In this work we pursue the study of manipulability of social choice functions through “liftings”, that is, mappings which extend orderings over points to orderings over subsets of points. We discover a very weak notion of monotony which is closely related to independence of irrelevant alternatives. This allows us to establish an interesting and general theorem on manipulability. We show that this theorem is indeed equivalent to Arrow-Sen Theorem in the class of nonmanipulable social choice functions. As a consequence of this general theorem we obtain a manipulation theorem for linear profiles in the style of Gibbard-Satterthwaite Theorem but for social choice functions instead of voting schemes. We introduce the notion of nominator, which is a natural generalization of the notion of pairwise nominator introduced by Kelly. Then, we establish that, in the presence of rational properties over liftings, a social choice function is either manipulable, or it admits a nominator. In addition, we do a comparative study on different types of powerful voters (dictators, nominators, pairwise nominators and weak-dictators) present in the literature. Although, in general, they are non-equivalent notions, we show that under some natural conditions, modulo nonmanipulability, the last three are equivalent or even all the notions are equivalent.
This is a preview of subscription content,to check access.
Access this article
Figure 2 summarizes all these results.
A linear order is a total preorder for which antisymmetry holds, and the flat order is that for which all the elements are indifferent between them.
These properties are also known as Gärdenfors principles.
Note that we have chosen to have only nonempty outputs, in order to allow the possibility of having real partial functions. If we admit the empty set as output, we could define as the empty set the output of undefined inputs and then all the functions would be total.
The name of Transitive explanations is the one used by Kelly (1988) for this axiom. This axiom is also called Transitive rationality by Taylor in his book of 2005 Taylor (2005). Moulin (1988) calls the social choice functions satisfying this property rationalizable choice functions. Sen (1971) gives an axiomatic characterization of these functions.
For the case of the indifferent function, it is enough to associate every profile with the flat order.
A voting scheme is a function V that maps a profile P into an element V(P) of X.
Many authors use the term strategy-proof for nonmanipulable.
This version of the plurality rule: is an adaptation of the plurality rule presented by Taylor (2005) in the setting of voting rules.
It is easy to see that this function coincides with the Borda rule when the profile is constituted by linear ballots.
A strong version of PI can be established if we consider \(x\succeq ^*y\) instead of \(x\simeq ^*y\).
Those social choice functions which give as output the Condorcet winner whenever it exists.
Arrow K (1951) Social choice and individual values, 1st edn. Wiley, New York
Arrow K, Kelly JS (2011) An interview with Kenneth Arrow. In: Arrow K, Sen AK, Suzumura K (eds) Handbook of Social Choice and Welfare, Handbooks in Economics, vol 2. North-Holland, pp 4–24 (chap. 13, Part II)
Arrow K, Sen AK, Suzumura K (2002) Handbook of Social Choice and Welfare, Handbooks in Economics, vol 1. North-Holland
Barberà S (1977) The manipulation of social choice mechanisms that do not leave too much to chance. Econometrica 45(7):1573–1588
Barberà S (2011) Strategy-proof social choice. In: Arrow K, Sen AK, Suzumura K (eds) Choice and Welfare, Handbooks in Economics, vol 2. North-Holland, pp 731–831 (chap. 25)
Barberà S, Bossert W, Pattanaik PK (2004) Ranking sets of objects, Handbook of Utility Theory, vol 2. Kluwer Publisher, pp 893–978 (chap. 17)
Barberà S (1977) Manipulation of social decision functions. J Econ Theory 15(2):266–278
