Basic Geometry of Voting

What could be easier than interpreting an election? You just count to determine which candidate receives the most votes. What can be difficult about something so elementary? After all, even nursery school children know how to vote to select their juice of choice before nap time.

It is trivial to tally a plurality election; just count how many voters have each candidate top-ranked. Surprisingly, when this elementary description is used to analyze the procedure, it quickly introduces mathematical complications that severely limit what can be learned.

The beverage brouhaha, which initiated the season of dissent for the hypothetical department of the fable, started with the winer’s discovery that the department’s plurality ranking conflicts with their rankings of pairs of beverages. The radical disagreement raises interesting theoretical questions. How does a majority vote ranking of a pair relate to its relative ranking within a plurality outcome? Can anything go wrong with pairwise rankings?

As we learned in Chap. 3, problems and paradoxes must be anticipated with procedures based on pairwise rankings. The subtle reason is that these methods manage to drop the central assumption of transitivity. This suggests using methods, such as the plurality vote and the BC, which admit only rational voters. Do they have problems? Rest assured; there are plenty of them! But, with a further decomposition of profiles, it becomes easy to analyze the problems and create profile examples.

In addition to the single profile consequences, there are fascinating voting properties which require several profiles. A natural illustration is the Dean’s Council controversy (in the fable) caused by combining the profiles for two subcommittees. Multiprofile issues are important because they help us understand what can happen if a voter votes strategically, or if he doesn’t vote, or whether there are problems should new voters arrive, or when groups combine to form coalitions, or if voters change preferences, or … In Sect. 5.1, problems such as the Dean’s council are explained. In Sect. 5.2, attention turns to how “more can be less.” In Sect. 5.3, the emphasis is on strategic voting. While all of these conclusions depend upon the geometry of profile sets, whenever possible, the simpler geometry of the representation triangle or cube is exploited. Then, to conclude, Sects. 5.4, 5.5 address the intriguing problems of the “list” methods and apportionment of Congressional seats.