What is the Nash Equilibrium?

The Nash equilibrium is a type of solution — proposed by John Forbes Nash in 1950 — commonly used in game theory… But what is it?

Concave’s Community Explains: What is the Nash Equilibrium?

Here’s what Coussins has to say about that:

“The Nash equilibrium is a type of solution — proposed by John Forbes Nash in 1950 — commonly used in game theory, whose very definition emphasizes its self-fulfilling character.

A Nash equilibrium is, in fact, a combination of individual decisions, called “strategies”, where each one correctly anticipates the choices of the others; there is self-realization since the outcome realized is the result of decisions made in the belief that it will be realized.

In fact, the big — and only — question a player asks himself when making his choice is, in game theory: what will the other player do?

The player’s beliefs about the behavior of the others are therefore essential to the decision. The diversity of beliefs can thus correspond to a multiplicity of equilibria.

What does the theory predict about the players’ choices?

Nothing very precise, since it depends on what each player thinks the other will do.

In fact, many outcomes of a game can result from the choices of “reasonable” individuals: so there is no particular reason to favor equilibria.

One might even question the use of the word “equilibrium” to refer to Nash solutions since the rules of the games exclude any process (they assume a unique and simultaneous choice on the part of the players). “

He then continues by giving us a bit more info, context, and even some examples:

John Nash defined an interaction situation as stable if no agent has an interest in changing its strategy.

The formalization of this simple observation has been essential for game theory.

Origin of the notion A game is a formal framework where several agents decide on a strategy, knowing that their utility depends on the choices of all.

Before Nash, the determination of a stable situation had no formal method. Even if the current translation of a Nash equilibrium may seem simplistic, the considerable possibilities of resolution opened up by Nash earned him the “Nobel Prize” in economics in 1994, jointly with Reinhard Selten and John Harsanyi.

This definition applies to games with any number of players. Nash showed that all results found before him lead to stable equilibria in his sense.

Optimality First example Two players simultaneously choose a number between 0 and 10. The player who announced the smaller number wins that number, and the other player wins the same minus two.

In case of a tie, both players suffer the penalty of two. The only Nash equilibrium in this game is when both announce zero.

In all other pairs of strategies, the player who announces more or the same can improve his result by declaring less. The Ice Cream Man Example Two ice cream vendors must choose a location on a given length of the land.

Since the prices and products are the same, each customer will go to the shop closest to him. It is easy to see that the only Nash equilibrium for these two merchants will be when they are both sides by side in the center of the beach, although this is the least suitable position for customer satisfaction.

This example is often cited as a negative counterpart to Adam Smith’s invisible hand. The limit of rationality Rational players can be expected to choose the Nash equilibrium.

Can we say that agents who declare that they want nothing are intelligent? Similarly, in the case of the prisoner’s dilemma, the single Nash equilibrium is the least desirable solution, when both betray.

Ian Stewart stated Experimental economics has shown that in certain simple situations, human beings do not spontaneously behave in a Nash-optimal strategy.

Uniqueness Any game can have many Nash equilibria or none at all — this is the case for the game of distinction.

Nevertheless, Nash managed to show that any game with a finite number of players and a finite number of strategies admits at least one Nash equilibrium in mixed strategy — that is if we consider as a possible strategy to draw randomly (with fixed probabilities) between several strategies.”

By this time, we usually have a lot of community contributions laid out for you, but this time, Philosopher Coussins really drove the ball forward and put down the work with an awesome contribution, the one you see above.

It feels like a big wall of text, but it’s totally worth reading as you’ll understand the concept pretty well with this explanation.

Now, let’s take a look at some other contributions…

____________________________

Nash Equilibrium and the Prisoner Dilemma:

Philosopher Rutrab was the first mind who brought the Prisoner Dilemma first.

Here’s what he had to say about that:

“Nash equilibrium is a decision-making theory. Have you ever heard about the prisoner’s dilemma? let me explain it to you:

The story goes like this; Two thieves plan to rob a store. As they approach the door, the police arrest them for trespassing.

The police suspect that the pair planned to rob the store but they lack the evidence to prove it.

They, therefore, require a confession to charge the suspects with a more serious crime. The interrogator separates the suspects and tells them each:

“We are charging you with trespassing which will land you a one-month jail sentence. I know you were planning to rob the store but I can’t prove it without your testimony.

Confess to me now, and I will dismiss your trespassing charge and set you free. Your friend will be charged with attempted robbery and face 12 months in jail.

I’m offering your friend the same deal. If you both confess, your testimony is no longer as valuable and you will both receive 8 months in jail.”

Both players are self-interested and want to minimize their jail time. What should they do? (cf: picture!).

Player 1’s available strategies are the rows (Quiet or Confess) and their corresponding payoffs are the first numbers in each cell.

Players 2’s available strategies are the columns and their corresponding payoffs are the second numbers in the cells.

If player 1 stays Quiet and player 2 stays Quiet the game ends in the top left corner of the matrix.

If both players Confess the game ends in the bottom right corner of the matrix and so on.

Now a lot of strategies come to our mind, but how can we all agree? Just communicate anon.”

_______________________________________

Some Context on The Prisoner Dilemma:

In the real world, people don’t always reach that mutually beneficial point.

Prisoner’s Dilemma is a famous example of why two completely “rational” individuals fail to reach an equilibrium point.

It’s about two accomplices (A & B) who are caught for a crime. The police have enough evidence to convict on a lesser charge.

The only problem, the police know, but can’t prove that the pair committed the crime.

They have a choice: confess or remain silent.

• Say if one confesses and the other remains silent, the one who confesses is let go, while the other is convicted of murder.

• Or if they both confess, they both serve time for the lesser charge.

• If they both remain silent, they will both serve time for the lesser charge.

The dilemma faced by each of the prisoners is obviously: which is the best option?

Following this, the Nash equilibrium is at the point where neither prisoner A nor B will benefit from changing strategy.

Conclusion

• If they stay silent, they remain in the dilemma, as there is a benefit to be had by confessing.

• But if both confess, then there’s no benefit in changing strategy (keeping silent again).

• So the equilibrium point for the Prisoner’s Dilemma is that both prisoners confess.