D2 June 2007 Q6

EdexcelOld spec8 marksGame Theory

6. Anna (A) and Roland (R) play a two-person zero-sum game which is represented by the following pay-off matrix for Anna.

Formulate the game as a linear programming problem for player R. Write the constraints as inequalities. Define your variables clearly. (8)

Mark scheme

Scheme

Marks

Alt 1

Game from R’s point of view.

Add 7

B1, B1

Let R play 1 with probability \(\text{P}_1\), 2 with probability \(\text{P}_2\), 3 with probability \(\text{P}_3\)
V = value of the game

Maximise \(\text{P} = \text{V}\)

Subject to \(\text{V} - \text{P}_1 - 9\text{P}_2 - 10\text{P}_3 \leqslant 0\)

M1 A1ft

\(\text{V} - 10\text{P}_1 - 6\text{P}_2 - 5\text{P}_3 \leqslant 0\)

A1ft

\(\text{V} - 2\text{P}_1 - 3\text{P}_2 - 8\text{P}_3 \leqslant 0\)

A1ft

\(\text{P}_1 + \text{P}_2 + \text{P}_3 \leqslant 1\) accept =
\(\text{V}, \text{P}_1, \text{P}_2, \text{P}_3 \geqslant 0\)

(8 marks)

Add 4 to all entries

Let R play 1 with probability \(\text{P}_1\), 2 with probability \(\text{P}_2\), 3 with probability \(\text{P}_3\); let V = value of game.

Let \(x_1 = \dfrac{\text{P}_1}{\text{V}},\ x_2 = \dfrac{\text{P}_2}{\text{V}},\ x_3 = \dfrac{\text{P}_3}{\text{V}}\)

Maximise \(\text{P} = x_1 + x_2 + x_3\)

Subject to \(10x_1 + 2x_2 + x_3 \leqslant 1\)

M1 A1ft

\(x_1 + 5x_2 + 6x_3 \leqslant 1\)

A1ft

\(9x_1 + 8x_2 + 3x_3 \leqslant 1\)
\(x_1, x_2, x_3 \geqslant 0\) accept \(\text{P}_i \geqslant 0\)

The codes listed for Alt 1 add up to 9, but the question total is 8 marks.