An island has two reefs that are suitable for fishing, and there are twenty
fishers who simultaneously and independently choose at which of the two reefs
( 1 or 2 ) to fish. Each fisher can fish at only one reef. The total number of
fish harvested at a single reef depends on the number of fishers who choose to
fish there. The total catch is equally divided between the fishers at the
reef. At reef 1 , the total harvest is given by \(f_{1}\left(r_{1}\right)=8
r_{1}-\frac{r_{1}^{2}}{2}\), where \(r_{1}\) is the number of fishers who select
reef 1 . For reef 2 , the total catch is \(f_{2}\left(r_{2}\right)=4 r_{2}\),
where \(r_{2}\) is the number of fishers who choose reef 2 . Assume that each
fisher wants to maximize the number of fish that he or she catches.
(a) Find the Nash equilibrium of this game. In equilibrium, what is the total
number of fish caught?
(b) The chief of the island asks his economics advisor whether this
arrangement is efficient (i.e., whether the equilibrium allocation of fishers
among reefs maximizes the number of fish caught). What is the answer to the
chief's question? What is the efficient number of fishers at each reef?
(c) The chief decides to require a fishing license for reef 1 , which would
require each fisher who fishes there to pay the chief \(x\) fish. Find the Nash
equilibrium of the resulting location-choice game between the fishers. Is
there a value of \(x\) such that the equilibrium choices of the fishers results
in an efficient outcome? If so, what is this value of \(x\) ?
