Pure-Strategy Nash Equilibria 🎮

Welcome, students! In this lesson, you will learn how to find pure-strategy Nash equilibria in finite strategic-form games. A pure-strategy equilibrium is a situation where each player chooses one specific action, and no one can improve their outcome by changing only their own choice. This idea appears in real life all the time, from traffic decisions to business competition to choosing routes in a video game map 🚗📈🎯.

What you will learn

By the end of this lesson, you will be able to:

• Locate pure-strategy Nash equilibria in a game.
• Check whether a strategy profile is an equilibrium.
• Explain the intuition behind why a pure equilibrium is stable.

A key idea in game theory is that players make decisions while considering what others may do. In a pure-strategy Nash equilibrium, everyone’s choice is “best” given the other players’ choices. No player has a profitable one-step switch.

What is a pure-strategy Nash equilibrium?

A strategic-form game lists the players, the actions available to each player, and the payoff each player receives for every possible combination of actions. A strategy profile is one complete set of choices, one for each player.

A strategy profile is a pure-strategy Nash equilibrium if each player’s action is a best response to the actions chosen by the other players. In simpler words, if everyone else keeps their choices fixed, you cannot do better by changing only your own action.

For two players, if Player 1 chooses action $a$ and Player 2 chooses action $b$, then $(a,b)$ is a pure-strategy Nash equilibrium when:

• $a$ is the best response to $b$ for Player 1, and
• $b$ is the best response to $a$ for Player 2.

This is an important stability concept. It does not mean the outcome is perfect for everyone. It only means no individual player wants to deviate alone. That is why equilibrium can still leave both players worse off than some other possible outcome. A famous example is the Prisoner’s Dilemma, where the equilibrium is stable but not socially optimal.

How to check whether a profile is an equilibrium

To test whether a strategy profile is a pure-strategy Nash equilibrium, follow this method:

1. Fix the other players’ actions.
2. Look at one player at a time.
3. Compare that player’s payoff from each available action.
4. See whether the chosen action gives the highest payoff.
5. Repeat for every player.

If every player is playing a best response, the profile is a Nash equilibrium.

Here is the logic behind it: suppose Player 1 is choosing between actions $A$ and $B$, while Player 2 is already choosing $C$. If Player 1 gets payoff $5$ from $A$ and payoff $3$ from $B$, then $A$ is Player 1’s best response to $C$. If Player 1 actually chose $A$, then Player 1 has no reason to switch.

The same idea applies in larger games with more players or more actions. The only difference is that you may need to check many more action profiles.

Example 1: A simple two-player game

Consider this payoff table, where the first number in each cell is Player 1’s payoff and the second number is Player 2’s payoff.

| | Player 2: Left | Player 2: Right |

|------------|-----------------|-----------------|

| Player 1: Up | $(2,2)$ | $(0,3)$ |

| Player 1: Down | $(3,0)$ | $(1,1)$ |

Let’s check each outcome.

Step 1: Best responses for Player 1

• If Player 2 chooses Left, Player 1 compares $2$ from Up with $3$ from Down. So Down is better.
• If Player 2 chooses Right, Player 1 compares $0$ from Up with $1$ from Down. So Down is better again.

So Player 1’s best response is always Down.

Step 2: Best responses for Player 2

• If Player 1 chooses Up, Player 2 compares $2$ from Left with $3$ from Right. So Right is better.
• If Player 1 chooses Down, Player 2 compares $0$ from Left with $1$ from Right. So Right is better again.

So Player 2’s best response is always Right.

Step 3: Find the intersection

The profile $(\text{Down}, \text{Right})$ is the only outcome where both players are using best responses. Therefore, it is the pure-strategy Nash equilibrium.

Notice something interesting: Player 1 gets $1$ and Player 2 gets $1$ at equilibrium, but the outcome $(\text{Up}, \text{Left})$ gives both players $2$. That better outcome is not an equilibrium because each player would want to change unilaterally. This shows that equilibrium is about stability, not always about the best total payoff.

