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This is Chapter 25 of a forthcoming book edited by K. Arrow, A. Sen, and K. Suzumura. It is intended only for participants of my 2007 MAA short course on the “mathematics of voting” in New Orleans. Chapter 25 GEOMETRY OF VOTING1 Donald G. Saari Director, Institute for Mathematical Behavioral Sciences Departments of Mathematics and Economics University of California, Irvine Irvine, California 92698 1. Introduction 2. Simple Geometric Representations 2.1 A geometric profile representation 2.2 Elementary geometry; surprising results 2.3 The source of pairwise voting problems 2.4 Finding other pairwise results 3. Geometry of Axioms 3.1 Arrow’s Theorem 3.2 Cyclic voters in Arrow’s framework? 3.3 Monotonicity, strategic behavior, etc. 4. Plotting all election outcomes 4.1 Finding all positional and AV outcomes 4.2 The converse; finding election relationships 5. Finding symmetries – and profile decompositions 6. Summary Abstract. I show how to use simple geometry to analyze pairwise and posi- tional voting rules as well as those many other decision procedures, such as runoffs and Approval Voting, that rely on these methods. The value of us- ing geometry is introduced with three approaches, which depict the profiles along with the election outcomes, that help us find new voting paradoxes, compute the likelihood of disagreement among various election outcomes, and explain problems such as the “paradox of voting.” This geometry even extends McGarvey’s theorem about possible pairwise election rankings to 1This research was supported by NSF grant DMI-0233798. My thanks to K. Arrow, N. Baigent, H. Nurmi, T. Ratliff, M. Salles, and K. Sieberg among others for comments and corrections on an earlier draft. 1 indicate all possible pairwise tallies. After using geometry to provide a be- nign interpretation for Arrow’s Theorem, an intuitive argument is described to analyze a variety of seemingly disparate topics such as strategic behav- ior, monotonicity, and the “no-show” paradox. Another geometric approach identifies all possible positional and Approval Voting election outcomes ad- mitted by a given profile: the converse becomes a geometric tool that identi- fies new election relationships. Finally, a geometric “profile decomposition” is described with which we can identify and explain all possible differences in positional and pairwise voting outcomes and generate illustrating profiles for any possible paradox. 1 Introduction “Geometry of voting” is intended to capture the sense that “a picture is worth a thousand words.” After all, geometry has long served as a powerful tool that provides a global perspective of whatever we happen to be studying while exposing unexpected relationships. This is why we graph functions, plot data, study the Edgeworth box from economics, and use diagrams to enhance lectures. Similarly, the geometry of voting seeks to create appro- priate geometric tools to capture global aspects about decision and voting rules while exposing new relationships. Since most, if not all voting rules in wide use involve pairwise or positional methods, or are based on them (such as runoffs), I emphasize these methods. Why is social choice so complex? In part, it is due to the “curse of dimensionality;” e.g., this is why standard geometric tools fail the challenges offered by social choice. After all, the large dimensions of a profile space alone make it impossible to graph relationships between profiles and their election outcomes. This is displayed already with three alternatives where the 3! = 6 dimensions of profile space overwhelm any hope to use standard graphs to connect profiles with election outcomes. (In this chapter, a profile is used in the traditional manner of specifying the number of voters whose preferences are given by each ranking.) Because standard approaches will not work, we must develop new geometric tools that will offer help. In the next section, for instance, three methods are described that geometrically depict profiles along with their associated procedural outcomes. Section 3 shows how to use geometry to analyze axiomatic issues ranging from Arrow’s Theorem to concerns about strategic behavior, monotonicity, and the “no-show” paradox (where a voter does better by not voting). Ge- ometry even demonstrates why many seemingly dissimilar concerns admit a 2 strikingly similar analysis. A different theme is motivated by the temptation—one that we all expe- rience whenever we encounter a “nail-biting” close election involving three or more candidates—to explore whether the outcome would have changed had a different election rule been used. (For instance, had a different elec- tion rule been used in the 2000 US presidential election, could Gore have beaten Bush?) As published results typically consider only the better known methods, we must wonder what would have happened had any of the infinite number of other rules been used. The geometric approach in Sect. 4 resolves this problem by showing how to depict all possible positional and Approval Voting outcomes for any specified profile. The converse creates an easily used tool to identify new election relationships and to compute probability estimates. In the final section, natural geometric symmetries within a profile are identified and extracted to create a “profile decomposition.” This decom- position permits us to construct, analyze, and describe all possible election paradoxes that can occur with all election rules that are based on positional and/or pairwise election outcomes. 