What is the difference between a dominant strategy and a Nash Equilibrium?

A dominant strategy is an action that yields the highest payoff for a player regardless of what the other players do. A Nash Equilibrium is a set of strategies (one per player) where no player can improve their payoff by unilaterally changing their strategy. A dominant strategy always leads to a Nash Equilibrium, but a Nash Equilibrium does not require any player to have a dominant strategy. Many games have Nash Equilibria where neither player has a dominant strategy, such as coordination games.

Can a game have more than one Nash Equilibrium?

Yes. Many games have multiple Nash Equilibria. For example, a coordination game like the Battle of the Sexes has two pure strategy Nash Equilibria and one mixed strategy Nash Equilibrium. When multiple equilibria exist, predicting which one players will actually reach can be challenging and may depend on focal points, communication, or historical precedent.

What is the difference between simultaneous and sequential games?

In a simultaneous game, all players choose their strategies at the same time (or without knowledge of others' choices). These are represented using payoff matrices in normal form. In a sequential game, players make decisions in a specific order, with later players observing earlier players' actions. These are represented using decision trees (extensive form) and are solved using backward induction.

Why does the Prisoner's Dilemma lead to a suboptimal outcome?

In the Prisoner's Dilemma, each player has a dominant strategy to defect (confess), because defecting yields a higher payoff regardless of the other player's choice. When both players follow their dominant strategy, the resulting Nash Equilibrium (both defect) is worse for both players than mutual cooperation. This occurs because each player maximizes their individual payoff without internalizing the cost imposed on the other player.

What is a mixed strategy, and when is it used?

A mixed strategy involves a player randomizing over their available actions with specific probabilities, rather than choosing a single action with certainty. Mixed strategies are used when a game has no pure strategy Nash Equilibrium, or when predictability would be exploitable. The probabilities are chosen so that each player makes their opponent indifferent among their own options. Nash's Theorem guarantees that every finite game has at least one Nash Equilibrium, possibly in mixed strategies.

How does repeated interaction change the outcome of the Prisoner's Dilemma?

When the Prisoner's Dilemma is played repeatedly with no known end point (infinitely repeated), cooperation can be sustained as an equilibrium outcome through strategies like tit-for-tat, where players cooperate initially and then mirror their opponent's previous action. The threat of future punishment (retaliation) can deter defection if players value future payoffs sufficiently. This is formalized by the Folk Theorem, which states that many cooperative outcomes can be sustained as equilibria in infinitely repeated games.

EconomicsCollege

Game Theory

Game theory is a branch of applied mathematics that provides a framework for analyzing strategic interactions — situations where the outcome for each participant depends not only on their own actions but also on the actions of others.

This guide covers the core concepts of game theory — Nash Equilibrium, dominant strategies, the Prisoner's Dilemma, sequential games, mixed strategies, and backward induction — with worked examples, memory aids, and a 10-question practice quiz.

1Introduction

Game theory has revolutionized how economists understand strategic decision-making. Far from being a mere academic curiosity, it provides powerful tools for analyzing situations ranging from the pricing decisions of oligopolistic firms to international trade negotiations, environmental policy, and auction design.

At its core, game theory formalizes interactive situations where rational players make interdependent decisions. Unlike traditional economic models that treat agents as isolated decision-makers, game theory explicitly accounts for the fact that each participant's optimal choice depends on what they expect others to do.

Why Game Theory Matters

Oligopoly analysis: Models pricing, output, and advertising decisions among interdependent firms.
Negotiation & bargaining: Provides frameworks for labor disputes, trade agreements, and contract design.
Public policy: Informs regulation, auction design, environmental agreements, and mechanism design.
Everyday decisions: Explains strategic behavior in auctions, voting, sports, and competitive situations.

In Practice

Game theory is not limited to economics. Its applications span political science, biology (evolutionary game theory), computer science (algorithm design), and military strategy. The 1994 Nobel Prize in Economics was awarded to John Nash, John Harsanyi, and Reinhard Selten for their contributions to game theory.

