Extensions of the Simpson voting rule to the committee selection setting

Committee selection rules are procedures selecting sets of candidates (committees) of a given size on the basis of the preferences of the voters. Two natural extensions of the well-known single-winner Simpson voting rule to the multiwinner setting have been identified in the literature. We propose an in-depth analysis of those committee selection rules, assessing and comparing them with respect to several desirable properties, among which are unanimity, fixed majority, non-imposition, stability, local stability, Condorcet consistency, some kinds of monotonicity, resolvability and consensus committee. We also investigate the probability that the two methods are resolute and suffer the reversal bias, the Condorcet loser paradox and the leaving member paradox. We compare the results obtained with the ones related to further well-known committee selection rules. The probability assumption on which our results are based is the widely used Impartial Anonymous Culture.