Exercises in Bargaining

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.

1.1. Calculate and draw the set of possible pairs of utilities that the players can get assuming that they may also divide amounts smaller than 1 dollar. That is, x1+x2≤1.
1.2. Assume now that the agents can also conduct lotteries over the money they divide. Does their set of possible utilities change?
1.3. Assume now that instead of player 1, another player (player 3) with the utility function u3(x3)=x32 is playing with player 2. Answer 1.1. and 1.2. for the new problem.
1.4. What is the maximum utility player 3 can get whenever player 2 receives the utility of a half. What is the agreement that gives these utilities?

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.

4.1 Show that fα satisfies INV (invariance) and IIA (independence of irrelevant alternatives).
4.2 Show that fα satisfies SYM (symmetry) if and only if α=0.5.

5

5.1 Show that the dictatorial solution w.r.t player 1 satisfies PAR (Pareto optimality) and IIA.
5.2 How are the bargaining solutions in the first question related to the dictatorial solution?

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.

6.1 Are all the players risk averse?
6.2 Can you tell which player is more risk averse than which? Prove your answer.
6.3 If players 1 and 2 would bargain over a dollar, who would receive a larger share? Base your answer on the Nash bargaining solution.
6.4 Is the set of possible utility pairs derived from dividing a dollar (without lotteries) between players 1 and 4 convex?

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.

8.1 Is the quantities chosen in the Nash equilibrium of the Cournot duopoly can be chosen in a Nash equilibrium in the above described game? If so, propose strategies that constitute a Nash equilibrium, and induce this outcome.
8.2 What are the subgame perfect equilibria of the game? Are there mixed strategy subgame perfect equilibria?

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.

10.1 What are the subgame perfect equilibria of this game?
10.2 Does the SPE payoff converge when n goes to infinity? If so, what are the limit payoffs?

11. Consider again the game of question 9. However assume now that δ12=1.

11.1 What are the subgame perfect equilibria of this game?
11.2 Does the SPE payoff converge when n goes to infinity? If so, what are the limit payoffs?

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).

12.1 Propose strategies that constitute a subgame perfect equilibrium. Distinguish between three cases 1. b<x; 2. b>x; and 3. b=x. Where x is the amount of money player 1 proposes to player 2 in the subgame perfect equilibrium of the usual game (without the modification).
12.2 In the alternating offers games we have studied, an agreement is always reached without delay. Under which circumstances an agreement may be reached with some delay? (Hint: this is a very difficult question; read the book Bargaining and Markets). Do not try to give a complete answer to the question.

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.

13.1 What is the coalitional game associated with this market?
13.2 What is the core of this coalitional game?
13.3 Draw supply and demand curves for cars. What are the competitive equilibria?
13.4 In the core of the above game player 1 always sells the car to one of the other players. What is the game that involves only player 1 and the player that buys the car? What is the core? Why it is different than the core of the three player game?