The T-power of a majority under multi-winner scoring rules

In multi-winner elections, the goal is to select a pre-given size committee of members from a given set of candidates. We define the concept of T-power of a majority-group of voters (Total majority-power or Total tyranny-power) as the ability of the group to guarantee, under a given voting rule, that all the committee members belong to its favorite candidates, regardless of the preferences of the other voters. In this case, we say also that such a voting rule is exposed to T-power. This situation can be considered extreme and worrying in democracy. In this paper, we examine the T-power under classes of multi-winner scoring rules. We focus on some common rules, and it turns out that BLOC rule (k-k-Approval) is totally exposed to T-power, that is, any majority-group of any size has T-power under this rule. On the other hand, CCR, k-Plurality and k-Inverse-Plurality rules are almost notexposed to it, that is, no majority of any size has T-power. These last rules may providesome protection for minoritygroups in the sense of preventing T-power.