On dynamic aggregation systems

N. L. Polyakov   ; M. V. Shamolin

Trudy Seminara im. I.G. Petrovskogo, Trudy Seminara imeni I. G. Petrovskogo, Tome 32 (2019) no. 32, pp. 257-282

Résumé

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.

Bibliographie

[1] Shelah S., “On the Arrow property”, Adv. Appl. Math., 34 (2005), 217–251 | DOI | MR | Zbl

[2] Polyakov N., Shamolin M., “On a generalization of Arrow's impossibility theorem”, Dokl. Math., 89:3 (2014), 290–292 | DOI | MR | Zbl

[3] Polyakov N., Functional Galois connections and a classification of symmetric conservative clones with a finite carrier, Preprint, 2018 | MR

[4] Aizerman M., Aleskerov F., “Voting operators in the space of choice functions”, Math. Soc. Sci., 11:3 (1986), 201–242 | DOI | MR | Zbl

[5] Aleskerov F. T., Arrovian Aggregation Models, Springer, New York, 1999 | MR

[6] Aleskerov F. T., “Local aggregation models”, Automat. Remote Control, 2000, no. 10, 3–26 | MR | Zbl

[7] Polyakov N. L., Shamolin M. V., “O zamknutykh simmetrichnykh klassakh funktsii, sokhranyayuschikh lyuboi odnomestnyi predikat”, Vestn. SamGU. Estestvennonauch. ser., 2013, no. 6 (107), 61–73

[8] Post E., Two-Valued Iterative Systems of Mathematical Logic, Ann. Math. Stud., 5, Princeton Univ., 1942 | MR

[9] Marchenkov S. S., Funktsionalnye sistemy s operatsiei superpozitsii, Fizmatlit, 2004

[10] Lau D., Function Algebras on Finite Sets. A Basic Course on Many-Valued Logic and Clone Theory, Springer, Berlin, 2006 | MR | Zbl

[11] Arrow K., Social Choice and Individual Values, Yale Univ. Press, 1963 | MR

[12] Brandt F., Conitzer V., Endriss U. et al., Handbook of Computational Social Choice, Cambridge Univ. Press, 2016 | Zbl