Find Solution of game theory problem using oddment method
| Player A\Player B | B1 | B2 |
| A1 | 9 | 7 |
| A2 | 5 | 11 |
Solution:
1. Saddle point testing
Players
| | | Player `B` | | |
| | | `B_1` | `B_2` | | |
| Player `A` | `A_1` | | 9 | 7 | | |
| `A_2` | | 5 | 11 | | |
We apply the maximin (minimax) principle to analyze the game.
| | | Player `B` | | |
| | | `B_1` | `B_2` | | Row
Minimum |
| Player `A` | `A_1` | | (9) | [7] | | `[7]` |
| `A_2` | | 5 | 11 | | `5` |
| Column
Maximum | | `(9)` | `11` | | |
Select minimum from the maximum of columns
Column MiniMax = (9)
Select maximum from the minimum of rows
Row MaxiMin = [7]
Here, Column MiniMax `!=` Row MaxiMin
`:.` This game has no saddle point.
2. Dominance rule to reduce the size of the payoff matrix
Using dominance property
| | | Player `B` | | |
| | | `B_1` | `B_2` | | |
| Player `A` | `A_1` | | 9 | 7 | | |
| `A_2` | | 5 | 11 | | |
Also, no course of action dominates the other
Reduced matrix is
| | | Player `B` | | |
| | | `B_1` | `B_2` | | |
| Player `A` | `A_1` | | 9 | 7 | | |
| `A_2` | | 5 | 11 | | |
Find oddments of individual strategies
1. Find absolute difference between the two values in the first row and put it against second row of the matrix
`|9-7|=2`
2. Find absolute difference between the two values in the second row and put it against first row of the matrix
`|5-11|=6`
3. Find absolute difference between the two values in the first column and put it against second column of the matrix
`|9-5|=4`
4. Find absolute difference between the two values in the second column and put it against first column of the matrix
`|7-11|=4`
So Oddment matrix is
| | | Player `B` | | |
| | | `B_1` | `B_2` | | Oddments |
| Player `A` | `A_1` | | 9 | 7 | | `|5-11|=6` |
| `A_2` | | 5 | 11 | | `|9-7|=2` |
| Oddments | | `|7-11|=4` | `|9-5|=4` | | |
Find probabilities of individual strategies
`P(A_1)=(6)/(2+6)=3/4`
`P(A_2)=(2)/(2+6)=1/4`
`P(B_1)=(4)/(4+4)=1/2`
`P(B_2)=(4)/(4+4)=1/2`
Value of game `V=(9*6+5*2)/(2+6)=64/8=8`
Value of game `V=(7*6+11*2)/(2+6)=64/8=8`
Value of game `V=(9*4+7*4)/(4+4)=64/8=8`
Value of game `V=(5*4+11*4)/(4+4)=64/8=8`
This material is intended as a summary. Use your textbook for detail explanation.
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