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- Exercise 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.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 5/67. 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?
- 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) to3. 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.] %3DConsider the following normal form game Player 2 L C R Player 1 5,4 4, 5 3,8 8,3 8,0 8,8 5,5 6,3 0,8 UMD
- 2. 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.A game is played as follows: First Player 1 decides (Y or N) whether or not to play.If she chooses N, the game ends. If she chooses Y, then Player 2 decides (Y or N) whetheror not to play. If he chooses N the game ends. If he chooses Y, then they go ahead and playanother game with the payoffs shown below. A player who opts out by choosing N gets 2 andthe other player gets 0. Draw the tree of this game and then find the two subgame-perfect Nashequilibria.Player 2 E F H A 6, 5 6, 7 9, 6 7,6 В Player 1 C 6, 7 6, 9 8, 5 9, 7 5, 8 5, 6 7,5 7,5 7,9 8, 7 11, 6 5, 6 (1) In the Unique Nash equilibrium of this game, which strategy does Player1 play? And why? (2) In the Unique Nash equilibrium of this game, which strategy does Player2 play? And why? (3) Is this game dominance solvable? And Why? (4) Does this game have at least one inadmissible Nash equilibrium? And Why?
- 1. A dealer decides to sell a rare book by means of an English auction with a reservation price of 54. There are two bidders. The dealer believes that there are only three possible values, 90, 54, and 45, that each bidder’s willingness to pay might take. Each bidder has a probability of 1/3 of having each of these willingnesses to pay, and the probabilities for each of the two bidders are independent of the other’s valuation. Assuming that the two bidders bid rationally and do not collude, the dealer’s expected revenue is approximately ______. 2. A seller knows that there are two bidders for the object he is selling. He believes that with probability 1/2, one has a buyer value of 5 and the other has a buyer value of 10 and with probability 1/2, one has a buyer value of 8 and the other has a buyer value of 15. He knows that bidders will want to buy the object so long as they can get it for their buyer value or less. He sells it in an English auction with a reserve price which he must…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. D3 0: she makes a pront of $9 An is ISK AVCise and aunty she makes. Let P be the probability that it rains the day of the concert. Find the number P tha makes Ann indifferent between organizing the concert in the Club or at the Park. 8. Compute the mixed strategy Nash equilibrium of the game below: Bungle F2 # Zippy E Left Right DOD F4 Up 2,-1 R I 20 F3 $ BAG 3 4 -2,0 HOT % 5 F5 Down 3,1 T 4,-2 VO MacBook Air 6 F6 Y & 7 F7 * U 8