1. Method & Example-1
Method
|
oddment method Steps (Rule)
|
|
Step-1:
|
Find column difference `C_1-C_2` and `C_2-C_3`
(Subtract each element of each column from the corresponding element on its left and placing the answer to the right of the matrix)
|
|
Step-2:
|
Find row difference `R_1-R_2` and `R_2-R_3`
(Subtract each element of each row from the corresponding element on its above and placing the answer to the below of the matrix)
|
|
Step-3:
|
Find oddments of individual strategies using determinants
(Line each players strategies in pairs and calculate the determinant formed by Column/Row differences, placing the absolute value alongside the remaining strategy)
|
|
Step-4:
|
If the sum of this row oddments = the sum of column oddments then Find probabilities of individual strategies and value of the game
If these sums are different, then the method fails to answer.
|
Example-1
1. Find Solution of game theory problem using oddment method
| Player A\Player B | B1 | B2 | B3 | | A1 | 7 | 1 | 7 | | A2 | 9 | -1 | 1 | | A3 | 5 | 7 | 6 |
Solution:
1. Saddle point testing
Players
| | | Player `B` | | | | | | `B_1` | `B_2` | `B_3` | | | | Player `A` | `A_1` | | 7 | 1 | 7 | | | | `A_2` | | 9 | -1 | 1 | | | | `A_3` | | 5 | 7 | 6 | | |
We apply the maximin (minimax) principle to analyze the game.
| | | Player `B` | | | | | | `B_1` | `B_2` | `B_3` | | Row
Minimum | | Player `A` | `A_1` | | 7 | 1 | 7 | | `1` | | `A_2` | | 9 | -1 | 1 | | `-1` | | `A_3` | | [5] | (7) | 6 | | `[5]` | | Column
Maximum | | `9` | `(7)` | `7` | | |
Select minimum from the maximum of columns
Column MiniMax = (7)
Select maximum from the minimum of rows
Row MaxiMin = [5]
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` | `B_3` | | | | Player `A` | `A_1` | | 7 | 1 | 7 | | | | `A_2` | | 9 | -1 | 1 | | | | `A_3` | | 5 | 7 | 6 | | |
Also, no course of action dominates the other
Reduced matrix is
| | | Player `B` | | | | | | `B_1` | `B_2` | `B_3` | | | | Player `A` | `A_1` | | 7 | 1 | 7 | | | | `A_2` | | 9 | -1 | 1 | | | | `A_3` | | 5 | 7 | 6 | | |
Find column difference `C_1-C_2`, `C_2-C_3` and row difference `R_1-R_2`, `R_2-R_3`
| | | Player `B` | | | | | | | `B_1` | `B_2` | `B_3` | | `C_1-C_2` | `C_2-C_3` | | Player `A` | `A_1` | | 7 | 1 | 7 | | 6 | -6 | | `A_2` | | 9 | -1 | 1 | | 10 | -2 | | `A_3` | | 5 | 7 | 6 | | -2 | 1 | | `R_1-R_2` | | `-2` | `2` | `6` | | | | | `R_2-R_3` | | `4` | `-8` | `-5` | | | |
Find oddments of individual strategies using determinants
Oddment for `A_1` = det `|[10,-2],[-2,1]|=|10-4|=6`
Oddment for `A_2` = det `|[6,-6],[-2,1]|=|6-12|=6`
Oddment for `A_3` = det `|[6,-6],[10,-2]|=|(-12)-(-60)|=48`
Oddment for `B_1` = det `|[2,-8],[6,-5]|=|(-10)-(-48)|=38`
Oddment for `B_2` = det `|[-2,4],[6,-5]|=|10-24|=14`
Oddment for `B_3` = det `|[-2,4],[2,-8]|=|16-8|=8`
So Oddment matrix is
| | | Player `B` | | | | | | `B_1` | `B_2` | `B_3` | | Oddments | | Player `A` | `A_1` | | 7 | 1 | 7 | | 6 | | `A_2` | | 9 | -1 | 1 | | 6 | | `A_3` | | 5 | 7 | 6 | | 48 | | Oddments | | `38` | `14` | `8` | | |
Sum of A Oddment `=6+6+48=60`
Sum of B Oddment `=38+14+8=60`
Here these sums are equal
Find probabilities of individual strategies
`P(A_1)=6/60=1/10`
`P(A_2)=6/60=1/10`
`P(A_3)=48/60=4/5`
`P(B_1)=38/60=19/30`
`P(B_2)=14/60=7/30`
`P(B_3)=8/60=2/15`
Value of game `V=(7*6+9*6+5*48)/60`
`=336/60`
`=28/5`
Optimal strategy for `A=(1/10,1/10,4/5)`
Optimal strategy for `B=(19/30,7/30,2/15)`
This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then
|
