Consider an entry game with the following timing: In the first stage an entrant decides whether or not to enter. In the second stage, if there is entry an incumbent decides whether or not to prey (fight the entrant by launching a price war—engage in predatory pricing) or accommodate. The payoff matrix is shown in Figure 21.9. Find the two Nash equilibria. Draw the game tree for this game. What is the subgame perfect equilibrium if the game is played once? Suppose that the incumbent operates in 10 different markets. There are 10 firms considering entry, one for each market. They enter sequentially and each can observe the past behavior of the incumbent towards other entrants ( i.e., the game is repeated 10 times). What is the unique subgame perfect equilibrium for the 10 potential entrants and the incumbent? Consider the same situation as in (b), except that the incumbent operates in and is confronted by an infinite number of markets and entrants. If the discount factor is near enough to one, show that the following strategies are a subgame perfect equilibrium: Incumbent: Always prey if entry, unless failed to prey in the past. Entrants: Stay out unless the incumbent has failed to prey in the past. Reconsider the game in (b). How might the equilibrium change if there is a reasonably high probability that the incumbent has payoffs such that he actually prefers to prey (prey is his dominant strategy)? Explain why the equilibrium strategy for a “sane” incumbent, one whose payoffs are as given in the payoff matrix found in (b), is no longer an equilibrium strategy.
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