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Vector representation of preferences on \(\sigma \)-algebras and fair division in saturated measure spaces. (English) Zbl 1471.91212

In the paper the author investigates the fair division problem. Let \((\Omega,\Sigma,\mu)\) be a finite measure space. A finite measure space \((\Omega,\Sigma,\mu)\) is said to be saturated if \(L^1(X,\Sigma_X,\mu)\) is nonseparable for every \(X\in\Sigma\), where \(\Sigma_X=\{A\cap X:A\in\Sigma\}\). A preference relation on the \(\sigma\)-algebra \(\Sigma\) is a complete transitive binary relation \(\succsim\) on \(\Sigma\). A utility function is a function \(\nu:\Sigma\to\mathbb{R}\). A utility function \(\nu\) represents \(\succsim\), if for every \(A,B\in\Sigma\), \(\nu(A)\ge\nu(B)\Leftrightarrow A\succsim B\). In the paper the author axiomatize the preference relation on the \(\sigma\)-algebra of a saturated finite measure space represented by a vector measure. The axioms guarantee a unique vector representation of the preference relation in terms of a nonadditive measure on \(\Sigma\) given by a continuous and quasiconcave transformation of a vector measure. The author investigates the fair division problems with nonadditive preferences and proves the existence results on fair partitions. He shows the existence of individually rational Pareto optimal partitions, the existence of Walrasian equilibrium partitions with a positive price, the existence of core partitions, and the existence of Pareto optimal envy-free partitions.

MSC:

91B32 Resource and cost allocation (including fair division, apportionment, etc.)
28B05 Vector-valued set functions, measures and integrals
28E10 Fuzzy measure theory
46B22 Radon-Nikodým, Kreĭn-Milman and related properties
46G10 Vector-valued measures and integration
52A07 Convex sets in topological vector spaces (aspects of convex geometry)
91B16 Utility theory
91B50 General equilibrium theory
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