Social Welfare Functions with Voters Qualifications: Impossibility Results

We consider the social welfare function a la Arrow, where some voters are not qualified to evaluate some alternatives. Thus, the inputs of the social welfare function are the preferences of voters on the alternatives that they are qualified to evaluate only. Our model is a generalization of the peer rating model, where each voter evaluates the other voters (except for himself/herself). We demonstrate the following three impossibility results. First, if a transitive valued social welfare function satisfies independence of irrelevant alternatives and the Pareto principle, then a dictator who is qualified to evaluate all alternatives exists. Second, a transitive valued function satisfying the Pareto principle exists if and only if at least one voter is qualified to evaluate all alternatives. Finally, if no voter is qualified to evaluate all alternatives, then under a transitive valued social welfare function satisfying the weak Pareto principle and independence of irrelevant alternatives, all alternatives are indifferent for any preference profile of voters.