# take a look at John Nash’s legacy

*“You don’t have to be a mathematician to have number sense – John** Forbes Nash Jr*

“He’s a mathematical genius,” wrote Nash’s advisor and former Carnegie professor Richard Duffin in the letter of recommendation for Nash’s entry to Princeton University, where he continued his graduate studies in math and science. John Nash was indeed that. He is recognized as one of the greatest mathematicians the world has ever seen.

Thanks to the masterpieces he gave us, in 1978 Nash received the John von Neumann Theory Prize for his discovery of uncooperative equilibrium in games (Nash Equilibrium, now, after him). He was awarded the Memorial Nobel Prize in Economics along with John Harsanyi and Reinhard Selten for the work around game theory he did while at Princeton as a graduate student.

As Ron Howard directed the Oscar-winning film “A Beautiful Mind,” which depicts the tumultuous life of John Nash (Rusell Crowe played it and received an Oscar nomination for the role) ends 20 years, let’s take a look. eye to Nash’s groundbreaking contributions. Even fighting serious mental illnesses did not overcome him; he became one of the best mathematicians in history.

**A gifted mind**

Nash attended Carnegie Mellon University on the George Westinghouse Fellowship. Although he first majored in chemical engineering, he moved on to chemistry and then eventually to mathematics on the advice of his teacher John Lighton Synge. He obtained a bachelor’s and master’s degree in mathematics, then went to Princeton University to continue his graduate studies. At Princeton, he began to work on the theory of equilibrium which would later become the famous “Nash equilibrium”. Nash received a doctorate in 1950 with a thesis on uncooperative games.

**Game theory**

Nash’s article “Points of Equilibrium in N-Person Games” introduced us to Nash’s concept of equilibrium. Game theory focuses on situations where decisions made interact. This means that in such situations the gain for a decision maker depends not only on his own decision, but also on the decisions others make.

This is of paramount importance in real life as there are several such instances (eg, auctions) where other people’s gambling decisions are equally important. It depends not only on the bid amount, but also on the bids of other interested buyers. Game theory finds many applications in economics, computer science, logic, and mathematics, among others.

**Nash equilibrium**

Nash Equilibrium is one of the ways to define the solution to a non-cooperative game involving two or more players. A non-cooperative game is a game where there is competition between individuals. Covenants cannot be applied from the outside and can only work if they apply on their own.

In Nash Equilibrium, it is assumed that each player knows the balance strategies of the other players. A player gains nothing by deviating from the initially chosen strategy. Of course, this works assuming the other players also keep their strategies unchanged. A game may have no Nash equilibrium or have several.

Nash’s equilibrium is one of the fundamental concepts of game theory and finds extensive use in decision-making even today. It works on behavior and interactions between participants to bring out the best outcomes and helps predict player decisions as they make decisions simultaneously.

**Deeply relevant, even today**

Game theory finds many applications in economics, computer science, logic, and mathematics, among others. In fact, game theory plays a fundamental role in AI wherever more than one person is involved in problem solving. It also finds wide applications in contradictory generative networks.

Recently, we saw the DeepMind AI Research Lab introduce a game theory-based approach to help solve fundamental machine learning problems. He reformulated a competitive multi-agent game called EigenGame. Here, the team worked on a mechanism to consider principal component analysis as a competitive game in which each approximate eigenvector is controlled by a player whose goal is to maximize its own utility function. The team analyzed the properties of the PCA game and the behavior of its gradient-based updates. It has been discovered that the resulting algorithm is naturally decentralized and parallelizable thanks to the passage of messages.

Nash was one of the pioneering figures in mathematics whose work finds use in modern mathematics, computing, and statistics. His rich legacy will continue to grow even in the future, as the masterpieces he left behind are truly timeless.