# Taxonomy of powerful voters and manipulation in the framework of social choice functions

## Abstract

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.

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## Notes

1. Figure 2 summarizes all these results.

2. 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.

3. These properties are also known as Gärdenfors principles.

4. 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.

5. Although Arrow’s conditions were stated in the context of social welfare functions, it is not hard to see that, modulo transitive explanations, they are equivalent to the versions presented here for social choice functions (cf. Kelly 1988; Taylor 2005).

6. 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.

7. For the case of the indifferent function, it is enough to associate every profile with the flat order.

8. A voting scheme is a function V that maps a profile P into an element V(P) of X.

9. Many authors use the term strategy-proof for nonmanipulable.

10. This version of the plurality rule: is an adaptation of the plurality rule presented by Taylor (2005) in the setting of voting rules.

11. It is easy to see that this function coincides with the Borda rule when the profile is constituted by linear ballots.

12. A strong version of PI can be established if we consider $$x\succeq ^*y$$ instead of $$x\simeq ^*y$$.

13. Those social choice functions which give as output the Condorcet winner whenever it exists.

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## Acknowledgements

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.

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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