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Oligopoly & Game Theory

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A few interdependent firms

An oligopoly is a market dominated by a few large firms, with high barriers to entry. Its defining feature is strategic interdependence: because each firm is large, one firm’s pricing or output decision noticeably affects the others, so each must anticipate rivals’ reactions. Firms may collude (secretly or as a cartel) to act like a monopoly, or compete. Products can be standardized (steel) or differentiated (automobiles). Because of interdependence, the standard supply-and-demand model does not fit — economists analyze oligopoly with game theory.

Payoff matrices, dominant strategies, and Nash equilibrium

A payoff matrix lays out the outcomes for two firms under each combination of strategies. A firm has a dominant strategy if one choice is best regardless of what the rival does. A Nash equilibrium is an outcome where neither firm can improve by unilaterally changing its own strategy, given the other’s choice — no one has an incentive to deviate. If both firms have a dominant strategy, the pair of dominant strategies is the Nash equilibrium. Reading a matrix by comparing each firm’s payoffs one player at a time is the core skill.

The prisoner’s dilemma and cheating on collusion

The prisoner’s dilemma shows why cooperation is hard to sustain: two firms would jointly earn more by colluding (both charging high prices), but each individually has an incentive to cheat (undercut) to grab market share. Since cheating is the dominant strategy for both, they end up at the Nash equilibrium where both cheat — worse for the pair than cooperating but individually rational. This is why cartels are unstable: the constant temptation to defect tends to break collusive agreements down.

Reading the game
Dominant strategy: best choice no matter what the rival does · Nash equilibrium: no player gains by changing strategy alone
Analyze one firm at a time: fix the rival’s choice, pick this firm’s best response. Where both firms’ best responses coincide is the Nash equilibrium.
Worked example

Two firms choose High or Low price. Payoffs (Firm A, Firm B): both High → (10, 10); A High/B Low → (2, 15); A Low/B High → (15, 2); both Low → (5, 5). Find each firm’s dominant strategy and the Nash equilibrium.

  1. 1.Firm A, if B plays High: A gets 10 (High) vs. 15 (Low) → A prefers Low.
  2. 2.Firm A, if B plays Low: A gets 2 (High) vs. 5 (Low) → A prefers Low. So Low is A’s dominant strategy.
  3. 3.By symmetry, Firm B: if A High, B gets 10 vs. 15 → Low; if A Low, B gets 2 vs. 5 → Low. Low is B’s dominant strategy.
  4. 4.Both play their dominant strategy Low, giving the outcome (5, 5) — the Nash equilibrium.
Answer: Both firms have a dominant strategy of Low price, so the Nash equilibrium is (Low, Low) with payoffs (5, 5). Note both would earn more at (High, High) = (10, 10), but each is individually tempted to undercut — the classic prisoner’s dilemma.
Checkpoint

In a two-firm payoff matrix, Firm X earns more by advertising than by not advertising no matter what Firm Y chooses. This means that for Firm X, advertising is:

Watch out

A dominant strategy is one firm’s best move no matter what the rival does; a Nash equilibrium is an outcome where neither firm wants to deviate. A game can have a Nash equilibrium with no dominant strategies — do not treat the two terms as interchangeable.

Checkpoint

Two colluding firms agree to both set high prices to earn high joint profits. Why is this agreement typically unstable?

On the exam

To solve a payoff matrix, analyze one firm at a time: hold the rival’s choice fixed and find that firm’s best response, then repeat. Where both best responses meet is the Nash equilibrium. Always identify dominant strategies before naming the equilibrium.

Answer the 2 checkpoints as you read.

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