exams3solution_2008 - Econ 101A Final exam Mo 18 May, 2009....

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Econ 101A — Final exam Mo 18 May, 2009. Do not turn the page until instructed to. Do not forget to write Problems 1 and 2 in the f rst Blue Book and Problems 3 and 4 in the second Blue Book. 1
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Econ 101A — Final Exam Mo 18 May, 2009. Do not forget to write Problems 1 and 2 in the f rst Blue Book and Problems 3 and 4 in the second Blue Book .Goodluckonso lv ingtheprob lems ! Problem 1. Prisoner Dilemma Game with Altruism (40 points) Consider the standard Prisoner Dilemma game that we discussed in class. As you remember, the story is one of two prisoners, each of which has to choose between defecting (that is, confessing) and not defecting (keeping the mouth shut). The payo f s indicate the number of years in prison: 1 \ 2 Defection No Defection Defection 4 , 4 1 , 5 No Defection 5 , 1 2 , 2 1. Write the general de f nition of Nash Equilibrium. (3 points) 2. Using this de f nition, compute all the pure-strategy Nash equilibria of the game (that is, do not allow for probability distributions). (3 points) 3. Now comes the interesting part of the problem. Unlike in the classroom discussion, the prisoners are altruistic. The utility function of player 1 is a function both of his own payo f in years, π 1 , but also of the payo f of player 2, π 1 . Thus, player one’s utility is: U 1 = π 1 + απ 2 , with α 0 . Similarly for player 2: U 2 = π 2 + απ 1 .B r i e F y explain why the parameter α captures altruism, and discuss the special cases α =0 and α =1 . (4 points) 4. This implies that the matrix can be rewritten in terms of utility as 1 \ 2 Defection No Defection Defection 4(1+ α ) , α ) 1 5 α, 5 α No Defection 5 α, 1 5 α 2(1+ α ) , α ) Compute all the mixed-strategy equilibria of this new game, as a function of α, assuming α 0 . Call u (for Up) the probability that player 1 defects and l (for Left) the probability that player 2 defects. Discuss intuitively why altruism makes a di f erence – or does not make a di f erence – in this game. (15 points) 5. Under what values of α is ( s 1 = No Defection, s 2 = No Defection) an equilibrium in dominant strategies for this game? State clearly the de f nition of Dominant Strategy equilibrium and how it di f ers from Nash Equilibrium. (7 points) 6. Now let’s go back to the standard case with no altruism ( α ), but assume that the Prisoner’s Dilemma game is played twice, instead of once. That is, the game is repeated once. Each time, the matrix of payo f s above applies. Debate this assertion: “If the prisoners meet twice, we will observe the No Defection equilibrium even in the absence of altruism because the repetition provides an opportunity for collusion” Use backwards induction and be as precise as you can about your statements. (8 points) Solution to Problem 1 .
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This note was uploaded on 03/15/2010 for the course ECON 101a taught by Professor Staff during the Fall '08 term at University of California, Berkeley.

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exams3solution_2008 - Econ 101A Final exam Mo 18 May, 2009....

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