10.1 Game Theory

Chapter Overview

In this chapter, you will learn to:

1. Solve strictly determined games.
2. Solve games involving mixed strategies.

Strictly Determined Games

Game theory is one of the newest branches of mathematics. It first came to light when a brilliant mathematician named Dr. John von Neumann co-authored with Dr. Morgenstern a book titled Theory of Games and Economic Behavior. Since then it has played an important role in decision making in business, economics, social sciences and other fields.

In this chapter, we will study games that involve only two players. In these games, since a win for one person is a loss for the other, we refer to them as two-person zero-sum games. Although the games we will study here are fairly simple, they will provide us with an understanding of how games work and how they are applied in practical situations. We begin with an example.

In Example 1, since there is only one fixed optimal strategy for each player, regardless of their opponent's strategy, we say the game possesses a pure strategy and is strictly determined.

Next, we formulate a method to find the optimal strategy for each player and the value of the game. The method involves considering the worst scenario for each player.

To consider the worst situation, the row player considers the minimum value in each row, and the column player considers the maximum value in each column. Note that the maximum value really represents a minimum value for the column player because the game matrix depicts the payoffs for the row player. We list the method below.

Finding the Optimal Strategy and the Value for Strictly Determined Games

1. Put an asterisk(*) next to the minimum entry in each row.
2. Put a box around the maximum entry in each column.
3. The entry that has both an asterisk and a box represents the value of the game and is called a saddle point.
4. The row that is associated with the saddle point represents the best strategy for the row player, and the column that is associated with the saddle point represents the best strategy for the column player.
5. A game matrix can have more than one saddle point, but all saddle points have the same value.
6. If no saddle point exists, the game is not strictly determined. Non-strictly determined games are the subject of.

Non-Strictly Determined Games

In this section, we study games that have no saddle points. Which means that these games do not possess a pure strategy. We call these games non-strictly determined games. If the game is played only once, it will make no difference what move is made. However, if the game is played repeatedly, a mixed strategy consisting of alternating random moves can be worked out.

We consider the following example.

Reduction by Dominance

Sometimes an $m\times n$ game matrix can be reduced to a $2\times 2$ matrix by deleting certain rows and columns. A row can be deleted if there exists another row that will produce a payoff of an equal or better value. Similarly, a column can be deleted if there is another column that will produce a payoff of an equal or better value for the column player. The row or column that produces a better payoff for its corresponding player is said to dominate the row or column with the lesser payoff.

We summarize as follows:

Reduction by Dominance

1. Sometimes an $m\times n$ game matrix can be reduced to a $2\times 2$ matrix by deleting dominated rows and columns.
2. A row is called a dominated row if there exists another row that will produce a payoff of an equal or better value. That happens when there exists a row whose every entry is larger than the corresponding entry of the dominated row.
3. A column is called a dominated column if there exists another column that will produce a payoff of an equal or better value. This happens when there exists a column whose every entry is smaller than the corresponding entry of the dominated row.