1. Consider a two player game in which player 1 can choose A or B. The game ends if he chooses A, while it continues to player 2 if he chooses B. After observing B, Player 2 can then choose C or D. If player 2 chooses C, the game ends. If player 2 chooses D the game
continues with player 1 after D. After observing D, Player 1 then can choose E or F. The game ends after each of these choices.
(a). Model this as an extensive form game tree. Is it a game of perfect or imperfect information?
(b). How many terminal nodes does the game have? How many information sets? How many pure strategies does each player have?
(c). Imagine that the payoffs following choice A by player 1 are (2, 0), following C by player 2 are (3, 1), following E by player 1 are (0, 0) and following F by player 1 are (1, 2). What are the Pure Strategy Nash Equilibria of this game? Does one strike you as more appealing than the other? If so, explain why.
(d). Characterize the Subgame Perfect Nash Equilibria of this game. Discuss the underlying assumptions made in the analysis.
2. Two staff managers in the ΠβΨ sorority, the house manager (player 1) and kitchen manager (player 2), are supposed to select a resident assistant (RA) from a pool of three candidates: {a, b, c}. Player 1 prefers a to b and b to c. Player 2 prefers b to a and a to c. The process that is imposed on them is as follows: First, the house manager vetoes one of the candidates, and announces the veto to the central office for staff selection, and to the kitchen manager. Next, the kitchen manager vetoes one of the remaining two candidates and announces it to the central office. Finally, the director of the central office assigns the remaining candidate to be an RA.
(a). Model this as an extensive form game where a player's most preferred candidate gives a payoff of 2, the second gives a payoff of 1, and the last gives 0.
(b). Find the subgame perfect equilibria of this game. Is it unique?
(c). Now assume that before the two players play the game, player 2 can send an alienating E-mail to one of the candidates, which would result in that candidate withdrawing her application. Would player 2 choose to do this, and if so, with which candidate?
3. Consider the following Extensive-Form Game with Imperfect Information. Characterize the Subgame Perfect Nash Equilibria.
4. Suppose there is a single good that is owned by a single seller who values it at c > 0 (he can consume the good and get a payoff of c). There is a single buyer who has a small transportation cost k > 0 to get to and back from the seller's store, and he values the good at v > c + k. The buyer first decides whether to make the commute or stay at home, not buy the good and receive a payoff of 0. If the buyers commutes to the store, the seller observes this and can then make the buyer a Take-It-Or-Leave-It price offer p ≥ 0. After observing this offer, the buyer can then accept the offer, pay p and get the good, or he can walk out and not buy the good. Assume that c, v, and k, are common knowledge to all players.
(a). As best as you can, draw the extensive form of this game.
(b). Find the Subgame Perfect Nash Equilibrium of the game. Could the players construct an outcome that would make both players strictly better-off (yielding a Pareto Improvement)?
(c). Now assume that before the game is played, the seller can, at a small cost < v-c-k send the buyer a postcard that commits the seller to a certain price at which the buyer can buy the good (e.g., "bring this coupon and get the good at a price p" ). Would the seller choose to do so? Justify your answer with an equilibrium analysis.