greatest probability of survival

1. Suppose that in a market there are only two drug dealers, A and B, who sell a homogenous product. Each dealer has a constant marginal cost of 10. Consider the possibility that the two dealers will coordinate their activities in order to increase their profits. The cartel agreement would have each dealer restrict its output so that the market price will be high. However, each dealer might consider raising its output beyond the cartel quota (cheating on the agreement). Thus, for each dealer the two possible strategies are to comply and to cheat. Their possible payoffs are as follows:
- If both cheat, they each will earn profits of $10.
- If they both comply, they will each earn profits of $20.
- If one cheats and the other does not, the one who cheats will earn profits of $25; the one who doesn't cheat will earn profits of $5.
a. Do either of the dealers have a dominant strategy? Is there a Nash Equilibrium of this game? If so, what is it?
b. Suppose cartel agreements are contractible and contracts are enforced by a local mob boss. Would the dealers want to sign a contract? If so, how would the outcome change from part a?
c. Suppose that there is no mob boss. Instead, dealers A and B can regularly coordinate their activities on a daily basis. Dealer A has announced that he will "retire" in a year. Will the repeated interaction that they have until that time cause them to comply? Why or why not?
d. Suppose that instead of competing in the game described above, the two dealers were to compete in a Bertrand fashion in a single period. That is, the firms would simultaneously set their own price at any level. All of the demand for the good would go to the lower priced dealer. Do either of the dealers have a dominant strategy in this case? What is the Nash Equilibrium of the game?

2. Two teenagers, James and Dean, take their cars to opposite ends of Main Street, Middle-of-Nowhere, USA, at midnight and start to drive toward each other. The one who swerves to avoid a collision is the "chicken" and the one who keeps going straight is the winner. If both maintain a straight course, there is a collision in which both cars are damaged and both players injured. Suppose the payoffs are shown in the following table:

Dean
Swerve Straight
James Swerve 0, 0 -1, 1
Straight 1, -1 -2, -2

a. What is the Nash Equilibrium (or equilibria) of this game?
b. Suppose that James had access to some handcuffs that he could use to visibly lock his hands to the rearview mirror, completely inhibiting his ability to turn the steering wheel. Would James want to do this?

3. Consider the following game tree which can be considered a simplified model of the Cuban missile crisis. The Soviets have installed missiles in Cuba, and now the U.S. has the first move. It chooses between doing nothing and issuing a threat. Suppose the payoffs are as follows. If the United States does nothing, this is a major military and political achievement for the Soviets, so we score the payoffs as -2 for the United States and 2 for the Soviets. If the United States issues its threat, the Soviets get to move, and they can either withdraw or defy. Withdrawal is a humiliation (a substantial minus) for the Soviets and an affirmation of U.S. military superiority (a small plus), so we score it 1 for the United States and -4 for the Soviets. If the Soviets defy the U.S. threat, there will be a nuclear war. This is terrible for both, but particularly for the U.S., which as a democracy cares more for its citizens, so we score this -10 for the U.S. and -8 for the Soviets.

U.S., USSR
1, -4
withdraw
USSR

Defy -10, 8

US

-2, 2

a. What is the Subgame Perfect Nash Equilibrium of this game?
b. Is there an equilibrium to this game that is not subgame perfect (that is, one which involves a non-credible threat)?

4. Three antagonists, Larry, Mo, and Curly, are engaged in a three-way gunfight. There are two rounds. In the first round, each player is given one shot: first Larry, then Mo, and then Curly. After the first round, any survivors are given a second shot, again beginning with Larry, then Mo, and then Curly. For each person, the best outcome is to be the sole survivor. Next best is to be one of two survivors. In third place is the outcome in which no one gets killed. Dead last is that you killed.
Larry is a poor shot, with only a 30 percent chance of hitting a person at whom he aims. Mo is a much better shot, achieving 80 percent accuracy. Curly is a perfect shot - he never misses.
a. What is Larry's optimal strategy in the first round? (+1 point)
b. Who has the greatest chance of survival in this problem? Who has the lowest chance of survival? (+1 point)

See attached file for full problem description.