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repeated games

Microsoft Word - Game Theory and Applications Extra Credit Question 2.docx Game Theory and Applications Extra Credit Question #2 (ECO 4400; Fall 2021; Romano) Rules: This is worth up to 5 points to be added to your final exam. (Actually, if you also answer part v, you can get up to 8 points.) This is open-book, open-notes, but you are pledged to work alone. It is due on Friday, Dec. 3 by 3pm. Either put a hard copy in my mailbox in the economics department office (MAT 224), or email a pdf file to me at [email protected]. Make your answer is neat and show your work. Do not forget to put your name on the pdf file you provide. Question: A partnership has n members that split equally the revenues they generate from their work efforts in each period. Let xi denote the work effort of partner i, which can take on any value from 0 to 100, in increments of 10. (0 and 100 are included.) The revenues generated by the partnership equals: 1 2 n1.8 (x x ... x ), or n jj 1 1.8 x using summation notation. Each partner bears their own effort costs, which are equal to their x value. Thus, the payoff to partner i in a period is given by: n jj 1 i i x V 1.8 x . n (i) What is the single-period Nash equilibrium effort choice? (ii) Suppose the partners play the game repeatedly with an infinite time horizon, each period simultaneously choosing their effort levels and receiving payoffs each period (as usual in a repeated game). All partners have discount factor δ. Find the condition on the discount factor such that grim trigger strategies adopted by each partner imply xi = 100 for every player i results in SPNE. (Notes: (a) The condition depends on n. (b) Write down the inequality that shows the basic trade off before you simplify it to find the condition on δ.) (iii) Is the condition harder or easier to satisfy as n grows? (iv) Now suppose that there is a probability p each period of continuing the next period, or, put differently, there is a probability of (1-p) that any period that is reached is the last one. What is the modified condition for “cooperation,” i.e., for the same question as in part (ii)? (v) Extra Extra Credit (This is for potentially 3 more extra credit points.) Return to the case of the infinitely repeated game (i.e., with p = 1). Rather than playing the grim trigger strategy to potentially get players to cooperate on playing 100 each period, suppose the partners play a strategy with T periods of “punishment,” that punishment being a reversion to T periods of choosing the single-period Nash equilibrium effort if anyone does not choose x = 100, this followed by a return to choosing x = 100. Suppose that n = 3 and δ = .9. How many periods must T be for the strategy to get players to choose x = 100 in SPNE? Show how you get your answer.

Microsoft Word - Game Theory and Applications Extra Credit Question 2.docx Game Theory and Applications Extra Credit Question #2 (ECO 4400; Fall 2021; Romano) Rules: This is worth up to 5 points to be added to your final exam. (Actually, if you also answer part v, you can get up to 8 points.) This is open-book, open-notes, but you are pledged to work alone. It is due on Friday, Dec. 3 by 3pm. Either put a hard copy in my mailbox in the economics department office (MAT 224), or email a pdf file to me at [email protected]. Make your answer is neat and show your work. Do not forget to put your name on the pdf file you provide. Question: A partnership has n members that split equally the revenues they generate from their work efforts in each period. Let xi denote the work effort of partner i, which can take on any value from 0 to 100, in increments of 10. (0 and 100 are included.) The revenues generated by the partnership equals: 1 2 n1.8 (x x ... x ), or n jj 1 1.8 x using summation notation. Each partner bears their own effort costs, which are equal to their x value. Thus, the payoff to partner i in a period is given by: n jj 1 i i x V 1.8 x . n (i) What is the single-period Nash equilibrium effort choice? (ii) Suppose the partners play the game repeatedly with an infinite time horizon, each period simultaneously choosing their effort levels and receiving payoffs each period (as usual in a repeated game). All partners have discount factor δ. Find the condition on the discount factor such that grim trigger strategies adopted by each partner imply xi = 100 for every player i results in SPNE. (Notes: (a) The condition depends on n. (b) Write down the inequality that shows the basic trade off before you simplify it to find the condition on δ.) (iii) Is the condition harder or easier to satisfy as n grows? (iv) Now suppose that there is a probability p each period of continuing the next period, or, put differently, there is a probability of (1-p) that any period that is reached is the last one. What is the modified condition for “cooperation,” i.e., for the same question as in part (ii)? (v) Extra Extra Credit (This is for potentially 3 more extra credit points.) Return to the case of the infinitely repeated game (i.e., with p = 1). Rather than playing the grim trigger strategy to potentially get players to cooperate on playing 100 each period, suppose the partners play a strategy with T periods of “punishment,” that punishment being a reversion to T periods of choosing the single-period Nash equilibrium effort if anyone does not choose x = 100, this followed by a return to choosing x = 100. Suppose that n = 3 and δ = .9. How many periods must T be for the strategy to get players to choose x = 100 in SPNE? Show how you get your answer.

Dec 03, 2021

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