Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 37 / 5 / c

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- Solution c

Solution

False. Take three players of weight one and strict threshold one, so a coalition wins exactly when it contains at least two players. Set

S = {1, 2}

and

T = {2, 3}

. Then

v (S) = v (T) = v (S \cup T) = 1, v (S \cap T) = v ({2}) = 0.

The convex cooperative game inequality would require

1 \geq 1 + 1 - 0 = 2

. Thus even this elementary majority weighted voting game is not convex.

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