Recall the partnership game we discussed in class. Two law partners jointly own a firm and share equally in
its revenues. Each law partner individually decides how much effort to put into the firm. The firm's
revenue is given by 4 (s1+s2 + bs1s2) where sā and sā are the efforts of the lawyers 1 and 2 respectively.
The parameter b > 0 reflects the synergies between their efforts: the more one lawyer works, the more
productive is the other. The partner's effort level was Sā = [0, 4] and we initially assumed b = Ā¼. The payoff
for partners 1 and 2 are:
u1(s1, s2) = Ā½ [4 (s1+s2 + bs1s2)]-s1Ā²
u2(s1, s2) = Ā½ [4 (s1+s2 + bs1s2)] - s2Ā²
respectively, where the siĀ² terms reflect the cost of effort. (Notice that the cost of providing another unit of
effort is increasing in the amount of effort already provided). Assume the firm has no other costs. In
class, we showed that the only rationalizable strategies (i.e., those not deleted by the process of
iteratively deleting strategies that are never a best response) were s1* = S2* = 1/(1-b).
(a) Given the results of the above, what are the respective payoff for each of the partners (keep the
calculation in fractions)?
(b) What would each partner's payoffs be if they mutually agreed to put forth their highest effort, s1' = S2' =
4? If one partner went through with the agreement, to put forth their highest effort, should the other
partner adhere to the agreement? Provide numeric evidence of your answer.
(c) What coefficient of cooperation/teamwork (b) would be necessary to make s1* = S2* = 4 a Nash
equilibrium?
