If you want to master game theory, you need to master payoff matrices. They’re the single most important tool for analyzing two-player games, and once you understand them, you’ll see strategic situations everywhere.
A payoff matrix is simply a table that shows every possible outcome of a game and what each player gets in each scenario. But this simple visualization unlocks powerful insights about human behavior, business strategy, and why rational people sometimes make seemingly irrational choices.
The Anatomy of a Payoff Matrix
Let’s start with the structure. Here’s what a basic payoff matrix looks like:
(Player 1 payoff, Player 2 payoff)"] style A fill:#4c6ef5 style B fill:#51cf66 style C fill:#ff6b6b style D fill:#ffd43b
The basic format:
| Player 2: Strategy A | Player 2: Strategy B | |
|---|---|---|
| Player 1: Strategy A | (payoff₁, payoff₂) | (payoff₁, payoff₂) |
| Player 1: Strategy B | (payoff₁, payoff₂) | (payoff₁, payoff₂) |
Key points:
- Player 1 (row player) chooses the row
- Player 2 (column player) chooses the column
- Each cell shows the payoffs: (Player 1’s payoff, Player 2’s payoff)
- The first number is always Player 1’s payoff, the second is Player 2’s
Reading a Payoff Matrix: A Simple Example
Let’s revisit the coffee shop game from our previous post. Two coffee shops must decide whether to stay open late or close early.
The Payoff Matrix:
| Shop B: Stay Open | Shop B: Close Early | |
|---|---|---|
| Shop A: Stay Open | (3, 3) | (5, 1) |
| Shop A: Close Early | (1, 5) | (2, 2) |
How to read this:
If both stay open: (3, 3) → Each gets 3 units of profit If A stays open, B closes: (5, 1) → A gets 5, B gets 1 If A closes, B stays open: (1, 5) → A gets 1, B gets 5 If both close: (2, 2) → Each gets 2 units of profit
Split customers
Higher costs"] B[A Open, B Closed] --> B1["Profit: (5, 1)
A captures all customers
B saves costs"] C[A Closed, B Open] --> C1["Profit: (1, 5)
B captures all customers
A saves costs"] D[Both Close] --> D1["Profit: (2, 2)
Save costs
Miss customers"] style A fill:#4c6ef5 style B fill:#51cf66 style C fill:#51cf66 style D fill:#ffd43b
The Famous Prisoner’s Dilemma
The most famous game in game theory is the Prisoner’s Dilemma. It reveals a fundamental tension between individual and collective rationality.
The scenario: Two criminals are arrested. Police separate them and offer each the same deal:
- If you testify against your partner (defect) and they stay silent (cooperate), you go free and they get 10 years
- If both stay silent, you both get 1 year (minor charges)
- If both testify against each other, you both get 5 years
- If you stay silent and they testify, you get 10 years and they go free
The Payoff Matrix (negative numbers = years in prison):
| Prisoner B: Cooperate (Silent) | Prisoner B: Defect (Testify) | |
|---|---|---|
| Prisoner A: Cooperate (Silent) | (-1, -1) | (-10, 0) |
| Prisoner A: Defect (Testify) | (0, -10) | (-5, -5) |
Perspective} Choice1 --> Scenario1["If B cooperates:
A gets -1 (cooperate)
or 0 (defect)"] Choice1 --> Scenario2["If B defects:
A gets -10 (cooperate)
or -5 (defect)"] Scenario1 --> Result1["Defect is better!
(0 > -1)"] Scenario2 --> Result2["Defect is better!
(-5 > -10)"] Result1 --> Conclusion["No matter what B does,
A should DEFECT"] Result2 --> Conclusion Conclusion --> Tragedy["But if both defect: (-5, -5)
Both would be better off
if they cooperated: (-1, -1)"] style Start fill:#4c6ef5 style Conclusion fill:#ff6b6b style Tragedy fill:#ffd43b
The dilemma: From each player’s perspective, defecting is always better—no matter what the other player does. But when both follow this logic, they both get 5 years instead of the 1 year they’d get if both cooperated. Individual rationality leads to collective disaster.
Finding the Nash Equilibrium
A Nash Equilibrium is an outcome where no player can improve their payoff by unilaterally changing their strategy. It’s the stable point of the game—once you’re there, no one has an incentive to deviate.
