Combinatorial Aggregation

Social Choice Theory (SCT), and the study of collective methods for decision making in general, have received a lot of attention in the AI community in recent years. The reasons for this focus are clear: SCT provides tools for the analysis of collective choices of groups of agents, and as such is of immediate relevance to the study of multiagent systems. At the same time, studies in AI have led to a new and broadened perspective on classical results in SCT, e.g., via the use of knowledge representation languages for modelling preferences in social choice problems or via the complexity-theoretic analysis of the implementation of social choice rules (Chevaleyre et al., 2007). Particularly close to the interests of AI is the problem of social choice in combinatorial domains (Chevaleyre et al., 2008), where the space of alternatives the individuals have to choose from has a combinatorial structure. Definition 1. A combinatorial domain D is a finite product D = D1 × · · · ×Dm where each Dj is finite. My work aims at carrying out a complete analysis of problems of combinatorial aggregation, with particular attention to the binary case. In this general framework a set of individuals each make a choice over a finite number of issues, and such choices have to be aggregated into a collective one. Given a combinatorial domain D and a set of individuals N an aggregation procedure is a function F : DN → D. I believe that such problems of combinatorial aggregation are crucial to the study of both voting in combinatorial domains and more classical frameworks in SCT, like preference and judgment aggregation. First, several situations of decision making are instances of combinatorial aggregation, such as referenda and voting for committees. Second, aggregation problems where individuals express complex ballots, such as preferences or judgments, can be modelled as combinatorial aggregation problems for suitably enriched domains. This suggests that combinatorial aggregation is a general framework for the study of situations of collective decision making. Third, on a more practical level this work makes it possible for mechanism designers to analyse situations of decision making using a general framework to point out possible inconsistencies and avoid paradoxical situations. I intend to approach the study of combinatorial aggregation in the following way. First, carry out a thorough analysis of aggregation in binary combinatorial domains, proving several results in this general setting and apply them to classical frameworks of SCT. Second, extend these results to more complex combinatorial domains. Finally, explore possible uses of combinatorial aggregation in sequential voting, and discuss theoretical generalisations to more complex logical languages and practical applications.