# Nash Equilibrium

A game theory concept that determines the optimal solution in a non-cooperative game

## What is Nash Equilibrium?

Nash Equilibrium is a game theory concept that determines the optimal solution in a non-cooperative game in which each player lacks any incentive to change his/her initial strategy. Under the Nash equilibrium, a player does not gain anything from deviating from the initially chosen strategy, assuming the other players keep their strategies unchanged. A game may include multiple Nash equilibria or none of them.

Nash equilibrium is one the fundamental concepts in game theory. It conceptualizes the behavior and interactions between game participants to determine the best outcomes. It also allows predicting the decisions of the players if they are making decisions at the same time and the decision of one player takes into account the decisions of other players.

Nash equilibrium was discovered by American mathematician John Nash. He was awarded Nobel Prize in Economics in 1994 for his contributions to the development of game theory.

### Example of Nash Equilibrium

Imagine two competing companies: Company A and Company B. Both companies want to determine whether they should launch a new advertising campaign for their products. If both companies start advertising, each company will attract 100 new customers. If only one company decides to advertise, it will attract 200 new customers, while the other company will not attract any new customers. If both companies decide not to advertise, none of the companies will engage new customers. The payoff table is below:

Company A should advertise its products because the strategy provides a better payoff than the option of not advertising. The same situation exists for Company B. Thus, the scenario when both companies advertise their products is a Nash equilibrium.

### Example of Multiple Nash Equilibria

Under some circumstances, a game may feature multiple Nash equilibria.

John and Sam are registering for the new semester. They both have the option to choose either a  finance course or a psychology course. They have 30 seconds only before the registration deadline, so they do not have time to communicate with each other. If John and Sam register for the same class, they will benefit from the opportunity to study for the exams together. However, if they choose different classes, none of them will get any benefit.

In the example, there are multiple Nash equilibria. If John and Sam both register for the same course, they will benefit from studying together for the exams. Thus, the outcomes finance/finance and psychology/psychology are Nash equilibria in this scenario.