John F. Nash, one of the most influential mathematicians of the 20^{e} century and subject of the 2001 film, “A Beautiful Mind,” died last weekend with his wife, Alicia, in a taxi crash in New Jersey while returning from Oslo to receive the Abel Award in mathematics, one of the most prestigious honors in the field.

While groundbreaking academic work on the Nobel Laureate’s game theory has influenced many areas, including Politics (electoral and legislative rules), sports (football penalty kicks, running balance / football pass), and international relationships (nuclear deterrence), it left an indelible mark in the field of economics, completely changing the way economists think about how individuals and economic agents behave by considering how they react to the behavior and incentives of other individuals.

**The Nash Equilbrium and the 28-page doctorate of Nobel Laureate John F. Nash. thesis**

Nash’s short 28-page doctorate. thesis “Non-cooperative games, Published in 1951, redefines the definition of “balance” as a state where each player pursues an optimal strategy considering the strategies of all other players. He further proved that every finite game had at least one mixed strategy Nash equilibrium (brilliantly using a mathematical statement known as the fixed point theorem).

Simply put, a “mixed strategy” is a strategy in which, instead of choosing a single action known as a “pure strategy”, a player or individual chooses a set of different actions with a certain probability for each. .

In 2001’s “A Beautiful Mind,” Russell Crowe, playing Dr. Nash, explained Nash’s balance using an analogy of how a group of men should optimally approach a group of women while avoiding competition for. the most beautiful rather than pursuing it. together and ultimately lose. The film actually blurs Nash’s brilliant insight. The argument expressed by Russell Crowe implies that individuals can ignore the bad balance result of the game. Nash’s idea was that the bad balance is “uncooperative” (where they all compete on the most beautiful woman and lose) is the likely outcome (a type of ‘prisoner’s dilemma’ example that demonstrates that not all Nash equilibria are an optimal outcome).

**Nash Equilibria as Revolutionary Models of Imperfect Competition in Markets **

This phenomenon of individuals anticipating the possible actions of others is very important to economists concerned with describing individual economic behavior. Previously, the business world did not necessarily assume that economic agents viewed the incentives of other individuals, but rather viewed theirs as rational agents acting as “price takers” in a perfectly competitive market.

This original line of economic reasoning, which began with Adam Smith’s concept of “the invisible hand,” was part of what economists call the first welfare theorem, that a free competitive market (called by the economists a competitive or Walrasian equilibrium) will always lead to an economically optimal result (assuming that the markets are perfectly competitive, that the transaction costs are negligible and that the market players have perfect information).

When Nash and his contemporary game theorists like John von Neumann arrived in the mid-1920s^{e} century, they asked what would happen if they relaxed the assumption that markets were perfectly competitive and if monopolies and oligopolies existed in the model? Nash was not the first to ask this question. French mathematician Antoine Augustin Cournot examined duopolies in 1838 and was many years ahead of his time (his work was first discredited in France and it was not until 30 years later that another economist, Alfred Marshall, again plotted the supply and demand curves).

However, Nash was among the first to come up with a formal and comprehensive mathematical model describing such theoretical behavior of games, going beyond simple oligopoly and monopoly behavior.

**Nash’s sparkle: borrowing from the mathematical domain of topology**

For Nash to ultimately prove the existence of a “Nash equilibrium” in every finite situation (a very bold claim), it also required several previous breakthroughs in mathematics, especially in the seemingly unrelated realm of topology.

The Kakutani Fixed Point Theorem was developed by Japanese-born mathematician Shizuo Kakutani in 1941 and was used in John Nash’s doctorate. thesis to prove what has finally been called “Nash equilibrium”. Kakutani’s topological theorem showed the existence of “fixed points” for what are called “set-valued functions”, extending the work of Brouwer’s fixed point theorem which proved the existence of “fixed points. ”For continuous functions.

Nash’s ingenuity was to define any scenario (such as competition between monopolies or economic interaction between individuals) as a “defined-value function” with a discrete number of choices for each individual. According to Nash’s logic, Kakutani’s topological theorem then proved that any game or finite scenario must have at least one “fixed point”, or equilibrium.

This methodology was used again to mathematically formalize the first well-being theorem in competitive markets in what is today called the Arrow-Debreu model, first published in the groundbreaking 1954 article, entitled “Existence of an Equilibrium for a Competitive Economy”, which served as much as an analytical confirmation of the hypothesis of the “invisible hand” of Adam Smith under conditions of perfect competition (no monopolies or oligopolies ).

**The prisoner’s dilemma: an example of why a Nash equilibrium is not necessarily an optimal economic outcome **

The development of the Nash equilibrium fundamentally changed the economy. This methodology describes equilibria where a free market has not necessarily led to an economically optimal outcome (unless there is perfect competition).

The classic example is the Prisoner’s Dilemma, a game formalized by Nash’s thesis supervisor Albert Tucker, a scenario where two crime conspirators are arrested and come up with a deal to each prisoner where if they confess and testify against their accomplice, they ‘He will be released where the other will be sentenced to a full sentence (say 10 years) in prison. If the two prisoners remain silent, prosecutors cannot prove the most serious charges and both will serve reduced sentences (say just a year behind bars for less serious crimes). If both confess, prosecutors wouldn’t need their testimony and both would get the full sentence.

The two prisoners each have two choices, either betray their partner by confessing to the crime, or remain silent. If one prisoner confesses, while the other is released, they receive a reduced sentence, whereas if they are both, they are better off.

However, when calculating their optimal contingent strategies using theoretical game analysis, the two betraying suspects are Nash’s equilibrium (the likely outcome), although the best overall outcome is that both remain. quiet.

This is a small, simplified example of a type of game often seen elsewhere in economic analysis, including Greece’s decision to leave the euro, antitrust issues, public free-riding (also known as common goods tragedy) and environmental issues (all countries would theoretically benefit from a more stable climate, but a single country is often reluctant to reduce CO_{2} emissions).

**Nash’s beautiful spirit will continue to influence and inspire**

Nash’s style and method of problem analysis will undoubtedly continue to influence not only economics, but so many other areas he touched on, including mathematics, politics, sports, and international relations. . The human influence of John Nash and his wife, Alicia, who became advocates for mental health later in life as a result of his struggles with schizophrenia, documented in the 2001 film, “A Beautiful Mind” , will undoubtedly continue to inspire many as well.