PS19_solutions

PS19_solutions - Intermediate Microeconomics, Winter 2008...

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Unformatted text preview: Intermediate Microeconomics, Winter 2008 Problem Set No 19: Solutions Q1 A monopolist (Player A) has branches in infinitely many towns. He faces potential competitors in each town, who will be able to choose IN or OUT. They do so in sequential order and one at a time. If a potential competitor chooses to stay OUT, he receives a payoff of 1, while A receives a payoff of 5. If he chooses to go IN, he will receive a payoff of either 2 or 0, depending on the response of Player A to his action. Player A, in response to a choice of IN, must choose one of two pricing strategies, COOPERATIVE or AGGRES- SIVE. If he chooses COOPERATIVE, both player A and the competitor receive a payoff of 2, and if A chooses Aggressive, each player receives a payoff of 0. The monopolist discounts future payoffs with a discount factor δ = 0 . 9 . Find an equilibrium in which the competitors stay out of the market of the monopolist. Consider the following strategies: Player A plays AGGRESSIVE if the out- come { IN, COOPERATIVE } has not occurred before. And COOPERATIVE if he has played COOPERATIVE against a competitor playing IN before. Po- tential competitors play OUT unless a previous competitor has played IN and Player A has played COOPERATIVE. These strategies are complete because they tell each player what to do in each stage no matter what the history is. When evaluating this game keep in mind that it is the same monopolist playing in each period, the Player A cares about the payoffs in all periods. However, there is a different potential competitor in each period, so each entrant only gets one payoff. Therefore, a competitor will only play IN if Player A is going to play COOPERATIVE, no matter how this IN will change things in future periods. Do these strategies constitute a Nash Equilibrium? On the equilibrium path, no potential competitor plays IN, so each competitor faces the option of following their prescribed strategy (play OUT) and get a payoff of 1 or deviate (play IN) and get a payoff of 0. Player A is getting a payoff of 5 every period along the equilibrium path, so Player A certainly does not have an incentive to deviate. Therefore, all players are playing a best response, so we have found a Nash Equiblibrium. But is it a Subgame Perfect Equilibrium? In order to determine this, we need to check off the equilibrium path; that is, we need to check that all players are playing a Nash Equilibrium in the subgames after a potential competitor has played IN. In any such subgame, since Player A played AGGRESSIVE against the first potential competitor to play IN, all future potential competitors will play OUT. So, Player A gets a payoff of 0 in first stage, but then gets a payoff of 5 in every stage thereafter. Alternatively, Player A could deviate from his strategy and play COOPERATIVE against the first IN. Then, each future competitorand play COOPERATIVE against the first IN....
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This note was uploaded on 12/10/2011 for the course ECON 401 taught by Professor Burbidge,john during the Winter '08 term at Waterloo.

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PS19_solutions - Intermediate Microeconomics, Winter 2008...

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