Necessary and sufficient conditions for the strategyproofness of irresolute social choice functions

Abstract

While the Gibbard-Satterthwaite theorem states that every non-dictatorial and resolute, i.e., single-valued, social choice function is manipulable, it was recently shown that a number of appealing irresolute Condorcet extensions are strategyproof according to Kelly's preference extension. In this paper, we study whether these results carry over to stronger preference extensions due to Fishburn and Gärden-fors. For both preference extensions, we provide sufficient conditions for strategyproofness and identify social choice functions that satisfy these conditions, answering a question by Gärdenfors [15] in the affirmative. We also show that some more discriminatory social choice functions fail to satisfy necessary conditions for strategyproofness.

A weak version of Barberà-Kelly's Theorem

“…Brandt and Brill [11] give sufficient conditions in order to have strategyproofness with respect the liftings of Kelly, Fishburn and Gärdenfors. Putting together their results with Theorem 4.5 we obtain some information.…”

“…More recently, Brandt [10] considers manipulability with respect to the lifting of Kelly and establishes, among other interesting results, that the social choice functions which are Condorcet extensions are always manipulable. Also, Brandt and Brill [11] consider the Gärdenfors and Fishburn liftings. They establish necessary and sufficient conditions for manipulability of non resolute rules where the notions of preference over sets of alternatives involved in manipulability are given by the lifting of Gärdenfors and the lifting of Fishburn.…”

On the Discriminative Power of Tournament Solutions

“…Since alternatives are generally assumed to be mutually exclusive, it is typically understood that ties will eventually be broken by some procedure that is independent of the agents' preferences. This can for example be achieved by using a lottery (thus achieving ex ante fairness), letting a chairman (with known or unknown preferences) pick his most-preferred of the remaining alternatives, or simply assuming that a single alternative will be chosen according to a procedure that is completely unknown to the agents (see, e.g., Gärdenfors, 1979;Brandt and Brill, 2011;Brandt, 2011a). The uncertainty the agents face when it comes to the final selection process can be used as a powerful tool to satisfy certain formal criteria (such as impartiality, consistency, or strategyproofness) that would otherwise be impossible to attain.…”

Finding Strategyproof Social Choice Functions via SAT Solving

“…Existing results suggest that the more demanding notion of Fishburn-strategyproofness may only be satisfied by rather indiscriminating SCFs such as the top cycle (Feldman, 1979;Brandt & Brill, 2011;Sanver & Zwicker, 2012). 2 Using our computer-aided proving technique, we are able to confirm this suspicion by proving that, within the domain of majoritarian SCFs, Fishburn-strategyproofness is incompatible with Pareto-optimality.…”

“…5. Another natural and well-known set extension by Gärdenfors leads to an even stronger notion of strategyproofness, which cannot be satisfied by any interesting majoritarian SCF (Brandt & Brill, 2011). Note are defined as follows: Let R i be a preference relation over A and X, Y ⊆ A two nonempty subsets of A.…”