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\f. - {1 Consider a binary choice version of the moonlighting game 1]] which two players; 1 1:: er 2, are both endowed with 12. Player 1 can choose either G) to give away 2, III WhICh \"as P2 may 2 receives the quadrupled amount 8, or I) to take 3, in which case player 2 loses 3. Mayer . ' 1 then choose either R) to reward player I with 3 at a'cost to player 2 of 3, or P) W Pun-'Sh player and reduce 1's payoff by 6 at a cost of 2 to player 2. Thus, player 1's strategy set is {6359"} s and player 2's strategy set is {R, P}. 1. Calculate each player's payoffs, [1:2, n2}, for every strategy prole (GR, GP, TR, TP): and enter them in the game tree on the cover sheet. Egoistic preferences Suppose for problems 2 to 4 that players haVe egoistic preferences, specically, at: = x, ,1\" =12 . As usual, justify claims with reference to values. 2. What is 2'5 best reSponse (R or P) to 1 choosing G? 3. What is 2's best response (R or P) to l choosing T? 4. Which strategy prole (GR, GP, TR, or TP) is the subgame perfect Nash equilibrium? Levine Model For the remaining questions in this problem set, apply the Levine model, assuming that both players have the following preferences: (I, 'l' 1a,- 1 + it where, as usual, in denotes player i's material payoff, a,- player i's altruism coefcient, and ii the reciprocity parameter. For simplicity, also assume throughout that .1. = 1, so that the utility function becomes: - Hi=llft+ II} {II + 1} 2 Further, suppose there are three types of players. The altruistic type of player has an altruism coefcient an = (1.5, the neutral type of player (Levine calls this type \"selsh\") has an altruism coefcient an = U, and the spiteful type has an altruism coefcient as =. ~05. Assume that these types occur with probabilities pa = .5, Pa = .25, and p, = .25, and that this distribution is the same among both players 1 and players 2. Assume also common knowledge, i.e., all players know the rules of the game and the distribution of types. Applying the concept of perfect Bayesian Nash equilibrium, use backward induction and identify strategies consistent with beliefs about types based on Bayes' rule. Thus, we begin with player 2. u1=tti+ l1} 5. Suppose initially for problems 5 to 11 that player 1 chose G and that this signals with certainty that player 1 is altruistic, i.e., p(1 = all?) = 1. To determine whether player 2 should reward or punish, you have to calculate separately her utility for both strategies under all three assumptions about her type. To begin, calculate player 2's utility, assuming she rewards and is altruistic. Enter this and the next six values in the rst row (Give) of the table on the cover sheet. 6. Now calculate 2's utility assuming she rewards and is neutral. 7- - 1 - . . What Is 2 s utility, ifshe rewatda and is spiteful? 3. New calculate 2's utility, 9. What is 2' if she punishes and is altruistic. 8 utility, if she punishes and is neutral? 10. What is 2': utility, if she punishes and is spiteful? :9- Finallra ifplaycr 1 gives, what fraction ofplayers 2 will reward, i.e.. what is p-(RlG) equal 12. Now suppose for problems 12 to 19 that player 1 chose Taud that this signals with certainty that player 1 is NOT altruistic, p = all\") = O (i.e., player 1 must be neutral or spitetlj. First, calculate the expected value of player 1's altruism eoefeient and enter this in the space below the table. 13. Now nd player 2's expected utility, sssmning she rewards and is altniistie. Enter this and the next six values in the appropriate spots of the second row (T take) of the table on the cover sheet 14. Now calculate 2's expected utility assuming she rewards and is neutral. 15+ What is 2'5 expected utility, if she rewards and is spiteful? 16. Now calculate 2's expected utility, if she punishes and is altruistic. 1?. What is 2's expected utility. if she punishes and is neutral? 18. What is 2'5 expected utility, if she punishes and is spitefid? 19. Finally, if player 1 takes, what -action ofplayers 2 will reward. i.e.. what is p(R|T} equal to? 20. We turn new to player 1. From 1's position as the rst mover, what is the expected value of player 2's altruism coefcient? 21. Suppose for problems 21 to 25 that player 1 gives. What is the expected value of player 1's payoff, E(1)? Enter the answer to this and the following four questions in the appropriate spots of the first row (Give) of the table on the cover sheet. 27. From player 1's position, what is the expected value of player 2's payoff, E(12)? 23. What is player 1's expected utility, E(u,), if he gives and is altruistic? 24. What is player 1's expected utility, E(u,), if he gives and is neutral? 3 25. What is player I's expected utility, E(u, ), if he gives and is spiteful? 26. Suppose now for problems 26 to 30 that player 1 takes. What is the expected value of player I's payoff, E(n, )? Enter the answer to this and the following four questions in the appropriate spots of the second row (Take) of the table on the cover sheet. 27. From player I's position, what is the expected value of player 2's payoff, E(n2)? 28. What is player I's expected utility, E(u,), if he takes and is altruistic? 29. What is player I's expected utility, E(u, ), if he takes and is neutral? 30. What is player I's expected utility, E(u, ), if he takes and is spiteful? 31. What fraction of players I will give? 32. What are the probabilities of the various outcomes (GR, GP, TR, or TP)? Enter these in the table on the cover sheet