Game theory is a branch of mathematics that studies strategic decision making.

One of the key concepts in game theory is the “game,” which is a formal representation of a strategic situation. A game is typically defined by a set of players, a set of possible actions for each player, and a set of payoffs for each combination of actions. The payoffs represent the rewards or costs associated with each combination of actions.

Another important concept in game theory is the “strategy,” which is a plan of action for a player. A strategy is a function that assigns an action to each possible situation the player might face. A player’s choice of strategy determines their payoff, and the payoffs of the other players as well.

A third key concept in game theory is the “equilibrium,” which is a stable state in which no player can improve their payoff by changing their strategy. There are different types of equilibrium, such as Nash equilibrium, in which each player’s strategy is optimal given the strategies of the other players.

Game theory is used in a wide range of applications, including economics, political science, biology and computer science. In economics, game theory is used to study markets and industries, as well as individual decision making. In political science, it is used to study the behavior of nations and other political entities. In biology, it is used to study the evolution of animal behavior and the dynamics of ecosystems. In computer science, it is used to design algorithms for artificial intelligence and multi-agent systems.

Game theory is a way to understand the forces that govern our behavior

– Martin Shubik

The prisoner’s dilemma

The prisoner’s dilemma is a classic example of a game in game theory that illustrates the concept of rational individuals making decisions that lead to mutually suboptimal outcomes. The game is typically framed as a story about two suspects being arrested and questioned for a crime they may or may not have committed. The suspects are held in separate cells and are not able to communicate with each other.

The game is set up as follows:

• If both suspects remain silent, they will both be convicted of a minor crime and serve a short prison sentence.
• If one suspect confesses and the other remains silent, the confessing suspect will be set free while the silent suspect will be given a long prison sentence.
• If both suspects confess, they will both be given a medium-length prison sentence.

The payoffs for each suspect are the length of the prison sentence they will serve.

The game can be modeled as a matrix, with the columns representing the choice of one suspect and the rows representing the choice of the other suspect. The cells of the matrix contain the payoffs for each combination of choices.

The Nash equilibrium of this game is for both suspects to confess. This is because, regardless of the decision of the other suspect, confessing will result in the best outcome for each suspect. Even though the outcome of mutual confession is worse than the outcome of mutual silence, the suspects are unable to trust each other and so they both confess.

The prisoner’s dilemma is often used to model situations where cooperation is more beneficial than competition but where individual self-interest leads to mutual defection. It illustrates how game theory can be used to analyze situations where people have to make decisions based on incomplete information, and where trust and cooperation are important factors.