Give detailed explanation...be sure of your answers!!
1 (i) Two candidates A and B run for office in an election where an odd number of voters must vote for one or the other (abstentions are not allowed). Each voter is a supporter of exactly one of the candidates, and they assign higher utility to the victory of their candidate than to the victory of the other candidate. (a) Describe this situation as a game in strategic form. (6 marks) (b) Find all pure-strategy Nash-equilibria of this game. (5 marks) (ii ) Suppose that two firms produce similar but not identical products, and that the unit costs of these products are f4 for firm 1 and 66 for firm 2. The prices of each of these two products depend on the production profile (g1, q2) of both products: p1 = 20 - 391 - 4q2 and p2 = 30 - 491 - 592. Assume that each firm i controls its production profile qi. (a) Find each company's production which is the best response to the other company's production. (6 marks) (b) Find the production profile which is a Nash-equilibrium. (4 marks) (iii) Alice and Bob play a game given in strategic form as follows: L M R. 11 0, 1 1, 5 2, 2 2. 5 5, 4 4, 9 d 3, 0 7, 4 8, 3 Solve this game. (4 marks)[iii] Consider a nite, two-player, zerosum game (3,3210. Show that 1ninmaxu{s,t} 3 maxmjna}. be!" 365' mes 1-51" {5 marks} A nite zerosum game G = {SJ'11:} is symetrio if S = T and for all 31., 32 E 5,1432, 31} = u{31,sg]. Let A = {a{i,j}} be the matrix associated with a symmetric game G = [{1, . . .114}, {1, . . .114}, a}. [a] Show that the value V of a symmetric game is zero. {5 marks} [b] Show that an optimal strategy for one player is also an optimal strategy for the other player. {5 marks} Consider the following zero-sum game given in tabular form Find an optimal mixedstrategy prole for this game where each strategy includes all pure strategies with positive probability. {3 marks} Must all optimal mixedstrategy proles of nite, symmetric, zerosum games be of the form {191?}? Justify your answer. {2 marks} Country A will either attack country E or not attack it. IfA attacks, E can ght, resulting in payo's of l for both, or retreat, resulting in a payoff of 5 to A and 3 to E. If A does not attack, B will either attack, resulting in a payo' of 2 to E and 2 to A, or B will not attack, resulting in a payoff of 1D to both. [a] Describe this game using a tree, carefully labelling all its compo nents. {5 marks} [b] Find all subgame perfect Nashequilibria of the game. {5 marks} [c] Describe the game in strategic form. {5 marks} [d] Find a purestrategy Nash-equilibrium of this game which is not subgame perfect. {5 marks} Consider a game identical to chess, except that I each player may choose to pass and not make a move when it is their turn, and I the game ends with a draw after two consecutive passes. Prove that white [who is the rst player to move] has a strategy that guarantees victory or a draw. {5 marks} 4 (i) Consider a 2-player game given in strategic form as (S, T, u1, 12). (a) Define the minimar values of both players. (2 marks) (b) Define the cooperative payoff region of the game. (2 marks) (c) Find the minimax values and sketch the cooperative payoff region of the following 2-person game G given in tabular form as follows A B 1, -1 1, 0 II 3, 3 -2, 1 (4 marks) (d) Show that the point (2, 2) is in the cooperative payoff region of G by writing it as a convex combination of payoffs. (2 marks) (e) Consider now the game G" which consists of playing G repeatedly, and where the payoffs of the infinite game are the average payoffs of the individual matches. Describe, without proof, a Nash-equilibrium which results in average payoff of 2 for both players. (6 marks) (ii) Alice wants to buy a diamond from Bob. The value of the diamond is C1000k for an integer 1
