Incentives in Social Decision Schemes with Pairwise Comparison Preferences

Social decision schemes (SDSs) map the ordinal preferences of voters over multiple alternatives to a probability distribution over the alternatives. To study the axiomatic properties of SDSs, we lift preferences over alternatives to preferences over lotteries using the natural---but little understood---pairwise comparison (PC) preference extension. This extension postulates that one lottery is preferred to another if the former is more likely to return a preferred outcome. We settle three open questions raised by Brandt (2017) and show that (i) no Condorcet-consistent SDS satisfies PC-strategyproofness; (ii) no anonymous and neutral SDS satisfies both PC-efficiency and PC-strategyproofness; and (iii) no anonymous and neutral SDS satisfies both PC-efficiency and strict PC-participation. We furthermore settle an open problem raised by Aziz et al. (2015) by showing that no path of PC-improvements originating from an inefficient lottery may lead to a PC-efficient lottery