What Is Nash Equilibrium? Game Theory and the Prisoner's Dilemma
By Markets & Investing DeskUpdated Nov 10, 2026
Definition
A Nash equilibrium is a set of strategies, one per player, where no player can get a better result by changing only their own strategy while the others keep theirs.
Key takeaways
- John Nash published the concept in 1950 and shared the 1994 Nobel Memorial Prize in Economic Sciences.
- In the prisoner's dilemma, both players betray and serve 5 years each, though mutual silence would give 1 year each.
- A coordination game can have two equilibria, such as both firms picking standard X or both picking Y, with neither dominant.
- Every game with finitely many players and strategies has at least one equilibrium in mixed strategies.
Nash equilibrium is a set of strategies, one for each player in a game, in which no player can get a better result by changing only their own strategy while everyone else keeps theirs. Each player’s choice is a best response to the choices of the others, so no one has a reason to move first.
The idea comes from John Nash, who published it in 1950 as a Princeton graduate student. His work on non-cooperative games earned him a share of the 1994 Nobel Memorial Prize in Economic Sciences.
How Nash Equilibrium Works
Game theory studies situations where the outcome for each participant depends on the choices of all participants. A game is defined by its players, the strategies each can choose, and the payoffs each receives for every combination of choices.
Formally, a strategy profile s* is a Nash equilibrium if, for every player i:
uᵢ(sᵢ*, s₋ᵢ*) ≥ uᵢ(sᵢ, s₋ᵢ*) for every strategy sᵢ available to player i
Here uᵢ is player i’s payoff, sᵢ* is the equilibrium strategy of player i, and s₋ᵢ* is the equilibrium strategies of everyone else. In plain terms: holding the others fixed, player i’s equilibrium choice does at least as well as any alternative.
How to Read a Game Theory Matrix
Two-player games with a few choices are usually shown as a payoff matrix. One player’s strategies label the rows and the other’s label the columns. Each cell lists two payoffs, conventionally the row player’s first and the column player’s second.
To find pure-strategy equilibria in a matrix:
- For each column, find the row player’s best payoff and mark it.
- For each row, find the column player’s best payoff and mark it.
- Any cell where both payoffs are marked is a Nash equilibrium, because each player is already playing a best response to the other.
Example: The Prisoner’s Dilemma
The prisoner’s dilemma is the standard illustration. It was developed at the RAND Corporation in 1950 by Merrill Flood and Melvin Dresher.
Two suspects are arrested and held separately with no way to communicate. Prosecutors lack enough evidence for a full conviction, so each prisoner is offered a choice: betray the other by testifying, or stay silent. Payoffs are years in prison, so lower is better.
| B stays silent | B betrays | |
|---|---|---|
| A stays silent | 1, 1 | 10, 0 |
| A betrays | 0, 10 | 5, 5 |
Work through A’s options. If B stays silent, A serves 1 year by staying silent or 0 by betraying, so betraying is better. If B betrays, A serves 10 years by staying silent or 5 by betraying, so betraying is better again. B faces the same numbers. The only cell where both are playing a best response is both betray, 5 years each. That is the Nash equilibrium.
Yet both staying silent would give each prisoner just 1 year. That outcome is not stable: if either prisoner expects the other to stay silent, they can walk free by betraying. This gap between individual incentives and the collective result is why the prisoner’s dilemma is often used to describe incentive problems such as price wars, arms races and overuse of shared resources.
Nash Equilibrium vs. Dominant Strategy
A dominant strategy gives a player a better result than any alternative no matter what the opponent does. A Nash equilibrium only requires each strategy to be the best response to the strategies the others are actually playing.
In the prisoner’s dilemma, betraying is a dominant strategy for both players, so the equilibrium is easy to find. Many games have no dominant strategy but still have equilibria. Consider a hypothetical coordination game in which two companies each choose a technical standard, X or Y. Each earns 10 if they match and 0 if they don’t. Neither standard is dominant, since the right choice depends on the other company. Both (X, X) and (Y, Y) are Nash equilibria. Every dominant-strategy outcome is a Nash equilibrium, but not every Nash equilibrium involves dominant strategies.
Is Nash Equilibrium the Best Outcome?
Not necessarily. Equilibrium describes stability, not efficiency. The relevant comparison is Pareto optimality: an outcome is Pareto optimal if no one can be made better off without making someone worse off. Mutual silence in the prisoner’s dilemma is Pareto optimal but not an equilibrium; mutual betrayal is an equilibrium but not Pareto optimal. In the coordination game, both equilibria happen to be efficient, which shows the two ideas can overlap without being the same.
Mixed Strategies and Why Every Finite Game Has an Equilibrium
Some games have no pure-strategy equilibrium. In matching pennies, one player wins if the coins match and the other wins if they differ. Whatever pair of choices you pick, one player wants to switch.
The way out is a mixed strategy, where a player chooses randomly according to set probabilities. If each player picks heads and tails with 50% probability, neither can improve their expected payoff by changing their mix, so that pair of mixed strategies is a Nash equilibrium. Nash’s central theorem shows this generalizes: every game with finitely many players and strategies has at least one equilibrium in mixed strategies.
How Nash Equilibrium Is Used in Economics
Markets with a few large competitors fit the framework well because each firm’s profit depends on rivals’ choices. Antoine Augustin Cournot’s 1838 model of two firms choosing output levels is now read as an early Nash equilibrium: each firm’s quantity is the best response to the other’s.
As a hypothetical example, suppose two airlines on one route can charge a high or low fare. If both charge high, each earns more than if both charge low, but either can win market share by undercutting a high-priced rival. If the payoffs mirror the prisoner’s dilemma, both charging low is the equilibrium, which helps explain why firms may compete prices down even when keeping them high would pay more. The same reasoning appears in auction design, bargaining, trade negotiations and political strategy.
Limitations of Nash Equilibrium
- Rationality assumptions. Real people may misjudge payoffs, act on emotion, or care about fairness, so observed behavior can depart from the predicted equilibrium.
- Multiple equilibria. When a game has several equilibria, the concept alone does not say which one players will reach.
- Stability, not process. The definition describes a resting point, not how players get there or how long it takes.
- One-shot vs. repeated games. In a single prisoner’s dilemma, betrayal is the equilibrium. When the game repeats indefinitely, players can reward cooperation and punish betrayal over time, and cooperative outcomes can be sustained as equilibria.
- Unilateral deviations only. A Nash equilibrium can still be upset if a group of players changes strategies together.
The Bottom Line
Nash equilibrium matters to anyone studying how firms, bidders or negotiators act when each outcome depends on rivals' choices. It identifies stable outcomes, which can still leave everyone worse off, as when airlines compete fares down. Because real people may act on emotion or fairness, and games can have several equilibria, the concept works best as a model of incentives rather than an exact forecast of behavior.
