April 23, 2025

Aashna Chavan

11th Grade

DuPont Manual High School

Game theory is a branch of mathematics that studies strategic interactions among rational decision-makers. It provides a framework for understanding competitive and cooperative behaviors in various fields, including economics, political science, psychology, and artificial intelligence. At its core, game theory models situations where the outcome for each participant depends not only on their own choices but also on the choices of others. 

Game theory operates by modeling real-world situations as mathematical games, which consists of players, strategies and payoffs. Players represent decision-makers, strategies are the choices available to each player, and payoffs are the outcomes associated with different strategy combinations. The goal is to analyze these elements to predict the most likely outcomes. 

There are several types of games in game theory:

1. Zero-Sum vs. Non-Zero-Sum Games: In a zero-sum game, one player’s gain is another’s loss (chess or poker). In a non-zero-sum game, players can achieve mutually beneficial outcomes (trade negotiations)

2. Cooperative vs. Non-Cooperative Games: Cooperative games involve players forming alliances to achieve better outcomes, while non-cooperative games focus on individual strategy.

3. Simultaneous vs. Sequential Games: Simultaneous games require players to make decisions at the same time (Rock Paper Scissors), whereas sequential games involve turns (chess).

4. Perfect vs. Imperfect Information Games: In perfect information games, all players know previous moves (chess), while in imperfect information games, some information is hidden (poker).

A fundamental concept in game theory is the Nash Equilibrium, named after mathematician John Nash. A Nash Equilibrium occurs when no player has an incentive to change their strategy given the strategies of others. This means that each player’s decision is optimal based on the choices of their opponents. 

The mathematics of game theory involves probability, algebra, and calculus. Some key mathematical tools include:

1. Payoff Matrices: In a simple two-player game, payoffs are often represented in matrix form. For example, in the Prisoner’s Dilemma. The choices and outcomes for two prisoners are displayed in a table:

Each cell contains the payoffs for Prisoner A and Prisoner B. Prisoners can analyze these payoffs to determine the best course of action. 

2. Mixed Strategy Probability: In some games, players randomize their choices. If there is no pure strategy Nash Equilibrium, players may adopt mixed strategies by assigning probabilities to different actions. The expected payoff is calculated as: 

E(U) = p1U1 + p1U2 + … + pnUn, where pi represents the probability of choosing strategy i and Ui is the payoff for that strategy.

3. Best Response Functions: Players determine their optimal strategy using mathematical functions that describe the best response to an opponent’s strategy. 

4. Minimax Theorem: In zero-sum games, players seek to minimize the maximum possible loss. The Minimax theorem ensures that the best strategy is to minimize the worst-case scenario.

Game theory continues to influence fields like economics (auctions, market strategies), biology (evolution), and artificial intelligence. By leveraging mathematical models, game theory provides deep insights into strategic behavior and decision-making in complex environments.