Game theory is a widely used concept in economics, political science, psychology, and other fields. It is a mathematical approach that helps to analyze the behavior of individuals and groups in strategic situations.
Strategies are the key elements of game theory that determine the outcome of a game. In this article, we will discuss how to list strategies in game theory.
What is Game Theory?
Before we dive into the strategies, let’s have a brief overview of game theory. Game theory is the study of decision-making in situations where two or more individuals have conflicting interests. In these situations, each individual tries to maximize their own payoff while anticipating the actions of others.
How to List Strategies in Game Theory
Listing strategies is an essential part of game theory analysis. Here are some ways to list strategies:
1. Normal Form Representation
In normal form representation, a matrix is used to represent all possible combinations of actions and payoffs for each player. The rows represent the actions taken by one player, while the columns represent the actions taken by another player.
For example, consider a simple game between two players: Player A and Player B. Each player has two choices: cooperate (C) or defect (D). The possible outcomes and payoffs for each combination are as follows:
- If both players cooperate (C,C), they both get a payoff of 3.
- If Player A cooperates and Player B defects (C,D), then Player A gets a payoff of 0, and Player B gets a payoff of 5.
- If Player A defects and Player B cooperates (D,C), then Player A gets a payoff of 5, and Player B gets a payoff of 0.
- If both players defect (D,D), they both get a payoff of 1.
The normal form representation of this game is as follows:
2. Extensive Form Representation
In extensive form representation, the game is represented as a tree-like structure that shows the sequence of moves and the possible outcomes at each stage. The nodes in the tree represent decision points, while the edges represent the possible actions.
For example, consider a game between two players: Player A and Player B. Player A has two choices: to wait (W) or to attack (A).
If Player A waits, then Player B has two choices: to wait or to attack. The possible outcomes and payoffs for each combination are as follows:
- If both players wait (W,W), they both get a payoff of 2.
- If Player A waits and Player B attacks (W,A), then Player A gets a payoff of -1, and Player B gets a payoff of 4.
- If Player A attacks and Player B waits (A,W), then Player A gets a payoff of 4, and Player B gets a payoff of -1.
- If both players attack (A,A), they both get a payoff of -2.
The extensive form representation of this game is as follows:
A (4,-1) W (2,2) / \ / \ A (-2,-2) W (-1,4) A (-1,4) W (2,2)
3. Pure Strategy and Mixed Strategy
In game theory, a strategy is a set of actions that a player chooses in different situations to achieve a specific goal. There are two types of strategies: pure strategy and mixed strategy.
A pure strategy is a specific action taken by a player in every situation. For example, if a player always chooses to cooperate, regardless of the other player’s action, then that is a pure strategy.
A mixed strategy is a probability distribution over the set of pure strategies. In other words, it is a combination of different actions that a player takes with specific probabilities. For example, if a player chooses to cooperate with 60% probability and defect with 40% probability, then that is a mixed strategy.
Game theory provides an analytical framework for understanding strategic decision-making in various fields. Listing strategies is an essential part of game theory analysis.
Normal form representation and extensive form representation are two common ways to list strategies in game theory. In addition, pure strategies and mixed strategies are two types of strategies used in game theory analysis. By using these techniques to list and analyze strategies in games, we can make better decisions and achieve better outcomes in strategic situations.