We study a model of proportional representation, in which the policy space is multidimensional. We first show, via an example, that the assumption of quasi-concavity of the utility function is not sufficient to obtain the result that only the extreme parties get votes, contrary to the unidimensional case. We, then, study two cases in which stronger assumptions on voters' preferences assure that voters essentially vote, in any equilibrium, only for the extreme parties.
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