1. Method & Example-1
Method
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arithmetic method Steps (Rule)
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Step-1:
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Find the difference between the two values of Row-1 and put this value against the Row-2, ignore the sign.
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Step-2:
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Find the difference between the two values of Row-2 and put this value against the Row-1, ignore the sign.
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Step-3:
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Find the difference between the two values of Column-1 and put this value against the Column-2, ignore the sign.
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Step-4:
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Find the difference between the two values of Column-2 and put this value against the Column-1, ignore the sign.
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Step-5:
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Find probabilities of each by dividing their sum
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Step-6:
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Find value of the game by algebraic method.
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Example-1
Find Solution of game theory problem using arithmetic method
| Player A\Player B | B1 | B2 | B3 | | A1 | 10 | 5 | -2 | | A2 | 13 | 12 | 15 | | A3 | 16 | 14 | 10 |
Solution:
1. Saddle point testing
Players
| | | Player `B` | | | | | | `B_1` | `B_2` | `B_3` | | | | Player `A` | `A_1` | | 10 | 5 | -2 | | | `A_2` | | 13 | 12 | 15 | | | `A_3` | | 16 | 14 | 10 | |
We apply the maximin (minimax) principle to analyze the game.
| | | Player `B` | | | | | | `B_1` | `B_2` | `B_3` | | Row
Minimum | | Player `A` | `A_1` | | 10 | 5 | -2 | | `-2` | | `A_2` | | 13 | [12] | 15 | | `[12]` | | `A_3` | | 16 | (14) | 10 | | `10` | | Column
Maximum | | `16` | `(14)` | `15` | | |
Select minimum from the maximum of columns
Column MiniMax = (14)
Select maximum from the minimum of rows
Row MaxiMin = [12]
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` | | 10 | 5 | -2 | | | `A_2` | | 13 | 12 | 15 | | | `A_3` | | 16 | 14 | 10 | |
row-1 `<=` row-3, so remove row-1
| | | Player `B` | | | | | | `B_1` | `B_2` | `B_3` | | | | Player `A` | `A_2` | | 13 | 12 | 15 | | | `A_3` | | 16 | 14 | 10 | |
column-1 `>=` column-2, so remove column-1
| | | Player `B` | | | | | | `B_2` | `B_3` | | | | Player `A` | `A_2` | | 12 | 15 | | | `A_3` | | 14 | 10 | |
| | | Player `B` | | | | | | `B_2` | `B_3` | | | | Player `A` | `A_2` | | 12 | 15 | | | `A_3` | | 14 | 10 | |
Using arithemetic method to get optimal mixed strategies for both the firms.
| | | Player `B` | | | | | | `B_2` | `B_3` | | | | Player `A` | `A_2` | | 12 | 15 | | `|14-10|=4` `:. p_1=(4)/(3+4)=4/7` | | `A_3` | | 14 | 10 | | `|12-15|=3` `:. p_2=(3)/(3+4)=3/7` | | | | `|15-10|=5`
`:. q_1=(5)/(2+5)=5/7` | `|12-14|=2`
`:. q_2=(2)/(2+5)=2/7` | | |
1. Find absolute difference between the two values in the first row and put it against second row of the matrix
`|12-15|=3`
2. Find absolute difference between the two values in the second row and put it against first row of the matrix
`|14-10|=4`
`:. p_1=(4)/(3+4)=4/7`
`:. p_2=(3)/(3+4)=3/7`
3. Find absolute difference between the two values in the first column and put it against second column of the matrix
`|12-14|=2`
4. Find absolute difference between the two values in the second column and put it against first column of the matrix
`|15-10|=5`
`:. q_1=(5)/(2+5)=5/7`
`:. q_2=(2)/(2+5)=2/7`
Hence, firm `A` should adopt strategy `A_2` and `A_3` with 57% of time and 43% of time respectively.
Similarly, firm `B` should adopt strategy `B_2` and `B_3` with 71% of time and 29% of time respectively.
Expected gain of Firm A
`(1)` `12 xx 4/7+14 xx 3/7 = 90/7 ,` Firm `B` adopt `B_2`
`(2)` `15 xx 4/7+10 xx 3/7 = 90/7 ,` Firm `B` adopt `B_3`
Expected loss of Firm B
`(1)` `12 xx 5/7+15 xx 2/7 = 90/7 ,` Firm `A` adopt `A_2`
`(2)` `14 xx 5/7+10 xx 2/7 = 90/7 ,` Firm `A` adopt `A_3`
This material is intended as a summary. Use your textbook for detail explanation.
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