1 Signaling - Reputation Game Consider the following duopoly consisting of Firm 1 (The Incumbent) and Firm 2 (The Entrant) that operate on a market with two time periods. Profits are denoted in thousands of dollars (K). In period 1: Both firms are on the market, but only Firm 1 takes an action. Firm 1 (the Incumbent ) can either "prey" or "accommodate". In period 2: Only Firm 2 takes an action: Firm 2 (the Entrant) can either "stay" in the market or "exit " the market. Information problem: Firm 1 has either a sane or crazy CEO, which is a priori not known to the CEO of Firm 2. The probability that Firm 1's CEO is sane is 20% (with the remaining 80% the CEO is crazy). Payoff of Firm 2: In period 1, Firm 2 makes $100K profits if Firm 1 accommodates, but -$50K if Firm 1 preys. In period 2, if Firm 2 stays, it gets a payoff of $100K if Firm 1's CEO is sane, but -$50K if the CEO is crazy (assuming that in period 2 a crazy CEO at Firm 1 remains hostile toward Firm 2). If it exits, Firm 2 makes zero profit. Payoff of Firm 1: In period 1, Firm 1 with a sane CEO makes $120K when it accommodates, and $80K when it preys. Firm 1 with a crazy CEO always enjoys predation (preying brings $100K versus $0K for accommodation). In period 2, Firm 1 gets $120K if Firm 2 stays, but monopoly profits of $200K if Firm 2 exits. 1

1. Write down the structure of the game in its extensive form. [Hint: When you write down the payoffs, consider the profits of both periods.] 2. What is the equilibrium concept for such a game? 3. Suppose Firm 2 believes that the crazy CEO of Firm 1 always preys, but the sane CEO always accommodates. (a) Write down Firm 2's updated beliefs after observing Period 1's outcomes. That is, what are the Bayesian beliefs in the case of predation and accommodation? Justify your answer. (b) What is Firm 2's best response to its Bayesian beliefs in Period 2? [Hint: Consider the profits of period 2 in Firm 2's best response analysis.] (c) Given Firm 2's best response has Firm 1 an incentive to deviate from the above strategy? Justify your answer. 4. Suppose Firm 2 believes that both the crazy and the sane CEO of Firm 1 will prey, but if by any chance it observes accommodation it believes that the CEO is sane. (a) Write down Firm 2's updated beliefs after observing Period 1's outcomes. That is, compute Bayesian beliefs in the case of preda- tion and accommodation. (b) What is Firm 2's best response to its Bayesian beliefs in Period 2? (c) Given Firm 2's best response has Firm 1 an incentive to deviate from the above strategy? Justify your answer.) 5. Considering your answers to the two scenarios above, specify the equi- librium/a that you have found so far in this game. What type of equi- librium/a have you found so far (separating, pooling, semi-separating)? 2

Hint: Remember, an equilibrium must specify the players' best responses and their beliefs in the equilibrium. 6. Does is it make sense for the sane CEO to mimic a crazy one?