrant is indifferent between In and Out, and hence randomizes. • In the last round, the Incumbent Concedes. You should convince yourself that the equilibrium described in OR achieves precisely this. In particular, you should see that, whenever µ(h)(T) = b K−k+1, the probability of a Fight in the following round is exactly b (so the next entrant is indeed willing to randomize). Also, the posterior probability of T given F is b K−k , as per the equilibrium beliefs. Finally, as claimed in the previous section, it is easy to see that, following entry at or after Stage k ∗ , the continuation payoff to the incumbent if he plays F is indeed 0, as it must be to justify randomization. Plausible Beliefs The preceding arguments go a long way (but not all the way) towards proving that reputation can be maintained in a PBE. However, the original result due to Kreps and Wilson is stronger: first, they use the notion of Sequential Equilibrium; second, they show that any SE which satisfies a certain restriction on beliefs induces the reputation outcome. The restriction on beliefs has the following form: consider two histories h and h 0 such that h 0 differs from h only in that, at some subhistories following entry, the incumbent 3