If you are interested in game theory, you might have heard of saddle point. A saddle point is a specific point in a matrix game that represents the optimal strategy for both players. It is a unique solution where neither player can improve their payoff by changing their strategy.
But how do you find the saddle point in a game theory problem? In this tutorial, we will go through the steps to identify the saddle point in a matrix game.
Step 1: Represent the game as a matrix
The first step is to represent the game as a matrix. A matrix is simply an array of numbers arranged in rows and columns. Each row represents one player’s strategy, while each column represents the other player’s strategy.
Suppose we have a two-player zero-sum game where Player 1 can choose between two strategies (A and B), while Player 2 can choose between three strategies (X, Y, and Z). We can represent this game as a 2×3 matrix:
| X | Y | Z | |
|---|---|---|---|
| A | 4 | 6 | 5 |
| B | 7 | 2 | 3 |
Explanation:
The number in each cell represents the payoff for Player 1 if they choose the corresponding row strategy and Player 2 chooses the corresponding column strategy. For example, if Player 1 chooses A and Player 2 chooses X, then Player 1 gets a payoff of 4.
Step 2: Find the minimum value in each row
The next step is to find the minimum value in each row of the matrix. We can do this by scanning each row and picking the smallest number.
For our example, the minimum values are 4 for row A and 2 for row B:
- Minimum value in Row A = 4
- Minimum value in Row B = 2
Step 3: Find the maximum value among the minimum values
The third step is to find the maximum value among the minimum values we found in Step 2. This maximum value represents Player 1’s best choice of strategy given that Player 2 knows their strategy.
In our example, the maximum of the minimum values is 4. This means that Player 1’s best strategy is to choose A, and they will get a payoff of at least 4 no matter what Player 2 chooses.
Step 4: Find the maximum value in each column
Now we need to find the maximum value in each column of the matrix. We can do this by scanning each column and picking the largest number.
For our example, the maximum values are:
- Maximum value in Column X = 7
- Maximum value in Column Y = 6
- Maximum value in Column Z = 5
Step 5: Find the minimum value among the maximum values
Finally, we need to find the minimum value among these maximum values we found in Step 4. This minimum represents Player 2’s best choice of strategy given that Player1 knows their strategy.
In our example, both Players have a common saddle point at (A,Y) with a payoff of 6.
Conclusion:
In summary, finding the saddle point in a game theory problem involves representing the game as a matrix, finding the minimum value in each row, finding the maximum value among the minimum values, finding the maximum value in each column, and finding the minimum value among the maximum values. The intersection of the row and column that have these respective maximum and minimum values represent the saddle point.
By following these steps, you can identify the saddle point of any matrix game and determine the optimal strategy for both players.