(A fight) Each of two people has one unit of a resource. Each person chooses how much of the… 1 answer below »

(A fight) Each of two people has one unit of a resource. Each person chooses how much of the resource to use in fighting the other individual and how much to use productively. If each person i devotes yi to fighting then the total output is f(y1, y2) ≥ 0 and person i obtains the fraction pi(y1, y2) of the output, where

The function f is continuous (small changes in y1 and y2 cause small changes in f(y1, y2)), is decreasing in both y1 and y2 (the more each player devotes to fighting, the less output is produced), and satisfies f(1, 1) = 0 (if each player devotes all her resource to fighting then no output is produced). (If you prefer to deal with a specific function f, take f(y1, y2) = 2 − y1 − y2.) Each person cares only about the amount of output she receives, and prefers to receive as much as possible. Specify this situation as a strategic game and find its Nash equilibrium (equilibria?). (Use a direct argument: first consider pairs (y1, y2) with y1 = y2, then those with y1 = y2 1 = y2 = 1.)