Game Theory Basics: Worked Examples

Game theory basics examples trip up more students in Microeconomics than any other topic, not because the math is hard but because the method needs to be applied in a precise order. A 2x2 payoff matrix holds four cells and two players, and every exam question that asks you to find the equilibrium rewards the same systematic process: check for dominance, apply the best-response method, state the outcome. The numbers change; the method does not. The three fully worked examples below rise from the simplest case (one player has a dominant strategy) through a symmetric game (no dominant strategy, one Nash equilibrium) to the prisoner's dilemma, which is the canonical game theory basics practice problem in virtually every Microeconomics course worldwide.

What Is Game Theory?

Game theory studies how rational players make decisions when the outcome each player receives depends not only on their own choice but also on the choices of others. The field originated in the 1944 work of von Neumann and Morgenstern and was extended into equilibrium analysis by John Nash in the early 1950s, a history the Stanford Encyclopedia of Philosophy's entry on game theory traces in detail. MIT OpenCourseWare's Principles of Microeconomics course covers game theory as a core unit because its tools apply to pricing decisions, market entry, public goods, and bargaining problems that appear throughout intermediate economics.

At the introductory level, game theory basics involve two players, each choosing from a finite set of strategies, and a payoff matrix that records every outcome. The tools you need for most exam questions are: reading a payoff matrix correctly, identifying dominant strategies, and finding Nash equilibria via the best-response method. Mixed strategies, extensive-form games, and repeated games are covered in later modules.

How to Read a Payoff Matrix

A 2x2 payoff matrix places Player 1 as the row chooser (strategies labeled in rows) and Player 2 as the column chooser (strategies labeled in columns). Each cell shows two numbers in parentheses: the left number is Player 1's payoff and the right number is Player 2's payoff when that specific combination of strategies occurs.

The comma-separated convention is universal in introductory game theory. Read each cell as “(row player payoff, column player payoff).” Payoffs can represent profits, utility, years in prison, or any outcome; the analysis depends only on their relative sizes, not their units.

Dominant Strategies: Step-by-Step Method

A dominant strategy gives a player a higher payoff than any alternative, regardless of what the other player does. Identifying one cuts through the matrix immediately: if Player 1 has a dominant strategy, you know what Player 1 will play without needing to know Player 2's choice. The method has three steps: fix the opponent's strategy to one option, compare the row player's payoffs, then repeat for the other option.

Worked Example 1: Dominant Strategy Elimination

Two competing firms, Firm A (rows) and Firm B (columns), each choose a pricing strategy: High price or Low price. The payoff matrix below shows annual profit in millions.

Step 1: Find Firm A's dominant strategy.

Fix Firm B at High Price (left column). Firm A earns 8 at High Price versus 12 at Low Price. Low Price is better. Fix Firm B at Low Price (right column). Firm A earns 2 at High Price versus 4 at Low Price. Low Price is still better. Low Price dominates for Firm A: it wins in both comparisons.

Step 2: Find Firm B's dominant strategy.

Fix Firm A at High Price (top row). Firm B earns 8 at High Price versus 12 at Low Price. Low Price is better. Fix Firm A at Low Price (bottom row). Firm B earns 2 at High Price versus 4 at Low Price. Low Price is again better. Low Price dominates for Firm B too.

Step 3: State the equilibrium.

Both firms have Low Price as a dominant strategy. The equilibrium is (Low Price, Low Price) with payoffs (4, 4). Both firms earn 4 million. Notice that (High Price, High Price) would give both 8 million, but neither firm can commit to it without a binding agreement, because each faces an incentive to undercut. This structure is the prisoner's dilemma, which appears as Worked Example 3.

Common Errors in Dominant Strategy Problems

The most common error is comparing payoffs across rows (comparing Firm A's payoff to Firm B's payoff in the same cell). That comparison is meaningless: the two payoffs belong to different players. Only compare a player's payoffs across their own strategies while holding the opponent fixed.

A second error is concluding that no dominant strategy exists after checking only one of the opponent's strategies. You must check every strategy the opponent has. A strategy that dominates under one column may fail under another.

