Abstract

Abstract We consider a classic social choice problem in an online setting. In each round, a decision maker observes a single agent's preferences over a set of m candidates, and must choose whether to irrevocably add a candidate to a selection set of limited cardinality k. Each agent's (positional) score depends on the candidates in the set when he arrives, and the decisionmaker's goal is to maximize average (over all agents) score. We prove that no algorithm (even randomized) can achieve an approximation factor better than O( log log m log m ). In contrast, if the agents arrive in random order, we present a (1− 1 e −o(1))-approximate algorithm, matching a lower bound for the offline problem. We show that improved performance is possible for natural input distributions or scoring rules. Finally, if the algorithm is permitted to revoke decisions at a fixed cost, we apply regret-minimization techniques to achieve approximation 1 − 1 e − o(1) even for arbitrary inputs.