four questions about eco 405
1. (15pts) Player 1 has strategies A. B and C. Player 2 has strategies D, E, and F. D E F A (2, 2) (1, 3) (0,1) B (3, 1) (2, 1) (1, 1 ) C (1,0) (0, 0) (3, 0) (a) (2pts) Does Player I have any strictly dominated strategy? If yes, find them and state what strategy strictly dominates them. (b) (3pts) Does Player 2 have any weakly dominated strategy? If yes, find them and state what strategy weakly dominates them. (c) (2pts) Is there any weakly dominant strategy for player 1? If yes, find it. (d) (2pts) Is there any weakly dominant strategy for player 2? If yes, find it. (e) (3pts) Find all pure-strategy Nash equilibria. Write each equilibrium as a strategy profile, for example (A, D), rather than payoff pair (2,2). (f) (3pts) Determine if each Nash equilibrium is Pareto efficient.2. (12pts) Consider a variant of the Stag Hunt game. There are 10 hunters. Every hunter has two strategies: Stag or Hare. The value of a stag is V and the value of a hare is 1. If a player chooses Hare then the player always captures a hare and receives payoff 1 no matter what other players do. On the other hand, a stag can be caught if and only if at least 7 players choose Stag. The captured stag is equally shared by the players who chose Stag: for example, if seven hunters catch one stag, then each player will receive a payoff equal to =. Assume each hunter prefers 1/8 share of a stag to a hare, but prefer a hare to 1/9 (or smaller) share of a stag. In other words, - > 1> ?. (a) (2pts) Is all-players-choose-Stag a Nash equilibrium? Explain. (b) (2pts) Is all-players-choose-Hare a Nash equilibrium? Explain. (c) (2pts) Find all other Nash equilibria and explain why they are equilibria. (d) (4pts) Explain why the rest of the strategy profiles are not Nash equilbria. Be sure to offer pertinent reasons for different strategy profiles.3. (12pts) Consider a market with two firms and each firm produces the same product. The price of the product is determined by P = 60-Q, where O = Q1 + 02 is the total output of both firms. Both firms have zero marginal cost and zero fixed cost. (a) (3pts) First, find Firm I's best response function, namely optimal output Q, as a function of Firm 2's output Q2- (b) (1pts) Next, use symmetry to get Firm 2's best response function. (c) (4pts) Now, find the Cournot-Nash equilibrium for these two firms. How much profit does each firm earn in the equilibrium? (d) (4pts) If the two firms could collude how much would the total output be? How much is the total profit in collusion?4. (9ts) Two firms A and B are engaged in an advertising contest. Each firm can choose its advertising spending a 2 0 and b 2 0. Given their ad spendings a, b, Firm A's profit equals 1 =9a-a- - ab and Firm B's profit equals Mb = 12b- by - ab (a) (3pts) Find firm A's best response function, namely find A's optimal strategy a as a function of B's strategy b. (b) (3pts) Find firm B's best response function. (c) (3pts) Use the two best response functions to find the Nash equilibrium
