Game theory (mathematics)
Game theory is a mathematical framework used to analyze strategic interactions among rational decision-makers. It explores scenarios where the outcomes of individual choices depend on the actions of others, making it applicable in a wide range of fields from economics to biology and political science. Although game theory is often associated with games like poker or chess, it also addresses significant real-world issues, such as military strategies and negotiations.
A key concept in game theory is the "minimax theorem," introduced by John von Neumann, which identifies strategies for two-player zero-sum games, where one player's gain is exactly matched by the other's loss. The well-known "Prisoner's Dilemma" illustrates how individuals might act against their better interests due to lack of communication and trust, revealing important implications for cooperation and competition.
Applications of game theory extend to military doctrines, such as mutual assured destruction, highlighting its impact on foreign relations. Additionally, the study of perfect information games, such as Nim, offers insights into winning strategies based on mathematical principles. Overall, game theory provides a robust framework for understanding complex decision-making processes in various contexts.
Game theory (mathematics)
Summary: Game theory models various real-world and hypothetical situations as “games,” the play and strategy of which can be analyzed mathematically.
Game theory is the branch of mathematics dedicated to analyzing strategic behavior in different situations. It attempts to describe situations in which several people or entities must make choices even when the outcomes of their decisions rely on the choices made by others. While game theory can be used to address situations typically thought of as games, such as checkers and poker, it can also be used to study situations that are extremely practical and important, such as strategies to use in military operations or auctions and the evolution of species. As with many areas of mathematical modeling, approaching a problem in game theory first typically involves quantifying the objectives and options in terms of algebraic equations and then finding the choice that gives the highest possibility of maximal success.
History of Game Theory
People have studied games and strategies for centuries, but game theory came into its own as a branch of applied mathematics when John von Neumann proved what is known as the “minimax theorem” in 1928. This theorem considers games played between two players in which each player chooses one of a finite number of options, and—depending on the choice made by each player—one of the players gives a certain amount of money to the other player. This is commonly referred to as a “zero-sum game,” as the losses incurred by one player exactly equal the gains won by the other player. Von Neumann was able to prove that there is a unique strategy that will maximize a player’s winnings (or minimize losings), and one can find this strategy by considering the worst-case outcomes of each of the player’s choices and choosing the best-possible, worst-possible outcome. In particular, one would typically like to choose an option that leaves the player indifferent to the choice made by his or her opponent. This work was later expanded by von Neumann and Oskar Morgenstern in their book Theory of Games and Economic Behavior, which introduced game theory as a valuable tool for economists.
Rock-Paper-Scissors
An example of the type of strategy that von Neumann wrote about comes up when playing the children’s game of rock-paper-scissors. In this game, each of two players chooses one of three possible options (rock, paper, or scissors), and—depending on the choice made by each player—one of the two is declared the winner. In particular, rock beats scissors, scissors beats paper, and paper beats rock. If the two players make the same choice, the game is declared a tie. No matter which choice an opponent makes, one of a player’s three options will result in a win, one will result in a loss, and one will result in a tie. Therefore, if the player does not have any inside knowledge of what the opponent will choose, the player will do best by choosing one of the three options at random, each with a probability of one-third.
The Prisoner’s Dilemma
The most famous problem in game theory is the Prisoner’s Dilemma. The Prisoner’s Dilemma is a non-zero-sum game in which there are two participants, each choosing one of two possible outcomes. It is most often described by the following type of story: two criminals, Alice and Bob, are arrested after committing a crime. The police isolate the two prisoners and interrogate them separately. Each criminal must choose whether to confess or to deny the crime, without communicating with the other prisoner. If both confess, they will each get three years in jail. If both deny the crime, there will not be enough evidence to convict them of the felony, but both will get one year in jail. If Alice confesses and Bob denies the crime, then Alice will go free and Bob will go to jail for five years, but if Bob confesses and Alice denies the crime, then Bob will go free and Alice will go to jail for five years. One can see that no matter what Alice chooses to do, Bob will be better off confessing and no matter what Bob chooses to do, Alice will be better off confessing. Because they cannot communicate, one is led to suspect that they will both end up confessing, even though they would both be better off if they both chose to deny the crime. This situation’s key principle is how much the criminals trust their partner to deny the crime, rather than do what is in their own self-interest. While this story may seem contrived, it turns out to have many applications in areas such as economics, biology, and political science.
Applications of Game Theory
Much of the research on the Prisoner’s Dilemma, as well as other areas of game theory, has taken place at the RAND Institute, a nonprofit think tank originally set up by the United States Army and the Douglas Aircraft Company with a mission “to help improve policy making through research and analysis.” Along with then defense secretary Robert McNamara, they developed the game theoretic concept of mutually assured destruction (MAD), which leads to a military doctrine of nuclear deterrence. The idea is that if one country launches a nuclear attack on another, then the conflict quickly escalates until the whole planet is destroyed, and, therefore, such an attack will never take place. This concept has been critiqued by many scholars, but is still an influence on foreign relations today.
While most games in the real world deal with situations in which the players do not have full information or in which there is an element of chance, there is also a strong mathematical study of perfect information games such as checkers and Go. One famous example of such a game is Nim, a game played between two players starting with a number of objects in different piles. On each player’s turn, they can remove any number of objects from a single pile. The players alternate turns, and the player to remove the final object loses. This game has been extensively studied and written about by game theorists, such as Elwyn Berlekamp and John H. Conway. It turns out that one of the two players is guaranteed to have a winning strategy, but which player it is depends on the number of piles and the number of objects.
Bibliography
Berlekamp, Elwyn, John H. Conway, and Richard Guy. Winning Ways for Your Mathematical Plays. 2nd ed. Natick, MA: A K Peters, Ltd., 2001.
Nash, John. “Equilibrium Points in N-Person Games.” Proceedings of the National Academy of Sciences of the United States of America 36, no. 1 (1950).
von Neumann, John, and Oskar Morgenstern. Theory of Games and Economic Behavior. 4th ed. Princeton, NJ: Princeton University Press, 2007.
Poundstone, William. Prisoner’s Dilemma: John von Neumann, Game Theory and the Puzzle of the Bomb. New York: Anchor, 1992.