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

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Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 40 / 4 / a / Solution by

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

A payoff allocation is Pareto efficient if no feasible allocation weakly increases every player's payoff and strictly increases at least one. Here feasibility requires total payment at most the available estate. The proposed payments have

\frac{8}{3} + \frac{14}{3} + \frac{23}{3} = 15.

(1)

Any such improvement would have total payment greater than

15

, which is impossible. Each payment is also between zero and its claim, and exceeds its singleton coalition value. Hence the allocation is an imputation and

The allocation is Pareto optimal: (a) is true.

(2)

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