2Key Definitions

Essential vocabulary for understanding game theory at the university level.

Game

A formal model of an interactive situation where rational players make interdependent decisions

Players

The individuals or entities making strategic decisions within the game (e.g., firms, governments, consumers)

Strategy

A complete plan of action for a player in all possible contingencies within the game

Payoff

The utility or value a player receives from a particular outcome (e.g., profits, utility, years in prison)

Nash Equilibrium

A set of strategies where no player can improve their payoff by unilaterally changing their strategy

Dominant Strategy

A strategy that yields a higher payoff for a player regardless of what other players do

Payoff Matrix

A table showing payoffs for each player for every combination of strategies; used for simultaneous games

Zero-Sum Game

A game where the sum of payoffs for all players is always zero; one player's gain equals another's loss

Backward Induction

Solving a sequential game by starting at the end of the game tree and working backward to the first move

Mixed Strategy

Randomizing over actions with specific probabilities; used when no pure strategy NE exists

Subgame Perfect NE

A Nash Equilibrium that is also a NE in every subgame; found via backward induction in sequential games

Best Response

The strategy that yields the highest payoff for a player given the strategies chosen by all other players

3Types of Games

Games can be classified along several dimensions. Understanding these distinctions is essential for selecting the correct analytical framework.

Simultaneous vs. Sequential

Simultaneous Games

Players choose their strategies at the same time, without knowing the other players' choices.

Representation: Payoff matrix (normal form).

Solution: Find Nash Equilibria via best-response analysis or dominant strategy elimination.

Sequential Games

Players make decisions in a specific order, with later players observing earlier moves.

Representation: Decision tree (extensive form).

Solution: Backward induction to find Subgame Perfect Nash Equilibrium.

Zero-Sum vs. Non-Zero-Sum

Zero-Sum Games

The sum of payoffs for all players is always zero. One player's gain is exactly another's loss.

Examples: Poker, chess, penalty kicks, many competitive sports.

Non-Zero-Sum Games

The sum of payoffs can vary. Players can both gain or both lose — cooperation may be mutually beneficial.

Examples: Prisoner's Dilemma, coordination games, trade negotiations.

Cooperative vs. Non-Cooperative

Cooperative Games

Players can form binding agreements to coordinate their strategies. Focus is on coalition formation and surplus division.

Non-Cooperative Games

Players cannot form binding agreements; each acts independently to maximize their own payoff. Most economic game theory analysis uses this framework.

4The Prisoner's Dilemma & Nash Equilibrium

The Prisoner's Dilemma is the most famous game in game theory. It illustrates a fundamental conflict between individual rationality and collective welfare.

The Setup

Two suspects are arrested and interrogated separately. Each can Confess or Remain Silent. The payoffs represent years in prison (lower is better).

	B: Confess	B: Stay Silent
A: Confess	A: 5 yrs, B: 5 yrs	A: 0 yrs, B: 10 yrs
A: Stay Silent	A: 10 yrs, B: 0 yrs	A: 1 yr, B: 1 yr

Analysis

Dominant Strategy: For Suspect A, confessing is always better (5 < 10 if B confesses; 0 < 1 if B stays silent). The same logic applies to Suspect B.

Nash Equilibrium: (Confess, Confess) is the unique Nash Equilibrium, producing payoffs of (5 years, 5 years).

The Paradox: The individually rational choice leads to a collectively worse outcome (5 years each) than if both had cooperated and stayed silent (1 year each). This is a non-zero-sum game where self-interest undermines collective welfare.

Finding Nash Equilibria: Best-Response Method

To find pure strategy Nash Equilibria in a payoff matrix, use the best-response method:

Step-by-Step Method

For each of Player 2's strategies, identify Player 1's best response and underline that payoff.
For each of Player 1's strategies, identify Player 2's best response and underline that payoff.
Any cell where both payoffs are underlined is a Nash Equilibrium.

