We use cookies to improve your experience on our site and to show you relevant advertising. By browsing this website, you agree to our use of cookies. Learn more   

AtoZmath.com
Support us
(New) All problem can be solved using search box
I want to sell my website www.AtoZmath.com with complete code
 
 
Home What's new College Algebra Games Feedback About us
Algebra Matrix & Vector Numerical Methods Statistical Methods Operation Research Word Problems Calculus Geometry Pre-Algebra
Home > Operation Research calculators > Game Theory >> Matrix method example
7. Matrix method example ( Enter your problem )
( Enter your problem )
  1. Method & Example-1
  2. Example-2
 		Other related methods
  1. Saddle point
  2. Dominance method
  3. Oddment method
  4. Algebraic method
  5. Calculus method
  6. Arithmetic method
  7. Matrix method
  8. 2Xn Games
  9. Graphical method
  10. Linear programming method
  11. Bimatrix method
6. Arithmetic method
(Previous method)		2. Example-2
(Next example)

1. Method & Example-1


Method
Step-1: Player A's optimal strategies `=([[1,1]] xx P_(Adj))/([[1,1]] xx P_(Adj) xx [[1],[1]]) = [[p_1,p_2]]`
Step-2: Player B's optimal strategies `=([[1,1]] xx P_(Cof))/([[1,1]] xx P_(Adj) xx [[1],[1]]) = [[q_1,q_2]]`
Step-3: Value of the game = (Player A's optimal strategies) `xx` (Payoff matrix P) `xx` (Player B's optimal strategies)
`V = [[p_1,p_2]] xx P xx [[q_1],[q_2]]`
Here `P_(Cof)` = Cofactor matrix of P and `P_(Adj)` = Adjoint matrix of P
Example-1
1. Find Solution of game theory problem using matrix method
Player A\Player BB1B2B3
A1172
A2627
A3516


Solution:
1. Saddle point testing
Players
Player `B`
`B_1``B_2``B_3`
Player `A``A_1` 1  7  2 
`A_2` 6  2  7 
`A_3` 5  1  6 


We apply the maximin (minimax) principle to analyze the game.

Player `B`
`B_1``B_2``B_3`Row
Minimum
Player `A``A_1` 1  7  2 `1`
`A_2` (6)  [2]  7 `[2]`
`A_3` 5  1  6 `1`
Column
Maximum		`(6)``7``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.


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` 1  7  2 
`A_2` 6  2  7 
`A_3` 5  1  6 


row-3 `<=` row-2, so remove row-3

Player `B`
`B_1``B_2``B_3`
Player `A``A_1` 1  7  2 
`A_2` 6  2  7 


column-3 `>=` column-1, so remove column-3

Player `B`
`B_1``B_2`
Player `A``A_1` 1  7 
`A_2` 6  2 



reduced matrix
Player `B`
`B_1``B_2`
Player `A``A_1` 1  7 
`A_2` 6  2 


For this reduced matrix, calculate `P_(Adj)` and `P_(Cof)`

`P_(Adj) = [[2,-7],[-6,1]]`

and `P_(Cof) = [[2,-6],[-7,1]]`

Player A's optimal strategies `=([[1,1]] xx P_(Adj))/([[1,1]] xx P_(Adj) xx [[1],[1]])`

`=([[1,1]][[2,-7],[-6,1]])/([[1,1]][[2,-7],[-6,1]][[1],[1]])`

`=([[-4,-6]])/(-10)`

`=[[2/5,3/5]]`

`p_1=2/5` and `p_2=3/5`, where `p_1` and `p_2` represent the probabilities of player A's, using his strategies `A_1` and `A_2` respectively.


Similarly,
Player B's optimal strategies `=([[1,1]] xx P_(Cof))/([[1,1]] xx P_(Adj) xx [[1],[1]])`

`=([[1,1]][[2,-6],[-7,1]])/([[1,1]][[2,-7],[-6,1]][[1],[1]])`

`=([[-5,-5]])/(-10)`

`=[[1/2,1/2]]`

`q_1=1/2` and `q_2=1/2`, where `q_1` and `q_2` represent the probabilities of player B's, using his strategies `B_1` and `B_2` respectively.


Hence, Value of the game `V` = (Player A's optimal strategies) `xx` (Payoff matrix `P_(ij)`) `xx` (Player B's optimal strategies)

`V=[[2/5,3/5]][[1,7],[6,2]][[1/2],[1/2]]=4`

This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then Submit Here


6. Arithmetic method
(Previous method)		2. Example-2
(Next example)


Share this solution or page with your friends.
 
 
Home What's new College Algebra Games Feedback About us
 
Copyright © 2025. All rights reserved. Terms, Privacy
 
 

.