## Fair Proportional Representation Problems with Mixture Operators

Hugo GilbertADT 2017

Abstract : This paper deals with proportional representation problems in which a set of winning candidates must be selected according to the ballots of the voters. We investigate the use of a new class of optimization criteria to determine the set of winning candidates, namely mixture operators. In a nutshell, mixture operators are similar to weighted means where the numerical weights are replaced by weighting functions. In this paper: (1) we give the mathematical condition for which a mixture operator is fair and provide several instances of this operator satisfying this condition; (2) we show that when using a mixture operator as optimization criterion, one recovers the same complexity results as in the utilitarian case (i.e., maximizing the sum of agentâ€™s utilities) under a light condition; (3) we present solution methods to find an optimal set of winners w.r.t. a mixture operator under both Monroe and Chamberlin-Courant multi-winner voting rules and test their computational efficiency.