# Nash Equilibrium Definition

WRITTEN BY **PAUL BOYCE** | Updated 21 December 2020

## What is the Nash Equilibrium

Nash equilibrium refers to the situation whereby a group of individuals choose the most optimal strategy and do not deviate from that initial decision. Individuals stick to the initial decision in the knowledge that all other options are inferior.

The Nash equilibrium is often used in a game setting but is also applicable to real-life scenarios. Originally conceptualized by mathematician John Forbes Nash Jr., it was designed to explain the decision making in non-cooperative games involving two or more players.

There is Nash equilibrium when Player A chooses strategy 1 and Player B chooses strategy 2 – if Player B has no other option that would maximize their outcome. In other words, Player B chooses the next best option – thereby maximizing the outcome for all parties.

##### Key Points

- The Nash equilibrium is where all players choose the option that is the most positive outcome for each party.
- A Nash equilibrium occurs when players do not change their decision even after knowing the other players.
- Adam Smith stated that individuals following their own interest would produce socially optimal results. However, the Nash equilibrium states that the optimal results are only achieved when the group is considered.

The Nash equilibrium is a core part of Game Theory whereby players choose the best option available to them. This is an important theory in economics as it has wider-reaching effects. For example, at traffic lights, participants operate at the Nash equilibrium. Those who are on green will go as that is the best option for them, whilst those who are on red will stop – as the alternative would be to crash into another car. According to the substitution effect, consumers will go to a competitor such as Burger King – assuming its prices stay the same.

When applied to economics, the Nash equilibrium can help us decide whether government intervention is necessary or whether the market will be able to self-regulate. For instance, do subsidies really help provide lower costs to consumers, or, do they prevent businesses from going bankrupt and thereby prevent new, more efficient businesses from taking their place.

## Nash Equilibrium Example

### Prisoner Dilema

The prisoner’s dilemma is a common example of the Nash equilibrium. There are two criminals who have been arrested, but the prosecutors have little evidence against them. They separate both criminals into their own cell and ask them to confess. In return, the prosecutors wont press charges and they will be allowed to go free. However, if the other confesses, then they will spend 10 years in prison. For example, if Prisoner 1 confesses, then Prisoner 2 will go to jail for 10 years if they too do not confess. If both confess, then they both go to prisoner for 7 years. Yet if neither confess, then they only go to prisoner for 1 year each as there is insufficient evidence.

The Nash equilibrium is only achieved when there is an optimal outcome for both parties. In other words, both parties do not deviate from their original decision once they know the other players choice. In this case, the Nash equilibrium occurs when they both confess. Both prisoners do 7 years, but would do 10 years if they lied. Therefore, when considering the other players decision, this is the most optimal outcome.

It must be said that both prisoners would be better off lying and only receiving one year each. However, that is sub-optimal when the prisoners are aware of the others decision. They could then lie and avoid prison altogether. So from a selfish point of view, it would make sense for them to lie. In turn, this demonstrates the importance of co-operation to achieve the most desired outcome for both parties. They would both benefit if the other lied, yet they cannot be sure the other will do so. Both prisoners could end up with 10 years if the other confesses, but this is reduced to 7 if both confess. So whilst confessing it is sub-optimal, it guarantees the prisoner will not spend 10 years in prison.

## General FAQs on Nash Equilibrium

**How do you find Nash equilibrium?**

Nash equilibrium can be found where neither participant can better their current position based on the other players position.

**Why is Nash equilibrium important?**

Nash equilibrium is important as it expands on Adam Smiths theory that individuals pursuing their own individual interests benefit society. Instead, it shows that the most socially optimal point is where individuals meet at the Nash equilibrium, which often involves individuals coming together and co-operating to bring this about.

**What is a Nash equilibrium example?**

The most common example of the Nash equilibrium is that of the prisoner’s dilemma. There are two prisoners who have been taken in and separated by the prosecutors. There isn’t enough evidence to charge them of serious crimes, but there is enough to sentence them to one year in prison. The prosecutors offer both prisoners a get out of jail pass if they confess and implicate the other prisoner. There are four possible outcomes:

– Prisoner 1 confesses and receives 0 years. Prisoner 2 lies and receives 10 years.

– Prisoner 1 lies and receives 10 years. Prisoner 2 confesses and receives 0 years.

– Both Prisoners 1 and 2 confess, with both receiving 7 years.

– Both Prisoners 1 and 2 lie, only receiving 1 year each.

The most optimal result would occur if they both lie and only receive 1 year. However, that requires a level of co-operation between the two which isn’t achievable in the circumstances.

The Nash equilibrium is where both prisoners could not improve on their position given the other chosen course of action.