The logic of collective choice

This paper presents a modal logic for modelling individual and collective choices over a set of feasible alternatives. The logic extends propositional logic with a binary modality so that a formula can express not only properties of alternatives but also priorities of individuals over the properties. More importantly, each formula of this logic determines a preference ordering over alternatives based on the priorities over properties that the formula expresses. In such a way, preferences of multiple agents can be represented by a set of formulas in the same logic. This allows us to treat the problem of collective choice in a multi-agent system as aggregation of logical formulas. We further use this language to express a few plausible collective choice rules. Similar to preference aggregation, we specify collective choice rules by Arrow's conditions. Interestingly, all Arrowian conditions are plausible under the new setting except Independence of Irrelevant Alternatives. This gives us a natural way to avoid Arrow's impossibility result. Finally, we develop a model checking algorithm to automatically generate individual and collective choices in the logic.