Find Solution of game theory problem using calculus method
| Player A\Player B | B1 | B2 |
| A1 | 2 | -1 |
| A2 | -1 | 1 |
Solution:
1. Saddle point testing
Players
| | | Player `B` | | |
| | | `B_1` | `B_2` | | |
| Player `A` | `A_1` | | 2 | -1 | |
| `A_2` | | -1 | 1 | |
We apply the maximin (minimax) principle to analyze the game.
| | | Player `B` | | |
| | | `B_1` | `B_2` | | Row
Minimum |
| Player `A` | `A_1` | | 2 | [-1] | | `[-1]` |
| `A_2` | | -1 | (1) | | `-1` |
| Column
Maximum | | `2` | `(1)` | | |
Select minimum from the maximum of columns
Column MiniMax = (1)
Select maximum from the minimum of rows
Row MaxiMin = [-1]
Here, Column MiniMax `!=` Row MaxiMin
`:.` This game has no saddle point.
Solution using calculus method
Here `a=2,b=-1,c=-1,d=1`
`p_1=(d - c)/((a + d) - (b + c))=(1 +1)/((2 +1) - (-1 -1))=(2)/(3 +2)=2/5`
`p_2=1-p_1=1-2/5=3/5`
`q_1=(d - b)/((a + d) - (b + c))=(1 +1)/((2 +1) - (-1 -1))=(2)/(3 +2)=2/5`
`q_2=1-q_1=1-2/5=3/5`
`V=a*p_1*q_1+b*p_1*q_2+c*p_2*q_1+d*p_2*q_2`
`=2*2/5*2/5 -1*2/5*3/5 -1*3/5*2/5 +1*3/5*3/5`
`=8/25 -6/25 -6/25 +9/25`
`=1/5`
This material is intended as a summary. Use your textbook for detail explanation.
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