Voting Power, Hierarchical Pivotal Sets, and Random Dictatorships

In many traditional social choice problems, analyzing the voting power of the voters in a given profile is an important part. Usually the voting power of an agent is measured by whether the agent is pivotal. In this paper, we introduce two extensions of the set of pivotal agents to measure agents’ voting power in a given profile. The first, which is called hierarchical pivotal sets, captures the voting power for an agent to make other agents pivotal. The second, which is called coalitional pivotal sets, is based on the fact that each agent is given a weight that is computed similarly to the Shapley-Shubik power index. We also introduce random dictatorships induced by the two types of pivotal sets to approximate full random dictatorships. We show that the random dictatorships induced by the hierarchical pivotal sets are strategic-pivot-proof, that is, no agent can make herself become one of the possible dictators by voting differently. We then focus on the hierarchical pivotal sets when the hierarchical level goes to infinity. We prove that for any voting rule that satisfies anonymity and unanimity, and for any given profile, the union of the hierarchical pivotal sets are a sound and complete characterization of the non-redundant agents. We also show that if the voting rule does not satisfy anonymity, then this characterization might not be complete. Finally, we investigate algorithmic aspects of computing the hierarchical pivotal sets.