What did John Nash do?
Many remember the mathematician John Nash from the 2001 Oscar-winning film “A Beautiful Mind” starring Russell Crowe. The movie was based on Sylvia Nasar’s biography of the same name, which won the 1998 National Book Critics Circle Award for biography.
“A Beautiful Mind” documents Dr. Nash’s struggle with mental illness through the 1960s and 1970s and how he overcame the illness through his own strong will and with support from his wife and his friends. He returned to work in the mid-1980s, and in 1994, Dr. Nash shared the Nobel Prize in economics with two economists working in game theory.
Dr. Nash and his wife were killed in a traffic accident on May 23 while traveling in a taxi from an airport to their home. They had been in Oslo, Norway, where Dr. Nash and Dr. Louis Nirenberg received the Abel Prize for 2015 in mathematics from the Norwegian Academy of Science and Letters. The Abel Prize was established in 2002 because Alfred Nobel didn’t include a prize for mathematics.
Dr. Nash is best known for his advances in game theory. Game theory is essentially the study of how to come up with a winning strategy when the best choice of action depends on the action chosen by others. The best way to understand Dr. Nash’s work is by looking at a strategy game called the Prisoner’s Dilemma.
Here’s one version of the story: The police take two suspects into custody and interview them in separate rooms so they can’t talk to each other. Let’s call the suspects Ace and Snake. The police need help from one suspect to get a conviction of the other on the primary charges. The police give Ace and Snake the same options regarding confessing.
Each suspect is told that he can either confess, thereby implicating the other, or keep silent. If one breaks his silence and the other doesn’t, the snitch goes free as a reward for helping to prosecute the silent one, who gets a five-year prison sentence. If both confess, each will serve a two-year prison term. If neither confesses, the police must settle for lesser criminal charges for both suspects, and each will serve a one-year prison sentence.
As Ace thinks about whether to confess or remain silent, he needs to consider what option Snake may choose. If Snake confesses, Ace would be better off also confessing and taking the two-year sentence rather than facing a five-year sentence for remaining silent. But what if Snake keeps silent? Ace reasons that he should confess and go scot-free rather than taking a one-year sentence for remaining silent. So Ace concludes that it is in his best interest to confess, no matter what Snake does. Because Snake is given the same options, he also should decide it is best to confess.
A dilemma exists for Ace because of his inability to communicate with Snake. If they could talk to each other, they would choose to work together by remaining silent and taking one-year prison sentences. That beats the option of both confessing and getting twoyear sentences. However, the strategy for Ace to remain silent isn’t stable, as Snake could confess and force Ace into a five-year sentence.
Dr. Nash showed that in a game, there is always at least one stable point where neither player has an advantage. For Ace and Snake, it is for both to confess. This stability point is called a Nash equilibrium.
Over the years, the Prisoner’s Dilemma and Nash equilibrium have been used to explore the balance between cooperation and competition in diverse areas such as economics, biology, politics, military theory and computing.
For example, they have been used to explain the arms race between NATO and its opponent, the Warsaw Pact, during the Cold War following World War II. As both sides continued their arms buildup, neither side gained an advantage, but they racked up high costs maintaining their arsenals. If one side had decided to unilaterally disarm, they would face annihilation. The best outcome would have been for both sides to disarm but keep equal arsenals. That way, neither side had an advantage in war, and both sides avoided the costs of expensive weaponry.
However, the equality maintained while disarming could easily be erased if one side rearmed. The rational decision was for both sides to continue maintaining their costly arsenals. Both sides poured money into weaponry in a war of attrition that continued until July 1991, when the Warsaw Pact disbanded.
The Prisoner’s Dilemma has also been used to help explain the behavior of companies in advertising. When two businesses compete against each other for dominance in the marketplace, an essential part of the competition is advertising. If neither company advertises, they keep the money that would have been used for ads. However, if one company decides to spend on advertisements, it could increase its market share. To counteract this advantage, the other company must spend on advertising. Eventually the two companies will settle on a level of spending for advertising needed to maintain its share of the marketplace. You now know that level is called a Nash equilibrium.
Dr. George Boger is assistant dean and associate professor of management at the Texas A&M University-Texarkana College of Business. He can be reached at george.boger@tamut.edu.