How to find it:
- For each player, underline their best response to each of the opponent’s strategies
- A cell with both payoffs underlined is a Nash Equilibrium
Example with the Prisoner’s Dilemma:
| B: Cooperate | B: Defect | |
|---|---|---|
| A: Cooperate | (-1, -1) | (-10, 0) |
| A: Defect | (0, -10) | (-5, -5) |
- If B cooperates, A’s best response is defect (0 > -1)
- If B defects, A’s best response is defect (-5 > -10)
- If A cooperates, B’s best response is defect (0 > -1)
- If A defects, B’s best response is defect (-5 > -10)
The Nash Equilibrium is (Defect, Defect), even though (Cooperate, Cooperate) would give both players better outcomes!
Dominant Strategies
Sometimes a strategy is best no matter what the opponent does. This is called a dominant strategy.
In the Prisoner’s Dilemma:
- “Defect” is a dominant strategy for both players
- It’s the best choice whether the opponent cooperates or defects
regardless of opponent"] A --> C[Example: Prisoner's Dilemma] C --> C1["Defect dominates Cooperate"] A --> D[When it exists:] D --> D1["Rational players choose it"] D --> D2["Predicts game outcome"] D --> D3["Creates Nash Equilibrium"] style A fill:#4c6ef5 style B fill:#51cf66 style C1 fill:#ff6b6b
Real-World Applications
Payoff matrices help us understand countless real situations:
1. Business Competition (Pricing Game)
Two companies decide on pricing:
| Company B: High Price | Company B: Low Price | |
|---|---|---|
| Company A: High Price | (10, 10) | (2, 15) |
| Company A: Low Price | (15, 2) | (5, 5) |
This is also a Prisoner’s Dilemma! Both companies would profit more with high prices (10, 10), but each has an incentive to undercut the other, leading to (5, 5).
2. Climate Negotiations
Countries deciding on emissions:
| Country B: Reduce Emissions | Country B: Pollute | |
|---|---|---|
| Country A: Reduce Emissions | (8, 8) | (2, 10) |
| Country A: Pollute | (10, 2) | (3, 3) |
Same structure: Everyone benefits from reduction, but each country has an incentive to pollute while others reduce. The Nash Equilibrium is (Pollute, Pollute).
3. Coordination Games (Technology Standards)
Two companies choosing technology standards:
| Company B: Standard X | Company B: Standard Y | |
|---|---|---|
| Company A: Standard X | (5, 5) | (0, 0) |
| Company A: Standard Y | (0, 0) | (5, 5) |
Here we have two Nash Equilibria: (X, X) and (Y, Y). Both players want to coordinate, but which standard should they choose?
Both prefer different outcome
Example: Defect/Defect"] C --> C1["Multiple Nash Equilibria
Players want to coordinate
Example: Tech standards"] D --> D1["Multiple Nash Equilibria
Players want different things
Example: Battle of Sexes"] style A fill:#4c6ef5 style B fill:#ff6b6b style C fill:#51cf66 style D fill:#ffd43b
Building Your Own Payoff Matrix
When analyzing a real-world situation, follow these steps:
Step 1: Identify the players Who are the decision-makers?
Step 2: List the strategies What choices does each player have?
Step 3: Determine the payoffs For each combination of strategies, what does each player get?
Step 4: Construct the matrix Create the table with one player’s strategies as rows, the other’s as columns.
Step 5: Find equilibria Look for Nash Equilibria by finding best responses.
Why Payoff Matrices Matter
Once you can visualize a game in matrix form, you can:
Predict behavior: Find Nash Equilibria to see what rational players will do
Design incentives: Change payoffs to guide players toward better outcomes
Understand conflicts: See why individually rational choices can lead to bad outcomes
Spot opportunities: Identify situations where cooperation could benefit everyone
The Limits of the Matrix
Payoff matrices are powerful but have limitations:
- They only work well for two players (though they can be extended)
- They assume simultaneous moves (both players choose at the same time)
- They’re less useful for sequential games (where timing matters)
- They require quantifiable payoffs (you need to assign numbers)
For sequential games, we need a different tool: game trees. But for two-player simultaneous games, payoff matrices are unbeatable.
What’s Next?
Now you know how to visualize and analyze two-player games. But this raises a troubling question: if everyone acts rationally and follows the Nash Equilibrium, why do we sometimes end up with terrible outcomes?
In our next post, we’ll explore the paradox of game theory: why rational players sometimes lose, and how understanding this paradox explains everything from traffic jams to financial crises to arms races.
This is the second post in our Game Theory Series. Master payoff matrices, and you’ll start seeing the hidden game theory structure in everyday decisions.