two people are engaged in a joint project if each

Question

Two people are engaged in a joint project. If each person i puts in the e ort xi, a nonnegative number equal to at most 1, which costs her c(x_i), the outcome of the project is worth f(x₁; x₂). The worth of the project is split equally between the two people, regardless of their effort levels.

(a) Formulate this situation as a strategic game.

(b) Find its Nash equilibria when i. f(x₁; x₂) = 3x₁x_2, c(x_i) = x²_i , for i = 1; 2.

ii. f(x₁; x₂) = 4x₁x₂, c(x_i) = x_i, for i = 1; 2.

(c) In each case, is there a pair of e ort levels that yields both players higher payoffs than the Nash equilibrium e ort levels?