A fisherman's catch and payoff No. of Boats Other Two No. of Boats Fishermen Size of Cost of of Sent Out Sent Out Catch Boats Payoff 1 2 40 15 25 1 3. 35 15 20 1 4 30 15 15 2 75 30 45 3. 65 30 35 2 4 50 30 20 2.
There are three fishermen, and each day they individually decide how
many boats to send out to catch fish in the local lake. A fisherman can send
out one or two boats, and the daily cost of a boat is $15. The more boats
sent out, the more fish are caught. However, since there are only so many
fish to be caught on a given day, the more boats another fisherman sends
out, the fewer fish the remaining fishermen can catch. The accompanying
table reports the size of a fisherman’s catch, depending on how many boats each fisherman sends out.
A fisherman’s current-period payoff is the value of his catch (assume
that each fish sells for a price of 1), less the cost of the boats. For example,
if a fisherman sends out two boats and the other two fishermen each send
out one boat, then a fisherman’s payoff is 75 - 30 = 45. The stage game is
symmetric, so the table is to be used to determine any fisherman’s payoff.
The fishermen play an infinitely repeated game where the stage game has
them simultaneously choose how many boats to send out. Each fisherman’s
payoff is the present value of his payoff stream, where fisherman i’s discount factor is di. Find a collection of actions—one for each player—which results in a payoff higher than that achieved at the Nash equilibria for the stage game. Then construct a grim-trigger strategy that results in those actions being implemented, and derive conditions for that strategy to be a symmetric SPNE.
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