de Borda JC (1784) Mémoire sur les élections au scrutin. Histoire de l’Académie Royale des Sciences
Bossert W, Pattanaik PK, Xu Y (1994) Ranking opportunity sets: an axiomatic approach. Journal of Economic theory 63(2):326–345
Brandt F (2011) Group-strategyproof irresolute social choice functions. In: Walsh T (ed.) IJCAI 2011, Proceedings of the 22nd International Joint Conference on Artificial Intelligence, Barcelona, Catalonia, Spain, July 16-22, 2011, pp 79–84
Brandt F, Brill M (2011) Necessary and sufficient conditions for the strategyproofness of irresolute social choice functions. In: Apt KR (ed.) Proceedings of the 13th Conference on Theoretical Aspects of Rationality and Knowledge (TARK-2011), Groningen, The Netherlands, July 12-14, 2011, pp 136–142. ACM
Brandt F, Conitzer V, Endriss U, Lang J, Procaccia AD (2016) (eds.): Handbook of Computational Social Choice. Cambridge University Press
Camacho F, Pino Pérez R (2011) Leximax relations in decision making through the dominance plausible rule. In: Liu W (ed) Symbolic and Quantitative Approaches to Reasoning with Uncertainty, vol 6717. ECSQARU 2011, Lecture Notes in Computer Science. Springer, Berlin, Heidelberg, pp 569–581
Camacho F, Pino Pérez R (2021) Decision-making through dominance plausible rule: New characterizations. Mathematical Social Sciences 113:107–115
Ching S, Zhou L (2002) Multi-valued strategy-proof social choice rules. Social Choice and Welfare 19(3):569–580
de Finetti B (1937) La prévision: Ses lois logiques, ses sources subjectives. Annales de l’Institut Henri Poincaré 17:1–68
Dubois D, Fargier H (2004) A unified framework for order-of-magnitude confidence relations. In: UAI ’04, Proceedings of the 20th Conference in Uncertainty in Artificial Intelligence, July 7-11 2004, Banff, Canada, pp 138–145
Dubois D, Lang J, Prade H (1994) Possibilistic logic. In: Gabbay D, Hogger C, Robinson J (eds.) Handbook of Logic in Artificial Intelligence and Logic Programming, vol. 3: Nonmonotonic Reasoning and Uncertain Reasoning, pp 439—513. Oxford University Press, Inc
Duggan J, Schwartz T (2000) Strategic manipulability without resoluteness or shared beliefs: Gibbard-satterthwaite generalized. Social Choice and Welfare 17(1):85–93
Endriss U (2017) (ed.): Trends in Computational Social Choice. AI Access
Feldman A (1979) Nonmanipulable multi-valued social decision functions. Public Choice 34:177–188
Fishburn PC (1972) Even-chance lotteries in social choice theory. Theory and Decision 3(1):18–40
Fishburn PC (1973) The theory of social choice. Princeton University Press
Friedman N, Halpern J (1998) Plausibility measures and default reasoning. arXiv preprint cs/9808007
Gärdenfors P (1976) Manipulation of social choice functions. Journal of Economic Theory 13(2):217–228
Gärdenfors P (1979) On definitions of manipulation of social choice functions. In: Laffont JJ (ed) Aggregation and revelation of preferences. North-Holland, pp 29–36
Geist C, Endriss U (2011) Automated search for impossibility theorems in social choice theory: Ranking sets of objects. Journal of Artificial Intelligence Research 40:143–174
Gibbard A (1973) Manipulation of voting schemes: A general result. Econometrica 41(4):587–601
Halpern JY (1997) Defining relative likelihood in partially-ordered preferential structures. Journal of Artificial Intelligence Research 7:1–24. https://doi.org/10.1613/jair.391
Hansson B (1973) The independence condition in the theory of social choice. Theory and Decision 4:25–49
Kelly JS (1977) Strategy-proofness and social choice functions without singlevaluedness. Econometrica 45(2):439–446
Kelly JS (1988) Social Choice Theory: An Introduction. Springer-Verlag, Berlin
Lang J, van der Torre LW (2008) From belief change to preference change. In: Ghallab M, Spyropoulos C, Fakotakis N, Avouris N (eds.) Proceedings of 18th European Conference on Artificial Intelligence. ECAI 2008, Frontiers in Artificial Intelligence and Applications, vol. 178, pp 351–355. IOS Press
