Consider an ultimatum game. The social norm is a 60:40 split, meaning the Proposer keeps 60 and the Responder receives 40. Let R represent how strong the Responder's private reciprocity motive is, such that his satisfaction at rejecting an offer of y < 40 is (40 - y ).R. Based on this information ,evaluate the four statement and explain ,using one or two sentence ,whether the statement are true or false A. The responder will reject any offer below 60R/(1+R)B. If R=0, then the responder will accept any offer C. R=1,the minimum rent that the Responder require to accept an offer is 40.D. If R = 1.5 , then the maximum rent that the Proposer can claim is 24 .

Suppose players A and B play a discrete ultimatum game where A proposes to split a $5 surplus and B responds by either accepting the offer or rejecting it. The offer can only be made in $1 increments. If the offer is accepted, the players' payoffs resemble the terms of the offer while if the offer is rejected, both players get zero. Also assume that players always use the strategy that all strictly positive offers are accepted, but an offer of $0 is rejected. A. What is the solution to the game in terms of player strategies and payoffs? Explain or demonstrate your answer. B. Suppose the ultimatum game is played twice if player B rejects A's initial offer. If so, then B is allowed to make a counter offer to split the $5, and if A rejects, both players get zero dollars at the end of the second round. What is the solution to this bargaining game in terms of player strategies and payoffs? Explain/demonstrate your answer. C. Suppose the ultimatum game is played twice as in (B) but now there…

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Consider the following coordination game: Player 2P1   Comedy Show Concert Comedy Show 11,5 0,0 Concert 0,0 2,2 a. Find the Nash equilibrium(s) for this game.b. Now assume Player 1 and Player 2 have distributional preferences.  Specifically, both people greatly care about the utility of the other person.  In fact, they place equal weight on their outcome and the other person’soutcome, ρ = σ = ½. Find the Nash equilibrium(s) with these utilitarianpreferences.c. Now consider the case where Player1 and Player2 do not like each other.  Specifically, any positive outcome for the other person is viewed as anegative outcome for the individual, ρ = σ = -1. Find the Nashequilibrium(s) with these envious preferences.

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Game Theory de Drew Fudenberg, Jean Tirole. Exercise 2.6**

Consider the game of Chicken in which each player has the option to “get out of the way” and “hang tough” with payoffs:       Get out of the way Hang tough Get out of the way 2,2 1,3 Hang tough 3,1 00   a. Find all pure strategy Nash equilibria, if they exist b. Let k be the probability that player 1 chooses “hang tough” and u be the probability that player two chooses “hang tough.” Find the mixed stragety Nash equilibria, if they exist

Consider a modified version of the ultimatum game, with altruism. There is a pie of size 50 to be split between Proposer (P) and Responder (R). In the first stage, P offers a share 0 ≤ x ≤ 1 of the pie to R. In the second stage, R accepts or rejects P's offer. If R accepts, the pie is split according to the proposed division, with P receiving (1 - x) x 50 and R receiving x x 50. If R rejects, both players receive none of the pie. But players' payoffs are not equal to how much of the pie they receive. In particular, payoffs in the event of an accepted offer are: Up (1x) x 50+ bx × 50 UR = x x 50+b(1-x) × 50 (Payoffs in the event of a rejected offer are 0 for both players.) Here, the parameter b > 0 measures how altruistic players are, i.e. how much "happier" a player is when his opponent receives more pie. a) Suppose b = 0.5. How much pie does P receive in the subgame perfect Nash equilibrium of the game? (specify the total amount of pie, not the share). b) Suppose b=0.5. What is P's…

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For the operating systems game, let us now assume the intrinsic superiorityof Mac is not as great and that network effects are stronger for Windows.These modifications are reflected in different payoffs. Now, the payoff fromadopting Windows is 50 X w and from adopting Mac is 15 + 5 X m;n consumers are simultaneously deciding between Windows and Mac.a. Find all Nash equilibria.b. With these new payoffs, let us now suppose that a third option exists,which is to not buy either operating system; it has a payoff of 1,000.Consumers simultaneously decide among Windows, Mac, and nooperating system. Find all Nash equilibria.