1. Introduction

Game theory is a branch of mathematics that analyzes strategic interactions among rational decision-makers. It provides a framework for understanding situations where the outcome for each participant depends on the actions of others. Initially developed to address problems in economics, game theory has since found applications in various fields, including political science, biology, computer science, and psychology.​

2. Historical Development

2.1 Early Foundations

The origins of game theory can be traced back to the 18th century. In 1713, a letter attributed to Charles Waldegrave analyzed a card game called "le her," providing a minimax mixed strategy solution to a two-person version of the game. This early work laid the groundwork for future developments in strategic decision-making. ​

2.2 Cournot's Duopoly Model

In 1838, Antoine Augustin Cournot introduced a model of competition in oligopolies, presenting a solution concept that aligns with what is now known as the Nash equilibrium. His work, "Researches into the Mathematical Principles of the Theory of Wealth," analyzed how firms compete on the quantity of output produced, influencing market prices. ​



2.3 Zermelo's Theorem

In 1913, Ernst Zermelo published "On an Application of Set Theory to the Theory of the Game of Chess," proving that optimal strategies exist in chess, a game with perfect information. This work demonstrated that such games are strictly determined, meaning one player has a winning strategy, or both have strategies ensuring at least a draw. ​


2.4 Von Neumann and Morgenstern

The formal foundation of game theory as a distinct field emerged with John von Neumann's 1928 paper, "On the Theory of Games of Strategy." He introduced the minimax theorem, establishing that in zero-sum games with perfect information, there exists a strategy that minimizes the maximum possible loss. This work culminated in the 1944 book "Theory of Games and Economic Behavior," co-authored with Oskar Morgenstern, which laid the groundwork for modern game theory. ​


2.5 Nash Equilibrium

In 1950, John Nash extended game theory to non-cooperative games involving multiple players. He introduced the concept of Nash equilibrium, a set of strategies where no player can benefit by unilaterally changing their strategy, provided the strategies of the others remain unchanged. Nash's work broadened the applicability of game theory to a wider array of strategic interactions. ​

3. Fundamental Concepts

3.1 Players

Players are the decision-makers in a game, which can be individuals, groups, firms, or any entities capable of making strategic choices. The number of players can vary, influencing the complexity and analysis of the game.​

3.2 Strategies

A strategy defines a complete plan of action a player will follow throughout the game, specifying the choices to be made in every possible situation. Strategies can be classified as:​


Pure Strategies: A deterministic plan where a player consistently chooses a specific action.​

Mixed Strategies: A probabilistic approach where a player selects among available actions according to a specific probability distribution. ​

3.3 Payoffs

Payoffs represent the outcomes resulting from the combination of strategies chosen by the players. They are typically quantified in terms of utility, profit, or other measures of benefit or loss. The structure of payoffs influences the incentives and strategic choices of the players.​

3.4 Information

The information available to players significantly impacts their strategic decisions. Games are categorized based on information availability:​


Complete Information: All players are fully informed about the game's structure, including the payoffs and strategies available to other players. ​


Incomplete Information: Some aspects of the game, such as payoffs or strategies of other players, are unknown to one or more participants.​

Perfect Information: All players observe all moves made previously by other players.​


Imperfect Information: Some moves are not observed by all players.​

3.5 Equilibrium Concepts

Equilibrium concepts help predict the outcomes of strategic interactions:​

Nash Equilibrium: A set of strategies where no player can gain by unilaterally deviating from their strategy.​


Subgame Perfect Equilibrium: An extension of Nash equilibrium applicable to dynamic games, ensuring that players' strategies constitute a Nash equilibrium in every subgame.​

Evolutionarily Stable Strategy (ESS): In evolutionary game theory, an ESS is a strategy that, if adopted by a population, cannot be invaded by any alternative strategy.​