Strategyproof Randomized Social Choice for Restricted Sets of Utility Functions

When aggregating preferences of multiple agents, strategyproofness is a fundamental requirement. For randomized voting rules, so-called social decision schemes (SDSs), strategyproofness is usually formalized with the help of utility functions. A classic result shown by Gibbard in 1977 characterizes the set of SDSs that are strategyproof with respect to all utility functions and shows that these SDSs are either indecisive or unfair. For finding more insights into the trade-off between strategyproofness and decisiveness, we propose the notion of U-strategyproofness which requires that only voters with a utility function in the set U cannot manipulate. In particular, we show that if the utility functions in 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. We also prove for rank-based SDSs that this large gap in the utilities is required to be strategyproof and that the gap must increase in k. On the negative side, we show that U-strategyproofness is incompatible with Condorcet-consistency if U satisfies minimal symmetry conditions and there are at least four alternatives. For three alternatives, the Condorcet rule can be characterized based on U-strategyproofness for the set U containing all equi-distant utility functions.