What is a Nash equilibrium?
Updated Feb 20, 2026
Short answer
A Nash equilibrium is a set of strategies, one per player, where no player can improve their own outcome by unilaterally changing strategy while the others hold theirs fixed. It is a stability concept, not an optimality one — the equilibrium can be worse for everyone than an outcome they could reach by cooperating, which is exactly what the Prisoner's Dilemma demonstrates.
Deep explanation
The definition is precisely about unilateral deviation. Player A's strategy must be a best response to B's, and B's to A's, simultaneously.
The Prisoner's Dilemma, where the equilibrium is bad for both:
B stays silent B betraysA stays silent -1, -1 -3, 0A betrays 0, -3 -2, -2Whatever B does, A is better off betraying: 0 beats −1, and −2 beats −3. Betraying is a dominant strategy, and symmetrically for B. So (betray, betray) is the unique Nash equilibrium at (−2, −2) — even though (silent, silent) gives (−1, −1) and is better for both.
That is the central lesson: Nash equilibrium is stable, not efficient. (silent, silent) is Pareto-superior but not an equilibrium, because either player gains by defecting from it.
Existence. Nash's theorem guarantees at least one equilibrium in any finite game — but possibly only in mixed strategies, where players randomise. Matching Pennies has no pure equilibrium; its unique equilibrium has both players playing each side with probability ½. The intuition is that any predictable pure strategy would be exploited, so the mixing probabilities must make the opponent indifferent.
Multiple equilibria are common, which limits the concept's predictive power. The Stag Hunt has both (stag, stag) and (hare, hare) — the first is better for both, the second is safer. Equilibrium analysis alone does not say which occurs; that depends on trust, communication, and precedent.
Where it applies in practice. Auction design, network routing (the price of anarchy measures how much worse selfish routing is than coordinated), congestion pricing, competitive pricing between firms, and the adversarial training dynamics in GANs.
Real-world example
Two competing shops choosing prices:
B prices high B prices lowA prices high £50k, £50k £10k, £70kA prices low £70k, £10k £30k, £30kBoth would prefer (high, high) at £50k each. But given B prices high, A gains by cutting to £70k; and given B prices low, A still gains by cutting, £30k against £10k. Pricing low is dominant for both, so the equilibrium is (low, low) at £30k each.
This is why cartels are unstable without enforcement — each member has a private incentive to defect — and why competition regulators watch for the communication and monitoring mechanisms that would make cooperation sustainable.
Common mistakes
- - Assuming a Nash equilibrium is the best outcome for the players
- it is merely stable against unilateral deviation, and is often Pareto-inferior.
- - Believing every game has a pure-strategy equilibrium — Matching Pennies does not, and requires mixed strategies.
- - Analysing only unilateral deviation and concluding the outcome is robust
- a *coordinated* deviation by several players may still be profitable.
- - Assuming the equilibrium is unique. Many games have several, and the concept alone does not predict which is reached.
- - Confusing a dominant strategy with an equilibrium
- a dominant strategy always yields one, but equilibria exist without dominance.
- - Applying single-shot analysis to repeated interactions, where cooperation can be sustained by the threat of future retaliation.
Follow-up questions
- Does every game have a Nash equilibrium?
- How does repetition change the Prisoner's Dilemma?
- What is the price of anarchy?
- What is a mixed strategy and why would a player randomise?