mathematics//game theory//Nash equilibrium

A Nash equilibrium is a combination of strategies, one per player, in which no player can improve their own payoff by changing only their own strategy while the others keep theirs, and it is the standard prediction of how self-interested players settle in a game: bidders in a spectrum or electricity auction, operators sharing a radio channel, firms setting prices. John Nash introduced it in 1950 and proved that every game with finitely many players and strategies has at least one, provided randomized (mixed) strategies are allowed.

A Nash equilibrium is a combination of strategies, one per player, in which no player can improve their own payoff by changing only their own strategy while the others keep theirs, and it is the standard prediction of how self-interested players settle in a game: bidders in a spectrum or electricity auction, operators sharing a radio channel, firms setting prices. John Nash introduced it in 1950 and proved that every game with finitely many players and strategies has at least one, provided randomized (mixed) strategies are allowed.

Two drone operators share a channel. Each can transmit politely, backing off when it hears the other, or aggressively, transmitting regardless. If both are polite both get decent throughput; if one is aggressive it gets more and the polite one less; if both are aggressive their packets collide and both do badly. Whatever the other does, aggression pays a little more, so both aggressive is the equilibrium, although both polite would serve each better. This is the structure of the prisoner's dilemma, and it is the first lesson of the concept: an equilibrium is stable, which says nothing about whether it is good.

In zero-sum games the Nash equilibrium coincides with the minimax solution, so the minimax theorem covers that case; Nash takes over outside it, where the players' interests are partly shared and partly opposed.

It is the basis of market design. Auction and tariff rules are chosen so that the equilibrium the participants drift toward (bidding true values, using a channel fairly) is the outcome the designer wants, which is how spectrum auctions and electricity markets are engineered.

It assumes rational players who know the game, and many games have several equilibria with nothing to pick between them. Against a real adversary, whose payoffs and rationality are guesses, the safer design is usually the worst case (minimax) rather than a prediction of the exact move.