Answer Happy

Introduction to Game Theory Spring 2022 Problem set 2 Due: February 15 Be sure to show all your work for partial credit. 1. (15 points) Two people are engaged in a joint project. If each person i puts in efforts the result of the project is worth fixi, x3). Each person's effort level x is a non-negative number and effort costs ex). The worth of the project is split equally between the two people, regardless of their effort levels, so the payoff of each player is RX, X2)2-(X.). Draw the players' Best Response functions and find the Nash equilibria when n. 2) 3x1x2 and x) = x2for i=1,2. Provide a brief interpretation 2. (15 points) Consider a model of Cournot competition as studied in class with 2 fims and a linear inverse demand function PQ) - 4-(where Q = 91 +92 is the total quantity produced by the two firms and a is a parameter). The firms have different marginal costs: c for Firm I and ca for Firm 2. (a) Find the Nash equilibrium (b) Assume Firm I's marginal cost is larger (c>c). Which firm produces more in equilibrium? How do the quantities produced in equilibrium change if Firm 1 improves its technology, leading to a slightly lower (while ez is unchanged)? (c) Find the total quantity produced and each firm's profit in equilibrium. Describe what happens to these when Firm I changes its technology as above. 3. (10 points) Consider the Hotelling model studied in class but assume the "city" is a square, and consumers are distributed uniformly over this square. Otherwise the game is the same as in class: 2 firms choose locations, consumers buy one unit from the closest firm, firms maximize the number of units sold. Find the Nash equilibrium 4. (10 points) Find all the Nash equilibria (both pure and mixed) of the 2-player game below. MR A-1,1 3.1 1.2 B 2,2 2,-1 0,0 C 0.0 0.2