Finding a Collective Set of Items: From Proportional Multirepresentation to Group Recommendation

We consider the following problem: There is a set of items (e.g., movies) and a group of agents (e.g., passengers on a plane); each agent has some intrinsic utility for each of the items. Our goal is to pick a set of $K$ items that maximize the total derived utility of all the agents (i.e., in our example we are to pick $K$ movies that we put on the plane's entertainment system). However, the actual utility that an agent derives from a given item is only a fraction of its intrinsic one, and this fraction depends on how the agent ranks the item among available ones (in the movie example, the perceived value of a movie depends on the values of the other ones available). Extreme examples of our model include the setting where each agent derives utility from his or her most preferred item only (e.g., an agent will watch his or her favorite movie only), from his or her least preferred item only (e.g., the agent worries that he or she will be somehow forced to watch the worst available movie), or derives $1/K$ of the utility from each of the available items (e.g., the agent will pick a movie at random). Formally, to model this process of adjusting the derived utility, we use the mechanism of ordered weighted average (OWA) operators. Our contribution is twofold: First, we provide a formal specification of the model and provide concrete examples and settings where particular OWA operators are applicable. Second, we show that, in general, our problem is NP-hard but---depending on the OWA operator and the nature of agents' utilities---there exist good, efficient approximation algorithms (sometimes even polynomial time approximation schemes). Interestingly, our results generalize and build upon those for proportional represented in multiwinner voting scenarios.