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 u_{1}(x_{1})=x_{1}^{0.5} and
of player 2 is u_{2}(x_{2})=x_{2}.

**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, x_{1}+x_{2}≤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 u_{3}(x_{3})=x_{3}^{2}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
f^{α}(S,d)=argmax(s_{1}-d_{1})^{α}(s_{2}-d_{2})^{1-α}
where the maximum is taken over all (s_{1},s_{2})∈S
that satisfy s_{i}≥d_{i}, i = 1,2.

**4.1**Show that f^{α}satisfies INV (invariance) and IIA (independence of irrelevant alternatives).**4.2**Show that f^{α}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
u_{1}(x_{1})=x_{1}^{a},
of player 2 is u_{2}(x_{2})=x_{2}^{b},
of player 3 is u_{3}(x_{3})=ln(x_{3}+1),
and of player 4 is u_{4}(x_{4})=x_{4}^{2},
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
(x_{1},x_{2}) such that
x_{1}≥0,x_{2}≥0,
and x_{1}+x_{2}=1. Then Player 2 says "Yes" or "No".
If player 2 says "Yes" the game ends and the outcome is
((x_{1},x_{2}),0); if Player 2 says "No",
Player 1 again makes an offer (y_{1},y_{2}),
and Player 2 says "Yes" or "No". If player 2 says "Yes" the
game ends and the outcome is ((y_{1},y_{2}),1);
if Player 2 says "No", Player 2 makes an offer
(z_{1},z_{2}), and Player 1 says "Yes" or "No".
If Player 1 says "Yes" the game ends and the
outcome is ((z_{1},z_{2}),2); if Player 1 says
"No" the outcome is D. The utility functions of the
players are u_{1}((x_{1},x_{2}),t)=δx_{1}
and u_{2}((x_{1},x_{2}),t)=δx_{2},
and u_{1}(D)=u_{2}(D)=0. Both δ_{1}
and δ_{2} 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 δ_{1}=δ_{2}=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?