Consider the game below. How many best responses does Player 1 have to strategy L of Player 2? Player 1 T M B L 2,0 2,2 2,3 Player 2 с 2,1 3,2 1, 2 R 3, 2 0,1 0,3
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- Consider again the normal form of the Prisoner's dilemma game. Determine any Nash Equilibrium. Player 2 R (-10,-10) (-1,-25) (-25,-1) (-3,-3) C Player 1 RBased on the following game, what are the secure strategies for player 1 and player 2? Player 1 S1 and 12 S1 and t1 S2 and 12 S2 and 11 s1 S2 Player 2 tl 10,15 -10,7 t2 15,8 10,20In the following game, find the dominated strategies for each player and the reduced game and solve the pure equilibrium strategy. Also solve the mixed equilibrium strategy, if it doesn't exist, explain Y1 Y2 Y3 |(10,-10) (2,-2) X1 |(4,-4) (5,-5) X2 (3,-3) (4,-4) X3 (8,-8) (6,-6) (9,-9)
- Consider the following game. Firm 1 can implement one of two actions, A or B. Firm 2 observes the action chosen by Firm 1 and then decides whether to fight it or not. (-10, 20) F2 Fight A Don't fight -(30, 10) Firm 1 (-10, 0) Fight B Don't fight -(20, 15) (a) Consider the following strategy profile: Firm 1 chooses A; Firm 2 chooses fight if A, and fight if B. • this strategy profile is [Select] (b) Consider the following strategy profile: Firm 1 chooses B; Firm 2 chooses fight if A, and don't fight if B. • this strategy profile is [Select] O (c) Consider the following strategy profile: Firm 1 chooses B; Firm 2 chooses don't fight if A and don't fight if B. • this strategy profile is [Select] 00 F2 ()Consider the following sequential game. Player 1 plays first, and then Player 2 plays after observing the choice of Player 1. At the bottom of the decision tree, the first number represents the payoff of Player 1, while the second number represents the payoff of Player 2. For player 2, A stands for Accommodate and F stands for Fight. Player 1 Enter Player 2 A F A Stay out Player 2 F (16,30) (-6,18) In the Nash equilibrium of this game, player 2 earns Player 2 player 1 would play (0,40) (0,20) ✓an incentive to threaten F because, if player 1 believed him then so that player 2 would earn ✓. However, this threat isRefer to Table Left-Right-ZigZag. How many pure strategy Nash equilibria does this game have? Table: Left-Right-ZigZag Player 1 Drive left Driver right Zigzag 1) 1 2) 2 3) 3 4) 4 5) None of the above. Drive left (1,1) (-1,-1) (0,0) Player 2 Driver right (-1,-1) (1,1) (0,0) Zigzag (0,0) (0,0) (0,0)
- A game involves two players: player A and player B. Player A has three strategies a1, a2 and a3 while player B has three strategies b1, b2 and b3. Player B b1 b2 b3 a1 -40,30 70,20 -10,120 Player A a2 40,60 80,80 60,20 a3 -30,40 -50,110 150, -70 Assuming that this is a one-time game, answer the following questions: Is there any dominant strategy for each player? What is the secure strategy of each player. What is the Nash equilibrium of the game?14. You have baked a cake, but your two dear daughters won't stop fighting on who gets the biggest slice. To settle the dispute, to ask your dear daughter one (DD1) to cut the cake and your dear daughter two (DD2) to choose which piece she wants. (a) Draw the extensive form of the game. Let dear daughter one's strategies be "Cut Evenly" or "Cut Unevenly"; depending on what is on the platter, dear daughter two's strategies might in- clude "Take Big Slice", "Take Small Slice", or "Take Equal Slice". Assign payoffs to dear daughter one and dear daughter two that grow with the size of the slice that they receive. (b) Use backward induction to find the equilibrium outcome of this game. (c) Is the promise to take a small slice by DD2, if DD1 cuts unevenly, credible? Explain carefully. (d) After the rules are announced, dear daughter two says "It is not fair! I want to be the one who gets to cut the cake, not the one who chooses the slice!". Is dear daughter two's complaint valid? You are…W X Y Z 47, 15| 39, 41 45, 53 12, 56 In equilibrium, what is the probability that player 1 will use the pure strategy X in this game?
- (d) Consider a simultaneous-move game between two firms choosing to sell their product at either £6, £7 or £8. The actions and payoffs are given in the matrix below. Firm 2's Prices £6 £7 £8 Firm 1's prices £6 4, 5 3, 5 2, 1 £7 0,4 2, 1 3,0 £8 -1, 1 4, 3 0, 2 What are the Nash equilibria of this game? Game theory is often used by firms competing under an oligopoly as a means of determining their best strategy. Why is game theory a useful tool and which characteristics of an oligopoly make it particularly useful for firms competing in this market structure? One outcome of an oligopoly is that firms may have an incentive to collude. Explain some of the conditions that make collusion more likely to occur and how game theory can explain why collusive agreements often break down.B block allow 5,9 3, 11 high A 6,2 4,4 low In this normal form game, there is a Nash equilibrium where A chooses high; block high; allow low; block low; allow and B chooses2- Consider the following game. Player 2 Player 1 U 12, 2 | 3, 9 5, 8 4, 2 D (a) Find all the Nash equilibria, pure and mixed. (b) Suppose that the payoff of the column player u:(D, L) is reduced from 8 to 6, but all other payoffs remain the same. Again, find all the pure- and mixed-strategy Nash equilibria. (c) Compare the mixed-strategy equilibria in parts (a) and (b). Did this worsening in one of player 2's payoffs change player 2's equilibrium mixed strategy? Did it change player l's? Give some intuition.
