hings are not quite so simple in actuality, but Kreps and wilson prove that something very close to this can be supported in a sequential equilibrium; indeed, while there exist other sequential equilibria in which no "reputation"is maintained, there exists a simple restriction on the entrants'beliefs which ensures that the reputation outcome obtains. OR analyze the situation using the notion of PBE, and do not investigate outcome uniqueness. I will follow their treatment, emphasizing certain key steps Model Payoffs in the stage game are as follows 1 In 2 C 0.a b-1, fe Figure 1:0=01=R, T), a>1>b>0, CR=fT=0, fR=CT=-1 To understand the equilibrium strategies, observe first that, if the entrant expect the incumbent to Fight with probability b, her expected payoff from entry equals b(1-b)+(6 )b=0; hence, she is indifferent between staying Out and going In Now, the Tough incumbent will always Fight in the Pbe under consideration. This implies, among other things, that as soon as an entrant observes C in any previous round, he will conclude that the incumbent is Regular, and therefore will enter; so will every subsequent entrant Consider the problem faced by the Regular incumbent who observes entry at some stage k< K. If she Concedes, she gets 0 in the current period, but faces entry thereafter-and hence, by best-responding, she can secure a payoff of no more than 0. Now suppose that she Fights, and that this leads the next entrant to revise her beliefs in such a way that he is indifferent between In and Out; suppose further that, this being the case, the next entrant chooses Out with probability Then, by Fighting the current entrant(and best-responding thereafter), the incumbent also gets a continuation payoff of 0 Thus, following entry, the incumbent is indifferent between F and C. Hence, he entrant will be such that he is indeed indifferent between going In and staying Out o randomize, and the probability of C can be chosen so that the posterior beliefs of th Notice that this argument relies heavily on the equilibrium assumption!Things are not quite so simple in actuality, but Kreps and Wilson prove that something very close to this can be supported in a sequential equilibrium; indeed, while there exist other sequential equilibria in which no “reputation” is maintained, there exists a simple restriction on the entrants’ beliefs which ensures that the reputation outcome obtains. OR analyze the situation using the notion of PBE, and do not investigate outcome uniqueness. I will follow their treatment, emphasizing certain key steps. Model Payoffs in the stage game are as follows: b, cθ q 1 0,a Out In q 2 F C b − 1, fθ Figure 1: Θ = Θ1 = {R, T}, a > 1 > b > 0, cR = fT = 0, fR = cT = −1. To understand the equilibrium strategies, observe first that, if the entrant expect the incumbent to Fight with probability b, her expected payoff from entry equals b(1 − b) + (b − 1)b = 0; hence, she is indifferent between staying Out and going In. Now, the Tough incumbent will always Fight in the PBE under consideration. This implies, among other things, that as soon as an entrant observes C in any previous round, he will conclude that the incumbent is Regular, and therefore will enter; so will every subsequent entrant. Consider the problem faced by the Regular incumbent who observes entry at some stage k < K. If she Concedes, she gets 0 in the current period, but faces entry thereafter—and hence, by best-responding, she can secure a payoff of no more than 0. Now suppose that she Fights, and that this leads the next entrant to revise her beliefs in such a way that he is indifferent between In and Out; suppose further that, this being the case, the next entrant chooses Out with probability 1 a . Then, by Fighting the current entrant (and best-responding thereafter), the incumbent also gets a continuation payoff of 0. Thus, following entry, the incumbent is indifferent between F and C. Hence, he can randomize, and the probability of C can be chosen so that the posterior beliefs of the next entrant will be such that he is indeed indifferent between going In and staying Out. [Notice that this argument relies heavily on the equilibrium assumption!] 2