The Impossibility of Extending Random Dictatorship to Weak Preferences

One of the most celebrated results in microeconomic theory is the Gibbard-Satterthwaite theorem (Gibbard, 1973; Satterthwaite, 1975), which states that every strategyproof and Pareto-optimal social choice function is a dictatorship. However, the theorem crucially relies on the assumption that outcomes are deterministic. Gibbard (1977) later considered social decision schemes, i.e., social choice functions that return lotteries over the alternatives, and showed that the class of strategyproof and ex post efficient functions extends to all random dictatorships. This class contains a unique rule that treats all agents equally: the uniform random dictatorship, henceforth random dictatorship (RD), where an agent is chosen uniformly at random and his favorite alternative is implemented as the social choice. Gibbard’s notion of strategyproofness is based on stochastic dominance and prescribes that no voter can obtain more utility by misrepresenting his preferences no matter what his utility function is (as long as it is consistent with his ordinal preferences). Another implicit assumption in Gibbard’s theorem is the anti-symmetry of individual preferences.