Rolling the Dice: Recent Results in Probabilistic Social Choice

When aggregating the preferences of multiple agents into one collective choice, it is easily seen that certain cases call for randomization or other means of tiebreaking. For example, if there are two alternatives, a and b, and two agents such that one prefers a and the other one b, there is no deterministic way of selecting a single alternative without violating one of two basic fairness conditions known as anonymity and neutrality. Anonymity requires that the collective choice ought to be independent of the agents’ identities whereas neutrality requires impartiality towards the alternatives.1 Allowing lotteries as social outcomes hence seems like a necessity for impartial collective choice. Indeed, most common “deterministic” social choice functions such as plurality rule, Borda’s rule, or Copeland’s rule are only deterministic as long as there is no tie, which is usually resolved by drawing a lot. The use of lotteries for the selection of officials interestingly goes back to the world’s first democracy in Athens, where it was widely regarded as a principal characteristic of democracy (Headlam, 1933), and has recently gained increasing attention in political science (see, e.g., Goodwin, 2005; Dowlen, 2009; Stone, 2011; Guerrero, 2014). It turns out that randomization—apart from guaranteeing impartiality—allows the circumvention of well-known impossibility results such as the GibbardSatterthwaite Theorem. Important questions in this context are how much “randomness” is required to achieve positive results and which assumptions are made about the agents’ preferences over lotteries. In this chapter, I will survey some recent axiomatic results in the area of probabilistic social choice. Probabilistic social choice functions (PSCFs) map collections of individual preference relations over alternatives to lotteries over alternatives and were first for-