Convex Rationing Solutions (Incomplete Version, Do not distribute)
نویسنده
چکیده
This paper introduces a notion of convexity on the rationing problem and characterizes the solutions that are convex. We study the composition of convexity with other traditional axioms explored in the literature such as upper and lower composition. We characterize the class of rules satisfying upper composition and convexity for any number of agents. The dual rules, satisfying lower compositon and convexity are also provided. A class of solutions meeting upper composition, convexity and consistency is provided. The combination of convexity, scale invariance, consistency and any one of the compositions axioms leads to a re-characterization of the priority solutions introduced by Moulin[2000]. 1 Notation and preliminaries Let N a finite subset of N, and R+ the positive Euclidean space of the dimension of N . Let S 2 the simplex in R+, that is {p ∈ R+ | p1 + p2 = 1}. We will denote as the order of the simplex such that for each x, y ∈ S , x y if and only if x1 > y1. For X,Y ⊂ S , we say X Y if x y for all x ∈ X, y ∈ Y. Also, we denote by (Y,X) ≡ (X,Y ) ≡ {z | x z y for all x ∈ X, y ∈ Y }. The definition of the closed and semiopened intervals for subset of S 2 is similarly given. For x ∈ R+ , xN = ∑ i∈N xi, x̃ = x xN , and ‖ x ‖ is the euclidean norm of x. Notice that x̃ is the multiple of x that belongs to the simplex S . For x ∈ R+ , A ⊂ R+ , the ray with apex x and slopes in A is given by rayx(A) = {x+ t · a | a ∈ A, t ≥ 0}. Given a function f : R → R, the left and right limits at x are denoted by f−(x) and f+(x) respectively. Definition 1 (i) A problem is a pair [c; b] ∈ R+ × R+ such that ∑
منابع مشابه
Partial and Asymmetric Information
This is a preliminary and incomplete version of a manuscript to be prepared for the CIME-EMS Summer School on Stochastic Methods in Finance, July, 2003. Please do not quote or distribute.
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