# An Introduction to Allocation Rules by Jens Leth Hougaard

By Jens Leth Hougaard

This e-book specializes in studying rate and surplus sharing difficulties in a scientific style. It deals an in-depth research of assorted forms of ideas for allocating a typical financial price (cost) among participants of a gaggle or community – e.g. contributors, agencies or items. the consequences may also help readers assessment the professionals and cons of a number of the equipment concerned about phrases of varied components akin to equity, consistency, balance, monotonicity and manipulability. As such, the e-book represents an updated survey of rate and surplus sharing tools for researchers, scholars and practitioners alike. The textual content is observed via functional situations and various examples to make the theoretical effects simply accessible.

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16) Clearly, if C is convex (resp. concave) then mixed serial cost sharing coincides with increasing (resp. decreasing) serial cost sharing. Hence, for any problem in D+ , mixed serial cost sharing results in non-negative cost shares. 7. Consider three agents √ (n = 3) with demands q = (1, 3, 5) and cost function C(Q) = Q2 + 64 Q which is concave on [0, 4) and convex on (4, ∞). This cost function C can be decomposed √ into a convex function R∗ (Q) = Q2 and a concave function S ∗ (Q) = 64 Q. 6 3 3 Sum 81 192 273 ˜ + S˜ where Alternatively, consider the CS-decomposition C = R ˜ R(Q) = 0 √ Q2 + 64 Q − 24Q − 48 and, ˜ S(Q) = √ Q2 + 64 Q 24Q + 48 if Q ≤ 4 if Q > 4, if Q ≤ 4 if Q > 4.

1 we see that the rules diﬀer considerably with respect to how they distribute the shares. It seems that ϕCEG results in distributions with the smallest spread whereas ϕCEL results in shares with the largest spread. In case E ≤ Q/2, shares given by the proportional rule seems more spread than shares given by the Talmud rule whereas when E ≥ Q/2 it appears to be the other way around. In fact, such characterizations in terms of economic inequality comparisons can be formalized using the notion of Lorenz-domination (also known as majorization).

X ¯2 ) with x ¯1 + x ¯ continuity of α implies that ¯=x x2 . Since α(0) = 0 ≤ x1 + x2 ≤ α(E) E ¯1 + n¯ ¯ and α(E) = x1 + x2 . By consistency and there exists E such that 0 ≤ E ≤ E equal treatment ϕ(˜ q , E) = (x1 , x2 , . . , x2 ) with E = x1 + nx2 . By choice of ¯ contradicting E ≤ E. ¯ We conclude that ϕ ¯1 + n¯ x2 = E n, E = x1 + nx2 > x satisﬁes resource monotonicity. Now suppose that ϕ satisﬁes strict resource monotonicity (only weak monotonicity was shown above) then the proof could continue like this: For every 2-agent problem and every λ ∈ [0, 1] deﬁne x1 = f (q1 , λ) if and only if (x1 , λ) = ϕ((q1 , 1), x1 + λ).