1. Method & Example-1 (pure strategy nash equilibrium)
Method
The values of payoff functions can be described by a bimatrix
| | | Player `B` | | | | | | `L` | `R` | | | | Player `A` | `U` | | `a_11`, `b_11` | `a_12`, `b_12` | | | `D` | | `a_21`, `b_21` | `a_22`, `b_22` | |
The values of payoff functions can be given separately for particular players
`A = [[a_11,a_12],[a_21,a_22]]`,
`B = [[b_11,b_12],[b_21,b_22]]`
Matrix `A` is called a payoff matrix for player-1, matrix `B` is called a payoff matrix for player-2.
`a_(ij)` is the maximum in the column `j` of the matrix `A`, underline this value `a_(ij)`.
`b_(ij)` is the maximum in the row `i` of the matrix `B`, underline this value `b_(ij)`.
If the matrix has 1 or more cells with both values underline, then it said that the game has pure strategy nash equilibrium,
Otherwise the game has no pure strategy nash equilibrium.
Example-1
1. Find Solution of game theory problem using Bimatrix method
| Player A\Player B | L | R | | U | 9,9 | 1,10 | | D | 10,1 | 2,2 |
Solution:
| | | Player `B` | | | | | | `L` | `R` | | | | Player `A` | `U` | | 9, 9 | 1, 10 | | | `D` | | 10, 1 | 2, 2 | |
| | | Player `B` | | | | | | `color{green}{L}` | `color{green}{R}` | | | | Player `A` | `color{red}{U}` | | 9 , 9 | 1 , 10 | | | `color{red}{D}` | | 10 , 1 | 2 , 2 | |
The cells with both entries underlined represents pure strategy nash equilibrium.
The game has pure strategy nash equilibrium : `{(D,R)}`
This material is intended as a summary. Use your textbook for detail explanation.
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