Game theory is a mathematical approach that analyses the decision-making process of individuals, organizations, and governments in various situations. It involves predicting the behavior of one person or group based on the actions of others as they interact with each other.
One of the most popular methods of solving game theory is through graphical representation. In this article, we will discuss how graphical methods solve game theory.
What is Game Theory?
Game theory is the study of strategic decision-making concerning how people behave in competitive situations where the outcome depends on the choices of all involved players. It uses mathematical models to represent and analyze these interactions between rational decision-makers.
Types of Games in Game Theory
There are two main types of games in game theory: cooperative and non-cooperative games. Cooperative games involve players working together to achieve a common goal, while non-cooperative games involve players making decisions independently without any communication or collaboration.
Non-Cooperative Game Theory
Non-cooperative games are analyzed using a technique called Nash Equilibrium. This technique provides a solution where no player has an incentive to change their strategy given what their opponents are doing.
Cooperative Game Theory
In cooperative games, players can communicate and make binding agreements about their strategies. The most popular method for solving cooperative games is through the Shapley value, which allocates payoffs to each player based on their contribution to the coalition.
Graphical Method in Game Theory
The graphical method is a visual tool used to solve two-player zero-sum games, where one player’s gain is equal to another player’s loss. The method plots all possible strategies for both players on a graph and identifies the optimal strategy for each player.
The Payoff Matrix
The first step in using graphical methods is creating a payoff matrix that shows all possible outcomes and payoffs for each player’s strategy. The matrix represents the possible combinations of strategies that each player can choose.
- The rows of the matrix represent Player 1’s strategies.
- The columns of the matrix represent Player 2’s strategies.
The values in the matrix represent the payoffs to each player. For example, if Player 1 chooses strategy A and Player 2 chooses strategy B, then the payoff is shown at the intersection of row A and column B.
Plotting the Graph
After creating a payoff matrix, we can plot a graph by connecting all points with equal payoffs. This graph is called the indifference curve. An indifference curve shows all possible combinations of strategies that give a certain payoff to each player.
Identifying Optimal Strategy
To identify the optimal strategy for each player, we find where their indifference curves intersect. This point is called Nash Equilibrium, where neither player has an incentive to switch their strategy given what their opponent is doing.
Game theory relies on mathematical models to analyze strategic decision-making in competitive situations. The graphical method is a visual tool used to solve two-player zero-sum games by plotting all possible strategies for both players on a graph and identifying the optimal strategy for each player. Using this method can help us understand how rational decision-makers interact in various situations and predict their behavior more accurately.