More Yes–No Voting

In this chapter, as in Chapter 2, our primary interest is in yes-no voting systems that are not weighted. We begin by returning to the theorem in Chapter 2 that characterized the weighted voting systems as precisely those that are trade robust (meaning that an arbitrary trade among several winning coalitions can never simultaneously render all of them losing). A natural question suggested by this result is whether trade robustness really needs to be stated in terms of “several winning coalitions.” That is, perhaps a yes-no voting system is weighted if and only if a (not necessarily one-for-one) trade between two winning coalitions can never simultaneously render both losing. Recall that in showing that the procedure to amend the Canadian constitution is not trade robust we needed only two winning coalitions.

In Section 8.2, we show that life, or at least mathematics, is not quite that simple. There, we present a yes–no voting system that is not trade robust, but which has the property that an arbitrary trade between two winning coalitions always leaves at least one of them winning. Although not without its charms (we use a so-called magic square to construct the system), this is not a real-world example and, indeed, we know of no such real-world example.