Strategic Abstention based on Preference Extensions: Positive Results and Computer-Generated Impossibilities

Abstract

Voting rules allow multiple agents to aggregate their preferences in order to reach joint decisions. A common flaw of some voting rules, known as the no-show paradox, is that agents may obtain a more preferred outcome by abstaining from an election. We study strategic abstention for set-valued voting rules based on Kelly's and Fishburn's preference extensions. Our contribution is twofold. First, we show that, whenever there are at least five alternatives and seven agents, every Pareto-optimal majoritarian voting rule suffers from the no-show paradox with respect to Fishburn's extension. This is achieved by reducing the statement to a finite - yet very large - problem, which is encoded as a formula in propositional logic and then shown to be unsatisfiable by a SAT solver. We also provide a human-readable proof which we extracted from a minimal unsatisfiable core of the formula. Secondly, we prove that every voting rule that satisfies two natural conditions cannot be manipulated by strategic abstention with respect to Kelly's extension and give examples of well-known Pareto-optimal majoritarian voting rules that meet these requirements.

Optimal bounds for the no-show paradox via SAT solving

“…When assuming that voters have incomplete preferences over sets or lotteries, participation and Condorcetconsistency can be satisfies simultaneously and positive results for common Condorcet-consistent voting rules have been obtained by Brandt [7] and Brandl et al [4,5,6]. Abstention in slightly different contexts than the one studied in this paper recently caught the attention of computer scientists working on voting equilibria and campaigning [16,1].…”

“…To achieve this, we encode these problems as formulas in propositional logic and then use SAT solvers to decide their satisfiability and extract minimal unsatisfiable sets (MUSes) in the case of unsatisfiability. This approach is based on previous work by Tang and Lin [35], Geist and Endriss [20], Brandt and Geist [8], and Brandl et al [4]. However, it turned out that a straightforward application of this methodology is insufficient to deal with the magnitude of the problems we considered.…”

Computational Social Choice: The First Ten Years and Beyond

“…Due to its rigorous axiomatic foundation and its emphasis on impossibility results, social choice theory is particularly well-suited for computer-aided theorem proving techniques. Apart from work that is directed towards formalizing and verifying existing results (see, e.g., [83,101]), a number of recent papers have proved new theorems with the help of computers [30,32,37,41,43]. This branch of research was initiated by Tang and Lin [128], who reduced well-known impossibility results such as Arrow's theorem to finite instances, which could then be checked by SAT solvers.…”

“…Fortunately, when relying on SAT solving, this criticism can be addressed by extracting a human-readable proof from an inclusion-minimal unsatisfiable set of clauses returned by the SAT solver. This approach, pioneered by Brandt and Geist [37], has been successfully applied in several recent papers [30,32,41,43].…”

On the Indecisiveness of Kelly-Strategyproof Social Choice Functions

“…2 In the context of strategic abstention (i.e., manipulation by deliberately abstaining from an election), even more positive results can be obtained. Brandl et al [9] have shown that all of the above mentioned SCFs that are strategyproof for strict preferences are immune to strategic abstention even when preferences are weak.…”