Home > Operation Research calculators > Game Theory >> Algebraic method example

3. Algebraic method example ( Enter your problem )
  1. Method & Example-1
  2. Example-2
Other related methods
  1. Saddle point
  2. Dominance method
  3. Algebraic method
  4. Calculus method
  5. Arithmetic method
  6. Matrix method
  7. 2Xn Games
  8. Graphical method
  9. Linear programming method
  10. Bimatrix method

1. Method & Example-1



Method
algebraic method Steps (Rule)
Step-1: A play’s (`p_1`, `p_2`)
`p_1=(d - c)/((a + d) - (b + c))` and `p_2 = 1 - p_1`
Step-2: B play’s (`q_1`, `q_2`)
`q_1=(d - b)/((a + d) - (b + c))` and `q_2 = 1 - q_1`
Step-3: Value of the game `V`
`V=(a * d - b * c)/((a + d) - (b + c))`

Example-1
1. Find Solution of game theory problem using algebraic method
Player A\Player BB1B2
A117
A262


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.
Any bug, improvement, feedback then Submit Here







 
Copyright © 2020. All rights reserved. Terms, Privacy
 
This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.
 





Adblocker detected!

Dear user,

We've detected that you are using AdBlock Plus or some other adblocking software which is preventing the page from fully loading.

We don't have any banner, Flash, animation, obnoxious sound, or popup ad. We do not implement these annoying types of ads!

We need money to operate the site, and almost all of it comes from our online advertising.

Please add atozmath.com to your ad blocking whitelist or disable your adblocking software.

After unblocking website please refresh the page and click on find button again.

Thanks for your support

×
  .