2 Simple geometric representations As it is not feasible to use standard graphs with choice theory, the first of three approaches described in this section addresses the problem by listing profiles in a manner that roughly mimics the structure of profile space. Advantages of using what I call the geometric profile representation are that it provides simpler and quicker ways to tally positional and pairwise ballots, it helps us develop intuition as to why the same profile can allow different rules to have conflicting election outcomes, and, later, it leads to natural profile relationships that explain the source of several problems from voting theory. The second geometric approach exposes surprisingly complex relation- ships that exist among voting rules. This approach probably can be used to analyze other decision rules because nothing more difficult than elementary geometry and algebra is needed. The third approach, which examines cycles, the paradox of voting, and other intricacies of pairwise voting, identifies all possible profiles that define specified pairwise outcomes; expect surprises in the interpretation of majority votes. Throughout the geometry unveils a common source for all voting problems: problems arise when a voting rule ignores crucial but available information about the profile. 3 2.1 Geometric profile representation A traditional way to describe a profile for the three alternatives A, B, C is to list how many voters prefer each of the six preference rankings; e.g., Number Ranking Number Ranking 7 A B C 12 C B A (1) 15 A C B 4 B C A 2 C A B 12 B A C The tedium of tallying ballots requires sifting through the data to find how many voters rank each candidate in different ways. This suggests searching for alternative ways to represent profiles that will simplify the tallying pro- cess. The following approach was developed in (Saari, 1994, 1995, 2001); for applications, see Nurmi (1999, 2000, 2002) and Tabarrok (2001). Assign each candidate to a vertex of an equilateral triangle (see Fig. 1a). Assign a ranking to each point in the triangle by its distance to each vertex where “closer is better.” This binary relationship divides the triangle into the thirteen regions depicted in Fig. 1: the six small open triangles represent strict rankings while the seven remaining ranking regions, which involve at least one tie, are portions of the lines. In Fig. 1a, for instance, the number 15 is in a region closest to the A vertex, next closest to C, and farthest from B, so it corresponds to an A C B ranking. Points on the vertical line are equal distance from the A and B vertices, so they represent the tie A ∼ B. This geometry positions the ranking regions in a manner similar to that of profile space in that adjacent “ranking regions” differ only by the ranking of an adjacent pair. 14 + 19s C C ... ... ....... ....... ....... ........ ... ... ...... ... ... ... .... ... ... ... ....... ... ... ... ... .. ... ... ... ......... ... ... ... .... ... ... ... ........... ... ... ... ............ ... ... ... .... .. ... 18 ... ... 29 .............. ... ... ... .... ... ... 2. 12 ... ............... ... ... ... ... ...... .. ... ........ ........ ........................ ........ ... ...... ...... ... ................ ..... ...... ... ... ....... ....... ... ........................... ....... ... ... ....... ....... ... .......................... .. ....... ... ... ............... ... ............................................ ... ... ......... ... ..................................... ... 34 ... 15............... 4 ... 23 ................................................. ... ... ....... ....... ... ..................................... ....... ... ... ....... ....... ... ........................................ ....... ... ... ...... ...... ... ...................................... .. ...... ... ... ....... ....... ... ........................................... ..... ....... ... ... ....... ....... ... ............................................. ..... ....... ... ... ....... ....... ... ..........................................
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