Game theory is the branch of mathematics that studies strategic interaction: situations in which the consequences of one participant’s decisions depend on decisions made by others. A game specifies participants, available strategies, information, and payoffs. Unlike decision theory concerned with an individual facing a given environment, game theory explicitly models other decision-makers as part of that environment. Its central questions concern incentives, expectations, coordination, and conflict, rather than recreational games alone. (ocw.mit.edu)
Historical development
The modern mathematical discipline took shape through John von Neumann and Oskar Morgenstern’s Theory of Games and Economic Behavior, published in 1944. Their work established a systematic framework for analyzing strategic choices and economic interaction. In 1950, John Nash introduced a general equilibrium concept and demonstrated its existence for finite games when randomized strategies are allowed. (nobelprize.org)
Subsequent developments addressed situations in which participants possess different information or make decisions sequentially. John Harsanyi developed methods for games of incomplete information, while Reinhard Selten introduced refinements that distinguish credible strategic behavior from equilibria supported by implausible threats. Nash, Harsanyi, and Selten jointly received the 1994 Nobel Memorial Prize in Economic Sciences for their analysis of equilibria in non-cooperative games. (doi.org)
Models, strategies, and information
The participants in a game are called players; they may represent individuals, firms, or other decision-making units. A strategy specifies how a player acts. In a sequential game, it is a complete contingent plan, including decisions at situations that might never occur. Payoffs represent players’ preferences over outcomes, not necessarily monetary rewards. (ocw.mit.edu)
A normal-form game lists players, strategy sets, and payoff functions. Two-player finite games are often displayed using a payoff matrix. An extensive-form game instead represents the order of decisions, chance events, and what players observe, usually through a game tree. Information sets group decision points that a player cannot distinguish. (ocw.mit.edu)
A pure strategy chooses a definite plan. A mixed strategy assigns a probability distribution over pure strategies; evaluating it usually involves the expected value of payoffs. Perfect information means that players observe previous moves when acting. Complete information concerns knowledge of the game’s structure, including players’ payoff functions. These are distinct conditions: a simultaneous-move game may have complete but imperfect information. (ocw.mit.edu)
Equilibrium and strategic incentives
A Nash equilibrium is a strategy profile in which no player can improve their payoff by changing their own strategy while the others keep theirs fixed. Each strategy is therefore a best response to the remaining strategies. Nash’s existence theorem guarantees at least one equilibrium for every game with finitely many players and finitely many pure strategies, allowing mixed strategies. It does not guarantee uniqueness or a pure-strategy equilibrium. (mit.edu)
A strictly dominant strategy gives a higher payoff than every alternative regardless of opponents’ choices. Such strategies need not exist. More generally, equilibrium depends on mutually consistent choices rather than a universally best action. A zero-sum game has payoffs that sum to zero at every outcome; one player’s gain exactly offsets another’s loss. Most economic interactions need not have this structure. (mit.edu)
In sequential settings, subgame-perfect equilibrium requires strategies to constitute a Nash equilibrium in every subgame. Backward induction solves finite perfect-information games by working backward from terminal decisions. In Bayesian games, private information is represented through player types and beliefs about others’ types, providing a framework for analyzing information asymmetry. (ocw.mit.edu)
Cooperation and repeated interaction
Non-cooperative game theory analyzes individual strategies and incentives; “non-cooperative” does not mean that cooperative behavior is impossible. Cooperative game theory instead commonly examines what coalitions can achieve and how their gains may be divided, often assuming enforceable agreements. Enforcement itself can also be incorporated into a non-cooperative model. (doi.org)
The prisoner’s dilemma illustrates a conflict between individual incentives and shared benefits. Each player chooses cooperation or defection. Defection yields a higher payoff against either choice by the other player, yet mutual cooperation gives both players more than mutual defection. The resulting equilibrium is therefore not the outcome both players jointly prefer. (ocw.mit.edu)
In repeated games, present actions can affect future responses. Cooperation may be sustained through credible rewards or punishments when future payoffs matter sufficiently. Repetition alone is not enough: outcomes depend on the horizon, monitoring, and preferences. In the standard finitely repeated prisoner’s dilemma with a known endpoint and complete information, backward induction produces defection in every round. (ocw.mit.edu)
Applications and limitations
In economics, game theory models bargaining, oligopoly, and strategic bidding. Auction theory studies how auction rules interact with bidders’ information and incentives. Mechanism design reverses the usual problem by selecting rules intended to produce specified outcomes despite participants’ private information and strategic choices. Engineering applications include routing, network resource allocation, and distributed decisions. (ocw.mit.edu)
An equilibrium is a mathematical consistency condition, not a guarantee of fairness, collective efficiency, or observed behavior. Predictions depend on how strategies, preferences, information, and expectations are modeled. Multiple equilibria can leave outcomes undetermined without additional assumptions, and equilibrium existence does not itself supply an efficient computational procedure. Research therefore also examines learning processes, experimental behavior, and computational complexity in finding equilibria. (doi.org)