ECONcepts Vol. 1: John Nash & Game Theory
By the Education & Research Core | June 2015
A brief timeline of Nash's life
A brilliant academic and mathematician, John Nash’s innumerable contributions to economics and other disciplines distinguishes him from his contemporaries.
John Forbes Nash Jr. was born in Bluefield, West Virginia on June 13, 1928. In 1948, Nash received his Master’s Degree in Mathematics from the Carnegie Institute of Technology, currently the Carnegie Mellon University. Two years later, at the age of 22, Nash received his PhD in Mathematics and taught in the Massachusetts Institute of Technology in 1951. After displays of erratic behavior, he was diagnosed with paranoid schizophrenia, a mental illness characterized by paranoia and and altered perception. This kept him from his academic pursuits until his recovery in 1954, an ordeal depicted by the biography and movie “A Beautiful Mind”. From here on, he would receive many honors for his past contributions as he continued his research in the field of Mathematics, until his untimely death last May 23, 2015. Nash is known for his numerous contributions to Mathematics that have been applied in other disciplines, among which, is Economics. He, along with John Harsanyi and Reinhard Selten, were awarded the Nobel Prize for their work on game theory. Aside from this, Nash is also recognized for his contributions to partial differential equations, being awarded the Abel Prize in 2015 for his body of work. 
Game Theory
Game Theory is the analysis of situations involving two or more interacting agents with differing agendas. This concept analyzes an agent’s possible strategies in a competitive situation, taking into account the actions of the others involved. In his research on game theory, John Nash explores noncooperative games: situations in which agents make decisions based independently, in absence of collusion or cooperation with other agents. From this, Nash creates the concept of the noncooperative equilibrium or the Nash equilibrium, where no change in an agent’s strategy will lead to any gains, given the strategy of the other agents. 
Probably Game Theory's most famous example - the Prisoner's Dilemma
Prisoner's Dilemma
This rather deep concept can be explained by a situation coined the Prisoner’s Dilemma. Two people suspected of theft, Adam and David, were brought under interrogation by the police. Arrest technicalities indicate that they will be detained for at least three months. The police however present each suspect, individually, with a deal: if one were to confess his crime (implicating his partner’s as well), and the other denies it, then the first would be granted a lighter sentence of one month and the latter, a heavier one of eight months. However, if both men confess their crime, then they both face a heavier sentence of five months. The situation then is illustrated as: (see infographic on the right) The Nash equilibrium in this situation is where they both confess their crime. From the table, it can be seen that the option that will yield the least months for both prisoners would be to confess. In this situation, the other strategy, denying the crime, will make the person worse off, since he will be getting a heavier sentence. With this in mind, the most rational and safe decision would be for both to confess their crimes. Note however, that this is not the most equitable situation for the prisoners, since they can serve less months if they both deny the crime. Unfortunately, with lack of knowledge of the choice of the other, to deny the crime would be a gamble. Legacy Nash’s study on noncooperative equilibrium proved to be a breakthrough. Before it, economists were under the impression that the actions of a firm’s competitors had no effect whatsoever on the firm’s interest. Nash equilibrium not only disproved this, but also demonstrated its versatility, branching out to a variety of disciplines and even everyday phenomenon. This is demonstrated, for example, in self-enforced laws like traffic lights, where there is no direct incentive in breaking the rule. Nash equilibrium is also extensively used in determining battle plans in war, as well as determining behavior of firms in an oligopoly, due to the concept’s analytic nature. An interesting and relevant application of Nash’s equilibrium is demonstrated in analyzing voter turnout, where studies show that, according to a model where political ideology is sorted in a line, the winner is determined among a group by his spot in a linear political spectrum in comparison with that of his competitors. References http://www.econlib.org/library/Enc/bios/Nash.html http://www.britannica.com/EBchecked/topic/403852/John-Nash http://econ.ucsb.edu/~grossman/teaching/Econ171/NE_applications-ho.pdf https://www.economics.utoronto.ca/osborne/igt/nashapp.pdf |