Example 2: A game with more than one equilibrium

Now consider another game.

| | Player 2: Left | Player 2: Right |

|------------|-----------------|-----------------|

| Player 1: Left | $(4,4)$ | $(1,2)$ |

| Player 1: Right | $(2,1)$ | $(3,3)$ |

Let’s identify best responses.

Player 1’s best responses

• If Player 2 chooses Left, Player 1 compares $4$ from Left with $2$ from Right, so Left is better.
• If Player 2 chooses Right, Player 1 compares $1$ from Left with $3$ from Right, so Right is better.

Player 2’s best responses

• If Player 1 chooses Left, Player 2 compares $4$ from Left with $2$ from Right, so Left is better.
• If Player 1 chooses Right, Player 2 compares $1$ from Left with $3$ from Right, so Right is better.

Now check the diagonal outcomes:

• $(\text{Left}, \text{Left})$ is a Nash equilibrium because both players are best responding.
• $(\text{Right}, \text{Right})$ is also a Nash equilibrium for the same reason.

This game has two pure-strategy Nash equilibria. Both are stable, but they are different stable points. Real-world examples of this kind of situation include choosing a common technology standard, meeting at a certain location, or driving on one side of the road. Once enough people coordinate on one option, switching alone is not helpful 🛣️.

Why pure equilibria make sense

Pure-strategy Nash equilibria are useful because they describe outcomes where no one has an incentive to make a small independent change. This is the basic idea of strategic stability.

Think about a restaurant choice with a friend. If both of you already decided on the same restaurant, and changing alone would make one person unhappy or lower their payoff, then the decision can be stable. The same logic works in economics, biology, politics, and computer science.

A pure equilibrium is not the same as:

• a socially best outcome,
• a fair outcome,
• or a guaranteed outcome that always exists.

Some finite games do not have any pure-strategy Nash equilibrium. In those cases, players may need to randomize, which leads to mixed strategies. But when a pure equilibrium does exist, it is often the simplest equilibrium to find and interpret.

A reliable way to search for pure equilibria

When working on homework or exams, students, it helps to use a careful checklist:

• Write down all possible strategy profiles.
• For each profile, ask whether Player 1 can improve by changing alone.
• Then ask whether Player 2 can improve by changing alone.
• If no player can improve, mark the profile as an equilibrium.

Another shortcut is to identify best responses first. In a payoff table, circle the highest payoff in each column for Player 1 and the highest payoff in each row for Player 2. Any cell that is a best response for both players is a pure-strategy Nash equilibrium.

This method works because equilibrium means mutual best response. It is a direct test of the definition.

Common mistakes to avoid

Here are some frequent errors:

• Confusing a high payoff with an equilibrium. A large payoff is not enough. The outcome must be stable against unilateral deviation.
• Checking only one player. Both players must be best responding.
• Thinking every game has a pure equilibrium. Some do, some do not.
• Ignoring ties. If a player gets the same highest payoff from more than one action, each of those actions can be a best response.

A tie means multiple best responses may exist. In that case, a strategy profile can still be an equilibrium as long as every chosen action is one of the best responses.

Conclusion

Pure-strategy Nash equilibria describe outcomes where each player chooses a specific action and no one can benefit by switching alone. To find them, compare each player’s payoffs across available actions while holding others fixed. If every player is choosing a best response, the strategy profile is a pure-strategy Nash equilibrium.

These equilibria matter because they help explain stable behavior in competitive and cooperative settings. Even when the outcome is not ideal for everyone, it can still persist because no single player wants to move away on their own. That is the core intuition behind pure-strategy equilibrium 🎯.

Study Notes

• A strategy profile is one complete set of actions, one for each player.
• A pure strategy means choosing one specific action, not randomizing.
• A pure-strategy Nash equilibrium occurs when every player’s action is a best response to the others’ actions.
• To test a profile, check whether any player can improve by changing only their own action.
• If no player can improve unilaterally, the profile is an equilibrium.
• A game may have zero, one, or multiple pure-strategy Nash equilibria.
• Equilibrium means stable, not necessarily best for everyone.
• Best-response tables are a helpful way to locate pure equilibria quickly.