D2 June 2012 Q5
5. Agent Goodie is planning to break into Evil Doctor Fiendish’s secret base.
He uses game theory to determine whether to approach the base from air, sea or land.
Evil Doctor Fiendish decides each day which of three possible plans he should use to protect his base.
Agent Goodie evaluates the situation. He assigns numbers, negative indicating he fails in his mission, positive indicating success, to create a pay-off matrix. The numbers range from −3 (he fails in his mission and is captured) to 5 (he successfully achieves his mission and escapes uninjured) and the pay-off matrix is shown below.
| Fiendish uses plan 1 | Fiendish uses plan 2 | Fiendish uses plan 3 | |
|---|---|---|---|
| Air | 0 | 4 | 5 |
| Sea | 2 | −3 | 1 |
| Land | −2 | 3 | −2 |
| Scheme | Marks |
|---|---|
| Row 1 (air) dominates row 3(land), (so Row 3 can be deleted) | B1 |
| (1) |
Notes
a1B1 CAO. Accept ‘air dominates land’ etc. Must have a named row dominating a named row
| Scheme | Marks | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
and row 2 with probability \(1 - p\). If F plays 1 G’s expected winnings are \(0 + 2(1 - p) = 2 - 2p\) If F plays 2 G’s expected winnings are \(4p - 3(1 - p) = 7p - 3\) If F plays 3 G’s expected winnings are \(5p + (1 - p) = 4p + 1\) | 1M1 1A1 | ||||||||||||
 | 2M1 2A1 | ||||||||||||
| \(7p - 3 = 2 - 2p\) \(9p = 5\) | 3DM1 | ||||||||||||
| \(p = \dfrac{5}{9}\) | 3A1 | ||||||||||||
| Goodie should play Row 1 (air) with probability \(\dfrac{5}{9}\), row 2 (sea) with probability \(\dfrac{4}{9}\) and never row 3 (land). | 4A1ft | ||||||||||||
| (7) |
Notes
b1M1 Setting up three probability equations, implicit definition of p.
b1A1 CAO
b2M1 Three lines drawn, accept \(p > 1\) or \(p < 0\) here. Must be functions of p.
b2A1 CAO \(0 \leqslant p \leqslant 1\), scale clear (or 1 line = 1), condone lack of labels. Rulers used.
b3DM1 Must have drawn 3 lines. Finding their correct optimal point, must have three lines and set up an equation to find \(0 \leqslant p \leqslant 1\). If solving each pair of SE’s must clearly select the correct one or M0, but allow recovery if their choice is clear from (c).
b3A1 CAO 5/9
b4A1ft All three options listed must ft from their p, check page 1, no negatives.
| Scheme | Marks |
|---|---|
| The value of the game to Goodie is \(\dfrac{8}{9}\). | B1 |
| (1) | |
| (9 marks) |
Notes
c1B1 CAO