We introduce a new method to rank single elements given an order over their sets. For this purpose, we extend the game theoretic notion of marginal contribution and of Banzhaf index to our ordinal framework. Furthermore, we characterize the resulting ordinal Banzhaf solution by means of a set of properties inspired from those used to axiomatically characterize another solution from the literature: the ceteris paribus majority. Finally, we show that the computational procedure for these two social ranking solutions boils down to a weighted combination of comparisons over the same subsets of elements.

We study the problem of finding a social ranking over individuals given a ranking over coalitions formed by them. We investigate the use of a ceteris paribus majority principle as a social ranking solution inspired from the classical axioms of social choice theory. Faced with a Condorcet-like paradox, we analyze the consequences of restricting the domain according to an adapted version of single-peakedness. We conclude with a discussion on different interpretations of incompleteness of the ranking over coalitions and its exploitation for defining new social rankings, providing a new rule as an example.