On Dynamic Aggregation Systems - Polyakov, N. L.; Shamolin, M. V. (Journal of mathematical sciences (New York, N.Y.), 2020)
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.