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.
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.
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 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.
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.
The graphical method is a simple way to solve zero-sum games with two strategies for each player. The steps are as follows:
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.