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 paper, 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 a number of 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 an SMT (satisfiability modulo theories) solver. In order 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.

Efficiency--no agent can be made better off without making another one worse off--and strategyproofness--no agent can obtain a more preferred outcome by misrepresenting his preferences--are two cornerstones of economics and ubiquitous in important areas such as voting, auctions, or matching markets. Within the context of random assignment, Bogomolnaia and Moulin have shown that two particular notions of efficiency and strategyproofness based on stochastic dominance are incompatible. However, there are various other possibilities of lifting preferences over alternatives to preferences over lotteries apart from stochastic dominance. In this paper, we give an overview of common preference extensions, propose two new ones, and show that the above-mentioned incompatibility can be extended to various other notions of strategyproofness and efficiency in randomized social choice.