Method
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calculus 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*p_1*q_1+b*p_1*q_2+c*p_2*q_1+d*p_2*q_2`
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Example-1
Find Solution of game theory problem using calculus method
| Player A\Player B | B1 | B2 |
| A1 | 1 | 3 |
| A2 | 5 | 2 |
Solution:
1. Saddle point testing
Players
| | | Player `B` | | |
| | | `B_1` | `B_2` | | |
| Player `A` | `A_1` | | 1 | 3 | |
| `A_2` | | 5 | 2 | |
We apply the maximin (minimax) principle to analyze the game.
| | | Player `B` | | |
| | | `B_1` | `B_2` | | Row
Minimum |
| Player `A` | `A_1` | | 1 | (3) | | `1` |
| `A_2` | | 5 | [2] | | `[2]` |
| Column
Maximum | | `5` | `(3)` | | |
Select minimum from the maximum of columns
Column MiniMax = (3)
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 calculus method
Here `a=1,b=3,c=5,d=2`
`p_1=(d - c)/((a + d) - (b + c))=(2 -5)/((1 +2) - (3 +5))=(-3)/(3 -8)=3/5`
`p_2=1-p_1=1-3/5=2/5`
`q_1=(d - b)/((a + d) - (b + c))=(2 -3)/((1 +2) - (3 +5))=(-1)/(3 -8)=1/5`
`q_2=1-q_1=1-1/5=4/5`
`V=a*p_1*q_1+b*p_1*q_2+c*p_2*q_1+d*p_2*q_2`
`=1*3/5*1/5 +3*3/5*4/5 +5*2/5*1/5 +2*2/5*4/5`
`=3/25 +36/25 +2/5 +16/25`
`=13/5`
This material is intended as a summary. Use your textbook for detail explanation.
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