What is Nash Equilibrium?
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There is an area of maths called “game theory”, which has a concept called a “game”, which is a general model of a situation that involves choosing between strategies.
A game has some number of players, who are each able to choose between some number of “pure strategies”. It’s assumed that they all choose at the same time, without knowing for certain what anyone else will choose. Depending on the pure strategy chosen by each player, each player gets given a score called a “payoff”, and their goal is to maximise their payoff. The players can also choose to employ “mixed strategies”, in which they pick between different “pure strategies” at random with specific probabilities.
A Nash equilibrium is a situation where all the players have chosen their strategies, and none of them can improve their payoff by changing their strategy alone.
For example, take the game “rock paper scissors”. In this game, there are two players and three pure strategies: rock, paper and scissors. If one player picks rock and the other plays scissors, the player who picked rock wins and gets a payoff of 1, while the player who picked scissors loses and gets a payoff of -1. Similarly, scissors beats paper and paper beats rock. If both players pick the same pure strategy, they both get a payoff of zero.
Suppose we’re in a situation where player A has picked the pure strategy “paper”, in other words they will always go with paper. And B has decided they will always go with rock. In this situation B always gets a payoff of -1. But if B changes their strategy to scissors while A stays the same, B’s payoff changes to 1. So this is not a Nash equilibrium.
In fact, the only Nash equilibrium in rock paper scissors is for each player to employ the mixed strategy in which they choose between the options at random, with each option having a probability of 1/3. In this situation the players’ *average* payoff is zero. And if either one of the players changes their strategy (while the other person stays the same), their average payoff will still be zero, since their opponent will still be just as likely to pick the option that beats them as the option that loses to them.
In practice, games are used to model all kinds of things: actual games like poker, financial markets, biological evolution, behaviour in social situations, computer networks, and so on. Usually these models involve a game being played repeatedly, with the players going through some kind of learning or evolutionary process. Nash equilibria are called “equilibria” becauase they often represent equilibrium points of these processes: if the players fall into these points, then they stay there.
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