7.
| \multirow{2}{*}{} | Player B |
| | Option W | Option X | Option Y | Option Z |
| \multirow{3}{*}{Player A} | Option Q | 4 | 3 | -1 | -2 |
| Option R | -3 | 5 | -4 | \(k\) |
| Option S | -1 | 6 | 3 | -3 |
A two person zero-sum game is represented by the pay-off matrix for player A shown above. It is given that \(k\) is an integer.
- Show that Q is the play-safe option for player A regardless of the value of \(k\).
Given that Z is the play-safe option for player B ,
- determine the range of possible values of \(k\). You must make your working clear.
- Explain why player B should never play option X. You must make your reasoning clear.
Player A intends to make a random choice between options \(\mathrm { Q } , \mathrm { R }\) and S , choosing option Q with probability \(p _ { 1 }\), option R with probability \(p _ { 2 }\) and option S with probability \(p _ { 3 }\)
Player A wants to find the optimal values of \(p _ { 1 } , p _ { 2 }\) and \(p _ { 3 }\) using the Simplex algorithm.
Given that \(k > - 4\), player A formulates the following objective function for the corresponding linear program.
$$\text { Maximise } P = V \text {, where } V = \text { the value of the original game } + 4$$ - Formulate the constraints of the linear programming problem for player A. You should write the constraints as equations.
- Write down an initial Simplex tableau, making your variables clear.
The Simplex algorithm is used to solve the linear programming problem. It is given that in the final Simplex tableau the optimal value of \(p _ { 1 } = \frac { 7 } { 37 }\), the optimal value of \(p _ { 2 } = \frac { 17 } { 37 }\) and all the slack variables are zero.
- Determine the value of \(k\), making your method clear.