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An Introduction to Allocation Rules by Jens Leth Hougaard

By Jens Leth Hougaard

This publication specializes in interpreting price and surplus sharing difficulties in a scientific model. It deals an in-depth research of assorted sorts of ideas for allocating a standard financial worth (cost) among contributors of a bunch or community – e.g. members, agencies or items. the implications can assist readers overview the professionals and cons of a few of the tools fascinated with phrases of assorted elements corresponding to equity, consistency, balance, monotonicity and manipulability. As such, the e-book represents an updated survey of price and surplus sharing equipment for researchers, scholars and practitioners alike. The textual content is observed via useful circumstances and various examples to make the theoretical effects simply accessible.

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11) where α is chosen such that the cost shares add up to total costs C(Q). This rule captures the spirit of the constrained equal gains rule of the rationing model. We may further define the restricted average cost rule by the following cost sharing scheme: First, calculate shares x1i = min C(qi ), qi C(Q) , Q i = 1, . . , n. 12) If some agents are bounded by their stand-alone cost the remaining agents n must further share C(Q) − i=1 x1i in proportion to their demand and so forth until total costs are fully allocated.

It can be shown that if C is convex then the cost share of agent i using the increasing serial rule φIS i is non-decreasing in the demand of agent j, qj , for any j = i. If C is concave then the cost share of agent i using the increasing (resp. the decreasing serial rule φDS serial rule φIS i i ) is non-increasing (resp. non-decreasing) in the demand of agent j, qj , for any j = i. In general, both rules have decreasing (resp. increasing) agent specific unit prices when the cost function is concave (resp.

Ui (qi , φi ) = αi qi − φi , where α ∈ R++ . That is, dept. 3 Cost Sharing with Joint Cost Function 45 max α1 q1 − 2q12 , q1 yielding q1∗ = α1 /4. Note, that the optimal demand level of dept. 1 is independent of the demand of dept. 2 (since the cost share of dept. 1 does not depend on the demand of dept. 2). Dept. 2 chooses its demand for service by solving max α2 q2 − (q1 + q2 )2 − 2q12 , q2 and knowing that dept. 1 demands q1∗ = α1 /4, we get that q2∗ = (α2 −α1 /2)/2. Hence, in this case the cost sharing game induced by the increasing serial rule has got a unique Nash equilibrium in demands (q1∗ , q2∗ ) = α1 2α2 − α1 , 4 4 , and total cost is shared as IS (φIS 1 , φ2 ) = α12 2α22 − α12 , 8 8 .

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