How to Find a Nash Equilibrium

A Nash equilibrium is a strategy profile where no player can increase their payoff by changing their strategy unilaterally, given what the other player is doing. The best-response method finds it systematically: underline the best response for each player to each opponent strategy, then look for cells where both payoffs are underlined.

Worked Example 2: Nash Equilibrium in a 2x2 Matrix

Two students (Student A in rows, Student B in columns) both simultaneously choose whether to work on a group project individually or collaborate. Payoffs represent grade units (higher is better).

Step 1: Best responses for Student A.

Fix Student B at Collaborate. Student A earns 7 if she collaborates versus 9 if she works solo. Underline 9 in cell (Work Solo, Collaborate). Fix Student B at Work Solo. Student A earns 3 if she collaborates versus 5 if she works solo. Underline 5 in cell (Work Solo, Work Solo).

Step 2: Best responses for Student B.

Fix Student A at Collaborate. Student B earns 7 collaborating versus 9 working solo. Underline 9 in cell (Collaborate, Work Solo). Fix Student A at Work Solo. Student B earns 3 collaborating versus 5 working solo. Underline 5 in cell (Work Solo, Work Solo).

Step 3: Identify the Nash equilibrium.

Check every cell for double underlines. Cell (Work Solo, Work Solo) has payoffs (5, 5). Both payoffs are underlined. That cell is the Nash equilibrium. Cell (Collaborate, Collaborate) gives (7, 7) which is better for both players, but neither player will stay there alone: if Student B deviates to Work Solo while A collaborates, B earns 9 instead of 7, so the cooperative outcome is not a Nash equilibrium.

Finding Nash Equilibrium When No Dominant Strategy Exists

Not every game has a dominant strategy, but most 2x2 games have at least one pure-strategy Nash equilibrium. The best-response method finds it even when no strategy dominates. Work through each column for Player 1 and each row for Player 2, underline best responses, and check for double-underlined cells. If two cells are double-underlined, the game has two Nash equilibria. If none are, the game has no pure-strategy Nash equilibrium and you would need mixed strategies, which are covered in advanced modules.

The Prisoner's Dilemma Explained

The prisoner's dilemma is a game theory basics practice problem where both players have a dominant strategy that leads to a worse collective outcome than the cooperative alternative. Each player acts rationally by individual standards, but their combined rational choices produce a result that neither would choose if they could commit to cooperation. This structure appears throughout microeconomics: oligopoly pricing, public goods provision, arms races, and international trade policy all share the same payoff logic.

Worked Example 3: The Classic Prisoner's Dilemma

Two suspects, A (rows) and B (columns), are held separately and must each decide whether to Cooperate (stay silent) or Defect (testify against the other). Payoffs represent years in prison with negative numbers indicating a worse outcome (more years served) and less-negative numbers indicating a better outcome.

Step 1: Dominant strategy for Suspect A.

If B Cooperates (left column): A earns -1 by cooperating versus 0 by defecting. Defecting is better (0 > -1). If B Defects (right column): A earns -8 by cooperating versus -5 by defecting. Defecting is still better (-5 > -8). Defect dominates for Suspect A.

Step 2: Dominant strategy for Suspect B.

By symmetry, the same analysis applies to Suspect B. If A Cooperates: B earns -1 cooperating versus 0 defecting. If A Defects: B earns -8 cooperating versus -5 defecting. Defect dominates for Suspect B as well.

Step 3: Equilibrium and interpretation.

Both suspects defect. The equilibrium is (Defect, Defect) with payoffs (-5, -5). Both serve 5 years. Yet (Cooperate, Cooperate) at (-1, -1) would give both suspects only 1 year. The equilibrium is Pareto dominated: both players would be better off cooperating. But without a binding agreement, neither can trust the other to cooperate, so both defect.

