Method
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saddle point Steps (Rule)
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Step-1:
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1. Select the minimum element from each row and write them in Row Minimum column.
2. Select the maximum element from Row Minimum column and enclose it in [ ]. It is called Row MaxiMin.
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Step-2:
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1. Select the maximum element from each column and write them in Column Maximum row.
2. Select the minimum element from Column Maximum row and enclose it in ( ). It is called Column MiniMax.
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Step-3:
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1. Find out the elements that is same in rectangle [ ] and circle ( ).
2. If Column MiniMax = Row MaxiMin then the game has saddle point and it is the value of the game.
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Example-1
Find Solution of game theory problem using saddle point
| Player A\Player B | B1 | B2 | B3 | B4 |
| A1 | 20 | 15 | 12 | 35 |
| A2 | 25 | 14 | 8 | 10 |
| A3 | 40 | 2 | 10 | 5 |
| A4 | -5 | 4 | 11 | 0 |
Solution:
1. Saddle point testing
Players
| | | Player `B` | | |
| | | `B_1` | `B_2` | `B_3` | `B_4` | | |
| Player `A` | `A_1` | | 20 | 15 | 12 | 35 | |
| `A_2` | | 25 | 14 | 8 | 10 | |
| `A_3` | | 40 | 2 | 10 | 5 | |
| `A_4` | | -5 | 4 | 11 | 0 | |
We apply the maximin (minimax) principle to analyze the game.
| | | Player `B` | | |
| | | `B_1` | `B_2` | `B_3` | `B_4` | | Row
Minimum |
| Player `A` | `A_1` | | 20 | 15 | [(12)] | 35 | | `[12]` |
| `A_2` | | 25 | 14 | 8 | 10 | | `8` |
| `A_3` | | 40 | 2 | 10 | 5 | | `2` |
| `A_4` | | -5 | 4 | 11 | 0 | | `-5` |
| Column
Maximum | | `40` | `15` | `(12)` | `35` | | |
Select minimum from the maximum of columns
Column MiniMax = (12)
Select maximum from the minimum of rows
Row MaxiMin = [12]
Here, Column MiniMax = Row MaxiMin = 12
`:.` This game has a saddle point and value of the game is 12
The optimal strategies for both players are
The player A will always adopt strategy 1
The player B will always adopt strategy 3
This material is intended as a summary. Use your textbook for detail explanation.
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