Game Theory
- Introduction to Game Theory
- Two-Person Zero-Sum Games
- Non-Zero-Sum and Cooperative Games
- Game Theory in Business and Economics
- Game Theory in Politics
- Psychological Game Theory
- Games of Chance and Risk
- Evolutionary Game Theory
- Games with Sequential Moves
- Game Theory in Social Interactions
- Ethics in Game Theory
- Technological Aspects of Game Theory
- Applying Game Theory in Everyday Life
Two-Person Zero-Sum Games
Solving Simple Zero-Sum Games
Stationary point that is not a local extremum.
Zero-sum games are a fundamental concept in game theory. They represent situations where the gain of one player is exactly balanced by the losses of the other player. In other words, the sum of total gains and losses in the game is zero. This article will guide you through the process of solving simple zero-sum games.
Understanding the Matrix Form of Zero-Sum Games
Zero-sum games can be represented in a matrix form, where each cell represents the outcome of a particular combination of strategies. The rows represent the strategies of one player, while the columns represent the strategies of the other player. The value in each cell is the payoff for the row player, which is the loss for the column player.
The Concept of Saddle Points
A saddle point in a zero-sum game is a position where the row player's minimum payoff is maximized, and the column player's maximum loss is minimized. In other words, it's a position where neither player would benefit from changing their strategy unilaterally. If a saddle point exists in a zero-sum game, it represents the optimal strategy for both players.
The Minimax and Maximin Strategies
The minimax and maximin strategies are fundamental to zero-sum games. The minimax strategy involves minimizing the maximum possible loss, while the maximin strategy involves maximizing the minimum possible gain. In a zero-sum game, these two strategies coincide at the saddle point.
Pure and Mixed Strategies in Zero-Sum Games
A pure strategy involves always choosing the same action, regardless of what the other player does. A mixed strategy involves choosing different actions with certain probabilities. In some zero-sum games, a mixed strategy can provide a better outcome than any pure strategy.
Solving Simple Zero-Sum Games Using Graphical Method
The graphical method is a simple way to solve zero-sum games with two strategies for each player. The steps are as follows:
- Plot the payoffs for each strategy combination on a graph.
- Identify the points where the row player's minimum payoff is maximized (the maximin point) and where the column player's maximum loss is minimized (the minimax point).
- If the maximin and minimax points coincide, that's the saddle point and the optimal strategy for both players. If they don't coincide, there's no saddle point and the game has no pure strategy solution.
By understanding these concepts and methods, you can effectively analyze and solve simple zero-sum games. This will provide a solid foundation for understanding more complex games and strategies in game theory.