Why Cooperation Breaks Down

The prisoner's dilemma reveals a structural problem: any individual player who cooperates while the other defects earns the worst payoff (-8). Rational players anticipate this and defect preemptively. The incentive to defect exists whether the other player cooperates or defects, so individual rationality guarantees defection. The MIT OpenCourseWare Microeconomics lecture notes on game theory explain how oligopoly pricing between competing firms reproduces this exact structure: undercutting rivals on price is individually rational and collectively destructive.

In repeated games, cooperation can emerge when players interact indefinitely and can punish defection in future rounds. That result, explored by Axelrod's 1984 tournament and analyzed formally in the folk theorem of repeated games, is covered in intermediate microeconomics. For the introductory payoff matrix problems that appear in most microeconomics assessments, the one-shot prisoner's dilemma always resolves to (Defect, Defect).

Applying These Methods in Exams

Exam questions on game theory basics step by step follow a predictable template. You receive a payoff matrix and a question that asks for one or more of: the dominant strategy for a named player, the Nash equilibrium, whether the outcome is Pareto efficient, and the payoffs at equilibrium. The following approach handles all of these in under five minutes.

1. Label the matrix. Confirm which player chooses rows, which chooses columns, and which payoff number in each cell belongs to which player. Write this at the top of your working.
2. Check for dominant strategies. For each player, hold each opponent strategy fixed and compare the player's payoffs. If one strategy always wins, write “[strategy] dominates for [player].”
3. Apply the best-response method. For each column, underline Player 1's highest payoff. For each row, underline Player 2's highest payoff. Do all of one player's underlines before switching to the other.
4. Identify double-underlined cells. Each is a Nash equilibrium. If none exist, state that no pure-strategy Nash equilibrium exists.
5. State the equilibrium in full. Write the strategy pair and the payoff pair: “The Nash equilibrium is (Defect, Defect) with payoffs (-5, -5).” Then state whether it is Pareto efficient.

The subject calculators hub supports the quantitative side of your Microeconomics module. For worked examples in related areas of economics, the elasticity of demand walkthrough and the monetary versus fiscal policy worked examples use the same step-by-step format. The university resources hub has tools for grade tracking and citation generation.

Game theory basics practice problems at exam level typically add one layer of complexity: asymmetric payoffs where the two players face different incentives, or a third strategy row. Both extensions use the same best-response method. The core skill, scanning payoffs systematically rather than guessing, transfers directly. The Gaussian elimination worked examples in the same subject-mastery cluster show a similar step-by-step approach applied to a different area of quantitative reasoning.

If you want to work through game theory payoff problems with guided feedback, explaining your reasoning before seeing the solution:

Key Takeaways

1. A payoff matrix records every strategy combination. Each cell holds two numbers: left for the row player, right for the column player. Reading the wrong player's payoff is the most common procedural error in game theory exam questions.
2. A dominant strategy beats every alternative regardless of the opponent's choice. Check it by holding the opponent fixed at each of their strategies and comparing the player's payoffs. A strategy must win in every comparison, not just one.
3. A Nash equilibrium is a strategy pair where neither player can improve by switching unilaterally. Use the best-response underline method: underline each player's best response to every opponent strategy, then find double-underlined cells.
4. A 2x2 game can have zero, one, or two pure-strategy Nash equilibria. If no cell has both payoffs underlined, the game has no pure-strategy Nash equilibrium.
5. The prisoner's dilemma has a dominant strategy equilibrium at (Defect, Defect) that is Pareto inferior to (Cooperate, Cooperate). Individual rationality produces collective irrationality. This structure models oligopoly pricing, public goods, and other coordination failures in microeconomics.
6. Pareto efficiency of an equilibrium is a separate question from the equilibrium itself. Check it by comparing the equilibrium payoffs to every other cell: if any cell improves at least one player without harming another, the equilibrium is Pareto inefficient.
7. Game theory basics worked examples in microeconomics exams reward systematic method over intuition. Label the matrix, check dominance, apply best-response underlines in two passes, state the equilibrium with payoffs, and assess Pareto efficiency.

For more practice with quantitative microeconomics methods, the grade calculators hub helps you track your module standing while you build these skills, and the maths exam revision guide covers the deliberate-practice methods that transfer to quantitative problem sets across disciplines.