Problem 2. Below-Nash punishment. Consider an infinite repeated game where the stage game is given as follows: Player 2 L C R T 10,10 3, 15 0,7 Player 1 M 15,3 7,7 -4,5 B 7,0 5,-4 -15, 15 (a) Find all pure-strategy Nash equilibrium of the stage game. (b) Using Folk Theorem, show that there exists a discount factor 8 < 1 such that a pure strategy subgame perfect equilibrium in which the players play strategy profile TL in every period along the equilibrium path. (c) Prove that off-equilibrium path punishment of MR, TC, TC,... for player 1, and BC, ML, ML,... for player 2 is a support for equilibrium.

parameter z= 1 In the unique subgame perfect Nash equilibrium, what is the sum of the payoffs tothe two players?

1) What are the Nash equilibria? Which one is unreasonable/non-credible threat? 2) What are the subgame perfect Nash equilibria? Does SPNE concept eliminate the unreasonable Nash equilibrium?

3) Consider the following two extensive form games and answer the following questions. How many subgames present in the following games? Find all pure strategies SPNE. Find all pure strategies NE of these games. Game A Game B u (5,3) u (0,2) 2 2 L L (-5,4) d (0,0) 1 1 1 (1,-5) (-10,-10) u R (2,3) A M 2 R 2 r d 1 (8,1) B (2,0) (2,2)

Suppose now we alter the game so that whenever Colin chooses "paper" the loser pays the winner 3 instead of 1: rock paper scissors rock 0. -3 1 1. раper scissors -1 -1 3 (a) Show that xT= (,) and yT= (5) together are not a Nash equilibrium 3'31 for this modified 3'3 game. (b) Formulate a linear program that can be used to calculate a mixed strategy x € A(R) that maximises Rosemary's security level for this modified game. (c) Solve your linear program using the 2-phase simplex algorithm. You should use the format given in lectures. Give a mixed strategy x E A(R) that has an optimal security level for Rosemary and a mixed strategy y E A(C) that has an optimal security level for Colin.

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.

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All parts s answer needed

Consider the following game. Which one of the following statements is TRUE?    1. There are 8 subgames for this extensive-form game.   2. There are 8 proper subgames for this extensive-form game.   3. There is no Subgame Nash Equilibrium for this game.   4. There exists some Subgame Nash Equilibrium that gives the payoff 7 to Player 1 and 3 to Player 2.