## Abstract

We consider consecutive aggregation procedures for individual preferences 𝔠 ∈ ℭ_{r} (*A*) on a set of alternatives *A*, |*A*| ≥ 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 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 𝔇 ⊆ ℭ_{r} (*A*) with respect to a symmetric set of lots 𝒥. On the basis of this classification, it is shown that a clone ℱ of local aggregation functions that preserves the set ℜ_{2} (*A*) of rational preferences with respect to a symmetric set 𝒥 contains nondictatorial aggregation functions if and only if 𝒥 is a set of maximal lots, in which case the clone ℱ is generated by the majority function. On the basis of each local aggregation function *f*, lot *J*, and an adaptation function 𝒜, one constructs a nonlocal (in general) aggregation function *f*_{J,A} that imitates a consecutive aggregation procesure. If *f* dynamically preserves a set 𝔇 ⊆ ℭ_{r} (*A*) with respect to a set of lots 𝒥, then the aggregation function *f*_{J,A} preserves the set 𝔇 for each lot *J* ∈ 𝒥. If 𝔇 = ℜ_{2}(*A*), then the adaptation function can be chosen in such a way that in any profile **c** ∈ (ℜ_{2}(*A*))^{n}, the Condorcet winner (if it exists) would coincide with the maximal element with respect to the preferences *f*_{J,A} (**c**) for each maximal lot *J* and *f* that dynamically preserves the set of rational preferences with respect to the set of maximal lots.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.S. Shelah, “On the Arrow property,”
*Adv. Appl. Math.*,**34**, 217–251 (2005).MathSciNetCrossRefGoogle Scholar - 2.N. Polyakov and M. Shamolin, “On a generalization of Arrow’s impossibility theorem,”
*Dokl. Math.*,**89**, No. 3, 290–292 (2014).MathSciNetCrossRefGoogle Scholar - 3.N. Polyakov,
*Functional Galois Connections and a Classification of Symmetric Conservative Clones with a Finite Carrier*, Preprint (2018).Google Scholar - 4.M. Aizerman and F. Aleskerov, “Voting operators in the space of choice functions,”
*Math. Soc. Sci.*,**11**, No. 3, 201–242 (1986).MathSciNetCrossRefGoogle Scholar - 5.F. T. Aleskerov,
*Arrovian Aggregation Models*, Springer, New York (1999).CrossRefGoogle Scholar - 6.F. T. Aleskerov, “Local aggregation models,”
*Autom. Remote Control*, No. 10, 3–26 (2000).Google Scholar - 7.N. L. Polyakov and M. V. Shamolin, “On closed symmetric classes of functions preserving an arbitrary one-place predicate,”
*Vestn. Samarsk. Univ. Estestv. Ser.*, No. 6 (107), 61–73 (2013).Google Scholar - 8.E. Post,
*Two-Valued Iterative Systems of Mathematical Logic*, Ann. Math. Stud., Vol. 5, Princeton Univ. (1942).Google Scholar - 9.S. S. Marchenkov,
*Functional Systems with Superposition*[in Russian], Fizmatlit, Moscow (2004).Google Scholar - 10.D. Lau,
*Function Algebras on Finite Sets. A Basic Course on Many-Valued Logic and Clone Theory*, Springer, Berlin (2006).zbMATHGoogle Scholar - 11.K. Arrow,
*Social Choice and Individual Values*, Yale Univ. Press (1963).Google Scholar - 12.F. Brandt, V. Conitzer, U. Endriss, et al.,
*Handbook of Computational Social Choice*Cambridge Univ. Press (2016).Google Scholar