Optimal bounds for the no-show paradox via SAT solving

Abstract

Voting rules allow multiple agents to aggregate their preferences in order to reach joint decisions. Perhaps one of the most important desirable properties in this context is Condorcet-consistency, which requires that a voting rule should return an alternative that is preferred to any other alternative by some majority of voters. Another desirable property is participation, which requires that no voter should be worse off by joining an electorate. A seminal result in social choice theory by Moulin [28] has shown that Condorcetconsistency and participation are incompatible whenever there are at least 4 alternatives and 25 voters. We leverage SAT solving to obtain an elegant human-readable proof of Moulin's result that requires only 12 voters. Moreover, the SAT solver is able to construct a Condorcet-consistent voting rule that satisfies participation as well as a number of other desirable properties for up to 11 voters, proving the optimality of the above bound. We also obtain tight results for set-valued and probabilistic voting rules, which complement and significantly improve existing theorems.

Finding Strategyproof Social Choice Functions via SAT Solving

“…Of particular interest could be such properties that link the behavior of SCFs for different domain sizes. As initial steps in this direction, we were able to extend the approach to cover the property of participation (Brandl et al, 2015;Brandt et al, 2016c) as well as a weak version of composition-consistency (cf. Section 4.1).…”

“…Results for different variants of the no-show paradox (Brandl, Brandt, Geist, & Hofbauer, 2015;Brandt, Geist, & Peters, 2016c) support this hypothesis. It should be noted, however, that-at least currently-an expert user or programmer is required to operate these systems.…”

Revisiting the Connection between the No-Show Paradox and Monotonicity

“…More precisely, Moulin [1988] proves that with four or more alternatives and with at least 25 voters, no Condorcet rule satisfies PART. Recently, Brandt et al [2016] proves that the minimal number of voters to obtain this incompatibility is exactly 12 using computational techniques 12 . As a consequence of these results, we can conclude that scoring rules satisfy PART in the fixed electorate setting as well or that the result of Moulin…”

Judgment aggregation in dynamic logic of propositional assignments

“…[22] later brought this approach to the level of theorem discovery in the field of ranking sets of objects, automatically testing all combinations of 20 axiomatic properties via SAT-solving, which combined with a general inductive lemma gave rise to 84 impossibility theorems (among which many non-trivial ones). In recent years, a number of open problems in classical social choice theory has then been tackled and solved [5][6][7] using a variety of techniques in automated reasoning from SAT-solvers, satisfiability modulo theory (SMT-solvers) and minimal unsatisfiable subset extraction.…”