Game theory

Game theory is the mathematical study of strategic interactions. It examines how different players make decisions when their success depends on the choices of others. This science acts as an umbrella term for the study of rational decision-making. It applies to humans, animals, and computers alike. Today, it is a vital tool in social sciences, economics, logic, systems science, and computer science.

To understand how these models work, one must look at the relationship between players. In a zero-sum game, any gain for one participant is exactly balanced by a loss for another. This means the total resources remain constant. For example, in poker, one player's winnings are equal to the opponents' losses. In contrast, non-zero-sum games allow for outcomes where the net result is not zero. In these scenarios, players might both gain or both lose depending on their strategies.

Games are also categorized by how players interact and the rules they follow. Cooperative games involve players who can form binding commitments, often enforced by external rules like contract law. Non-cooperative games occur when players cannot form alliances or must rely on self-enforcing agreements. Furthermore, games can be simultaneous or sequential. In simultaneous games, players move at the same time or without knowing previous actions. In sequential games, players take turns, and earlier decisions affect later ones.

Another way to classify games is through symmetry. A symmetric game is one where each player earns the same payoff for making the same choice. The identity of the player does not change the game's structure. Common examples include the prisoner's dilemma or the game of chicken. Asymmetric games involve different strategy sets for different players. An example is the ultimatum game, where players face different options.

History shows that mathematical interest in games predates modern theory. Around 1564, Cardano wrote about games of chance in his book *Liber de ludo aleae*. In 1657, Huygens published work on the concept of expectation. In 1713, Charles Waldegrave analyzed the card game "le her," providing a minimax mixed strategy solution. Later, in 1838, Antoine Augustin Cournot modeled competition in oligopolies. By 1913, Ernst Zermelo proved that optimal chess strategy is strictly determined.

Modern game theory emerged in the 20th century through the work of John von Neumann. In 1928, he published a paper on the theory of games of strategy. He used the Brouwer fixed-point theorem to prove results for two-person zero-sum games. In 1944, von Neumann and Oskar Morgenstern published *Theory of Games and Economic Behavior*. This work introduced cooperative games and an axiomatic theory of expected utility. This allowed economists to model decision-making under uncertainty.

In 1950, John Nash expanded the field significantly. He developed the Nash equilibrium, a concept for non-cooperative games. A Nash equilibrium occurs when no player can improve their payoff by changing their strategy alone. Nash proved this exists for every finite n-player, non-zero-sum, non-cooperative game. This was a major leap beyond von Neumann's earlier work. His contributions were so impactful that he, Reinhard Selten, and John Harsanyi were awarded the Nobel Prize in 1994.

The impact of game theory is visible in many prestigious awards. As of 2020, fifteen game theorists have won the Nobel Prize in economics. This includes recent winners Paul Milgrom and Robert B. Wilson. In the 1970s, the field moved into biology through evolutionary game theory. John Maynard Smith applied these ideas to evolution and won the Crafoord Prize in 1999. From designing markets to understanding nuclear strategy at the RAND Corporation, game theory remains essential.