 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/iii/paper-40/4/b/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

The Shapley value is the expected marginal contribution of a player when the players join in uniformly random order. Equivalently,

ϕ_{i} (v) = \sum_{S \subseteq N ∖ {i}} \frac{∣ S ∣ ! ( n - ∣ S ∣ - 1 )!}{n !} (v (S \cup {i}) - v (S)) .

For three players, the weights for predecessor sets of sizes

0, 1, 2

are

1/3, 1/6, 1/3

. Using the computed values,

ϕ_{1} ϕ_{2} ϕ_{3} = \frac{1}{3} (0) + \frac{1}{6} [(6 - 2) + (9 - 5)] + \frac{1}{3} (15 - 11) = \frac{8}{3}, = \frac{1}{3} (2) + \frac{1}{6} [(6 - 0) + (11 - 5)] + \frac{1}{3} (15 - 9) = \frac{14}{3}, = \frac{1}{3} (5) + \frac{1}{6} [(9 - 0) + (11 - 2)] + \frac{1}{3} (15 - 6) = \frac{23}{3} .

Thus

ϕ (v) = x : (b) is true.

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