Nash equilibrium

Picture two rival coffee shops on the same street, each deciding what price to charge. Once both have set their prices, ask a simple question of each owner: given what my competitor is actually doing, am I happy with my own choice, or would I want to change it? A Nash equilibrium is the situation where the answer is 'I'm happy' for everyone at once. No single player can do better by changing their move alone, so nobody has a reason to move. It is the resting point of a strategic tug-of-war.

More precisely, a set of strategies (one chosen by each player) is a Nash equilibrium if each player's strategy is a best response to the others' strategies. 'Best response' just means the move that gives you the highest payoff, taking everyone else's moves as fixed. Suppose two firms each choose 'high price' or 'low price'. If, when my rival prices low, my best move is also to price low — and the same is true for them — then 'both low' is a Nash equilibrium, because neither of us can gain by switching to high alone. The idea is named after mathematician John Nash, who proved in 1950 that every finite game has at least one such equilibrium (allowing mixed strategies).

Nash equilibrium is the central solution concept of modern game theory; it is the tool economists reach for to predict the outcome of any situation where players' fortunes depend on each other — pricing wars, auctions, arms races, traffic, even evolution. But two honest cautions. First, an equilibrium can be bad for everyone: the prisoner's dilemma has a Nash equilibrium where both confess, even though both staying silent would be better. 'Equilibrium' means stable, not fair or efficient. Second, a game can have several Nash equilibria, and the concept alone does not tell you which one will actually happen.