Everyone has heard of the famous Prisoner’s Dilemma: two criminals are taken to separate rooms and without communication with each other, must decide to either testify against their partner and convict them, or stay silent. If both criminals betray each other, they both serve two years in prison. If A testifies against B but B stays silent, then A will be released and B will serve ten years in prison.

If both criminals stay silent, they both only serve one year. The Nash equilibrium in this problem is to have both testify against each other. Even though having both remain silent has a better result, there is a chance that the partner will betray them. This is an example of a pure-strategy Nash equilibrium where there is one strategy profile for the best result.

But, what exactly is Nash equilibrium? Nash equilibrium is a concept in game theory and economic theory. In game theory, Nash equilibrium is when all players in a non-cooperative game can optimize their outcomes based on other players’ decisions. Nash equilibrium is achieved when no player has a reason to change their own strategy, even if they know other players’ strategies. In economic theory, Nash equilibrium is used to show that “decision-making is a system of strategy interactions based on the action of other players”. It is used to model economic behavior and predict the best response for any situation.

## History

Nash equilibrium is named after American mathematician John Nash. But, the concept of Nash equilibrium existed long before John Nash fully defined it. In 1838, Antione Augustin Cournot created a similar theorem called the Theory of Oligopoly, or the Cournot equilibrium. It used nursing’s firms to explore how the ideal output for a firm to maximize profit depends on the output of the other firm.

Next, in 1944, game theorists John von Neumann and Oskar Morgenstern introduced the mixed-strategy equilibrium. This concept proposed that “a Nash equilibrium existed for a finite game with a specific set of actions with players choosing probability distributions over pure strategies or specific strategy profiles. But, this concept was restricted to two-player games with rational players and zero-sum games (according to Brilliant, a zero-sum game is a game in which it is impossible for any player to help themselves without hurting another player).

Finally, in 1951, John Nash published multiple articles that outlined his Nash equilibria theory of games including “Equilibrium Points in N-person Games” and “Non-Cooperative Games”. Nash was able to prove that any fame with finite actions must have at least one mixed-strategy Nash equilibrium or multiple mixed-strategy Nash equilibria. Nash’s theorem applied to a wider variety of games compared to the past two theorems.

But, the development of the Nash equilibrium didn’t end here: in 1967, John Harsanyi developed Bayesian game models; in 1974, Robert Aumann introduced correlated equilibrium; in 1975, Reinhard Selten showed an issue in the normal-form model that was solved in 1982 by David M. Keeps and Robert Wilson with their definition of sequential equilibrium.

## Types of Nash Equilibrium

There are two types of Nash equilibrium: mixed and pure strategy.

In mixed strategy Nash equilibrium, the probability distribution is announced. This is the likelihood that you will use a particular strategy. If the opponent(s) don’t change their strategy based on the known probability distribution, then we have reached a mixed-strategy Nash equilibrium.

In pure strategy Nash equilibrium, you have one set strategy. It can be thought of as a strategy with a probability of 1 (100%). In pure strategy, there is no chance of something else happening.

## How to Find the Nash Equilibrium

While there is no set formula for finding the Nash equilibrium, it is possible to test for it. If the players in the game do not change their strategy after knowledge of other players’ strategies, then we have reached the Nash equilibrium. Keep in mind that not all games have a Nash equilibrium and some games have multiple Nash equilibria.

The Nash equilibrium is a fascinating theory that affects many decisions made in life: from the simplest of games to important economic strategies that will effect the futures of thousands of people.

## 1 comment

The Nash equilibrium comes up in real life in some surprising places – for instance, two of the most popular games on AoPS, Reaper and Greed Control, both use the concept. In Reaper, you want to ‘reap’ the highest possible time, but there’s just one problem – someone else may reap before you, or even the same time as you, resulting in a loss for you. In Greed Control, you pick a number every day, and your score is the number you picked divided by the number of people who picked it. Both of these games are extremely interesting and definitely require the application of Nash equilibriums.