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- a W 3,5 3,4 8,4 0,0 3,3 8,9 y 0,1 5,9 9,8 Describe a strategy for player 1 that dominates x. O (1/3.0. 2/3) 1.0,0) O01.1) toExercise 6.8. Consider the following extensive-form game with cardinal payoffs: 1 R O player pay 000 2 1 M 3 b 010 O player 3's payoff 1 2 221 2 000 0 0 (a) Find all the pure-strategy Nash equilibria. Which ones are also subgame perfect? (b) [This is a more challenging question] Prove that there is no mixed-strategy Nash equilibrium where Player 1 plays Mwith probability strictly between 0 and 1.There are 2 players. They take Ston eS From the Pilt of 6 Stones. Player 1 can takt only 2 or 3 Stones. piayer 2 can taKeS only 2 or 4 StonesS. P layers take turns, observe dil Previous moves, and player 1 mover first. A pla yer loses if She can not make a legal move, While another player ir declared the Winner. Let the pay oF OF Winning egval to 1 and the payoFF OF losing equal to 0. a) represent the in ɛxtemsive Form (depict Only legalmoes) b) Find all SPNE OF this game& explain your a mwwer. Who Will Win ? game Only legalmar)
- 7. N [0.75] B A [0.25] 1 E F 6 2 J K J K 12 3 9. 6 6. 1 In equilibrium, what is the probability that player 1 will use the pure strategy E in this game?Consider the following game. Player S DE F A 10, 10 8, 7 8, 8 Player R B 8,7 9, 8 9, 12 C 9,8 7, 10 12, 7 Find the best response of step-2 Player S assuming that step-0 players (either Ror S) choose the strategy at random with equal probability. Select one: O a. F O b. E O c. B O d. A O e. C O f. D2. Consider the following Bayesian game with two players. Both players move simultaneously and player 1 can choose either H or L, while player 2's options are G, M, and D. With probability 1/2 the payoffs are given by "Game 1" : GMD H 1,2 1,0 1,3 L 2,4 0,0 0,5 and with probability 1/2 the payoffs are according to "Game 2" : G |M|D H 1,2 1,3 1,0 L 2,4 0,5 0,0 (a) Find the Nash Equilibria when neither player knows which game is actually played. (b) Assume now that player 2 knows which one among the two games is actually being played. Check that the game has a unique Bayesian Nash Equilibrium.
- Player 2 C D A Player 1 2,3 В 6, —2 | 4, 3 7,8 Select the value corresponding to the probability with which player 2 plays C in the mixed strategy Nash equilibrium: O 1/8 O 1/6 1/5 O 1/4 1/3 1/2 2/3 O 3/4 4/5 O 5/6In the final round of a TV game show, contestantshave a chance to increase their current winnings of$1 million to $2 million. If they are wrong, theirprize is decreased to $500,000. A contestant thinkshis guess will be right 50% of the time. Should heplay? What is the lowest probability of a correctguess that would make playing profitable?please solve it in 30 minute please Consider an infinitely repeated game where the stage game is the game of chicken: Swerve Straight Swerve 0,0 -1,1 Straight 1, –1 -10, -10 Assume that the discount factor is very close to one. For concreteness, you can assume that d = 0.99. (a) Is there an SPE in which player l's payoff is 1/(1 – 6)? Explain. (b) Is there an SPE in which each player gets a payoff of 0? Explain. (c) Is there an SPE in which player l's payoff is -2/(1 – 8)? Explain.
- 3. Find the saddle point, if it exists, for the following game. (b) Solve the following game by using the principle of dominance and find the probabilities of strategies for each player and the value of the game. Player B Player A II III IV V 3 4 4 II 2 4 III 4 4 IV 4 4 20 2420 87603. Consider the game below. С1 C2 C3 R1 1, 1 4, 6 8, 5 1, 2 5, 4 R2 R3 2, 6 2, 7 7, 6 0, 7 3.1. Does the game have any pure strategy NEs? 3.2. Check whether a mixed strategy NE exists in which A is mixing R1 and R2 with positive probabilities, playing R3 with zero probability, while B is mixing C1 and C3 with positive probabilities while playing C2 with zero probability. [Let (p1,P2, P3) be the probabilities with which A plays (R1, R2,R3) and let (q1,92, 93) be the probabilities with which B plays (C1, C2,C3). Make use of the following NE test: m* is a NE if for every player i, u;(mị , m²¿) = u;(Si, m²¡) for every si E S¡|m¡(sji) > 0 and u¡(m¡ ,m²¡) > u¡(s¡,m;) for every si E S¡ |m¡ (s¡) = 0. Hint: Each player must be indifferent between those of her pure strategies that are used (with positive probability) in her mixed strategy, and unused strategies must not yield a payoff that is higher than the payoff a player gets with her NE (mixed) strategy.] %3D0 E F 3 2 FIGURE 2 23 D G 2 H 0 1 (2) Two friends, Albert (A) and Berta (B), have to decide on going to one of three bars, Xenia's (X), Yara's (Y), or Zana's (Z). They decide to select a bar by alternatively vetoing bars until one remains. First Albert vetoes a bar. If at least two bars remain, then Berta vetoes another bar. That process continues until a single bar remains unvetoed. Suppose Albert prefers Xenia's to Yara's to Zana's and Berta prefers Zana's to Yara's to Xenia's. Assume that, for each of the two, going to their most preferred bar gives a utility of 2, going to their next preferred bar gives a utility of 1, and going to their least preferred bar gives a utility of 0. Model this as an extensive form game and find its Nash equilibria. Which of the Nash equilibria are subgame perfect?