8.10 3.11 Playing It Safe: Consider the following dynamic game: Player 1 can choose to play it safe {denote this choice by S), in which case both he and player 2 get a payoff of 3 each, or he can risk playing a game with player 2 (denote this choice by R]. If he chooses R then they play the following simultaneous-move game: Player 2 a. Draw a game tree that represents this game. How many proper suh- games does it have? b. Are there other game trees that would work? Explain briefly. c. Construct the matrix representation of the normal form of this dynamic game. d Find all the Nash and suhgame-perfectequilihria of the dynamic game. Resident Assistant Selection with a Twist: Two staff managers in the "Ed! sorority the house manager (player 1} and kitchen manager (player 2) must select a resident assistant from a pool ofthree candidates: {0, b, c}. Player 1 prefers a to b, and b to c. Player2 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 ofce 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 ofce. Finally the director of 8.5 Exercises 0 173 the central ofce assigns the remaining candidate to be a resident assistant at l'leJ. a. Model this as an extensive-form game (using a game tree] in which a player's most-preferred candidate gives a payoff of 2, the second most-preferred candidate gives a payoff of 1, and the last candidate gives I]. . Find the suhgame-perfect equilibrium for this game. Is it unique? c. Are there Nash equilibria that are not subgame-perfect equilibria? d. 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? 8.9 Entry Deterrence 2: Consider the Cournot duopoly game with demand p = 100 - (91 + 92) and variable costs c;(q;) = 0 for i e {1, 2). The twist is that there is now a fixed cost of production k > 0 that is the same for both firms. a. Assume first that both firms choose their quantities simultaneously. Model this as a normal-form game. b. Write down the firm's best-response function for k = 1000 and solve for a pure-strategy Nash equilibrium. Is it unique? c. Now assume that firm 1 is a "Stackelberg leader" in the sense that it moves first and chooses q. Then after observing q, firm 2 chooses 92. Also assume that if firm 2 cannot make strictly positive profits then it will not produce at all. Model this as an extensive-form game tree as best you can and find a subgame-perfect equilibrium of this game for k = 25. Is it unique? d. How does your answer in (c) change for k = 225