5Advanced Concepts

Mixed Strategies

When a game has no pure strategy Nash Equilibrium, players randomize over their actions with specific probabilities. A mixed strategy Nash Equilibrium exists when no player can improve their expected payoff by changing their probabilities.

Key principle: Each player's mixing probabilities are chosen to make the opponent indifferent among their own pure strategies.

Expected payoff formula: E(payoff) = p × (payoff if action A) + (1 – p) × (payoff if action B), weighted by the opponent's probabilities.

Nash's Theorem: Every finite game has at least one Nash Equilibrium, possibly in mixed strategies.

Backward Induction & Subgame Perfect NE

For sequential games, a Subgame Perfect Nash Equilibrium (SPNE) refines the Nash Equilibrium concept by requiring that strategies form a Nash Equilibrium in every subgame — not just the overall game.

Backward induction: Start at the final decision nodes of the game tree. Determine the optimal choice at each terminal node. Then move backward to the preceding nodes, assuming future players will act optimally.

Why it matters: SPNE eliminates “non-credible threats” — strategies that are NE for the whole game but involve irrational behavior in subgames that would never actually occur.

Repeated Games & the Folk Theorem

When games are played repeatedly, new equilibrium outcomes become possible that are not achievable in the one-shot game.

Finitely Repeated

If the game has a known end point, backward induction from the final round often unravels cooperation, yielding the one-shot NE in every round.

Infinitely Repeated

With no known end point, strategies like tit-for-tat can sustain cooperation. The Folk Theorem states that many cooperative outcomes are sustainable if players are sufficiently patient.

6Worked Examples

Introductory

Oligopoly Pricing: Simultaneous Game with Dominant Strategies

Firms Alpha and Beta simultaneously choose High or Low Price. Payoffs are profits in millions.

	Beta: High	Beta: Low
Alpha: High	($50M, $50M)	( 0M, $70M)
Alpha: Low	($70M, 0M)	( 0M, 0M)

Step 1 – Alpha's best responses: If Beta plays High, Alpha prefers Low ($70M > $50M). If Beta plays Low, Alpha prefers Low (

0M >

0M). Low Price is Alpha's dominant strategy.

Step 2 – Beta's best responses: If Alpha plays High, Beta prefers Low ($70M > $50M). If Alpha plays Low, Beta prefers Low (

0M >

0M). Low Price is Beta's dominant strategy.

Nash Equilibrium: (Low, Low) with payoffs (

0M,

0M).

Key insight: This is a Prisoner's Dilemma — both firms would be better off at (High, High) = ($50M, $50M), but individual incentives lead to (Low, Low).

Intermediate

Market Entry: Sequential Game with Backward Induction

Firm A decides whether to Invest (I) or Don't Invest (DI). Firm B then decides whether to Enter (E) or Don't Enter (DE).

If A chooses DI: B enters ($0,

0) or doesn't (

00, $0). B chooses Enter (

0 > $0).

If A chooses I: B enters ($40,

0) or doesn't (

50, $0). B chooses Enter (

0 > $0).

A's decision: DI leads to $0 (B enters). I leads to $40 (B enters). A chooses Invest ($40 > $0).

SPNE: A: Invest; B: Enter regardless. Outcome: ($40,

0).

Key insight: A strategy in a sequential game must specify actions at every decision node, even those not reached in equilibrium.

Intermediate

Coordination Game: Multiple Nash Equilibria

TechCo and Innovate Inc. choose between Standard X or Standard Y. Payoffs in millions.

	Innovate: X	Innovate: Y
TechCo: X	( 00M, 00M)	( 0M, 0M)
TechCo: Y	( 0M, 0M)	($80M, $80M)

NE #1: (X, X) with payoffs (

00M,

00M) — Pareto superior.

