Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 31 / 6 / a

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- Solution a

Solution

For a transferable utility game, an imputation is an efficient and individually rational payoff allocation:

I (v) = {x \in R^{3} : \sum_{i} x_{i} = v (N), x_{i} \geq v ({i})} .

Here it is

I (v) = {x : x_{1} + x_{2} + x_{3} = 12, x_{1} \geq 1, x_{2} \geq 2, x_{3} \geq 3} .

z_{1} = x_{1} - 1

,

z_{2} = x_{2} - 2

,

z_{3} = x_{3} - 3

identifies it with a scaled standard simplex

z_{1} + z_{2} + z_{3} = 6

. Its three vertices are

(7, 2, 3)

,

(1, 8, 3)

and

(1, 2, 9)

.

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