2. Consider a Cournot competition environment with one good and two firms, Firm 1 and Firm 2. The (pure) strategy space of Firm 1 is S₁ = [0, 1], and the strategy s₁ of Firm 1 corresponds to the amount of the good they produce. Similarly, the (pure) strategy space of Firm 2 is S2 = [0, 1], and the strategy s2 of Firm 2 corresponds to the amount of the good they produce. If Firm 1 were to produce quantity s₁ and Firm 2 were to produce quantity s2, the prevailing price in the good market would be 1 — 81 — 82, the utility of Firm 1 would be their profit, u₁ (81, 82) = (1-81-82 - c)81, and the utility of Firm 2 would be their profit, u2($1, $2) = (1-81-82-c)s2, where 0 ≤ c< 1 is the marginal cost of production for both firms. (a) Find the pure-strategy Nash equilibria of this game. (b) Are there other Nash equilibria in this game. Consider now a Bertrand competition environment with one good and two firms, Firm 1 and Firm 2. The (pure) strategy space of Firm 1 is S₁ = [0,5], and the strategy s₁ of Firm 1 corresponds to the price they charge for the good. Similarly, the (pure) strategy space of Firm 2 is S₂ = [0,5], and the strategy s2 of Firm 2 corresponds to the price they charge for the good. If Firm 1 were to choose price S₁ and Firm 2 were to choose price s2, then the demand Firm 1 would face is given by D1 (81, 82) = if 81 < 82, 1 if 8₁ = 82, if 81 82, the demand Firm 2 would face is given by D2 (81, 82) = 1 12 0 the utility of Firm 1 would be their profit, u₁ (S1, S2) if s2 < $1, if 82 81, = if s2 > S1, = - (81 — 1)D1(81, 82), and the utility of Firm 2 would be their profit, u2(81, 82) = (82 − 1)D2(81, 82). (The marginal cost of production for both firms is 1.) - (c) Find the pure-strategy Nash equilibria of this game. Consider now an altered version of the Bertrand competition environment above in which the (pure) strategy space of Firm 1 is S₁ {0, 1, 2, 3, 4, 5}, the (pure) strategy space of Firm 2 is S₂ = {0, 1, 2, 3, 4, 5}, and otherwise the game is the same as the Bertrand competition environment above. = (d) Find the pure-strategy Nash equilibria of this game.