Political Power on a Line Graph

We consider situations of majority voting, where the players are ordered linearly. This order may be based on, for example, ideology or political preferences over economic policy, ethical principles, environmental issues, and so on. Winning and losing coalitions are given by a majority voting game, while restrictions on cooperation are determined by a line graph, where only connected coalitions are feasible and can form a (winning) coalition. Various solutions for line-graph games can then be viewed as power indices measuring the ability of political parties to turn losing coalitions into winning ones, taking into account the cooperation restrictions among the parties. Here, we start by observing that a number of existing power indices either are not core stable, or do not reward intermediate veto players. Then, we take a closer look at the average hierarchical outcome, called hierarchical index in the context of this paper, and the $$\tau $$ -index. These indices are core stable and, moreover, reward all veto players. Specifically, the $$\tau $$ -index rewards all veto players equally, while the hierarchical index always assigns higher power to the two extreme veto players than to intermediate veto players. We axiomatically characterize the (i) hierarchical index by core stability and a weaker version of component fairness and (ii) the $$\tau $$ -index by core stability and a weaker version of Myerson’s [Math Oper Res 2(3), 225–229 (1977)] fairness property.