6.
| \multirow{6}{*}{Player A} | Player B |
| \multirow[b]{2}{*}{Option Q} | Option X | Option Y | Option Z |
| | 1 | 5 | 3 |
| Option R | 4 | -3 | 1 |
| Option S | 2 | -4 | -2 |
| Option T | 3 | -2 | 0 |
A two person zero-sum game is represented by the pay-off matrix for player A, shown above.
- Explain, with justification, why this matrix may be reduced to a \(3 \times 3\) matrix by removing option S from player A's choices.
- Verify that there is no stable solution to the reduced game.
Player A intends to make a random choice between options \(\mathrm { Q } , \mathrm { R }\) and T , choosing option Q with probability \(p _ { 1 }\), option R 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 linear programme, writing the constraints as inequalities.
Maximise \(P = V\), where \(V =\) the value of original game + 3
$$\begin{aligned}
\text { subject to } & V \leqslant 4 p _ { 1 } + 7 p _ { 2 } + 6 p _ { 3 }
& V \leqslant 8 p _ { 1 } + p _ { 3 }
& V \leqslant 6 p _ { 1 } + 4 p _ { 2 } + 3 p _ { 3 }
& p _ { 1 } + p _ { 2 } + p _ { 3 } \leqslant 1
& p _ { 1 } \geqslant 0 , p _ { 2 } \geqslant 0 , p _ { 3 } \geqslant 0 , V \geqslant 0
\end{aligned}$$ - Explain why \(V\) cannot exceed any of the following expressions
$$4 p _ { 1 } + 7 p _ { 2 } + 6 p _ { 3 } \quad 8 p _ { 1 } + p _ { 3 } \quad 6 p _ { 1 } + 4 p _ { 2 } + 3 p _ { 3 }$$
- Explain why it is necessary to use the constraint \(p _ { 1 } + p _ { 2 } + p _ { 3 } \leqslant 1\)
The Simplex algorithm is used to solve the linear programming problem.
Given that the optimal value of \(p _ { 1 } = \frac { 7 } { 11 }\) and the optimal value of \(p _ { 3 } = 0\) - calculate the value of the game to player A .
(3)
Player B intends to make a random choice between options \(\mathrm { X } , \mathrm { Y }\) and Z , choosing option X with probability \(q _ { 1 }\), option Y with probability \(q _ { 2 }\) and option Z with probability \(q _ { 3 }\) - Determine the optimal strategy for player B, making your working clear.