On dynamic aggregation systems

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.