Proving the Incompatibility of Efficiency and Strategyproofness via SMT Solving

Two important requirements when aggregating the preferences of multiple agents are that the outcome should be economically efficient and the aggregation mechanism should not be manipulable. In this article, we provide a computer-aided proof of a sweeping impossibility using these two conditions for randomized aggregation mechanisms. More precisely, we show that every efficient aggregation mechanism can be manipulated for all expected utility representations of the agents’ preferences. This settles an open problem and strengthens several existing theorems, including statements that were shown within the special domain of assignment. Our proof is obtained by formulating the claim as a satisfiability problem over predicates from real-valued arithmetic, which is then checked using a satisfiability modulo theories (SMT) solver. To verify the correctness of the result, a minimal unsatisfiable set of constraints returned by the SMT solver was translated back into a proof in higher-order logic, which was automatically verified by an interactive theorem prover. To the best of our knowledge, this is the first application of SMT solvers in computational social choice.