Influence Vs. Utility in the Evaluation of Voting Rules: A New Look at the Penrose Formula

In this paper, we clarify the relationship between influence/power measurement and utility measurement, the most popular two social objective criteria used when evaluating voting mechanisms. For one particular probabilistic model describing the preferences of the electorate, the so-called Impartial Culture (IC) model used by Banzhaf, the Penrose formula show that the two objectives coincide. The IC probabilistic model assumes that voter preferences are independent. In this article, we prove a general version of the Penrose formula, allowing for correlations in the electorate. We show that in that case, the two social objectives no longer coincide, and qualitative conclusions can be very different.