When John Nash, the legendary mathematician who inspired the film A Beautiful Mind, began working on his doctoral thesis, game theory was still a brand-new field.

It was a powerful concept, but at the time the mathematics behind it were restricted to two-player I-win-you-lose situations. While realizing that real-life situations are not always so black-and-white doesn't take a math genius, translating that complexity into equations did. Nash's innovative contributions gave game theory wings, and it has since taken flight in fields as diverse as evolutionary biology, city planning, and even quantum mechanics.

Nash and his wife, Alicia, died in a car crash in New Jersey on May 23 while on their way back from Oslo, Norway, where he had just been awarded what is widely considered the highest honor in mathematics, the Abel Prize. He received this highly prestigious and tragically final award for work on partial differential equations, but it is his contributions to game theory that made him a mathematical superstar.

By extending the I-win-you-lose model that defined early game theory to include any number of players and more of the nuances involved in decision making, Nash took a field that was of interest only to mathematicians and made it an invaluable tool for not only economists, but also for scientists across an array of fields.

The core concept behind his work, the Nash equilibrium, is essentially based on the idea that the degree to which a change in strategy is advantageous for a player depends on whether the other players change their strategy, too. The Nash equilibrium of any given game is the combination of all the players' actions that produces the best possible outcome each player could expect if everyone knows everyone else's moves. Nash won the Nobel Memorial Prize in Economic Sciences for this work in 1994.

The classic example of the Nash equilibrium in action is the "Prisoner's Dilemma." In this scenario, two conspirators are offered a deal after committing a crime together: confess and you're off the hook, but the other guy gets 10 years. But without any confession, the prosecutors won't have enough evidence to convict either of them of the more serious crime and would only be able to put them in prison for a year. If both confess, however, they both end up with eight-year sentences because the prosecutors no longer need their testimony. While staying quiet provides the best outcome overall, it is a risky choice because the two criminals cannot communicate their intentions. Calculations using the Nash equilibrium show this lack of communication will most likely lead both criminals to confess out of fear that the other one will rat, but both would end up with longer sentences as a result.

The concepts involved in the Prisoner's Dilemma extend far beyond this one puzzling scenario. In honor of Nash, here are some examples that demonstrate the remarkable reach of his work on game theory.

1. Traffic Flow

Closely related related to the Nash equilibrium, Wardrop equilibria are a standard tool for modeling traffic flow and transportation efforts. Transportation planners have been using Wardrop equilibria for decades to optimize tolls, minimize congestion on roads, and more.

2. Ecology and Evolutionary Biology

Meerkats live in large groups and protect themselves via a lookout system. At least one meerkat keeps itself and the rest of its community safe by watching for predators, but misses out on food while on guard duty. As a result, it is less favorable to be the lookout than not to be the lookout as long as someone steps up to the job, but if no one fills the less-favorable position of lookout then everyone could suffer the far worse consequence of being eaten by a predator. This presents a scenario similar to that of the Prisoner's Dilemma, called the Volunteer's Dilemma, that has helped to shape ecological and evolutionary theories.

3. Quantum Physics

Of all of the areas of science for Nash's work to reach, quantum physics seems an unlikely one. Yet a paper published in Nature Communications in 2013 demonstrates a link between these two seemingly unrelated fields. The paper was titled "Connection between Bell nonlocality and Bayesian game theory."

4. Blackout Prevention

Power outages happen more often on hot summer days because more people run their energy-guzzling air conditioners. On such days, it is more desirable for each individual to turn on their air conditioner, but if too many people use them and the power goes out, then everyone must suffer through the sweltering heat. Researchers have used game theory to craft programs that offer incentives to people who save electricity when it is in high demand, as detailed in a 2013 paper in the journal Proceedings of the National Academy of Sciences.

5. Public Health

In 2009, a global pandemic of H1N1 influenza, or swine flu, broke out. Vaccines were in short supply, raising concerns that governments of wealthier countries would buy up so much of the supply that poorer countries would be left without enough. Using the Nash equilibrium and related concepts in game theory, researchers determined that under some conditions it would actually be in the wealthier countries' best interest to give their vaccine supplies to countries that do not have enough. This can help prevent the spread of the epidemic, according to the paper published in Operations Research in 2009.

6. Board Games

Yes, game theory has been used to develop games intended for recreational play. Nash came up with the idea for the game now known as Hex in 1948, although another scientist named Piet Hein came up with the same game independently several years earlier. Played on a diamond-shaped board that is divided into connected hexagons, Hex cannot possibly end in a draw. What else would you expect from a game theorist? You can play the game online here.

Photo: David Orban | Flickr