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Social Choice and Welfare

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01-08-2014 | Original Paper

Shapley–Shubik methods in cost sharing problems with technological cooperation

Authors: Eric Bahel, Christian Trudeau

Published in: Social Choice and Welfare | Issue 2/2014

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Abstract

In the discrete cost sharing model with technological cooperation (Bahel and Trudeau in Int J Game Theory 42:439–460, 2013a), we study the implications of a number of properties that strengthen the well-known dummy axiom. Our main axiom, which requires that costless units of demands do not affect the cost shares, is used to characterize two classes of rules. Combined with anonymity and a specific stability property, this requirement picks up sharing methods that allow the full compensation of at most one technological contribution. If instead we strengthen the well-known dummy property to include agents whose technological contribution is offset by the cost of their demand, we are left with an adaptation of the Shapley–Shubik method that treats technologies as private and rewards their contributions. Our results provide two interesting axiomatizations for the adaptations of the Shapley–Shubik rule to our framework.

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In the traditional model, Moulin and Vohra (2003) also describe a family of methods with a flow representation. As they do not use the dummy property, there are less restrictions on the flows.

Throughout the paper, we use the following convention for vector inequalities:

$\bar{x}=(\bar{x}_{1},\ldots ,\bar{x}_{m})\le \tilde{x}=(\tilde{x} _{1},\ldots ,\tilde{x}_{m})$ iff $\bar{x}_{i}\le \tilde{x}_{i}$, for every $i=1,\ldots ,m$;
$\bar{x}=(\bar{x}_{1},\ldots ,\bar{x}_{m})<\tilde{x}=(\tilde{x} _{1},\ldots ,\tilde{x}_{m})$ iff $\left[ \bar{x}\ne \tilde{x}\text { and }\bar{x}_{i}\le \tilde{x}_{i},\text { for every }i=1,\ldots ,m\right] $.

Formally, let $f(z_{x},\cdot )$ be a flow to $z_{x}$ and $x^{\prime }\le x$. A projection of $f(z_{x},\cdot )$ on $[ 0,z_{x^{\prime }}[$, denoted $p_{x^{\prime }}f(z_{x},\cdot )$, is defined as follows: for any $i\in N$ and $t\in [0,z_{x^{\prime }}[$ write $M=\left\{ j\in N\left| t_{j}=x_{j}^{\prime }+1\right. \right\} $ and let

$$\begin{aligned} p_{x^{\prime }}f_{i}(z_{x},t)&= 0\text { if }i\in M \\&= \sum \nolimits _{w_{M}\in \left[ z_{x^{\prime }}^{M},z_{x}^{M}\right] }f_{i} (z_{x},t^{N\backslash M}+w_{M})\text { otherwise,} \end{aligned}$$

with the convention that the sum is simply $f_{i}(z_{x},t)$ if $M=\emptyset $. Then, a cost sharing method $y$ is a fixed-flow method if for $x^{\prime }\le x$, there is a flow $f(z_{x},\cdot )$ representing $y(\cdot ,x)$ such that $p_{x^{\prime }}f(z_{x},\cdot )$ represents $y(\cdot ,x^{\prime })$.

Remember that the public Shapley–Shubik method is the Shapley value of the game where stand-alone costs are computed by assuming that a coalition has access to all technologies. The Average Single Rent method differs by defining the stand-alone cost of coalition $S$ as the average cost when it has access to all but one of the technologies in $N\backslash S.$ Therefore, the average between $ASR$ and $SS^{pub}$ is the Shapley value of the game where, to define the stand-alone cost of coalition $S,$ we assume that half the time it has access to all technologies (as for $SS^{pub})$ and half the time it has access to all but one technologies of agents in $N\backslash S$.

See Bahel and Trudeau (2013b).

An exception is made for agents $k$ such that $t_{k}=0,$ who do not share their technology or ask for any units. If they start sharing their technology, the cost drops to zero.

Starting from $C_{2}$ and removing the first $x_{j}-1$ units of demand of agent $j$ leaves us with the same cost function as if we start with $C_{1}$ and remove the last $x_{j}-1$ units of demand of agent $j.$

It is well known that any flow to $z_{x}$ can be represented as a (unique) convex combination of distinct paths to $z_{x}$; see Wang (1999) or Bahel and Trudeau (2013a).

Bahel E, Trudeau C (2013a) A discrete cost sharing model with technological cooperation. Int J Game Theory 42:439–460CrossRef

Bahel E, Trudeau C (2013b) Consistency requirements and pattern methods in cost sharing problems with technological cooperation. Virginia Polytechnic Institute and State University Working Paper e07–35

Friedman E (2004) Paths and consistency in additive cost sharing. Int J Game Theory 32:501–518CrossRef

Moulin H (1995) On additive methods to share joint costs. Jpn Econ Rev 46:303–332CrossRef

Moulin H, Sprumont Y (2005) On demand responsiveness in additive cost sharing. J Econ Theory 125:1–35CrossRef

Moulin H, Sprumont Y (2007) Fair allocation of production externalities: recent results. Rev Econ Polit 117:7–37

Moulin H, Vohra R (2003) A representation of additive cost sharing methods. Econ Lett 80:399–407CrossRef

Shubik M (1962) Incentives, decentralized control, the assignments of joint costs, and internal pricing. Manag Sci 8:325–343CrossRef

Sprumont Y (1998) Ordinal cost sharing. J Econ Theory 81:126–162CrossRef

Sprumont Y (2008) Nearly-serial sharing methods. Int J Game Theory 37:155–184CrossRef

Wang Y (1999) The additivity and dummy axioms in the discrete cost sharing model. Econ Lett 64:187–192CrossRef

Title: Shapley–Shubik methods in cost sharing problems with technological cooperation
Authors: Eric Bahel
Christian Trudeau
Publication date: 01-08-2014
Publisher: Springer Berlin Heidelberg
Published in: Social Choice and Welfare / Issue 2/2014
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-013-0775-6

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