Voting with Ties: Strong Impossibilities via SAT Solving

Voting rules allow groups of agents to aggregate their preferences in order to reach joint decisions. The Gibbard-Satterthwaite theorem, a seminal result in social choice theory, implies that, when agents have strict preferences, all anonymous, Pareto-optimal, and single-valued voting rules can be strategically manipulated. In this paper, we consider multi-agent voting when there can be ties in the preferences as well as in the outcomes. These assumptions are extremely natural—especially when there are large numbers of alternatives—and enable us to prove much stronger results than in the overly restrictive setting of strict preferences. In particular, we show that (i) all anonymous Pareto-optimal rules where ties are broken according to the preferences of a chairman or by means of even-chance lotteries are manipulable, and that (ii) all pairwise Pareto-optimal rules are manipulable, no matter how ties are broken. These results are proved by reducing the statements to finite—yet very large—problems, which are encoded as formulas in propositional logic and then shown to be unsatisfiable by a SAT solver. We also extracted human-readable proofs from minimal unsatisfiable cores of the formulas in question, which were in turn verified by an interactive higher-order theorem prover.