Strategyproof Randomized Social Choice for Restricted Sets of Utility Functions

Social decision schemes (SDSs) map the voters' preferences over multiple alternatives to a probability distribution over these alternatives. In a seminal result, Gibbard (1977) has characterized the set of SDSs that are strategyproof with respect to all utility functions and his result implies that all such SDSs are either unfair to the voters or alternatives, or they require a significant amount of randomization. To circumvent this negative result, we propose the notion of $U$-strategyproofness which postulates that only voters with a utility function in a predefined set $U$ cannot manipulate. We then analyze the tradeoff between $U$-strategyproofness and various decisiveness notions that restrict the amount of randomization of SDSs. In particular, we show that if the utility functions in the set $U$ value the best alternative much more than other alternatives, there are $U$-strategyproof SDSs that choose an alternative with probability $1$ whenever all but $k$ voters rank it first. On the negative side, we demonstrate that $U$-strategyproofness is incompatible with Condorcet-consistency if the set $U$ satisfies minimal symmetry conditions. Finally, we show that no ex post efficient and $U$-strategyproof SDS can be significantly more decisive than the uniform random dictatorship if the voters are close to indifferent between their two favorite alternatives.