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Saks and Yu (2005) proved that if D is convex then every monotone deterministic allocation rule is implementable. We prove in this appendix the following generalization of their result: Theorem 1 Every domain with a convex closure is a proper monotonicity domain. 1.1 Preparations First we recall the definitions of monotonicity and cyclic monotonicity. An allocation rule f is called monotone if 〈f(v) − f(w), v − w 〉 ≥ 0 for every v, w ∈ D, (1) and f is called cyclically monotone if for every k ≥ 2, for every k vectors in D (not necessarily distinct), v1, v2,..., vk the following holds: k∑ 〈vi − vi+1, f(vi) 〉 ≥ 0, (2) i=1 where vk+1 is defined to be v1. By taking k = 2 in (2) it can be seen that every cyclically monotone allocation rule is monotone

Year: 2013

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