1.   Credible Punishment or Repeated Cooperation. Recall that in the problem set 6, we   described two factories that can either cooperate or compete with each other. Now,          suppose that after calculation of their profits, we have found the following payoff matrix.

(1) If the upstream firm decides to find a lawyer to legalize their cooperation agreement, so the mutual-cooperation option will be the only favored option (and can be achieved as a Nash Equilibrium), find the appropriate  (the cost of the punishment, possible the fee  paid to the lawyer) and  (the punishment amount added to the cheat player) so the       punishment in credible.

(2) Now, instead of using credible punishments to the cheated players, the two factories         decide to collaborate in a long time. Assume that the above payoff matrix applies to every quarter (1 period), and they decide to sign a contract to cooperate for 10 years (40 periods in total). Find the subgame perfect equilibrium in every period.

(3) Suppose that instead of signing a contract to cooperate for 10 years, the two factories     agree to sign the contract at the beginning of every period. Neither of them knows when they will stop interacting. Assume the probability that they may end the cooperation in  the next period is  . What is the minimum  so that mutual cooperation will be the        equilibrium in every period?

(4) Now, suppose that we go back to the one-time game again, but instead, the punishment    changes to a $5,000 fine from the cheated player to a third party. Re-write the new payoff table and indicate the Risk Dominant Equilibrium when the probability distribution is       assumed uniform.

(5) Based on the setup in Q1(4) above, the factory must have a minimum belief at what probability about the other factory’s cooperation, to cooperate?

2.   Stackelberg. Two firms share an oligopoly market. The first firm, Albert Co., enters the market first. At the beginning, it sees the whole market and determine the production      level ! . However, it is aware that another firm, Bob Co., will be its follower and will    produce  "  when entering the market. Albert Co. takes this action into its consideration when it determines its !  in the first stage. Assume that the market’s demand curve is

 =  − (!  + " ), and the cost functions are (! ) = ! !  for Albert Co. and (" ) = "  "  for Bob Co., respectively.

(1) Find Bob Co.’s response function  "  =  " (! ).

(2) Determine the equilibrium of this model by finding !  and  "  as functions of  ,  , ! , " .

(3) Find the profits for Albert Co. and Bob Co., then explain whether the market has the leader advantage.

3.   BNE. You manage a firm developing two types of medicines and sharing the market with another bio firm. Your competitor has labs suitable to produce medicine X, while your     firm has labs suitable to produce medicine Y. However, your marketing skill is better to   sell medicine X while your competitor’s is to medicine Y.

If your relationship is tense (unknown probability, could be assumed to be ), you two  would agree to share each other’s lab, but not collaborating to develop medicines, and   sell the products separately. Otherwise (with probability 1 −  , then), you two can        develop and sell products collaboratively, and the payoffs would be better than working alone. The detailed payoff matrix showing profits (in million $) can be found below:

A: tense relationship ()                               B: nice relationship (1 − )

(1) What is your expected payoff when you choose medicine X, and your competitor chooses Y under the tense relationship and chooses X under the nice relationship?

(2) Find the Bayes Nash Equilibrium of this question.

(3) You are unsure about  , assume that your first hunch is  = 80%, a very tense                 relationship. But now, the manager in the other bio firm invites you to watch a baseball   game together. You know that about 75% under a nice relationship do people invite         others to watch game together ((|) = 0.75). Use Bayes’ Rule to calculate  .