Abstract
The fuzzy core is well-known in theoretical economics, it is widely applied to model the conditions of perfect competition. In contrast, the original author’s concept of fuzzy contractual allocation as a specific element of the fuzzy core is not so widely known in the literature, but it also represents a (refined) model of perfect competition. This motivates the study of its validity: the existence of fuzzily contractual allocations in an economic model; it also implies the existence (non-emptiness) of the fuzzy core and develops an approach from [15]. The proof is based on two well-known theorems: Michael’s theorem on the existence of a continuous selector for a point-to-set mapping and Brouwer’s fixed point theorem. In literature, only the non-emptiness of the fuzzy core was proven under essentially stronger assumptions—typically, it applies replicated economies and Edgeworth equilibria.
The study was supported by the Program of Basic Scientific Research of the Siberian Branch of the Russian Academy of Sciences (Grant no. FWNF-2022-0019).
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Notes
- 1.
A convex model with a compact set \(\mathcal{A}(X)\) of feasible allocations and preferences that are continuously extendable to a neighborhood of \(\mathcal{A}(X)\).
- 2.
Admitting some inaccuracy in formulas here and below, we identify a vector with a one-element set containing it.
- 3.
Clearly, for a dominating fuzzy coalition t one may always think that \(\sum _{i\in \mathcal I}t_i=1\).
- 4.
A linear segment with ends \(a, b\in L\) is the set \( [ a,b]={\textrm{co}}\{ a,b\}=\{\lambda a+(1-\lambda )b\mid 0\le \lambda \le 1\}\).
- 5.
Earlier in literature allocations from fuzzy core were interpreted only as Edgeworth’s equilibria and served as a technical tool more than an economic concept.
- 6.
According to the modern views, the term semi-continuous mapping is specifically applied for a function—point-to-point map—and hemicontinuous for a correspondence.
- 7.
Note that in original paper item (c) has a typo for the range of \(\phi :X\rightarrow \mathcal{K}(Y)\). Author denoted \(\mathcal{K}(Y)\) as a set of all convex subsets of Y, but speak and prove the result for a narrower class of sets \(\mathcal{D}(Y)\subset \mathcal{K}(Y)\), see p. 372. Here I present a less general result, to avoid a cumbersome specification of \(\mathcal{D}(Y)\).
- 8.
It is standardly defined as \(\rho (x,S)=\inf _{y\in S} \rho (x,y)\).
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Marakulin, V. (2023). On the Existence of Fuzzy Contractual Allocations, Fuzzy Core and Perfect Competition in an Exchange Economy. In: Khachay, M., Kochetov, Y., Eremeev, A., Khamisov, O., Mazalov, V., Pardalos, P. (eds) Mathematical Optimization Theory and Operations Research: Recent Trends. MOTOR 2023. Communications in Computer and Information Science, vol 1881. Springer, Cham. https://doi.org/10.1007/978-3-031-43257-6_23
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