Universitat Pompeu Fabra, 1995/6 and 1996/7
1. There are two players who may divide 1 dollar between them. The (von Neumann-Morgenstern) utility function of player 1 is u1(x1)=x10.5 and of player 2 is u2(x2)=x2.
2. Consider all pairs of players of question 1 above. Assume that if they do not reach an agreement both get 0 dollars. Calculate, for the two different cases, the utilities the players will get according to the Nash solution. How much money each player gets?
3. There are to possible events R (rain) and S (sun). The lottery that gives the decision maker x(R) dollars if it will rain tomorrow and x(S) dollars if it will be sunny is donated by [x(R),x(S)]. Assume that x(R) and x(S) cannot be negative. The preferences of the decision maker on lotteries is presented by the function V[x(R),x(S)]=[x(R)]0.75[x(S)]0.25. Can his preferences be presented by subjective probabilities and a von Neumann-Morgenstern utility function? If so what are the probabilities and the function?
4. Consider the following bargaining solution: Let 0<α<1. Let fα(S,d)=argmax(s1-d1)α(s2-d2)1-α where the maximum is taken over all (s1,s2)∈S that satisfy si≥di, i = 1,2.
6. There are four players. The (von Neumann-Morgenstern) utility function of player 1 is u1(x1)=x1a, of player 2 is u2(x2)=x2b, of player 3 is u3(x3)=ln(x3+1), and of player 4 is u4(x4)=x42, where 1>a>b>0.
7. Find the mixed strategy equilibria of the games "chicken" and the "prisoners dilemma."
8. Consider a game with two players similar to Cournot duopoly, in which the players choose the quantities they produce sequentially; i.e., Player 1 chooses a (nonnegative) quantity, and after that, player 2 chooses a nonnegative quantity. The payoffs functions are like those in the Cournot duopoly.
In the questions below consider only pure strategies.
9. Consider the following bargaining game. In the first round, Player 1 makes an offer (x1,x2) such that x1≥0,x2≥0, and x1+x2=1. Then Player 2 says "Yes" or "No". If player 2 says "Yes" the game ends and the outcome is ((x1,x2),0); if Player 2 says "No", Player 1 again makes an offer (y1,y2), and Player 2 says "Yes" or "No". If player 2 says "Yes" the game ends and the outcome is ((y1,y2),1); if Player 2 says "No", Player 2 makes an offer (z1,z2), and Player 1 says "Yes" or "No". If Player 1 says "Yes" the game ends and the outcome is ((z1,z2),2); if Player 1 says "No" the outcome is D. The utility functions of the players are u1((x1,x2),t)=δx1 and u2((x1,x2),t)=δx2, and u1(D)=u2(D)=0. Both δ1 and δ2 are larger than zero and smaller than 1.
What are the subgame perfect equilibria of this game?
10. Consider bargaining game similar to the above in which Player 1 makes offers for n rounds, and Player 2 makes an offer only in the n+1 round, which is the last round.
11. Consider again the game of question 9. However assume now that δ1=δ2=1.
12. Consider the infinite horizon bargaining game with the following modification. After every proposal of player 1, player 2 has three options: accept, reject, and quit. The first two are like in the original game; if player 2 chooses quit, the game ends, player 1 receives nothing and player 2 receives b dollars. 0<b<1. (After proposals of player 2, player 1 does not have the option of quit; i.e. this additional option is given only to player 2).
13. Consider a market with three players. There are two goods: cars and money. All players have 2 dollars each. Player 1 has a car which is equivalent according to his preferences to 0 dollars. Players 2 and 3 do not have cars; player 2's evaluation of player 1's car is 1 dollar, and player 3's evaluation is 2 dollars.