Leal J, Pino Pérez R (2017) A weak version of Barberà-Kelly’s theorem. Revista Colombiana de Matemáticas 51:173–194
Lewis D (1973) Counterfactuals and comparative possibility. In: Harper W, Stalnaker R, Pearce G (eds.) IFS. The University of Western Ontario Series in Philosophy of Science (A Series of Books in Philosophy of Science, Methodology, Epistemology, Logic, History of Science, and Related Fields), pp 57–85. Springer, Dordrecht
Mata Díaz A, Pino Pérez R (2017) Impossibility in belief merging. Artificial Intelligence 251:1–34
Mata Díaz A, Pino Pérez R (2018) Epistemic states, fusion and strategy-proofness. In: Fermé E, Villata S (eds.) Proceedings of the 17th International Workshop on Non-Monotonic Reasoning. NMR-2018, Tempe, Arizona, USA., pp 176–185
Mata Díaz A, Pino Pérez R (2019) Manipulability in logic-based fusion of belief bases: Indexes vs. liftings. Revista Iberica de Sistemas e Tecnologias de Informac̃ao (RISTI) 2019(E20):490–503
Mata Díaz A, Pino Pérez R (2021) Merging epistemic states and manipulation. In: Vejnarová J, Wilson N (eds.) Symbolic and Quantitative Approaches to Reasoning with Uncertainty - 16th European Conference, ECSQARU 2021, Prague, Czech Republic, September 21-24, 2021, Proceedings, Lecture Notes in Computer Science, vol. 12897, pp 457–470. Springer
Moulin H (1988) Axioms of cooperative decision making. Cambridge University Press
Pini M, Rossi F, Venable K, Walsh T (2009) Aggregating Partially Ordered Preferences. Journal of Logic and Computation 19(3):475–502
Satterthwaite MA (1975) Strategy-proofness and Arrow’s conditions: Existence and correspondence theorems for voting procedures and social welfare functions. Journal of Economic Theory 10(2):187–217
Sen AK (1969) Quasi-transitivity, rational choice and collective decisions. The Review of Economic Studies 36(3):381–393
Sen AK (1971) Choice functions and revealed preference. The Review of Economic Studies 38(3):307–317
Shackle GLS (1953) On the meaning and measure of uncertainty. Metroeconomica 5:97–115
Shackle GLS (1955) Uncertainty in economics and other reflections. Cambridge University Press
Spohn W (1988) Ordinal conditional functions: A dynamic theory of epistemic states. In: Harper W, Skyrms B (eds) The University of Western Ontario Series in Philosophy of Science (A Series of Books in Philosophy of Science, Methodology, Epistemology, Logic, History of Science, and Related Fields), vol 42. Springer, Dordrecht, pp 105–134
Taylor AD (2002) The manipulability of voting systems. The American Mathematical Monthly 109(4):321–337
Taylor AD (2005) Social Choice and the Mathematics of Manipulation. Cambridge University Press
van Benthem J, Girard P, Roy O (2009) Everything else being equal: A modal logic for ceteris paribus preferences. Journal of Philosophical Logic 38:83–125
Wilson R (1972) Social choice theory without the Pareto principle. Journal of Economic Theory 5(3):478–486
We thank the anonymous reviewers for their remarks, which have been very helpful for improving our work. We also thank Professor Olga Porras for her careful proof reading. The second author has benefited from the support of the AI Chair BE4musIA of the French National Research Agency (ANR-20-CHIA-0028) and has also been partially funded by the program PAUSE of Collège de France and by the Consejo de Desarrollo Científico Humanístico Tecnoló-gico y de las Artes de la Universidad de Los Andes (CDCHTA-ULA) through the Project N\(^\circ\) C-1855-13-05-AA.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Mata Díaz, A., Pino Pérez, R. & Leal, J.F. Taxonomy of powerful voters and manipulation in the framework of social choice functions. Soc Choice Welf 61, 277–309 (2023). https://doi.org/10.1007/s00355-022-01448-w