In , Guha gave a complete characterization of path independent social decision functions which satisfy the independence of irrelevant alternatives condition, the strong Pareto principle, and UII, i.e., unanimous indiﬀerence implies social indiﬀerence. These conditions necessarily imply that a path independent social decision function is neutral and monotonic. In this paper, we extend Guha’s characterization to the class of neutral monotonic social functions. We show that neutral monotonic social functions and their specializations to social decision functions, path independent social decision functions, and social welfare functions can be uniquely represented as a collection of overlapping simple games, each of which is deﬁned on a nonempty set of concerned individuals. Moreover, each simple game satisﬁes intersection conditions depending on the number of social alternatives; the number of individuals belonging to the concerned set under consideration; and the collective rationality assumption. We also provide a characterization of neutral, monotonic and anonymous social decision functions, where the number of individuals in society exceeds the (ﬁnite) number of social alternatives, that generalizes both the representation theorem of May  and the representation theorems of Ferejohn and Grether .
Blau, Julian H. and Brown, Donald J., "The Structure of Neutral Monotonic Social Functions" (1978). Cowles Foundation Discussion Papers. 718.