On the Computational Complexity of Non-dictatorial Aggregation
We investigate when non-dictatorial aggregation is possible from an algorithmic perspective, where non-dictatorial aggregation means that the votes cast by the members of a society can be aggregated in such a way that the collective outcome is not simply the choices made by a single member of the society. We consider the setting in which the members of a society take a position on a fixed collection of issues, where for each issue several different alternatives are possible, but the combination of choices must belong to a given set X of allowable voting patterns. Such a set X is called a possibility domain if there is an aggregator that is non-dictatorial, operates separately on each issue, and returns values among those cast by the society on each issue. We design a polynomial-time algorithm that decides, given a set X of voting patterns, whether or not X is a possibility domain. Furthermore, if X is a possibility domain, then the algorithm constructs in polynomial time such a non-dictatorial aggregator for X. We also design a polynomial-time algorithm that decides whether X is a uniform possibility domain, that is, whether X admits an aggregator that is non-dictatorial even when restricted to any two positions for each issue. As in the case of possibility domains, the algorithm also constructs in polynomial time a uniform non-dictatorial aggregator, if one exists.
The research of Lefteris Kirousis was partially supported by the Special Account for Research Grants of the National and Kapodistrian University of Athens. The work of Phokion G. Kolaitis is partially supported by NSF Grant IIS-1814152.
- 1.Arrow, K.J.: Social Choice and Individual Values. Wiley, New York (1951)Google Scholar
- 2.Bessiere, C., Carbonnel, C., Hebrard, E., Katsirelos, G., Walsh, T.: Detecting and exploiting subproblem tractability. In: IJCAI, pp. 468–474 (2013)Google Scholar
- 6.Carbonnel, C.: The meta-problem for conservative Mal’tsev constraints. In: Thirtieth AAAI Conference on Artificial Intelligence (AAAI-2016) (2016)Google Scholar
- 9.Endriss, U.: Judgment aggregation. In: Brandt, F., Conitzer, V., Endriss, U., Lang, J., Procaccia, A.D. (eds.), Handbook of Computational Social Choice, pp. 399–426. Cambridge University Press (2016)Google Scholar
- 11.Larose, B.: Algebra and the complexity of digraph CSPs: a survey. In: Dagstuhl Follow-Ups, vol. 7. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2017)Google Scholar
- 12.List, C., Puppe, C.: Judgment Aggregation: A Survey (2009)Google Scholar
- 13.Nehring, K., Puppe, C.: Strategy-proof social choice on single-peaked domains: possibility, impossibility and the space between. University of California at Davis (2002). http://vwl1.ets.kit.edu/puppe.php
- 14.Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings of the 10th Annual ACM Symposium on Theory of Computing, pp. 216–226 (1978)Google Scholar
- 16.Szegedy, M., Xu, Y.: Impossibility theorems and the universal algebraic toolkit. CoRR, abs/1506.01315 (2015)Google Scholar
- 17.Szendrei, Á.: Clones in Universal Algebra, vol. 99. Presses de l’Université de Montréal, Montreal (1986)Google Scholar