Source: Microeconomic Theory, Texas A&M University
Tags: game theory, prisoner's dilemma, dominant strategy, Nash equilibrium, duopoly, payoff matrix, strategic interaction, oligopoly pricing, interdependence
Game theory is the toolkit economists use to analyse oligopoly behaviour, because each firm's best move depends on what rivals do. The prisoner's dilemma shows why firms often end up in outcomes that are collectively worse than cooperation. Dominant strategies, Nash equilibrium, and payoff matrices are the core building blocks you need to be able to read and solve.
Game theory
A framework for analysing strategic decision-making where the outcome for each participant depends on the choices of all participants.
Payoff matrix
A table showing the payoffs (profits, sentences, etc.) for each player under every combination of strategies. The standard way to represent a simultaneous-move game.
Dominant strategy
A strategy that gives a player a better payoff than any other strategy, regardless of what the other player does. If you would choose the same action no matter what your opponent picks, that action is your dominant strategy.
Nash equilibrium
An outcome where no player can improve their payoff by unilaterally changing their strategy, given the other player's choice. Every dominant-strategy equilibrium is a Nash equilibrium, but not every Nash equilibrium involves dominant strategies.
Prisoner's dilemma
A game where two players acting in their own self-interest each choose a strategy that leads to a worse collective outcome than if they had cooperated. The individually rational choice is collectively irrational.
Duopoly
An oligopoly with exactly two firms. The simplest case for modelling strategic interaction.
Two suspects (Rose and Colin) are interrogated separately. Each can confess or stay silent:
Both confess: each gets 20 years.
Both stay silent: each gets 5 years.
One confesses, the other stays silent: the confessor gets 3 years, the silent one gets 30 years.
Confessing is a dominant strategy for both players. Whatever Colin does, Rose is better off confessing, and vice versa. The Nash equilibrium is (Confess, Confess) with 20 years each, even though (Don't Confess, Don't Confess) with 5 years each would be better for both.
This is the central tension: individual rationality leads to collective irrationality.
The chapter presents three variations of a pricing game between Filip and Gus, each illustrating a different strategic situation.
Payoffs:
Both choose Low Price: each earns $25,000.
Both choose High Price: each earns $50,000.
Filip picks Low, Gus picks High: Filip earns $75,000, Gus loses $10,000.
Filip picks High, Gus picks Low: Filip loses $10,000, Gus earns $75,000.
Low Price is a dominant strategy for both. No matter what Gus does, Filip does better choosing Low Price (and the same logic holds for Gus). The Nash equilibrium is (Low, Low) at $25,000 each.
This is a prisoner's dilemma applied to pricing. Both firms would prefer (High, High) at $50,000 each, but neither can trust the other not to undercut.
The payoffs are adjusted so that Gus still has Low Price as a dominant strategy, but Filip does not. Filip's best response depends on what Gus does.
Solving this requires two steps:
First, identify that Gus will choose Low Price (his dominant strategy).
Then, given that Gus will choose Low, determine Filip's best response to Low Price.
The Nash equilibrium is found by reasoning through one player's dominant strategy first, then working out the other player's best reply.
Neither Filip nor Gus has a strategy that is always best regardless of the other's choice. Each player's optimal move depends entirely on what the rival does.
To find the Nash equilibrium:
For each of Gus's possible choices, find Filip's best response.
For each of Filip's possible choices, find Gus's best response.
The Nash equilibrium is the cell where both players are simultaneously playing their best response to each other.
This is the most common real-world scenario. Pure dominance is rare; most strategic situations require checking best responses.
Step-by-step approach:
Fix one player's strategy and compare the other player's payoffs across their options. Circle or underline the higher payoff.
Repeat for the other player's strategy.
Repeat the whole process for the second player.
The cell where both players have their best-response payoff circled is the Nash equilibrium.
If a player's best response is the same regardless of what the rival does, that player has a dominant strategy.
No formulas in this section, but be comfortable reading 2x2 payoff matrices. Each cell contains two numbers: one for each player. Convention in this chapter places one player's payoff in the upper-left triangle and the other in the lower-right triangle of each cell.
⚠️ Be able to identify dominant strategies by checking each player's payoffs column by column (or row by row). This is the single most common exam question format for game theory.
⚠️ Understand that a Nash equilibrium can exist even when neither player has a dominant strategy. Check best responses for each player separately.
⚠️ The prisoner's dilemma result (both defect) is the equilibrium, not the cooperative outcome. Exams often test whether you can distinguish the equilibrium from the jointly preferred outcome.
⚠️ In a sequential or asymmetric game, solve for the player with the dominant strategy first, then find the other player's best response to that known choice.
Q: In the prisoner's dilemma, why don't both players simply cooperate to get the better outcome?
A: Each player has an individual incentive to defect regardless of what the other does. Cooperation would require a binding commitment mechanism, which the basic one-shot game does not allow. Each player's dominant strategy is to confess, so the equilibrium is mutual confession.
Q: Filip and Gus are in a pricing game. Filip earns $50,000 if both set High Price, $75,000 if he sets Low while Gus sets High, $25,000 if both set Low, and loses $10,000 if he sets High while Gus sets Low. Does Filip have a dominant strategy?
A: Yes. If Gus sets High, Filip prefers Low ($75,000 > $50,000). If Gus sets Low, Filip prefers Low ($25,000 > -$10,000). Low Price is Filip's dominant strategy because it yields a higher payoff in both scenarios.
Q: What is the difference between a dominant strategy equilibrium and a Nash equilibrium?
A: A dominant strategy equilibrium is a special case of Nash equilibrium where every player has a dominant strategy. A Nash equilibrium only requires that no player can improve by unilaterally changing strategy, which can occur even when players lack dominant strategies. Every dominant-strategy equilibrium is a Nash equilibrium, but the reverse is not true.
Q: In a 2x2 game, neither player has a dominant strategy. How do you find the Nash equilibrium?
A: For each of Player 1's possible actions, identify Player 2's best response. Then for each of Player 2's possible actions, identify Player 1's best response. The Nash equilibrium is the outcome where both players are simultaneously best-responding to each other.
game theory, oligopoly strategy, prisoner's dilemma, dominant strategy, Nash equilibrium, payoff matrix, duopoly game, strategic interdependence, best response, cooperative vs non-cooperative outcome, pricing game, 2x2 matrix, simultaneous move game, mutual defection