Method
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algebraic method Steps (Rule)
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Step-1:
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A play's (`p_1`, `p_2`)
`p_1=(d - c)/((a + d) - (b + c))` and `p_2 = 1 - p_1`
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Step-2:
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B play's (`q_1`, `q_2`)
`q_1=(d - b)/((a + d) - (b + c))` and `q_2 = 1 - q_1`
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Step-3:
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Value of the game `V`
`V=(a * d - b * c)/((a + d) - (b + c))`
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Example-1
Find Solution of game theory problem using algebraic method
| Player A\Player B | B1 | B2 |
| A1 | 1 | 7 |
| A2 | 6 | 2 |
Solution:
1. Saddle point testing
Players
| | | Player `B` | | |
| | | `B_1` | `B_2` | | |
| Player `A` | `A_1` | | 1 | 7 | |
| `A_2` | | 6 | 2 | |
We apply the maximin (minimax) principle to analyze the game.
| | | Player `B` | | |
| | | `B_1` | `B_2` | | Row
Minimum |
| Player `A` | `A_1` | | 1 | 7 | | `1` |
| `A_2` | | (6) | [2] | | `[2]` |
| Column
Maximum | | `(6)` | `7` | | |
Select minimum from the maximum of columns
Column MiniMax = (6)
Select maximum from the minimum of rows
Row MaxiMin = [2]
Here, Column MiniMax `!=` Row MaxiMin
`:.` This game has no saddle point.
Matrix size is 2`xx`2, so dominance rule is not required.
Solution using algebraic method
Here `a=1,b=7,c=6,d=2`
`p_1=(d - c)/((a + d) - (b + c))=(2 -6)/((1 +2) - (7 +6))=(-4)/(3 -13)=2/5`
`p_2=1-p_1=1-2/5=3/5`
`q_1=(d - b)/((a + d) - (b + c))=(2 -7)/((1 +2) - (7 +6))=(-5)/(3 -13)=1/2`
`q_2=1-q_1=1-1/2=1/2`
`V=(a * d - b * c)/((a + d) - (b + c))=((1 xx 2) - (7 xx 6))/((1 +2) - (7 +6))=(2 -42)/(3 -13)=4`
This material is intended as a summary. Use your textbook for detail explanation.
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