NE #2: (Y, Y) with payoffs ($80M, $80M).

No dominant strategy: TechCo's best response depends on Innovate's choice, and vice versa.

Key insight: Two pure strategy NE exist. The challenge is coordinating on the Pareto-superior equilibrium (X, X).

Advanced

Mixed Strategy Nash Equilibrium: Penalty Kicks

Kicker shoots Left (L) or Right (R). Goalie dives Left or Right. Payoffs are scoring probabilities for Kicker and saving probabilities for Goalie.

	Goalie: Left	Goalie: Right
Kicker: Left	(0.6, 0.4)	(0.9, 0.1)
Kicker: Right	(0.8, 0.2)	(0.5, 0.5)

Step 1 – Make Goalie indifferent: Let p = P(Kicker shoots Left). Set Goalie's expected payoff from diving L equal to diving R:

0.4p + 0.2(1–p) = 0.1p + 0.5(1–p)

0.2p + 0.2 = –0.4p + 0.5 → 0.6p = 0.3 → p = 0.5

Step 2 – Make Kicker indifferent: Let q = P(Goalie dives Left). Set Kicker's expected payoff from shooting L equal to shooting R:

0.6q + 0.9(1–q) = 0.8q + 0.5(1–q)

–0.3q + 0.9 = 0.3q + 0.5 → 0.6q = 0.4 → q = 2/3

MSNE: Kicker: L with p = 0.5, R with p = 0.5. Goalie: L with q = 2/3, R with q = 1/3.

Expected scoring probability: 0.7 (70%).

Key insight: In a mixed strategy NE, each player's probabilities are chosen to make the opponent indifferent, not to maximize their own expected payoff directly.

7Memory Aids

Nash Equilibrium

“No Regrets — at a Nash Equilibrium, no player wishes they had done something different, given what everyone else did.”

Dominant Strategy

“Always Best — a dominant strategy wins no matter what. If you have one, play it every time.”

Backward Induction

“Start at the End — to solve a sequential game, think like a chess player: figure out the last move first, then work your way back to the beginning.”

Prisoner's Dilemma

“Rational but Regrettable — each player does what's best for themselves, yet both end up worse off than if they had cooperated.”

Mixed Strategy

“Make Them Indifferent — you choose your probabilities not to help yourself, but to make your opponent unable to exploit you.”

8Common Mistakes

Confusing Dominant Strategy with Nash Equilibrium

Assuming every Nash Equilibrium involves dominant strategies

A dominant strategy always leads to a Nash Equilibrium, but many Nash Equilibria exist without either player having a dominant strategy. In coordination games, for example, neither player has a dominant strategy, yet multiple Nash Equilibria exist.

Misreading the Payoff Matrix

Mixing up which payoff belongs to which player

By convention, the first number in each cell is the row player's payoff and the second number is the column player's payoff. Reversing this leads to completely wrong conclusions about best responses and equilibria.

Incorrect Backward Induction Order

Solving sequential games from the beginning instead of the end

Backward induction requires starting at the terminal nodes and working backward. Analyzing earlier nodes before later ones yields incorrect results because it ignores how later players will actually respond.

Assuming Cooperation in Non-Cooperative Games

Expecting players to cooperate without binding agreements

In non-cooperative games, players act independently to maximize their own payoff. Unless the game is repeated with credible punishment mechanisms, you should not assume players will cooperate simply because cooperation would make both better off.

Forgetting Mixed Strategy Equilibria

Concluding that a game has no Nash Equilibrium when no pure strategy NE exists

Nash's Theorem guarantees that every finite game has at least one Nash Equilibrium. If no pure strategy NE exists, look for mixed strategy equilibria where players randomize over their actions.

Incomplete Strategies in Sequential Games

Only specifying a player's action at the node reached in equilibrium

A strategy in a sequential game must specify a player's action at every decision node, including those that are not reached on the equilibrium path. This is essential for defining credible threats and for SPNE analysis.

