Game Theory Concepts

Why This Matters

Game theory is the analytical backbone of oligopoly analysis—and oligopoly questions appear consistently on the AP Microeconomics exam. When firms recognize that their pricing and output decisions affect competitors (and vice versa), simple supply-and-demand analysis breaks down. You need game theory to explain why firms might collude, why cartels tend to collapse, and why oligopolies often produce outcomes somewhere between perfect competition and monopoly. The concepts here—dominant strategies, Nash equilibrium, payoff matrices, and the prisoner's dilemma—give you the tools to analyze any strategic interaction the exam throws at you.

The College Board specifically tests your ability to read payoff matrices, identify dominant strategies, find Nash equilibria, and calculate the incentive needed to change a player's behavior. These skills show up in both multiple-choice questions and FRQs, often in the context of two firms deciding whether to advertise, set high or low prices, or cheat on a collusive agreement. Don't just memorize definitions—know how to apply each concept to a payoff matrix and explain why rational players end up where they do.

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Core Equilibrium Concepts

Understanding equilibrium in games means identifying stable outcomes where no player wants to change their behavior. An equilibrium occurs when each player's strategy is a best response to what others are doing.

Nash Equilibrium

• A stable outcome where no player can improve their payoff by unilaterally changing strategy—this is the most tested concept in AP game theory
• Can occur in pure strategies (specific actions like "advertise" or "don't advertise") or mixed strategies (randomizing between actions)
• Multiple Nash equilibria can exist in a single game, requiring you to check each cell of the payoff matrix systematically

Dominant Strategy

• A strategy that yields the highest payoff regardless of what opponents choose—if you find one, the player will always select it
• Dominant strategy equilibrium occurs when all players have dominant strategies, making the outcome highly predictable
• Not every game has dominant strategies—when absent, you must use best-response analysis to find Nash equilibrium

Payoff Matrix

• A table displaying outcomes for all strategy combinations—rows represent one player's choices, columns represent the other's
• Each cell contains two numbers showing payoffs for both players (typically listed as row player first, column player second)
• Essential for identifying equilibria—circle best responses for each player, then find cells where both payoffs are circled

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The Prisoner's Dilemma and Collusion

The prisoner's dilemma explains why cartels collapse and why oligopolists struggle to maintain high prices. Individual rationality leads to collectively suboptimal outcomes when players can't make binding agreements.

Prisoner's Dilemma

• Both players have a dominant strategy to defect, even though mutual cooperation would yield higher combined payoffs
• Illustrates the tension between individual and collective rationality—each firm's incentive to cheat undermines collusion
• The classic oligopoly application: two firms agreeing to keep prices high both have incentives to secretly cut prices and steal market share

Cooperative vs. Non-cooperative Games

• Non-cooperative games assume no binding agreements—players act in self-interest, leading to outcomes like the prisoner's dilemma
• Cooperative games allow enforceable contracts, enabling players to commit to mutually beneficial strategies
• Oligopoly analysis typically uses non-cooperative frameworks because antitrust laws prohibit binding price-fixing agreements

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Game Timing and Structure

Whether players move simultaneously or sequentially fundamentally changes how you analyze strategic interactions. The timing of moves determines what information players have when making decisions.

Simultaneous Games

• Players choose actions without knowing opponents' choices—this is the standard setup for payoff matrix analysis
• Requires anticipating others' likely moves, often leading to dominant strategy or Nash equilibrium analysis
• Most AP exam questions use this format because payoff matrices naturally represent simultaneous decisions

Sequential Games

• Players move in order, with later movers observing earlier actions—creates first-mover or second-mover advantages
• Analyzed using game trees (decision diagrams showing branches for each possible action)
• Backward induction solves these games—start at the final decision and work backward to find optimal strategies

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Strategies in Repeated Interactions

When games are played multiple times, players can reward cooperation and punish defection, fundamentally changing equilibrium outcomes. The shadow of future interactions can sustain cooperation that would collapse in one-shot games.

Repeated Games

• Multiple rounds allow strategies based on history—players can condition current actions on past behavior
• Cooperation becomes sustainable when the present value of future cooperation exceeds the one-time gain from cheating
• Explains why long-term business relationships often feature more cooperative behavior than one-time transactions

Tit-for-Tat Strategy

• Start by cooperating, then mirror your opponent's previous action—cooperate if they cooperated, defect if they defected
• Combines niceness (starts cooperative), retaliation (punishes defection), and forgiveness (returns to cooperation)
• Highly effective in repeated prisoner's dilemmas—promotes stable cooperation while deterring exploitation

Mixed Strategies

• Randomizing over actions prevents opponents from exploiting predictable behavior—each action played with some probability
• Used when no pure strategy Nash equilibrium exists—the randomization makes opponents indifferent between their choices
• Less commonly tested on AP exams but important for understanding games like matching pennies or penalty kicks

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Quick Reference Table

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Self-Check Questions

1. Given a 2x2 payoff matrix, how do you systematically identify whether each player has a dominant strategy, and what do you conclude if only one player has one?

2. In a prisoner's dilemma, both players end up at the Nash equilibrium despite a better collective outcome existing. Explain why this happens and connect it to cartel instability in oligopoly markets.

3. Compare simultaneous and sequential games: which analytical tool (payoff matrix or game tree) applies to each, and how does the Stackelberg model differ from a standard Cournot duopoly in this regard?

4. A firm is considering cheating on a collusive agreement. The collusive payoff is $\$50$ per period forever, the cheating payoff is $\$80$ in the first period followed by $\$30$ forever after (due to retaliation). Under what conditions would the firm maintain cooperation?

5. Why might tit-for-tat sustain cooperation in a repeated prisoner's dilemma when defection is the dominant strategy in the one-shot version? What features of the strategy make it effective?