Voting with CP-nets using a Probabilistic Preference Structure Cristina Cornelio, Umberto Grandi, Judy Goldsmith, Nicholas Mattei, Francesca Rossi and K. Brent Venable Abstract Probabilistic conditional preference networks (PCP-nets) provide a compact repre- sentation of a probability distribution over a collection of CP-nets. In this paper we view a PCP-net as the result of aggregating a collection of CP-nets into a single structure. We use the resulting PCP-net to perform collective reasoning tasks, e.g. determining the most preferred alternative, when a group of agents expresses their preferences via CP-nets. We propose several PCP-net based methods to perform CP- net aggregation and evaluate these methods both axiomatically and experimentally. 1 Introduction The study of preferences plays an important role in the field of artificial intelligence [19] and machine learning [7]. The ability to express preferences in a faithful way, which can be handled efficiently, is essential in many reasoning tasks. In settings such as e-commerce, on demand video, and other settings where supply outstrips an individuals ability to view all the available choices, we require an efficient formalism to model and reason with complex, interdependent preferences [8]. We may also use these preferences to make decisions about joint plans, actions, or items in multi-agent environments [18]. Agents express their prefer- ences over a set of alternative decisions, these preferences are aggregated into one decision which satisfies as many agents as possible. Often multi-attribute preference modeling and reasoning causes a combinatorial explo- sion, often leading to high computational cost [5, 6, 9]. The set of alternatives is often described as a product of multiple features, for example, a user’s preferences over a set of cars, which can be described by their colors, technical specifications, cost, reliability, etc. A number of compact representation languages have been developed to tackle the com- putational challenges arising from these problems. Among others, we mention conditional preference structures (CP-nets) [3], soft constraints [2, 19], and GAI-nets [10]. In this paper we focus on CP-nets as the tool for modeling the preferences of a single agent. CP-nets are a qualitative preference modeling framework that allow for conditional preference statements. Preferences are often uncertain: we may be unsure about our preference ordering over certain items, or there could be noise in our preference structure due to lack of precision in elicitation or sensor collection. We may also need to represent preferences that are di- rectly conflicting, such as disagreement in multi-agent systems or voting settings [14, 15, 20]. PCP-nets model uncertain preferences natively while using the same preferential dependency structure used with CP-nets [1, 4]. However, in a PCP-net, a preference ordering over a vari- able’s domain is replaced by a probability distribution over all possible preference orderings of the variable’s domain. Thus, a PCP-net defines a probability distribution over a collection of CP-nets: all those CP-nets that can be obtained from the PCP-net by choosing a vari- able ordering from the distribution over all orderings. Given a PCP-net, one can define the optimal variable assignment in two natural ways: as the most probable optimal outcome, or as the optimal outcome of the most probable CP-net induced by the PCP-net. If the