Two law partners jointly own a firm and share equally in its revenues. Each law

Question

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 s1 and s2 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. Assume that 0 b 1/4, and that each effort level lies in the interval Si = [0, 4]. The payoffs for partners 1 and 2 are:

u1(s1; s2) = 1[4(s1 + s2 + bs1s2)] s212

u2(s1; s2) = 1[4(s1 + s2 + bs1s2)] s22

respectively, where the s2i terms reflect the cost of effort. Assume the firm has no other costs.

Show that the only rationalizable strategies (those not deleted by the process of iteratively deleting strategies that are never a best response) are s1 = s2 = 1/(1b)

Is s a Nash equilibrium?

If the partners agree to work the same amount as each other and they write a contract specifying that amount, what common amount of effort s should they agree each to supply to the firm if their aim is to maximize revenue net of total effort costs? How does this amount compare to the rationalizable effort levels?

Explanation / Answer

Nash Equilibrium is a concept within game theory where the optimal outcome of a game is where there is no incentive to deviate from their initial strategy. More specifically, the Nash Equilibrium is a concept of game theory where the optimal outcome of a game is one where no player has an incentive to deviate from his chosen strategy after considering an opponent's choice. Overall, an individual can receive no incremental benefit from changing actions, assuming other players remain constant in their strategies. A game may have multiple Nash Equilibria or none at all.

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