Fair Multiwinner Elections with Allocation Constraints
We consider the classical multiwinner election problem where the goal is to choose a subset of k unit-sized candidates (called committee) given utility functions of the voters. We allow arbitrary additional constraints on the chosen committee, and the utilities of voters to belong to a very general class of set functions called β-self bounding. When β = 1, this class includes XOS (and hence, submodular and additive) utilities as special cases. We define a novel generalization of core stability called restrained core to handle constraints on the committee, and consider multiplicative approximations on the utility under this notion.Our main result is the following: If a smooth version of Nash Welfare is globally optimized over committees that respect the constraints, then the resulting optimal committee lies in the eβ-approximate restrained core for β-self bounding utilities and arbitrary constraints. As a result we obtain the first constant approximation for stability with arbitrary additional constraints even for additive utilities (factor of e), as well as the first analysis of the stability of Nash Welfare with XOS functions even in the absence of constraints.We complement this positive result by showing that the c-approximate restrained core can be empty for c < 16/15 even for additive utilities and one additional constraint. Furthermore, the exponential dependence on β in the approximation is unavoidable for β-self bounding functions even in the absence of any constraints.We next present improved and tight approximation results for multiwinner elections with simpler classes of utility functions and simpler types of constraints. We also present an extension of restrained core to extended justified representation with constraints, and show an existence result for the special case of matroid constraints. We finally generalize our results to the setting when candidates have arbitrary sizes (Participatory Budgeting) and there are no additional constraints. Our proof techniques are different from previous analyses of Nash Welfare and are of independent interest.
