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 Tr. Semim. im. I. G. Petrovskogo, 2019, Issue 32, Pages 257–282 (Mi tsp110)

On dynamic aggregation systems

N. L. Polyakov, M. V. Shamolin

Abstract: We consider consecutive aggregation procedures for individual preferences $\mathfrak c\in \mathfrak C_r(A)$ on a set of alternatives $A$, $|A|\geq 3$: on each step, the participants are subject to intermediate collective decisions on some subsets $B$ of the set $A$ and transform their a priori preferences according to an adaptation function $\mathcal{A}$. The sequence of intermediate decisions is determined by a lot $J$, i.e., an increasing (with respect to inclusion) sequence of subsets $B$ of the set of alternatives. An explicit classification is given for the clones of local aggregation functions, each clone consisting of all aggregation functions that dynamically preserve a symmetric set $\mathfrak D\subseteq \mathfrak C_r(A)$ with respect to a symmetric set of lots $\mathcal{J}$. On the basis of this classification, it is shown that a clone $\mathcal{F}$ of local aggregation functions that preserves the set $\mathfrak{R}_2(A)$ of rational preferences with respect to a symmetric set $\mathcal{J}$ contains nondictatorial aggregation functions if and only if $\mathcal{J}$ is a set of maximal lots, in which case the clone $\mathcal{F}$ is generated by the majority function. On the basis of each local aggregation function $f$, lot $J$, and an adaptation function $\mathcal{A}$, one constructs a nonlocal (in general) aggregation function $f_{J,A}$ that imitates a consecutive aggregation procesure. If $f$ dynamically preserves a set $\mathfrak D\subseteq \mathfrak C_r(A)$ with respect to a set of lots $\mathcal{J}$, then the aggregation function $f_{J,A}$ preserves the set $\mathfrak{D}$ for each lot $J\in\mathcal{J}$. If $\mathfrak D=\mathfrak R_2(A)$, then the adaptation function can be chosen in such a way that in any profile $\mathbf c\in (\mathfrak R_2(A))^n$, the Condorcet winner (if it exists) would coincide with the maximal element with respect to the preferences $f_{J, \mathcal A}(\mathbf c)$ for each maximal lot $J$ and $f$ that dynamically preserves the set of rational preferences with respect to the set of maximal lots.

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English version:
Journal of Mathematical Sciences (New York), 2020, 244:2, 278–293

UDC: 510.6+510.633

Citation: N. L. Polyakov, M. V. Shamolin, “On dynamic aggregation systems”, Tr. Semim. im. I. G. Petrovskogo, 32, 2019, 257–282; J. Math. Sci. (N. Y.), 244:2 (2020), 278–293

Citation in format AMSBIB
\Bibitem{PolSha19}
\by N.~L.~Polyakov, M.~V.~Shamolin
\paper On dynamic aggregation systems
\serial Tr. Semim. im. I. G. Petrovskogo
\yr 2019
\vol 32
\pages 257--282
\mathnet{http://mi.mathnet.ru/tsp110}
\elib{https://elibrary.ru/item.asp?id=43209420}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2020
\vol 244
\issue 2
\pages 278--293
\crossref{https://doi.org/10.1007/s10958-019-04619-w}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85075857577}