What is Nash Equilibrium?


What is Nash Equilibrium?

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It’s a game theory term.

It means, a situation where you’re in a situation where all players (people involved in whatever situation you’re in, but ‘game theory’, so) are at a point where they cannot make a better play by themselves. Essentially, they’ve gained the most out of the situation that they’re going to unless someone else does something different that gives them an opportunity.

Nash Equilibrium is an idea that if everybody follows a process then no one person can break away from that process to succeed on their own unless at least one other person agrees to go along with them in breaking away.

Think of it like: if everybody buys their food at the supermarket, and farmer Joe sells his produce only to the supermarket, I can’t go buy farmer Joe’d stuff at a better price than everybody else buys it for at the supermarket .

If farmer Joe is willing to break from norms and not only sell to the supermarket, but also to me, only then can I get a better deal.

Also in cards, nobody can know what cards anybody else has unless someone breaks from norms and shows their cards and if someone else peeks at those shown cards, or if a player has a partner who can signal the cards they have.

In school you get in trouble if you cheat and the person you cheated off of also gets in trouble, and the Nash Equilibrium says this is soundly reasoned.

Basically: nobody can cheat unless somebody helps them cheat.

As long as everybody else follows the rules, no one person can successfully break the rules.

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.

Situation where anyone making a move would worsen his own personal outcome.

Take the prisoner dilemna example. Two accomplices committed a crime together and are sentenced to prison for 5 years. Each one is told “if you snitch on your accomplice, your time in prison will be reduced by 5 years, and his will be increased by 10 years”

If they snitch on each other, they both get jailed for 10 years total. If they don’t, they both get 5 years total (which is better). If one does but not the other, the one who snitched gets 0 year and his accomplice gets 15.

The situation where both of them plan to snitch on each other (thus 10 years each) is a Nash equilibrium: because if any of them modifies his decision (and chooses not to snitch on his accomplice), he will worsen his personal outcome by getting 5 more years.

You’d tell me “yeah what if they agree to both not snitch on each other”. Yeah that example is a bit limited indeed, but the point is, there are contexts where that kind of agreement isn’t feasible.

Take the “split or steal” example. It’s a TV game show: [https://en.wikipedia.org/wiki/Golden_Balls#Split_or_Steal](https://en.wikipedia.org/wiki/Golden_Balls#Split_or_Steal)

Well, the case where both contestants choose “steal” is a Nash equilibrium. If one of them changes his decision he’s just shooting himself in the foot with no personal gain from it. Although the case where they’d both choose “split” would be better for everyone.