Frequently Asked Questions

What is the difference between a dominant strategy and a Nash Equilibrium?: A dominant strategy is an action that yields the highest payoff for a player regardless of what the other players do. A Nash Equilibrium is a set of strategies (one per player) where no player can improve their payoff by unilaterally changing their strategy. A dominant strategy always leads to a Nash Equilibrium, but a Nash Equilibrium does not require any player to have a dominant strategy. Many games have Nash Equilibria where neither player has a dominant strategy, such as coordination games.
Can a game have more than one Nash Equilibrium?: Yes. Many games have multiple Nash Equilibria. For example, a coordination game like the Battle of the Sexes has two pure strategy Nash Equilibria and one mixed strategy Nash Equilibrium. When multiple equilibria exist, predicting which one players will actually reach can be challenging and may depend on focal points, communication, or historical precedent.
What is the difference between simultaneous and sequential games?: In a simultaneous game, all players choose their strategies at the same time (or without knowledge of others' choices). These are represented using payoff matrices in normal form. In a sequential game, players make decisions in a specific order, with later players observing earlier players' actions. These are represented using decision trees (extensive form) and are solved using backward induction.
Why does the Prisoner's Dilemma lead to a suboptimal outcome?: In the Prisoner's Dilemma, each player has a dominant strategy to defect (confess), because defecting yields a higher payoff regardless of the other player's choice. When both players follow their dominant strategy, the resulting Nash Equilibrium (both defect) is worse for both players than mutual cooperation. This occurs because each player maximizes their individual payoff without internalizing the cost imposed on the other player.
What is a mixed strategy, and when is it used?: A mixed strategy involves a player randomizing over their available actions with specific probabilities, rather than choosing a single action with certainty. Mixed strategies are used when a game has no pure strategy Nash Equilibrium, or when predictability would be exploitable. The probabilities are chosen so that each player makes their opponent indifferent among their own options. Nash's Theorem guarantees that every finite game has at least one Nash Equilibrium, possibly in mixed strategies.
How does repeated interaction change the outcome of the Prisoner's Dilemma?: When the Prisoner's Dilemma is played repeatedly with no known end point (infinitely repeated), cooperation can be sustained as an equilibrium outcome through strategies like tit-for-tat, where players cooperate initially and then mirror their opponent's previous action. The threat of future punishment (retaliation) can deter defection if players value future payoffs sufficiently. This is formalized by the Folk Theorem, which states that many cooperative outcomes can be sustained as equilibria in infinitely repeated games.

Practice Quiz

Test your understanding of game theory — select the correct answer for each question.

1.What is a dominant strategy?

2.In a Nash Equilibrium, which of the following is true?

3.The Prisoner's Dilemma illustrates a situation where:

4.Which method is typically used to solve sequential games and find a Subgame Perfect Nash Equilibrium?

5.A game where players choose their actions without knowing what the other players have chosen is called a:

6.Consider the following payoff matrix. What is Player A's best response if Player B chooses "Up"? | | B: Up | B: Down | | A: Left | (5, 5) | (0, 6) | | A: Right | (6, 0) | (1, 1) |

7.In a zero-sum game, if one player gains

0, the other player(s) must collectively:

8.If a player uses a mixed strategy, they are:

9.Which of the following best describes a coordination game?

10.Game theory is particularly useful for analyzing which market structure?

Study Tips

Practice payoff matrices: Draw 2×2 matrices and systematically find best responses for each player. Underline payoffs to identify Nash Equilibria quickly.
Draw game trees: For sequential games, sketch the extensive form and practice backward induction step by step from the terminal nodes.
Work through mixed strategies: Set up and solve the indifference conditions algebraically. Verify your answer by computing expected payoffs.
Connect to real-world examples: Relate the Prisoner's Dilemma to oligopoly pricing, arms races, and environmental agreements to build intuition.