14 02 2023 E TS

Pagina 1

GAME THEORY - February 14, 2023 Last name: First name: ID #: SOLVE THE EXERCISES AND ANSWER THE QUESTIONS ON THESE SHEETS Exercise 1 5 points Find x such that the following bimatrix represents a potential game:   (x, 0) (2 , 1) (5 , 3) (3, 1) (2 , 2) (4 , 3) (4, 0) (3 , 1) (3 , 0)   Write the potential P such that p11 = 0. Using the potential, find all Nash equilibria in pure strategies, and explain what is the relation between the potential and the Nash equilibria of the original game. Answer of exercise 1 If x = 3, then the potential with p11 = 0 is P =   0 1 3 0 1 2 1 2 1   Since the potential represents, up to a constant, the utility of both players, we see that both the choice of strategies 1,3 (for the first and second player respectively) and the choice of strategies 3,2 are NE. 1 Exercise 2 5 points In a school board there are 10 professors that rank four students: Anne, Beth, Cecil and Dan. Their ranks are: 4 professors : A ≻ B ≻ D ≻ C 3 professors : D ≻ C ≻ B ≻ A 3 professors : B ≻ D ≻ C ≻ A Find the best student according to the simple majority rule, the Condorcet rule and the Borda rule. Explain your answer. Answer of exercise 2 By the simple majority rule Anne is the best student, since she is the first one according to 4 professors. The Condorcet winner is Beth (6 professors think she is better than Anne, and 7 professors think she is better than Cecil and Dan). The Borda winner is Beth (assuming that 4 points are assigned to the best student, 3 to the second best and so on, Ann gets 22 points, Beth 30, Cecil 19 and Dan 29). 2 Theory Questions Answer one and only one question (7 points). Only the question 2 may lead to the top grade 30 e lode. 1. Describe Nash bargaining theory, including the statement (but not the proof) of Nash bargaining…