In a non-cooperative, imperfect information, simultaneous-choice, one-period game, a Nash equilibrium

Question

BlackTom AI · Accepted Answer

A Nash equilibrium is a strategy profile in which every player's choice is a best response to the choices of the others. Relevant knowledge points: non-cooperative games mean no binding agreements; imperfect information and simultaneous choice imply players choose without observing others' actions; one-period (static) games mean a single decision point. Nash equilibrium requires mutual best responses but does not impose joint profit maximization or Pareto efficiency.

Step-by-step reasoning:
1) Define payoff structure: firms choose strategies (e.g., quantities or prices) simultaneously and independently. Each firm maximizes its own profit given the rivals' actions.
2) Best-response condition ensures no unilateral profitable deviation, but coordinated deviations (binding collusion) are not available in a non-cooperative game.
3) Demonstration with a standard Cournot duopoly (linear demand, zero marginal cost) shows the NE need not maximize joint profits:
   - Demand: P = a − Q where Q = q1 + q2. Profit for firm i: πi = qi*(a − q1 − q2).
   - First-order condition for firm 1: ∂π1/∂q1 = a − 2q1 − q2 = 0 → q1 = (a − q2)/2.
   - By symmetry q1 = q2 = q*, so q* = (a − q*)/2 → 3q* = a → q* = a/3.
   - Total NE output Q_NE = 2a/3; price P_NE = a − 2a/3 = a/3.
   - Each firm's profit π_NE = (a/3)*(a/3) = a^2/9; total profit = 2a^2/9 ≈ 0.222a^2.
   - Collusive (joint-profit-maximizing) output solves maximize Π = Q(a − Q) → a − 2Q = 0 → Q_coll = a/2; total profit = (a/2)*(a/2) = a^2/4 = 0.25a^2.
   - Since 0.222a^2 < 0.25a^2, the NE yields lower total profit than the cooperative optimum.

Why the correct answer is correct: Nash equilibrium focuses on unilateral incentives, not on maximizing the sum of firms’ profits. In many non-cooperative settings (Prisoner’s Dilemma, Cournot), the NE is suboptimal for the group.

Why other plausible statements are incorrect: any claim that a Nash equilibrium must maximize joint profits or must be Pareto efficient is false—non-cooperative incentives can produce inefficient equilibria. Uniqueness is not guaranteed either; NE can be multiple.

Conclusion: Therefore, a Nash equilibrium in such a game may not maximize the sum of the firms' profits, which matches the provided answer.

In a non-cooperative, imperfect information, simultaneous-choice, one-period game, a Nash equilibriumSingle choice

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