Consider the following stage game: Player 2 C2 D2 A2 С1| 8,8 | 0,9 | 0,0 D1 9,0 0,0 3,1 Player 1 А1 | 0,0 | 1,3 | 3,3 Assuming that it is played twice, describe a subgame-perfect equilibrium in which (C1, C2) is selected in the first period.
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- Consider the following two-player game.First, player 1 selects a number x≥0. Player 2 observes x. Then, simultaneously andindependently, player 1 selects a number y1 and player 2 selects a number y2, at which pointthe game ends.Player 1’s payoff is: u1(x; y1) = −3y21 + 6y1y2 −13x2 + 8xPlayer 2’s payoff is: u2(y2) = 6y1y2 −6y22 + 12xy2Draw the game tree of this game and identify its Subgame Perfect Nash Equilibrium.parameter z= 1 In the unique subgame perfect Nash equilibrium, what is the sum of the payoffs tothe two players?Paramter y = 0 If ⟨a, d⟩ is played in the first period and ⟨b, e⟩ is played in the second period, whatis the resulting (repeated game) payoff for the row player?
- Consider the following game in strategic or normal form. A2 B2 C2 A1 1,0 1,2 -2,1 B1 6,2 0,3 2,3 C1 2,2 -2,1 2,3 Use the iterative elimination of strictly dominated strategies to reduce the game as much as possible. What is the set of rationalizable strategies for each player? What is/are Nash equilibrium(s) in this game?4. Correlated EquilibriaConstruct an example (not one from class or the reading) of a Normal form game with a correlated equilibrium that is not a Nash equilibrium.Jesse and Janie are playing a game. On the table in front of them, there are 50 coins. They take turns removing some of these coins from the table. At each turn, the person moving can remove either 1,2,3,4,5,6, or 7 coins (they cannot remove 0 or 8, etc.). The person to remove the last coin wins. Suppose Janie is the first player to have a turn (a) In the subgame perfect equilibrium, Janie begins by taking _________coins (b) Suppose Janie makes a mistake in play and removes a number of coins that leaves 27 coins on the table when it is jesses turn. To have a chance at winning, Jesse should remove _________ Coins
- . Find the Nash equilibrium of the following modified Rock-Paper-Scissors game: • When rock (R) beats scissors (S), the winner’s payoff is 10 and the loser’s payoff is −10. • When paper (P) beats rock, the winner’s payoff is 5 and the loser’s payoff is −5. • When scissors beats paper, the winner’s payoff is 2 and the loser’s payoff is −2. • In case of ties, both players receive 0 payoff. You are suposed to create a system of equations and then solve for them and find 3 probabilities- please show how to do thatH3. Cooperation in a Finite Game We have seen that if we play the Prisoners' dilemma with finite repetition, there will be no cooperation in the subgame-perfect equilibrium. There are other games, however, when cooperation can be enforced for at least some time in a finite repetition as well. Give an example with explanation. (Hint: consider games with more than one Nash equilibria.)E3 Bayesian Game]. Consider a Bayesian game described by a following payoff matrix. Please solve (show your solution). 1. Enumerate all pure strategies for each player. 2. Suppose that player 1 observes his type ?1 = 3. How does player 1 think of the probability of ?2? 3. Find a (pure strategy) Bayesian Nash equilibrium.
- Question 1 Consider a first-price sealed bid auction of a single object with two biddersj = 1,2 and no reservation price. Bidder 1′s valuation is v1 = 2, and bidder 2′s valuation isv1 = 5. Both v1 and v2 are known to both bidders. Bids must be in whole dollar amounts.In the event of a tie, the object is awarded by a flip of a fair coin.(a) Find an equilibrium of this game.(b) Is the allocation of your answer to (a) efficient?John enjoys playing two-player zero-sum games. The matrix below shows the losses to John in a particular two-player zero-sum game. His strategies are denoted by I, II, and III, whereas the strategies for his opponent are denoted by A, B, and C. I II III A 3 2 4 B 0 1 1 C 1 3 0 (a) Explain which of John’s strategies is dominated.The opponent now has the option of a fourth strategy, D, which results in none ofJohn’s strategies being dominated.(b) Suggest possible values for the strategy D.Consider a game where each player picks a number from 0 to 60. The guess that is closest to half of the average of the chosen numbers wins a prize. If several people are equally close, then they share the prize. The game theory implies that (A) all players have dominant strategies to choose 0 (B) all players have dominant strategies to choose 30 (C) there is a Nash equilibrium where all players pick 0 (D) there is a Nash equilibrium where all players pick positive numbers Behavioral data in such games suggests that (A) most subjects choose 0; (B) most subjects choose 30;(C) common answers include 30, 15, 7.5, and 0; (D) most subjects use randomization.
