Geometry, Voting, and Paradoxes

Surprisingly subtle, unexpected election behaviors can arise when voters are restricted to only three kinds of preferences. of particular interest is that the questions raised in Section 1 about potential paradoxical election behavior can be answered by using elementary geometric arguments. As shown, conflict between pairwise and positional methods occurs in abundance and, when it occurs, it is supported by an open set of profiles. (This answers the robustness question.) Problems about the likelihood of strange behavior, or finding supporting profiles with the minimum number of voters, reduce to elementary arguments. Moreover, the geometry allows us to "see" where conflict occurs and to determine whether paradoxical outcomes are, or are not, isolated. For instance, FIGURE 6 identifies the profiles where each candidate wins with an appropriate wA method. So, when preferences are restricted as indicated, we must expect such pathological behavior in about 1 in 40 elections (with a sufficient number of voters). As shown by FIGURE 7, other settings increase the likelihood of this behavior to about 3 in 20 elections. Although we emphasized those election surprises that occur when voters' preferences come from only three possible types, other surprises already occur when preferences are restricted to only two types. Indeed, this is a special case of our analysis because it just requires setting one of x, y, or Z equal to zero; it is the behavior on one of the edges of the triangles T1, T2 or T3. For instance, by considering the vertical leg (where x = 0) of the triangles in Figure 5, we discover how this highly restrictive case allows two strict pairwise rankings to be accompanied with conflicting wλ outcomes. Without question, elections admit surprising behavior.