1. (Cournot's game with many firms) Consider Cournot's game
in the case of an arbitrary number n of firms; retain the assumptions that the inverse demand function p(Q) = a - Q if Q ≤ a and 0 if Q > a. The cost function of each firm i is ci(qi) = (c/2)*qi^2 with c < a. Find the best response function of each firm and set up the conditions for(q*1, . . . , q∗n) to be a Nash equilibrium, assuming that there is a Nash equilibrium in which all firms' outputs are positive. Solve these equations to find the Nash equilibrium. (First show that in an equilibrium all firms produce the same output, then solve for that output. If you cannot show that all firms produce the same output, simply assume that they do.) Find the price at which output is sold in a Nash equilibrium and show that this price decreases as n increases as the number of firms increases without bound.
2. (Common pool resource game) Consider a common pool resource game with two appropriators. (If you don't know what is a common pool resource, read the Wikipedia article about the "Tragedy of the Commons".) Each appropriator has an endowment e > 0 that can be invested in an outside activity with marginal payoff c > 0 or into the common pool resource. Let x ∈ X ⊆ [0, e] denote the player's investment into the common pool resource (likewise y denotes the opponent's investment). The return from investment into the common pool resource is (x/(x+y))*(a(x+y)-b(x+y)^2 ), with a, b > 0. So the symmetric payoff function is given by π(x, y) = c(e-x)+ (x/(x+y))*(a(x+y)-b(x+y)^2 ) if x, y > 0 and c • e otherwise. Find Nash equilibrium of the game. Proceed by deriving the best response correspondences first. How does Nash equilibrium depend on parameters a, b, and c (varying one at a time and keeping the others fixed)?