Extensive Form Game

The Extensive Form Game is a fundamental concept in game theory, which is a branch of mathematics that studies strategic decision making. It provides a graphical representation of a game, illustrating the sequence of actions and the possible outcomes. In this article, we will delve into the world of extensive form games, exploring their definition, characteristics, and applications.

Definition and Characteristics

An extensive form game is a mathematical model that represents a game as a tree, where each node represents a decision point, and the branches represent the possible actions. The game starts at the root node, and the players take turns making decisions, moving through the tree until a terminal node is reached, which determines the outcome of the game. The extensive form game is characterized by the following components: players, actions, information sets, and payoffs.

Players and Actions

In an extensive form game, the players are the decision-makers who take turns making actions. Each player has a set of available actions, and the choice of action determines the next node in the tree. The actions can be strategic, meaning that they are chosen based on the player’s preferences and beliefs about the game, or non-strategic, meaning that they are chosen randomly or based on external factors.

Information Sets

Information sets are the sets of nodes that a player cannot distinguish between when making a decision. In other words, the player does not know which node they are at, but they know that they are at one of the nodes in the information set. The information sets are crucial in determining the player’s strategy, as they need to consider all possible nodes in the information set when making a decision.

Payoffs

The payoffs are the rewards or penalties that the players receive at the terminal nodes of the tree. The payoffs can be positive, indicating a gain, or negative, indicating a loss. The payoffs are used to evaluate the desirability of each outcome, and the players aim to maximize their payoffs.

Applications of Extensive Form Games

Extensive form games have a wide range of applications in various fields, including economics, politics, and computer science. They are used to model and analyze complex decision-making situations, such as negotiations, auctions, and conflicts. The extensive form game provides a framework for evaluating different strategies and predicting the outcomes of actions, allowing players to make informed decisions.

Examples of Extensive Form Games

Some examples of extensive form games include the ultimatum game, where one player proposes a division of a resource and the other player can either accept or reject the proposal, and the prisoner’s dilemma, where two players must decide whether to cooperate or defect, with the payoffs depending on the combination of actions. These games illustrate the complexity of decision-making situations and the importance of strategic thinking.

The extensive form game has also been used to model and analyze real-world situations, such as international relations and business negotiations. By representing the game as a tree, players can evaluate different strategies and predict the outcomes of their actions, allowing them to make informed decisions and achieve their goals.