Game Theory

Game theory is the mathematical study of strategic interactions among rational decision-makers. It was formalized by John von Neumann and Oskar Morgenstern in 1944 and revolutionized by John Nash in the 1950s. A game consists of players (the decision-makers), strategies (the choices available to each player), and payoffs (the outcomes each player receives for every combination of strategies). Games can be cooperative (players can form binding agreements) or non-cooperative (each player acts independently). They can be zero-sum (one player's gain is another's loss) or non-zero-sum (mutual gains or losses are possible).

A Nash equilibrium is a situation where no player can benefit by unilaterally changing their strategy, assuming the other players keep their strategies unchanged. It is a stable state of mutual best responses. Every finite game has at least one Nash equilibrium, possibly in mixed strategies (where players randomize over pure strategies according to probabilities). The famous Prisoner's Dilemma illustrates why individually rational choices can lead to collectively poor outcomes: both prisoners confessing is the Nash equilibrium, even though both staying silent would give a better outcome for each.

Game theory encompasses many game types. Simultaneous games: all players move at once without knowing others' choices (Rock-Paper-Scissors). Sequential games: players move in turns with knowledge of previous moves (Chess, poker). These are represented as game trees. Bayesian games: players have private information (incomplete information) and must reason about what others know. Repeated games: the same game is played multiple times, enabling strategies like tit-for-tat that punish defection. Evolutionary game theory applies these ideas to biology, modeling how strategies evolve in populations without assuming rationality.

Adjust payoffs in a 2x2 game matrix to see how equilibrium strategies change. Explore the Prisoner's Dilemma and coordination games.