A situation in which no participant can do better by changing strategy alone. It is the central solution concept in game theory, it exists in every finite game, and its most famous illustration is a case where everyone acting rationally leaves everyone worse off.
A game is any situation where outcomes depend on the choices of several parties. A Nash equilibrium is a combination of strategies such that each is the best response to the others.
The definition is about stability rather than optimality. Nobody can improve their own position by deviating unilaterally, which does not mean the outcome is good, efficient, or fair.
John Nash proved in 1950, in a doctoral thesis of about twenty-seven pages, that every finite game with a finite number of players has at least one such equilibrium, provided mixed strategies are allowed, meaning players may randomise across their options.
The existence proof was the breakthrough. Earlier work by von Neumann and Morgenstern had solved zero-sum two-player games, where one player's gain is another's loss. Nash's result covers games with any number of players and any structure of interests, which is what made game theory applicable to economics generally.
He shared the 1994 Nobel Memorial Prize in Economic Sciences with John Harsanyi and Reinhard Selten.
The standard illustration is worth working through because its lesson is frequently misstated.
Two people are arrested and held separately. Each may stay silent or implicate the other. If both stay silent, both get a short sentence. If both implicate, both get a long one. If one implicates and the other stays silent, the informer goes free and the silent one gets the longest sentence.
For each individual, implicating is better regardless of what the other does. So both implicate, and both do worse than if both had stayed silent.
That outcome is the unique Nash equilibrium. It is also worse for both parties than an available alternative, which is the point: individually rational choices can produce a collectively bad result, and no appeal to rationality escapes it.
This is the structure underlying arms races, overfishing, price wars and free-riding on public goods. It is closely related to the tragedy of the commons, treated in its own capsule.
Repetition changes it. In a game played repeatedly with an indefinite horizon, cooperation can be sustained, because defecting invites retaliation in later rounds. Robert Axelrod's tournaments in the early 1980s found that tit for tat, cooperating first and then copying the opponent's last move, performed well against a wide field of submitted strategies.
A known final round destroys this, by backward induction: in the last round there is no future to protect, so defection is optimal, which makes the second-to-last round effectively final, and so on.
Communication, enforceable contracts, reputation and altruistic punishment all shift outcomes, and none of them changes the mathematics. They change the game.
The concept is a theorem and the theorem is not in question. Whether people play equilibrium strategies is an empirical matter and the answer is mixed.
The ultimatum game is the standard counterexample. One player proposes a split of a sum and the other accepts or rejects; rejection gives both nothing. The equilibrium is to offer the smallest possible amount and for it to be accepted, since something beats nothing. Actual players offer close to half and reject low offers routinely, accepting a personal loss to punish an unfair proposal.
Results vary substantially across cultures, which argues that the deviation reflects real preferences about fairness rather than a uniform cognitive limitation.
Multiple equilibria are a further practical problem. Many games have several, and the theory does not say which will occur, so predicting behaviour requires an additional account of how one is selected. Thomas Schelling's work on focal points addresses this and is not a general solution.
Auction design is the clearest success. Spectrum auctions raising very large sums have been designed by economists reasoning explicitly about equilibrium behaviour, and design failures have produced measurable losses.
Market design more broadly, including matching medical graduates to hospitals and students to schools, uses closely related machinery and has changed how those systems operate in several countries.
Oligopoly pricing, entry deterrence, bargaining and contract structure are all analysed this way, and antitrust cases turn on such arguments regularly.
Evolutionary biology uses a variant. An evolutionarily stable strategy is one that cannot be invaded by a rare alternative, which is a refinement of Nash equilibrium applied to populations rather than reasoning agents, and it works without anyone deciding anything.
