Based on courses taught at Brown University, Fall 1999, and Universitat Pompeu Fabra, 1995/6 and 1996/7

The course discusses game theoretic models of bargaining. Both the axiomatic and the strategic approaches are considered. The Nash bargaining problem and alternating offers models will be the main representatives of the the two approaches respectively.

The course applies game theory to study bargaining, however, no prior knowledge of game theory is required in order to participate. The necessary tools from game theory will be taught as a part of the course.

- Martin J. Osborne and Ariel Rubinstein,
*Bargaining and Markets,*Academic Press, San Diego CA, 1990. - Martin J. Osborne and Ariel Rubinstein,
*A Course in Game Theory*, MIT Press, Cambridge MA, 1994.

- Introduction: game theoretic modelling of economic situations.
- Decision making under risk: the St. Petersburg paradox, and the expected utility hypothesis.
- The axiomatic approach: Nash's bargaining problem and solution.
- Applications of the Nash solution
- Strategic games in normal form: Nash equilibrium.
- Strategic games in extensive form: Nash equilibrium, backward induction and subgame perfect equilibrium.
- The strategic approach: a finite horizon alternating offers bargaining game.
- The relationships between the axiomatic and strategic approaches.

**1.** 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?

**2.** A rich man is forced to play a version of Russian Roulette: The
cylinder of a six-shooter revolver containing *two* bullets is spun and the barrel
is then pointed at his head. He is now offered the opportunity of paying money to have the
two bullets removed before the trigger is fired. It turns out that the payment can be made
as high as $10,000,000 before he becomes indifferent between paying and taking the risk of
getting shot.

**2.1**Why would the rich man also be indifferent between having the trigger pulled when the revolver contains*four*bullets and paying $10,000,000 to have*one*of the bullets removed before the trigger is pulled? (Assume that his preferences satisfy the expected utility axioms.)**2.2**Why would the rich man not be willing to pay as much as $10,000,000 to have*one*bullet removed from a revolver containing only one bullet?

**3.** 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}.

**3.1.**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.**3.2.**Assume now that the agents can also conduct lotteries over the money they divide. Does their set of possible utilities change?**3.3.**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.**3.4.**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?

**4.** 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?

**5.** 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.

**5.1**Show that f^{α}satisfies INV (invariance) and IIA (independence of irrelevant alternatives).**5.2**Show that f^{α}satisfies SYM (symmetry) if and only if α=0.5.

**6.** The dictatorial solution is defined as follows:
s*_{1} = argmax {s_{1} | s_{2}≥d_{2}, s∈S}

s*_{2} = argmax {s_{2} | s_{1}=s*_{1}, s∈S}

Show that the dictatorial solution w.r.t player 1 satisfies Pareto efficiency and IIA.

**7.** 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.

**7.1**Are all the players risk averse?**7.2**Can you tell which player is more risk averse than which? Prove your answer.**7.3**If players 1 and 2 would bargain over a dollar, who would receive a larger share? Base your answer on the Nash bargaining solution.**7.4**Is the set of possible utility pairs derived from dividing a dollar (without lotteries) between players 1 and 4 convex?

**8.** Find the mixed strategy equilibria of the games
"chicken" and the "prisoners dilemma."

**9.** 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.

**9.1**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.**9.2**What are the subgame perfect equilibria of the game? Are there mixed strategy subgame perfect equilibria?

In the questions below consider only pure strategies.

**10.** 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?

**11.** 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.

**11.1**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 again the game of question 9. However
assume now that δ_{1}=δ_{2}=1.

**12.1**What are the subgame perfect equilibria of this game?**12.2**Does the SPE payoff converge when n goes to infinity? If so, what are the limit payoffs?

**13.** 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).

**13.1**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).**13.2**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.