## Cost sharing in production economies.(English)Zbl 1455.91126

Sotomayor, Marilda (ed.) et al., Complex social and behavioral systems. Game theory and agent-based models. New York, NY: Springer. Encycl. Complex. Syst. Sci. Ser., 421-462 (2020).
Summary: Throughout we will use a fixed set of agents $$N = \{1, 2, \dots, n\}$$ where $$n$$ is a given natural number. For subsets $$S, T$$ of $$N$$, we write $$S \subset T$$ if each element of $$S$$ is contained in $$T$$; $$T\backslash S$$ denotes the set of agents in $$T$$ except those in $$S$$. The power set of $$N$$ is the set of all subsets of $$N$$; each coalition $$S \subset N$$ will be identified with the element $$1_S \in \{0, 1\}^N$$, the vector with $$i$$-th coordinate 1 precisely when $$i \in S$$. Fix a vector $$x \in \mathbb{R}^N$$ and $$S \subset N$$. The projection of $$x$$ on $$\mathbb{R}^S$$ is denoted $$x_S$$, and $$x_{N\backslash S}$$ is sometimes more conveniently denoted $$x_{-S}$$. For any $$y \in \mathbb{R}^S$$, $$(x_{-S}, y)$$ stands for the vector $$z \in \mathbb{R}^N$$ such that $$z_i = x_i$$ if $$i \in N\backslash S$$ and $$z_i = y_i$$ if $$i \in S$$. We denote $$x(S)= \sum_{i\in S}x_i$$. The vector in $$\mathbb{R}^S$$ with all coordinates equal zero is denoted $$\mathbf{0}_S$$. Other notation will be introduced when necessary.
This entry focuses on different approaches in the literature through a discussion of a couple of basic and illustrative models, each involving a single facility for the production of a finite set $$M$$ of outputs, commonly shared by a fixed set $$N :=\{1, 2, \dots, n\}$$ of agents. The feasible set of outputs for the technology is identified with a set $$X \subset \mathbb{R}^M_+$$. It is assumed that the users of the technology may freely dispose over any desired quantity or level of the outputs; each agent $$i$$ has some demand $$x_i \in X$$ for output. Each profile of demands $$x \in X\!^N$$ is associated with its cost $$c(x)$$, i.e., the minimal amount of the idiosyncratic input commodity needed to fulfill the individual demands. This defines the cost function $$c : X\!^N \rightarrow \mathbb{R}_+$$ for the technology, comprising all the production externalities. A cost sharing problem is an ordered pair $$(x, c)$$ of a demand profile $$x$$ and a cost function $$c$$. The interpretation is that $$x$$ is produced, and the resulting cost $$c(x)$$ has to be shared by the collective $$N$$. Numerous practical applications fit this general description of a cost sharing problem.
In mathematical terms, a cost sharing problem is equivalent to a production sharing problem where output is shared based on the profile of inputs. However, although many concepts are just as meaningful as they are in the cost sharing context, results are not at all easily established using this mathematical duality. In this sense consider [J. Leroux, Games Econ. Behav. 62, No. 2, 558–572 (2008; Zbl 1137.91306)] as a warning to the reader, showing that the strategic analysis of cost sharing solutions is quite different from surplus sharing solutions. This monograph will center on cost sharing problems. For further reference on production sharing, see [L. D. Israelseni, “Collectives, communes, and incentives”, J. Comp. Econ. 4, No. 2, 99–124 (1980; doi:10.1016/0147-5967(80)90024-4); J. Leroux, Econ. Lett. 85, No. 3, 335–340 (2004; Zbl 1254.91149); 2008, loc. cit.; H. Moulin and S. Shenker, Econometrica 60, No. 5, 1009–1037 (1992; Zbl 0766.90013)].
For the entire collection see [Zbl 1457.91008].

### MSC:

 91B32 Resource and cost allocation (including fair division, apportionment, etc.) 91A10 Noncooperative games 91A12 Cooperative games 91A80 Applications of game theory

### Citations:

Zbl 1137.91306; Zbl 1254.91149; Zbl 0766.90013
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