Judgement aggregation by quota rules: majority voting generalized

The widely discussed `discursive dilemma' shows that majority voting in a group of individuals on logically connected propositions may produce irrational collective judgments. We generalize majority voting by considering quota rules, which accept each proposition if and only if the number of individuals accepting it exceeds a given threshold, where different thresholds may be used for different propositions. After characterizing quota rules, we prove necessary and sufficient conditions on the required thresholds for various collective rationality requirements. We also consider sequential quota rules, which ensure collective rationality by adjudicating propositions sequentially and letting earlier judgments constrain later ones. Sequential rules may be path dependent and strategically manipulable. We characterize path independence and prove its essential equivalence to strategy proofness. Our results shed light on the rationality of simple-, super-, and sub-majoritarian decision making.