Econ 521 - Week 14
1. Dynamic Game - Consider a game in which player 1 first selects between I and O. If player 1 selects O, then the game ends with payoff vector (x, 1) (x for player 1), where x is some positive number. If player 1 selects I, then this selection is revealed to player 2 and then players play the battle-of-the-sexes game in which they simultaneously and independently choose between A and B. If they coordinate on A, then the payoff vector is (3, 1). If they coordinate on B, then the payoff vector is (1, 3). If they fail to coordinate, then the payoff vector is (0, 0).
(a) Represent this game in the extensive form.
(b) Find the reduced normal form and find the pure-strategy Nash equilibria of this game.
(c) What are the pure-strategy subgame perfect equilibria of the game? Can you find any Nash equilibria that are not subgame perfect?
2. Entry decision - Imagine a market setting with three firms. Firm 2 and 3 are already operating as monopolists in two different industries (they are not competitors). Firm 1 must decide whether to enter firm 2's industry and thus compete with firm 2, or enter firm 3's industry and thus compete with firm 3. Production in firm 2's industry occurs at zero cost, whereas the cost of prouduction in firm 3's industry is 2 per unit. Demand in firm 2's industry is given by p = 9 - Q, whereas demand in firm 3's industry is given by p' = 14-Q', where p and Q denote the price and total quantity in firm 2's industry and p' and Q' denote the price and total quantity in firm 3's industry.
The game runs as follows: First, firm 1 chooses between E2 and E3. (E2 means "enter firm 2's industry and E3 means "enter firm 3's industry). This choice is observed by firm 2 and 3. Then, if firm 1 chooses E2, firm 1 and firm 2 compete as Cournot duopolists, where they select quantities q1 and q2 simultaneously. In this case, firm 3 automatically gets the monopoly profit of 36 in its own industry. On the other hand, if firm 1 chooses E3, then firms 1 and 3 compete as Cournot duopolists, where they select quantities q'1 and q'3 simultaneously; and in this case, firm 2 automatically gets its monopoly profit of 81/4.
(a) Calculate the SPNE of this game. In the equilibrium, does firm 1 enter firm 2's industry or firm 3's industry?
(b) In there a NE (not necessarily subgame perfect) in which firm 1 selects E2? If so, describe it. If not, briefly explain why.
3. Contracts in Finitely Repeated Games - Consider the following stage game,
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2
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X
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Y
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1
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A
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5,6
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0,0
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B
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8,2
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2,2
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(a) Find all of the pure strategy Nash equilibria of this game.
(b) Consider the two-period repeated game in which this stage game is played twice and the repeated game payoffs are simply the sum of the payoffs in each of the two periods. Is there a subgame perfect equilibrium of this repeated game in which (A, X) is played in the first period? If so, fully describe the equilibrium. If not, explain why.