 Solution

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

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

The core of a cooperative game consists of efficient allocations satisfying

\sum_{i \in S} x_{i} \geq v (S)

for every coalition, so no coalition can obtain a greater total payoff by leaving. Efficiency has already been checked. For singletons,

\frac{8}{3} \geq 0, \frac{14}{3} \geq 2, \frac{23}{3} \geq 5.

\frac{22}{3} \geq 6, \frac{31}{3} \geq 9, \frac{37}{3} \geq 11.

The empty constraint is automatic and the grand constraint is equality. Thus every core inequality holds and

x \in core (v) : (d) is true.

All four assertions about this allocation are true, although their definitions impose different requirements.

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