John F. Nash, one of the most influential mathematicians of the 20th century and subject of the 2001 film, “A Beautiful Mind”, died this past weekend along with his wife, Alicia, in a New Jersey taxi accident while returning from Oslo to accept the Abel Prize in mathematics, one of the most prestigious honors in the field.
While the groundbreaking game theory academic work of the Nobel laureate influenced so many fields including politics (electoral and legislative rules), sports (soccer penalty kicks, football run/pass balance), and international relations (nuclear deterrence), it left an indelible mark in the field of economics, completely changing how economists think about how individuals and economic agents behave by considering how they respond to the behavior and incentives of other individuals.
John Nash at the 2011 Nobel Laureates Beijing Forum. (Photo Credit: The Associated Press)
The Nash Equilbrium and John F. Nash’s 28-page Nobel Prize-winning Ph.D. thesis
Nash’s short 28-page Ph.D. thesis “Non-Cooperative Games”, published in 1951, redefined the definition of “equilibrium” as a state where each player pursues a strategy that is optimal given the strategies of all the other players. He further proved that every finite game had at least one mixed strategy Nash equilibrium (brilliantly using a mathematical statement known as a fixed-point theorem).
To put in layman’s terms, a "mixed strategy" is one where, 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, famously explained the Nash equilibrium using an analogy about how a group of gentlemen should optimally approach a group of women by avoiding competition over the most beautiful one rather than all pursuing her together and ultimately losing. The film actually muddles Nash's brilliant insight. The argument that Russell Crowe expresses implies that individuals can ignore the bad equilibrium outcome of the game. Nash's insight was that the bad "non-cooperative" equilibrium (where they all compete over the most beautiful female and lose) is the likely outcome (a type of "Prisoners Dilemma" example which demonstrates not every Nash equilibrium is an optimal outcome).
Nash Equilibria as breakthrough models of imperfect competition in markets
This phenomena of individuals anticipating the possible actions of others matters a lot to economists who are concerned with describing individual economic behavior. Previously, the economic world did not necessarily assume that economic agents considered the incentives of other individuals, but rather only considered their own 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’s known to economists as the First Welfare Theorem, that a free competitive market (referred to by economists as a competitive or Walrasian equilibrium) will always lead to an economically optimal outcome (assuming that markets are perfectly competitive, transaction costs are negligible, and that market participants have perfect information).
When Nash and his game theorist contemporaries like John von Neumann came along in the mid-20th 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. The French mathematician Antoine Augustin Cournot examined duopolies in 1838 and was many years ahead of his time (his work was initially discredited in France and it was not until 30 years later that another economist, Alfred Marshall, drew supply and demand curves again).
However, Nash was among the first to offer formal and comprehensive mathematical model describing such game theoretic behavior, going beyond just oligopoly and monopoly behavior.
Nash’s brilliance: borrowing from the mathematical field of topology
For Nash to ultimately prove the existence of a “Nash equilibrium” in every finite situation (a very bold claim), this also required several previous breakthroughs in mathematics, namely in the seemingly unrelated field of topology.
The Kakutani fixed point theorem was developed by the Japanese-born mathematician Shizuo Kakutani in 1941, and was used in John Nash’s Ph.D. thesis to prove what ultimately was called the “Nash Equilbrium”. The Kakutani topological theorem showed the existence of “fixed points” for what are known as “set-valued functions”, extending the work of the Brouwer fixed point theorem which proved the existence of “fixed points” for continuous functions.
Nash’s brilliance was defining any scenario (like competition between monopolists or economic interaction between individuals) as a “set-valued function” with a discrete number of choices for each individual. According to Nash’s logic, the Kakutani topological theorem then proved that any finite game or scenario had to have at least one “fixed point”, or equilibrium.
This methodology was used again to mathematically formalize the First Welfare Theorem about competitive markets in what is known today as the Arrow-Debreu model, first published in the 1954 breakthrough paper, entitled “Existence of an Equilibrium for a Competitive Economy”, which served very much as an analytical confirmation of Adam Smith's “invisible hand” hypothesis 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 economics. This methodology described equilibria where a free-market did not necessarily lead to an economically optimal outcome (unless there was perfect competition).
The classic example of this is the Prisoner’s dilemma example, a game formalized by Nash’s thesis advisor Albert Tucker, a scenario where two conspirators in a crime are arrested and offered a deal to each prisoner where if they confess and testify against their accomplice, they’ll be let go where the other will get a full sentence (say 10 years) in prison. If both prisoners stay quiet, the prosecutors cannot prove the more serious charges and both would serve a reduced time (say just a year behind bars for lesser crimes). If both confess, the prosecutors would not need their testimony, and both would get the full sentences.
The two prisoners each have two choices, either to betray their partner by confessing to the crime or remain silent. If one prisoner confesses, while the other is let go, they receive a reduced sentence, where if both, they are best off.
However, when calculating their optimal contingent strategies using game theoretic analysis, the two suspects betraying each other is the Nash equilibrium (the likely outcome), despite the best overall outcome being for both to remain silent.
This is a small and simplified example of a type of game often seen elsewhere in economic analysis including Greece’s decision to leave the euro, anti-trust issues, public good free riding (also known as tragedy of the commons), and environmental issues (all countries theoretically would benefit from a more stable climate, but any single country is often hesitant to curb CO2 emissions).
Nash’s beautiful mind will continue to influence and inspire
Nash’s style and method of analyzing problems will undoubtedly continue to influence not just economics, but so many other fields that it touched including mathematics, politics, sports, and international relations. The human influence of John Nash and his wife, Alicia, who became mental health advocates later in life following his struggles with schizophrenia, documented in the 2001 film, “A Beautiful Mind”, will undoubtedly continue to inspire many as well.
