Online (Budgeted) Social Choice

We consider classic social choice problems in an online setting. In the problems we consider, a decision-maker must select a subset of candidates in accordance to reported preferences, e.g. to maximize the value of a scoring rule. However, agent preferences cannot be accessed directly; rather, agents arrive one at a time to report their preferences, and each agent cares only about those candidates that have been selected by the time she arrives. On each step, the decision maker must choose whether to irrevocably add candidates to the final selection set given the preferences observed so far, with the goal of maximizing the average score over all agents. We show that when preferences are arbitrary but agents arrive in uniformly random order, an online selection algorithm can approximate the optimal value of an arbitrary positional scoring function to within a factor of (1−1/e)−o(1) as the number of agents grows large, nearly matching the performance of the best offline polynomialtime algorithm. When agent preferences are drawn from a Mallow’s model distribution, a different selection algorithm achieves approximation factor that limits to 1 as the number of agents grows large. Our methods are straightforward to implement, and draw upon connections to online computation and secretary problems.