Nash Equilibrium: Existence and Multiplicity

Welcome, students 👋 In this lesson, you will learn when a finite strategic-form game has one Nash equilibrium, many Nash equilibria, or no pure-strategy Nash equilibrium at all. This matters because Nash equilibrium is one of the main tools game theorists use to predict outcomes when people, firms, or teams make choices at the same time.

What you will learn

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

• describe what equilibrium multiplicity means,
• recognize why some games have several equilibria,
• explain why some games do not have any equilibrium in pure strategies.

A Nash equilibrium is a strategy profile where no player can improve their payoff by changing only their own strategy while the others keep theirs fixed. In finite games, equilibrium outcomes can be simple or surprisingly complicated. Sometimes there is exactly one equilibrium, sometimes there are several, and sometimes there is none in pure strategies. 🎯

What does existence mean in a finite game?

A finite strategic-form game has a limited number of players and a limited number of strategies for each player. Because there are only finitely many possible strategy profiles, you can, in principle, check each one to see whether it is a Nash equilibrium.

A pure-strategy Nash equilibrium is a strategy profile where each player’s choice is a best response to the others’ choices. Formally, if player $i$ chooses strategy $s_i$ and the other players choose $s_{-i}$, then $s_i$ is a best response if it gives player $i$ at least as much payoff as any other available strategy. In symbols, for every alternative strategy $s_i'$, we have

$$u_i(s_i,s_{-i}) \ge u_i(s_i',s_{-i}).$$

This idea is the key to checking existence in pure strategies. To find all pure equilibria, you look for strategy profiles where every player is already making a best response.

Example: coordination on a meeting place

Imagine two friends, Alex and Brooke, choosing whether to meet at the park or the library. They both prefer to meet together, but they must choose the same place or else they miss each other.

A payoff table might look like this:

• If both choose the park, each gets $2$.
• If both choose the library, each gets $1$.
• If they choose different places, each gets $0$.

Here, both $(\text{Park}, \text{Park})$ and $(\text{Library}, \text{Library})$ are Nash equilibria. Why? Because if Alex expects Brooke to go to the park, Alex does best by going to the park too. The same logic works for the library. This game has multiple pure-strategy equilibria.

When a game has one equilibrium

Some games have exactly one pure-strategy Nash equilibrium. This usually happens when one strategy is clearly the best response to all possible choices of the other players, or when only one strategy profile survives the best-response test.

Example: a dominant strategy game

Suppose each player in a two-player game has a strategy that is always better than the alternative, no matter what the other player does. That strategy is called a strict dominant strategy. If every player has a strict dominant strategy, then the game has exactly one pure-strategy Nash equilibrium: the profile where everyone uses their dominant strategy.

For instance, imagine two firms deciding whether to adopt a cheaper technology. If using the cheaper technology always gives a higher profit for each firm, regardless of what the other firm does, then both firms adopt it. That outcome is the unique pure-strategy Nash equilibrium.

This kind of uniqueness is easy to understand because the best response does not depend on guessing what the other player will do. In real life, however, dominant strategies are not always present.

Why multiple equilibria arise

Multiple equilibria happen when players’ best responses reinforce different outcomes. In other words, several different strategy profiles can each satisfy the Nash condition.

Coordination games

A classic source of multiplicity is a coordination game. These are games where players do better when they choose the same action, or when their choices fit together in the right way.

A famous example is choosing which side of the road to drive on. In countries where everyone drives on the right, the best response is to drive on the right. In countries where everyone drives on the left, the best response is to drive on the left. Both outcomes can be stable, even though only one is used in a particular place. The reason is that each outcome is self-reinforcing.

Strategic complements

Games with strategic complements often have multiple equilibria. Strategic complements mean that when one player increases their action, it makes a higher action more attractive for the other player too. In such settings, several consistent patterns of behavior may exist.

A simple example is a technology standard battle. Suppose two companies choose between technology A and technology B. If customers, suppliers, or partners support the same technology, both companies benefit more. Then $(A,A)$ and $(B,B)$ can both be equilibria.

What multiplicity means for prediction

Multiplicity matters because Nash equilibrium no longer gives just one prediction. Instead, it gives a set of possible outcomes. To choose among them, analysts may need extra information such as:

• pre-play communication,
• historical precedent,
• focal points,
• small differences in payoffs,
• or learning dynamics.

This is important in economics and politics because the same game can lead to different stable outcomes depending on how players coordinate. 📌

Why some games have no pure-strategy equilibrium

Not every finite game has a pure-strategy Nash equilibrium. This may seem surprising, but it is a normal feature of many games.

Matching pennies

A famous example is matching pennies. Two players each choose Heads or Tails. One player wants the choices to match, while the other wants them to differ. No matter what one player does, the other always has a profitable reason to switch.

There is no pure-strategy Nash equilibrium here because every strategy profile leaves at least one player wanting to change their choice. If both choose Heads, the player who wants mismatches would switch. If one chooses Heads and the other Tails, the player who wants matches would switch. There is no stable pure outcome.

Best-response cycles

Another way to understand the lack of a pure equilibrium is through a best-response cycle. Suppose player 1’s best response to player 2’s action is to switch, and player 2’s best response to player 1’s new action is also to switch. Then the game can keep cycling through improving responses without ever settling on one profile.

This shows that a finite game may have no pure equilibrium even though each player is always choosing rationally from their own point of view. The issue is not irrationality; it is the conflict of incentives.

Interpreting equilibrium outcomes in finite games

When you analyze a finite game, students, you should ask three questions:

1. How many pure-strategy Nash equilibria exist?
2. Are they unique or multiple?
3. Is there any pure-strategy equilibrium at all?

If there is exactly one equilibrium, the model gives a sharp prediction. If there are several, the model says the game has multiple stable outcomes, but it does not say which one will happen without extra information. If there is none in pure strategies, then players may need to randomize, which leads to mixed strategies. That topic goes beyond this lesson, but it is important to know that lack of a pure equilibrium does not mean the game has no solution at all.

A quick checklist for finding pure equilibria

To find pure-strategy Nash equilibria in a finite game:

• list all possible strategy profiles,
• find each player’s best responses,
• check which profiles are mutual best responses,
• count how many such profiles exist.

This method is straightforward in small games and very useful for understanding larger ones.

Conclusion

students, the big idea is that finite games can have different equilibrium patterns. Some games have one pure-strategy Nash equilibrium, often when a dominant strategy leads everyone to the same outcome. Some games have multiple equilibria, especially coordination games where several outcomes are self-reinforcing. Some games have no pure-strategy equilibrium at all, as in matching pennies, because players’ incentives keep pushing them away from any fixed outcome. Understanding existence and multiplicity helps you see not just whether a game has a solution, but also how many stable solutions it can have and why. ✅

Study Notes

• A pure-strategy Nash equilibrium is a strategy profile where each player is choosing a best response to the others.
• In a finite game, you can check every strategy profile to find pure equilibria.
• A game has unique equilibrium when there is exactly one pure-strategy Nash equilibrium.
• Multiplicity means there is more than one pure-strategy Nash equilibrium.
• Coordination games often have multiple equilibria because players prefer matching or compatible choices.
• Strict dominant strategies often produce a unique equilibrium.
• Some games, like matching pennies, have no pure-strategy Nash equilibrium because at least one player always wants to deviate.
• If no pure equilibrium exists, players may need to randomize, which leads to mixed strategies.