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- II. Determine which of the following two-person zero-sum games are strictly determinable and fair. Give the optimum strategies for each player in the case of it being strictly determinable. a- Player A Player B B1 B2 B3 A1 8 2 4 A2 5 -1 3 b. II II V V I -4 -2 -2 1 A II 1 -1 III -6 -5 -2 -4 4 IV 3 1 -6 0 -8 3.Solve for the Nash equilibrium (or equilibria) in each of the following games. (a) The following two-by-two game is a little harder to solve since firm 2’spreferred strategy depends of what firm 1 does. But firm 1 has a dominantstrategy so this game has one Nash equilibrium. Firm 2 Launch Don’tFirm 1 Launch 60, -10 100, 0 Don’t 80, 30 120, 0 What is the Nash equilibrium of this simultaneous-move game? (b) What would the outcome of this game be if instead firm 1 moved first and then, after seeing what firm 1 chose, firm 2 chose it strategy? In this case firm 1 doesn’t necessarily need to choose a best response, but firm 2 must choose a best response since it moves second.3. Determine the Nash Equilibrium for the following normal form of the game. A) MARY B) UP DOWN PLAYER 1 HARLEY UP (10, 10) (4,8) LEFT (3,0) UP MIDDLE (1, 1) DOWN (0,1) DOWN (8,4) (5,5) PLAYER 2 CENTER RIGHT (2,1) (1,1) (4,2) (0,0) (5,0) (0,1)
- Explain the concept of two-person zero- sum game and saddle point. Also provide a suitable example. Determine which of the following two-person zero-sum games are strictly determinable and fair. Give the optimum strategies for each player in the case of it being strictly determinable. Player A Player B B2 вз 4 B1 A1 5 -1 2 A220) Given the payoff matrix in the figure, the Nash equilibrium outcome of this game is for: Ajinomoto Produce 30 Produce 40 million pounds million pounds Ajinomoto makes $180 million Ajinomoto makes $200 million Produce 30 million ADM makes $180 million ADM makes $150 million pounds Ajinomoto makes $150 million Ajinomoto makes $160 million Produce 40 million ADM makes $200 million ADM makes $160 million pounds O A. each firm to produce 30 million pounds. B. each firm to produce 40 million pounds. C. ADM to produce 30 million pounds and for Ajinomoto to produce 40 million pounds. D. ADM to produce 40 million pounds and for Ajinomoto to produce 30 million pounds. ADMConsider the following sequential game: 1 O D,L U.L U,R U D,R D (0,2) 2 L What is the subgame perfect Nash equilibrium of this game? R. (-1,-1) (1,1)
- 2. Consider the following game: u 1 r 6,0 0,6 d 3,2 6,0 Are there any Nash equilibria in pure strategies for this game? Find the mixed strategy Nash equilibrium.Problem 4 Consider the following two-person game Player 2 R 0,1 2, 3 Player 1 M 5, 9 7, 3 1, 7 5, 5 10, 10 1, 5 a) Find the dominant strategies for both players. b) Explain why (B, L) is not a Nash Equilibrium. c) Find the Nash equilibrium of the above game2. Provide the normal-form expression to the extensive-form game below, and find all pure-strategy Nash equilibria there. Find the subgame-perfect Nash equilibrium then. A B X E Y B A B F t G F G BF B D A 0 3. Find the subgame perfect Nash equilibria (may or may not be unique) in the game below. A B 100 30 F 60 9 5 B 3 5 12 10 60 10 80 10 180 60 200 40
- Suppose now we alter the game so that whenever Colin chooses "paper" the loser pays the winner 3 instead of 1: rock paper scissors rock 0. -3 1 1. раper scissors -1 -1 3 (a) Show that xT= (,) and yT= (5) together are not a Nash equilibrium 3'31 for this modified 3'3 game. (b) Formulate a linear program that can be used to calculate a mixed strategy x € A(R) that maximises Rosemary's security level for this modified game. (c) Solve your linear program using the 2-phase simplex algorithm. You should use the format given in lectures. Give a mixed strategy x E A(R) that has an optimal security level for Rosemary and a mixed strategy y E A(C) that has an optimal security level for Colin.2. Consider the two-by-two bimatrix game listed below: Left Right (2,-2) (1,-1) Up Down (3,-3) (-1,1) where the player "Row" chooses the action "up" or "down" and the player "Column" chooses the action "left" or "right". Describe a mixed strategy Nash Equilibrium for this game.12) What is a Nash equilibrium? Why do we generally think that Nash equilibria will be likely outcomes of games?
