Credible Punishment or Repeated Cooperation. Recall that in the problem set 6, we described two factories that can eithe

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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.
Profit ($)
Upstream Factory Cooperate Separate
Profit


Downstream Factory
Cooperate
Separate
Upstream Factory Cooperate $17420, $17420
$20445, $13790
Separate $13790, $20445 $16380,
$16380
(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 𝛼 (the cost of
the punishment, possible the fee paid to the lawyer)
and 𝛽 (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 𝑝. What is the
minimum 𝑝 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?