Consider the repeated game in which the following stage game is played twice in succession, with payoffs averaged between the two rounds (and no discounting). d e 0,0 b| 0,10 18, 18 48,10 10, 48 40, 40 a 4y, 4y 10,0 0,0 Set the parameter y equal to 1 (a) If (a, d) is played in the first period and (b, e) is played in the second period, what is the resulting (repeated game) payoff for the row player? (b) What is the highest payoff any player can receive in any subgame perfect Nash equilibrium of the repeated game?

Consider the attached extensive-form game tree, where player 1 moves first, then player 2. The top payoff accrues to player 1, the bottom payoff to player 2. (a) Draw the strategic form for this extensive form game.(b) Find all of the Nash equilibria, including any mixed.(c) Which of these Nash equilibria do you think would be actually played? Why?

1. Consider a sequential continuum game with two players, in which Player 1 first chooses a continuous variable £₁, and then Player 2 selects £2, with payoff functions ₁ and 7₂. Find the best response for P2, as the function î2 = BR₂(x1), for each given #₂: (a) (b) (c) T₂(x1, x₂) ) = (1400 — x1 — x2)x2 − 13x² T₂(x1, x₂) T₂(X1, X2₂) = : 100x2 + 3x1x2 − 14x1 – 30x1x² = 15x2 x² + x² +1

Please try to solve it in 30 minute

a) Clearly indicate the eliminated dominated strategies. Apply the EIE algorithm, and determine the solution set of the game in normal form. b) Assuming that each player would not choose strategies that are strictly dominated, determine the Nash equilibria of the reduced game J2 (3,1) (4, 2) (2, 0) (0, 4) (1, 5) (3,0) (1, 3) (2, 2) a J1 (4, 1) (2, 5) (2,6) (5, 2)

2, 2 6,1 1,1 Player 1 B 1,6 5,5 1,1 C 1,1 1,1 4,4 Consider the simultaneous move game represented in normal form by this payoff matrix. Suppose that the game is repeated for two periods and the players know that the game will end at the end of two periods. They observe the first period outcome before they move to the second period. Assume that there is no discounting, i.e. 2nd period payoffs are not discounted. Which of the following outcomes could occur in some subgame perfect equilibrium (SPE) of this repeated game? Choose True if you think the outcome can be a SPE, otherwise choose False. i. (B, B) is played in the first period, (C, C) is played in the second period. ii. (A, A) is played in both periods. iii. (C, C) is played in both periods.

In the normal form game below, the payoff matrix depends on the parameter a. 1 1 2 ABC a 0,2 6,4 -2,4 b 0, -24 -2a, 2a 8, 2 C 4,0 0,4 0, 14 Find the values of a, for which at least one pure strategy Nash equilibrium exists. Compute the value of a for which the expected payoff is the same for both players when the mixed strategy profile (01,02) = ((1/3,2/3,0), (0,1/4,3/4)) is played. Find the best response of player 1 (as a function of the parameter a) to player's 2 mixed strategy o2 = (1/2,1/2,0). Assuming a = 0 eliminate iteratively all dominated strategies and find a mixed strategy Nash equilibrium in this game.

Problem 4: Consider an infinitely repeated game, where the base game is the following 2-person 2x2 game: A A 0,0 10, 10 S1: choose A always S2: choose B always B 10, 10 0,0 Assume both players discount the future at the same rate of r, 0 < r < 1. Limiting each player's strategies to the following six possibilities, S3: Choose A then mimic the other player's previous choice S4: Choose B, then mimic the other player's previous choice S5: Choose A, then choose the opposite of the other player's previous choice S6: Choose B, then choose the opposite of the other player's previous choice a. present the strategic form of this game, b. identify all pure-strategy Nash equilibria c. does repetition with these strategies "solve" the coordination dilemma that confronts the players in the single play of the above game.

Is the set of SPE of any extensive form game with perfect information is identical to the set of Nash Equilibria of the induced normal form game? If yes, provide a proof. If no, provide an explanation by illustrating a game. [Maximum word limit for this question is 80 words.]

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?