Game theory is a fascinating field that studies human behavior in strategic situations. It is widely used in economics, political science, and even biology.
One of the most important concepts in game theory is the saddle point, which represents a unique equilibrium point in a matrix game. In this article, we will discuss how to find saddle points in game theory.
What is a Matrix Game?
A matrix game is a two-player game where each player has a set of possible strategies. The outcome of the game depends on the combination of strategies chosen by both players. The payoff for each player is determined by the rules of the game and the choices made by both players.
What is a Saddle Point?
A saddle point is a unique equilibrium point in a matrix game where both players have found their optimal strategy. At this point, neither player can improve their payoff by changing their strategy because any change will result in a lower payoff.
How to Find Saddle Points?
To find saddle points in a matrix game, we need to perform two steps:
Step 1: Find the Minimum Value in Each Row
The first step involves finding the minimum value in each row of the matrix. We can do this by scanning each row and identifying the smallest number.
- For example, consider the following matrix:
S1 | S2 | S3 | |
P1 | 4 | 5 | 6 |
P2 | 1 | 2 | 3 |
P3 | 7 | 8 | 9 |
- The minimum values in each row are:
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P1: | 4 |
P2: | 1 |
P3: | 7d |
Step 2: Find the Maximum Value in Each Column using the Row Minimums from Step One.
The second step involves finding the maximum value in each column using the row minimums from step one. We can do this by scanning each column and identifying the largest number that corresponds to a row minimum.
- In our example, the maximum values in each column are:
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S1:d | 4 (from P2)d |
S2:d | 5 (from P1)d |
S3:b | 6 (from P1) |
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The Saddle Point
The saddle point is the intersection point of the row minimum and column maximum. In our example, the saddle point is (P2, S1) with a value of 4.
Conclusion:
Saddle points are important concepts in game theory that represent unique equilibrium points in matrix games. To find saddle points, we need to first find the minimum value in each row and then find the maximum value in each column using the row minimums. The intersection point of the row minimum and column maximum is the saddle point.