The Privacy-Welfare Trade-off: Effects of Differential Privacy on Influence & Welfare in Social Choice

This work studies a fundamental trade-off between privacy and welfare in aggregation of privatized preferences of individuals. It presents the precise rate at which welfare decreases with increasing levels of privacy. Trade-offs in achieving privacy while maintaining accuracy or more recently in maintaining fairness have been studied so far in prior works. Social choice functions help aggregate individual preferences while differentially private mechanisms provide formal privacy guarantees to release answers of queries operating on data. However, once differential privacy inducing noise is introduced into a voting system, the deterministic social choice function used to release an aggregated choice may not be ideal anymore. It might change the power balances between the voters. It could be the case that an alternate social choice function becomes more ideal to aggregate the preferences. There could be a constraint to operate the voting system at a specific level of either privacy or influence or welfare, and one would like to know the effects on rest of the unconstrained choices. In this paper, we introduce notions of welfare and probabilistic influence of privatization mechanisms to help precisely answer such questions through the proposed results. Throughout the paper, we restrict our work to social choice functions where several voters vote for either of two candidates. We present two different ways of proving each of our results that connects privacy with welfare & probabilistic influence: i.) by using first principles from combinatorics/probability, ii.) by using Fourier analysis of Boolean functions. Finally, we analyze the accuracy of the private mechanism on various social choice functions. The results in this paper thereby help bridge two prominent fields of social choice theory and differential privacy.