Game theory
Nash Equilibrium
Why does everyone stay put even when they could all do better? The strategy point nobody wants to leave alone.
A Nash equilibrium is a combination of strategies, one for each player, in which no player can increase their own payoff by unilaterally switching strategy while every other player keeps theirs fixed.
TL;DR
- A Nash equilibrium is a set of strategies where no player can do better by changing only their own move, given everyone else's choices.
- You find it by checking each outcome cell and asking whether either player could switch and gain; if neither can, that cell is an equilibrium.
- Strategies can be pure (a single definite choice) or mixed (a probability distribution over choices), and every finite game has at least one equilibrium once mixed strategies are allowed.
- A game can have zero pure-strategy equilibria, exactly one, or many, so finding one equilibrium does not guarantee it is unique.
- A Nash equilibrium is stable, not necessarily good: the Prisoner's Dilemma equilibrium leaves both players worse off than an outcome they could have reached together.
When to use it
- Predicting how self-interested players will settle when each knows the others are also optimizing and no one can commit their move in advance.
- Analyzing one-shot strategic situations such as pricing, bidding, or market entry, where you need the stable outcome rather than the desirable one.
- As the baseline solution concept before layering on refinements like subgame perfection or repeated play for more complex games.
When it fails
- It assumes players are rational payoff-maximizers with common knowledge of the game; real people misjudge payoffs, care about fairness, or make mistakes, so behavior drifts from the predicted equilibrium.
- When a game has multiple equilibria, the concept alone cannot say which one occurs, so you need extra ideas like focal points or risk-dominance to select among them.
- It describes a static resting point and ignores how players get there; in one-shot play with no coordination, people often fail to reach any equilibrium at all.
Worked example
Consider two arrested suspects questioned in separate rooms, each able to stay silent or confess. If both stay silent they get one year each; if both confess they get three years each; if one confesses while the other stays silent, the confessor walks free while the silent partner serves five years. The pair of choices where both confess is a Nash equilibrium: given that the other confesses, switching to silence would raise your own sentence from three years to five, so neither prisoner will change alone.
To find a Nash equilibrium you inspect each possible outcome and ask one question for every player: could this player raise their own payoff by changing only their own move while everyone else stays fixed? If no player can, the outcome is an equilibrium. In the suspects' case only the both-confess cell survives this test, because in every other cell at least one prisoner can gain by switching to confess, which is why confessing is each player's dominant strategy.
Not every game has such a clean single answer. A coordination game where two friends want to meet but prefer different venues has two pure-strategy Nash equilibria, and the concept alone cannot say which they will pick. Some games, like matching pennies, have no equilibrium in pure strategies at all, yet John Nash proved that every finite game has at least one equilibrium once players may use mixed strategies, playing each option with a fixed probability rather than committing to one.
| Prisoner B | ||
|---|---|---|
| Prisoner A | Stay silent (cooperate) | Confess (defect) |
| Stay silent (cooperate) | −1 / −1 | −5 / 0 |
| Confess (defect) | 0 / −5 | −3 / −3 |
Related concepts
- Prisoner's Dilemma — The most famous game whose Nash equilibrium leaves everyone worse off than an available cooperative outcome.
- Dominant Strategy — When every player has a dominant strategy, the resulting profile is automatically the Nash equilibrium.
- Tit for Tat — A repeated-game strategy that can sustain the cooperation the one-shot Nash equilibrium destroys.
- Schelling Point — How players select among multiple Nash equilibria when the theory alone cannot pick one.
FAQ
What is a Nash equilibrium in simple terms?
A Nash equilibrium is a situation where everyone is doing the best they can given what everyone else is doing, so no one wants to change their move alone. It is a stable standoff: not necessarily the best outcome for the group, just one nobody can improve by acting unilaterally.
What is an example of a Nash equilibrium?
The Prisoner's Dilemma is the classic example: two suspects each confess because, whatever the other does, confessing shortens their own sentence. Both confessing is the Nash equilibrium, since neither can improve by staying silent alone, even though both staying silent would have left them better off.
Nash equilibrium vs dominant strategy — what's the difference?
A dominant strategy is a player's best move no matter what others do; a Nash equilibrium is an outcome where each player's move is best given the others' actual moves. Every dominant-strategy outcome is a Nash equilibrium, but most equilibria arise without any dominant strategy existing.
Why does a Nash equilibrium happen even when it's bad for everyone?
Because each player optimizes only their own payoff, not the group's. In the Prisoner's Dilemma, confessing is individually safer whatever the partner does, so both rationally confess and reach a worse joint outcome. The equilibrium is stable precisely because no single player can escape it without help from the other.
How do you find a Nash equilibrium in a payoff matrix?
Check every cell for profitable deviations. For each outcome, ask whether the row player could switch rows to a higher payoff, and whether the column player could switch columns to a higher payoff. If neither can improve by moving alone, that cell is a Nash equilibrium; a game may have zero, one, or several.