Assignment 1. * by ECON6901 † Problem 1. Suppose there are 100 citizens and three candidates for a mayor: A, BandC, and no citizen is indifferent between any two of them. Each citizen can cast a vote for one of the candidate, the candidate who gets the majority wins. In case there is a tie for the first place there is a second round. Each citizen prefers his favorite candidate winning to a tie and prefers a tie to some other candidate winning. Show that a citizen’s only weakly domi- nated action is a vote for her least preferred candidate. Find a Nash Equilibrium in which some citizen does not vote for her favorite candidate, but the action she takes is not weakly dominated. Problem 2. (Finding Nash Equilibria using best response functions) Find the Nash Equlibria of the two-player strategic game in which each player’s set of actions is the set of nonnegative numbers and the players’ payoff functions are u 1 (a 1 ,a 2 )= a 1 (a 2 - a 1 ) and u 2 (a 1 ,a 2 )= a 2 (1 - a 1 - a 2 ). Problem 3. (Games with mixed strategy equlibria) Find all the mixed strategy Nash Equi- libria of the strategic game in Figure 1. L R L R T 6,0 0,6 T 0,1 0,2 B 3,2 6,0 B 2,2 0,1 Figure 1. Problem 4. (A coordination game, solve for the case of c< 1) Two people can perform a task if, and only if, they both exert effort. They are both better off if they both exert effort and perform the task than if neither exerts effort (and nothing is accomplished); the worst outcome for each person is that she exerts effort and the other person does not (in which case again nothing is accomplished). Specifically, the players’ preferences are represented by the expected value of the payoff functions in Figure 2, where c is a positive number less than 1 that can be interpreted as the cost of exerting effort. Find all the mixed strategy Nash equilibria of this game. How do the equilibria change as c increases? Explain the reasons for the changes. No effort Effort No effort 0, 0 0, - c Effort - c, 0 1 - c, 1 - c Figure 2. Problem 5. Consider an all pay auction with two bidders who value the object at 2 and 1 respectively. Find the mixed strategy equilibrium in this setting. *. This document has been written using the GNU T E X MACS text editor (see www.texmacs.org). †. Document transcribed by Jorge Rojas. 1