7.
| \multirow{2}{*}{} | Player B |
| | Option X | Option Y | Option Z |
| \multirow{3}{*}{Player A} | Option R | 3 | 2 | -3 |
| Option S | 4 | -2 | 1 |
| Option T | -1 | 3 | 6 |
A two person zero-sum game is represented by the pay-off matrix for player A, shown above.
- Verify that there is no stable solution to this game.
Player A intends to make a random choice between options \(\mathrm { R } , \mathrm { S }\) and T , choosing option R with probability \(p _ { 1 }\), option S with probability \(p _ { 2 }\) and option T with probability \(p _ { 3 }\)
Player A wants to find the optimal values of \(p _ { 1 } , p _ { 2 }\) and \(p _ { 3 }\) using the Simplex algorithm.
Player A formulates the following objective function for the corresponding linear programme.
$$\text { Maximise } P = V \quad \text { where } V = \text { the value of the game } + 3$$ - Determine an initial Simplex tableau, making your variables and working clear.
After several iterations of the Simplex algorithm, a possible final tableau is
| b.v. | \(V\) | \(p _ { 1 }\) | \(p _ { 2 }\) | \(p _ { 3 }\) | r | \(s\) | \(t\) | \(u\) | Value |
| \(p _ { 3 }\) | 0 | 0 | 0 | 1 | \(\frac { 1 } { 10 }\) | \(- \frac { 3 } { 80 }\) | \(- \frac { 1 } { 16 }\) | \(\frac { 33 } { 80 }\) | \(\frac { 33 } { 80 }\) |
| \(p _ { 2 }\) | 0 | 0 | 1 | 0 | \(- \frac { 1 } { 10 }\) | \(\frac { 13 } { 80 }\) | \(- \frac { 1 } { 16 }\) | \(\frac { 17 } { 80 }\) | \(\frac { 17 } { 80 }\) |
| V | 1 | 0 | 0 | 0 | \(\frac { 1 } { 2 }\) | \(\frac { 5 } { 16 }\) | \(\frac { 3 } { 16 }\) | \(\frac { 73 } { 16 }\) | \(\frac { 73 } { 16 }\) |
| \(p _ { 1 }\) | 0 | 1 | 0 | 0 | 0 | \(- \frac { 1 } { 8 }\) | \(\frac { 1 } { 8 }\) | \(\frac { 3 } { 8 }\) | \(\frac { 3 } { 8 }\) |
| \(P\) | 0 | 0 | 0 | 0 | \(\frac { 1 } { 2 }\) | \(\frac { 5 } { 16 }\) | \(\frac { 3 } { 16 }\) | \(\frac { 73 } { 16 }\) | \(\frac { 73 } { 16 }\) |
- State the best strategy for player A.
- Calculate the value of the game for player B.
Player B intends to make a random choice between options \(\mathrm { X } , \mathrm { Y }\) and Z .
- Determine the best strategy for player B, making your method and working clear.
(3)