During the second world war, a Hungarian mathematician was invited to join the infamous Manhattan Project. The scientists and engineers were having trouble coming up with a mechanism to compress a uranium core enough to start a nuclear chain reaction, igniting the bomb. The mathematician, who in the 1930’s already saw a war coming and quickly developed an expertise in modelling explosions, proposed an implosion mechanism; detonating regular explosives with explosive lenses placed around the uranium core. The engineers went ahead with the design and in the summer of 1945 the first prototype was tested. The bomb worked.
The mathematician was John von Neumann, namesake of the List of things named after John von Neumann, a polymath and considered one of the brightest minds of the 20th century. During the war, besides designing the first electronic computer and working on the bomb, von Neumann also became interested in strategy and games. Together with economist Oskar Morgenstern, he set out to describe all games in a rigorous, mathematical way. They only partially succeeded, but the work the two would publish, titled The Theory of Games and Economic Behavior, would have a profound impact on economics, other social sciences and biology.
To tackle games mathematically, Von Neumann first invents the concept of utility to order a player’s preferences and defines a rational player as someone who seeks to maximize utility. Then, he is able to show that each two-player zero-sum game (a term which he coined) has a solution, which consists of a set of strategies that each player will adopt. Von Neumann was able to extend this theorem, the so-called minimax theorem, to games with more than two players by making the players form coalitions, acting as one player, which reduces the game to a two player game. He isn’t entirely successful in elegantly describing non-zero-sum games, though. One of the games he describes is a simplified version of poker and he finds how often players should optimally bluff in certain situations. Since then, game theory has been contributed to by people such as John Nash, who developed noncooperative game theory, and Lloyd Shapley, who came up with the Shapley value, which solves the problem (for certain games at least) of how players in a coalition will divide up utility.
Millitary Applications
Game theory is nowadays perhaps most commonly associated with economics, but in the time right after its inception, the theory was heavily researched for its military applications. After the war, the RAND Corporation was founded. This think tank hired mathematicians to analyze combat situations. Using a metric like utility, called military worth, they analyzed different scenarios, like two aircraft or two tanks closing in on each other. It was also the place where the famous prisoner’s dilemma was first studied. Inspired by his own theory and by now a man of influence, von Neumann also designed and promoted the policy of mutually assured destruction (MAD). The policy led the US to stockpile thousands of nuclear weapons during the cold war. This was to ensure a Nash equilibrium between the US and Soviet union; if both superpowers had the ability to annihilate each other with nuclear weapons, neither side would have an incentive to attack or to disarm.
Biology
After military applications came evolutionary biology. Biologist William Hamilton used the theory to produce a mathematical model of altruism based on the degree of relatedness between different organisms. He showed that genes for self-sacrificing behaviors would spread as long as they benefitted blood relatives, who are also likely to carry the same genes. Chemist George Price took this idea even further and extended the model of Hamilton to account for all evolutionary change, not just traits that benefitted family members. This theory is now known as inclusive fitness. Price eventually became so disgusted by the idea that altruistic behavior could be explained by selfishness that, in an attempt to prove himself wrong, he gave away his belongings to the homeless and got so depressed that he ended his own life.
Economics
In economics, game theory has been used to model monopolistic competition and in understanding auctions. The US government has made billions of dollars on the auctioning of radio frequency bands by consulting game theorists, avoiding the failures of other countries, like a company picking up a licence for a mere 5000 dollars in New Zealand. Game theory is also heavily used by internet companies. Google makes a lot of its money through carefully designed keyword auctions. Game theorists have also been hired to price things like taxi rides and to develop addictive reward systems that get users hooked. In 2009, Elinor Ostrom became the first woman to win the Nobel prize in economics. She used the theory to analyze how the commons (collective resources) are governed and how rules could be set up to prevent depletion of these resources.
The fact that a single mathematical framework can be used to analyze things from a game of sports to predator-prey relationships and auctions makes game theory one of the most important tools in the social sciences today. And although the theory has been contributed to and refined over decades, at its heart it still embodies the view of human behavior of its creator. Von Neumann is quoted saying, “It is just as foolish to complain that people are selfish and treacherous as it is to complain that the magnetic field does not increase unless the electric field has a curl. Both are laws of nature.”