Nash Equilibrium

A Nash equilibrium is a set of strategies, one for each player in a game, in which no player can improve their own outcome by changing strategy while everyone else keeps theirs unchanged. Each player's choice is a "best response" to the others' choices, so once the players reach it nobody has a reason to move alone. It is the most widely used solution concept in game theory and a basic tool for analysing competition, bargaining, auctions and policy.

The concept is named after the American mathematician John Forbes Nash Jr., who defined it in a short 1950 paper in the Proceedings of the National Academy of Sciences and in his Princeton doctoral thesis the same year. Nash proved that every game with a finite number of players and strategies has at least one equilibrium if players are allowed to randomize ("mixed strategies"). He shared the 1994 Nobel Memorial Prize in Economic Sciences with John Harsanyi and Reinhard Selten. A Nash equilibrium is a stable point, not necessarily a good one: it need not be Pareto efficient.

Picture drivers choosing which side of the road to use. If everyone drives on the right, no single driver gains by switching to the left; doing so would be dangerous. Driving on the left is equally stable if everyone else does it. Each arrangement is an equilibrium because nobody wants to break from it alone.

The same logic applies to two petrol stations at a crossroads. If both charge similar prices, neither can win by raising prices (customers walk across the road) and cutting further may not pay either. They settle where each is doing the best it can given what the other is doing. The arrangement may not be the best for drivers, or even for the two stations together, but it holds because no one has a reason to change course on their own.

For each player $i$, let $s_i^*$ be their equilibrium strategy and $s_{-i}^*$ the equilibrium strategies of everyone else. The strategy profile is a Nash equilibrium if, for every player and every alternative strategy $s_i$:

• $u_i$ is player $i$'s payoff.
• $s_i^*$ is the strategy player $i$ actually uses in equilibrium.
• $s_i$ is any other strategy available to player $i$.
• $s_{-i}^*$ stands for the equilibrium strategies of all other players, held fixed.

The French mathematician Antoine Augustin Cournot described an early version of this idea in 1838 for a duopoly. Suppose two identical firms sell the same product, market price is $P = 100 - Q$, where $Q = q_1 + q_2$ is total output, and each unit costs $10 to make. Firm 1's profit is $(100 - q_1 - q_2 - 10) \times q_1$. Maximizing that for any given $q_2$ gives its best response:

Firm 2's best response is the mirror image. The Nash equilibrium is where each firm's output is a best response to the other's: $q_1 = q_2 = 30$. Total output is 60, the price is $100 - 60 = 40$, and each firm earns $(40 - 10) \times 30 = 900$. If they formed a cartel and acted like a single monopoly, they would produce 45 in total, charge 55 and share 2,025, about 1,012 each; but each would then be tempted to raise its own output, which is why the cartel outcome is not an equilibrium. Under perfect competition, price would fall to the $10 marginal cost.

• Pure and mixed strategies. In some games, such as rock-paper-scissors or a penalty kick in football, no pure-strategy equilibrium exists; players do best by randomizing, and the equilibrium is in mixed strategies.
• Multiple equilibria. Coordination games like the driving example have several equilibria. Which one emerges depends on history, conventions or "focal points", a term from Thomas Schelling.
• Subgame perfection. In sequential games, Reinhard Selten's refinement removes equilibria that depend on threats a player would not actually carry out.
• Bayesian Nash equilibrium. John Harsanyi extended the concept to games where players hold private information, which underpins auction theory and models of adverse selection.

Nash equilibrium is used to predict prices and output in oligopoly markets, which competition authorities rely on in antitrust merger reviews. It explains why the prisoner's dilemma ends in mutual defection, and why a bank run can be one of two equilibria: if depositors expect others to stay, staying is best; if they expect others to withdraw, withdrawing is best. It guides bidding strategies in auctions, the design of tax and subsidy schemes, and international agreements on tariffs or climate, where each country's best policy depends on what others do. In zero-sum games, Nash equilibrium coincides with von Neumann's earlier minimax solution.

The concept assumes that players are rational and correctly anticipate each other's behavior. Experiments often show people reasoning only a step or two ahead, or caring about fairness, so actual play can differ from the equilibrium; models of bounded rationality and learning try to capture this. When there are several equilibria, the concept does not by itself say which will occur. Computer scientists have also shown that finding a Nash equilibrium can be computationally very hard in large games, which raises the question of how real players could reach it. Supporters respond that equilibrium is best read as a consistency condition, a description of where behavior settles after learning, rather than a claim that people solve equations in their heads.

Nash's life, including his long struggle with schizophrenia and his recovery, was described in Sylvia Nasar's 1998 biography A Beautiful Mind, later a film. He also received the Abel Prize in mathematics in 2015, shortly before he and his wife died in a car accident.

• Nobel Prize: John F. Nash Jr., facts
• Stanford Encyclopedia of Philosophy: Game Theory