Quota rules for incomplete judgments

Suppose that a group of individuals are asked to aggregate their judgments on different—possibly logically interconnected—propositions in order to reach a collective decision. Quota rules are natural aggregation rules requiring that a proposition be collectively accepted if and only if the number of individuals that agree with it exceeds a given threshold. In cases where the individuals may also abstain on some of the issues at stake and report incomplete judgments, there are several ways for determining the relevant threshold, depending on the number of abstentions or the margin between those that agree and those that disagree with a given proposition. In this paper I systematically design quota rules for incomplete inputs, within the framework of judgment aggregation, and explore their formal properties. In particular, I characterise axiomatically three distinct classes of quota rules, extending known results of the literature that so far only applied to complete inputs.