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.

Social Combination Models

This chapter studies the historical development of social combination models. The social combination approach assumes that groups combine the group member preferences by some process to formulate a single collective group response. A social decision scheme formalizes any assumption about the group process that assigns probabilities of each group response given each distribution of member preferences. The assumptions may come from the constitutions or bylaws of a group, from previous research, or any other hypothesized group process. Different social decision schemes or social combination models may then be tested competitively against actual group performance as a test of the assumptions formalized by the social decision schemes. Stasser gives an excellent overall presentation of social decision scheme theory, including model formation, model testing, and using the equations for prospective modeling.