1. Prove that, if v is the minimax representation of a strategic-form game, then v must be superadditive.

2. Suppose that there are three firms producing the same product. Firm 1 is a small firm and can produce either zero units or one unit per day. Firms 2 and 3 are large firms, and each can produce either two or three units per day. The market price P depends on the total daily output, according to the formula.

P = 8 - (c₁ + c₂ + c₃).

where c, is the daily output of firm i. The transferable payoff to firm i is its daily revenue Pc,.

a. Model this game in strategic form.

b. Calculate the minimax representation in coalitional form of this game with transferable utility. Calculate the Shapley value of this coalitional game and check whether it is in the core

c. Calculate the rational-threats representation in coalitional form of this game with transferable utility. Is this coalitional game superadditive? Calculate the Shapley value of this coalitional game. Is this Shapley value individually rational, in the sense that each player i gets at least the worth of his one-person coalition v({i})?

d. What are the defensive-equilibrium representations in coalitional form of this game with transferable utility?