D2 June 2007 Q2
2. Denis (D) and Hilary (H) play a two-person zero-sum game represented by the following pay-off matrix for Denis.
| H plays 1 | H plays 2 | H plays 3 | |
|---|---|---|---|
| D plays 1 | 2 | −1 | 3 |
| D plays 2 | −3 | 4 | −4 |
(a) Show that there is no stable solution to this game. (3)
(b) Find the best strategy for Denis and the value of the game to him. (10)
| Scheme | Marks |
|---|---|
| \(\begin{bmatrix}2&-1&3\\-3&4&-4\end{bmatrix}\quad\) Row min \(-1 \leftarrow,\ -4\) | M1 A1 |
| col max \(2\ (\uparrow)\ \ 4\ \ 3\qquad 2 \neq -1\ \therefore\) not stable | A1 |
| (3) |
| Scheme | Marks |
|---|---|
| Let Denis play 1 with probability \(p\) So he’ll play 2 with probability \(1 - p\) | |
| If Hilary plays 1 Denis wins: \(2p - 3(1 - p) = 5p - 3\) If Hilary plays 2 Denis wins: \(-p + 4(1 - p) = 4 - 5p\) If Hilary plays 3 Denis wins: \(3p - 4(1 - p) = 7p - 4\) | M1 A2, 1, 0 |
 | M1 A2, 1, 0 |
| \(5p - 3 = 4 - 5p\) \(10p = 7\) \(p = \dfrac{7}{10}\) | M1 A1ft |
| Denis should play 1 with probability \(\dfrac{7}{10}\), 2 with probability \(\dfrac{3}{10}\) the value of the game is \(\dfrac{1}{2}\) | B1ft B1 |
| (10) | |
| (13 marks) |