Opinion Aggregation: Borda and Condorcet Revisited

In a world admitting a fixed finite set of alternatives, an opinion is an ordered pair of alternatives. Such a pair expresses the idea that one alternative is superior to another in some sense, and an opinion aggregator assigns an aggregate relation on the set of alternatives to every possible state of opinion. Our primary motivation is to extend the standard model of social choice theory to a more general one in which no specific reference to agents generating or holding opinions is needed. Although our analysis has some bearing on those cases where opinions reflect the goodness relations of agents in a society, it is not limited to them. In addition to that interpretation, opinions can also be used to represent other forms of comparative assessments emerging from different sources. The main results of the paper provide characterizations of two essential aggregation methods that remain well-defined in our larger context. These are the Borda rule and Condorcet's majority rule formulated in terms of opinion aggregation. Moreover, we show that these two opinion aggregators agree with the plurality rule and with the approval-voting rule when opinions are generated by ballots that are specific to these two well-established voting rules.