In this chapter we make a few remarks on the relationship between the existing theory and our work, and on possible generalizations of our results. First, in Section 7.1, we observe that all known restrictions on individual preferences guarantee that cores of voting games are nonempty. Thus, we may have a richer theory of representation (of committees) under the (usual) assumptions of restricted preferences. Next, we remark that restriction of preferences does not eliminate manipulability. Then we argue that there is a need to generalize the notion of exact and strong consistency to allow restricted preferences, and we notice that Dutta [1980b] contains such a generalization (see Section 7.2).

In Section 7.3 we discuss the problem of the existence of faithful and neutral representations of committees. We show how that problem can be resolved by the use of even-chance lotteries on alternatives. Section 7.4 is devoted to a systematic generalization of our results to weak orders. Finally, we discuss possible extensions of our results to infinite sets of alternatives (see Section 7.5).