A beautiful and restless mind at 86

John Nash, winner of the Nobel Prize in Economics and subject of a biographical film, talks about game theory and his new research.

By Diego Freire

Agência FAPESP – The beautiful mind that won the 1994 Nobel Prize in Economics for revolutionizing the field of mathematics known as game theory continues to contribute to new revolutions in science and social life. American mathematician John Nash, 86, visited São Paulo in late July and spoke about his current research at Princeton University.

Nash came to Brazil to deliver a lecture at the International Workshop on Game Theory and Economic Applications of the Game Theory Society (IWGTS) held at the School of Economics, Business Administration and Accounting at the University of São Paulo (FEA-USP) as part of the São Paulo School of Advanced Science (SPSAS), a FAPESP grant program.

Also taking part in the July 25-31, 2014 event were three other Nobel laureates in Economic Sciences: mathematician Robert Aumann (2005) from the Hebrew University of Jerusalem in Israel and economists Eric Maskin (2007) from Harvard University and Alvin Roth (2012) from Stanford University.

Well before obtaining fame in the eyes of the general public for having his story depicted in the 2001 film A Beautiful Mind, John Forbes Nash Jr. gained notoriety in the academic world for his contributions to game theory, a field developed in 1944 by mathematician John von Neumann (1903-1957) and economist Oskar Morgenstern (1902-1977).

Originally, work in this field examined games in which participants had to make choices based on the decisions of their opponents, and researchers studied mathematical functions that could explain competition or cooperation among the players. Nash’s research determined the equilibrium point of this relationship, which later became known as the Nash Equilibrium.

“Before, it used to be understood that, starting from an established value, whatever one person won, the other person lost. Because of this, playing was seen only as a formality. Over time, the fact of having won or lost and the value that had been in play became a subject of interest for studies,” he said in an interview given to Agência FAPESP.

One of the most famous applications of the Nash Equilibrium is toward the game known as the Prisoner’s Dilemma, in which two men are arrested on the suspicion of having committed a crime together. There is no evidence against them, and they are questioned separately and encouraged by the police to betray the other in exchange for their own freedom.

Two options are presented: keep quiet or accuse the other. If the two accuse each other, they are both punished; if they remain silent, they are both freed. However, the distrust of an accused regarding the decision that the other might make increases the probability that both will accuse each other, which leads to the worst outcome: prison for both of them.

The best solution for the two players is the least probable because it requires blind cooperation, given that they are not able to communicate with one another. For this reason, the most probable outcome is that they accuse each other because each has more to gain by betraying the other.

The Nash Equilibrium is the solution in which neither of the players could improve their outcome with a unilateral action. In this case, if one of the accused tends to betray the other and then unilaterally changes his strategy and decides to collaborate with police, he “loses” the game and is imprisoned.

This proposed concept is considered to be fundamental to game theory and is one of the methods most employed in social science to estimate the outcome of a strategic interaction.

Building upon this insight, his work has promoted the application of purely mathematical concepts to various fields that involve situations analogous to games, such as economics, anthropology, political science and biology.
 

Transfers of power

In the lecture given at the SPSAS, Nash described one of his recent experiments. “In an experimental game, the various player participants were not instructed about how they should react to the behavior of those with whom they were interacting. The interaction kept being repeated throughout the game until, naturally, the participants began to promote cooperation among themselves, forming coalitions,” he said.

During the process, Nash observed the behavior of the players according to the transfers of power that occurred when forming the coalitions, the acceptance expressed by some individuals and the distribution of rewards on the part of those favored.

“Acceptance depended on the bonuses. And players with various levels of powers could accept a transfer of power to another participant if they were compensated for it,” he stated.

The experiment allowed Nash to fit the actual process of forming coalitions and the associated patterns of acceptance into a mathematical model, which he demonstrated at the event.

His work with repetitive games, the formation of coalitions and methods of acceptance also revealed an apparent paradox that was noted by the mathematician: cooperative behavior develops naturally, even between organisms or species that interact only for selfish reasons, which is a phenomenon that he has studied in recent years.

Similarly to his design of the Nash Equilibrium, the motivation for these new research studies came from his restless observation of the world.

“The idea of the methods of acceptance came about while I was contributing to a science camp for young people, giving a lecture about evolution and how it naturally occurs in models of cooperation between two or more species,” he said.

This episode reveals the mathematician’s interest in new things, which was also demonstrated during the event. Not only did Nash attend the event to deliver his lecture, but he also participated as an observer of several other presentations and intently reviewed the poster exhibitions, expressing interest in the work developed in Brazil on game theory.

“Right now in my studies, I’m also trying to move forward into a more complex field that addresses games that can be partly competitive and partly cooperative,” he said. “I can’t be sure what they’ll call this in the future; if they’ll call it game theory, statistics or econometrics. What is certain is that there is a lot to be done.”