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Intermediate Microeconomics Unit 9: Game Theory and Oligopoly

When a few firms interact, each one's best choice depends on what the others do; game theory provides the tools, and the Cournot, Bertrand and Stackelberg models show how oligopoly outcomes fall between monopoly and competition.

Covers: Dominant strategies and the prisoners' dilemma, Nash equilibrium in pure strategies, Mixed strategies, Sequential games and backward induction, Repeated games and collusion, Cournot and Stackelberg quantity competition, Bertrand price competition

Study guide

A game lists players, their strategies and the payoffs from every combination of strategies. A strategy is strictly dominant if it beats every alternative whatever the others do; a dominated strategy can be crossed out because a rational player never uses it. A Nash equilibrium is a set of strategies in which each player's choice is a best response to the others', so no one gains by deviating alone. The prisoners' dilemma shows that individually rational play can lead to an outcome that is worse for everyone, such as both firms advertising heavily or both cutting prices. Some games have several equilibria, as in coordination games.

Some games have no pure-strategy equilibrium, such as matching pennies. Then players randomize. In a mixed-strategy equilibrium each player chooses probabilities that make the other player indifferent between her pure strategies, so the other has no reason to deviate. This means a player's probabilities are pinned down by the opponent's payoffs, not her own. In sequential games, players move in order and later players see earlier moves. Backward induction solves them from the end, giving a subgame perfect equilibrium that rules out threats a player would not carry out. Commitments that change one's own later incentives, such as building capacity, can make threats credible.

Repeated play changes incentives. If a prisoners' dilemma is repeated a known, finite number of times, backward induction predicts defection in every round. If it is repeated indefinitely, strategies such as grim trigger, which cooperate until someone cheats and then punish forever, can sustain cooperation when players are patient enough. Cooperation holds when the one-time gain from cheating is smaller than the discounted future loss from punishment, which gives a minimum discount factor. This explains both why cartels form and why they often break down: each member can profit by secretly producing more than its quota.

Oligopoly models differ in what firms choose and when. In Cournot competition, firms choose quantities simultaneously; each firm's reaction function gives its best output for any rival output, and with linear demand P = a − bQ and cost c, each of n firms produces (a − c)/((n + 1)b). In Stackelberg competition, a leader commits to output first and earns more than in Cournot, while the follower earns less. In Bertrand competition, firms choose prices; with identical products and equal costs, price falls to marginal cost even with two firms. Product differentiation or capacity limits let Bertrand firms keep prices above cost.

Key ideas

• A Nash equilibrium is a set of mutual best responses; it need not be efficient.
• Dominated strategies can be eliminated; dominant strategies are always played.
• In a mixed equilibrium each player's mix makes the other indifferent.
• Backward induction gives subgame perfect equilibria and removes non-credible threats.
• Finite repetition does not sustain cooperation in the standard model; indefinite repetition can.
• Cournot output lies between monopoly and competition and approaches competition as firms are added.
• Bertrand competition with identical goods and costs drives price to marginal cost.

Formulas and rules

Common mistakes

• Using a player's own payoffs to find her own mixing probabilities instead of the opponent's.
• Treating a threat as credible when the threatening player would not carry it out.
• Calling the jointly best outcome a Nash equilibrium when someone has an incentive to deviate.
• Confusing Cournot (quantity) with Bertrand (price) competition.
• Forgetting that the Stackelberg follower reacts to the leader's actual output.

Before the test, make sure you can

• Find dominant and dominated strategies and solve games by elimination.
• Identify all pure-strategy Nash equilibria in a two-player game.
• Compute mixed-strategy equilibrium probabilities and expected payoffs.
• Solve sequential games by backward induction and judge whether threats are credible.
• Find the discount factor needed to sustain collusion with grim trigger strategies.
• Solve Cournot (equal and unequal costs, n firms) and Stackelberg models.
• Explain Bertrand outcomes with identical and differentiated products.

Words to know

Nash equilibrium

A strategy profile in which each player's strategy is a best response to the others'.

Dominant strategy

A strategy that gives a higher payoff than any other no matter what opponents do.

Dominated strategy

A strategy that does worse than another strategy against every choice by opponents.

Prisoners' dilemma

A game in which dominant strategies lead to an outcome worse for all players than cooperation.

Mixed strategy

A strategy that randomizes over pure strategies with set probabilities.

Best response

The strategy that gives a player the highest payoff given the others' strategies.

Backward induction

Solving a sequential game by starting with the last decision and working back.

Subgame perfect equilibrium

A Nash equilibrium that is also an equilibrium in every subgame; it rules out non-credible threats.

Credible threat

A threat the player would actually want to carry out when the time comes.

Grim trigger strategy

Cooperate until the other player defects, then defect forever.

Discount factor

The weight δ placed on next period's payoff relative to today's.

Cartel

A group of firms that coordinate output or prices to raise joint profit.

Cournot model

Oligopoly in which firms choose quantities simultaneously.

Reaction function

A firm's best output (or price) as a function of its rival's choice.

Stackelberg model

Oligopoly in which a leader commits to output before the follower chooses.

Bertrand model

Oligopoly in which firms choose prices simultaneously.

Bertrand paradox

With identical products and equal costs, two price-setting firms drive price to marginal cost.

Sample practice questions

30 more questions like these, from level 1 to 5, are in the practice set.