Chaotic elections! A mathematician looks at voting (Q2723259)

The mathematics of voting has been gaining in visibility over the past years. The field has benefitted greatly from the attentions of several researchers who have brought powerful new insights and techniques to bear on centuries-old problems. Among those who have shaped the field with their particular vision is the author of this book. He has published numerous research papers on aspects of voting theory and also has a history of making his work accessible to beginning students in textbook form [Basic geometry of voting. Berlin: Springer. (1995; Zbl 0873.90006)]. Here, partly spurred on by the interest in voting theory generated by the 2000 U.S. Presidential election, the author gives a non-technical account of the background and results of his work, which culminated in two recent important research papers [Mathematical structure of voting paradoxes. I: Pairwise votes. Econ. Theory 15, No. 1, 1-53 (2000; Zbl 1080.91022); and II: Positional voting. Econ. Theory 15, No. 1, 55-102 (2000; Zbl 1081.91007)]. NEWLINENEWLINENEWLINEThe author's primary interest is in the setting where a finite set of voters linearly order a finite collection of alternatives or candidates and a selection procedure constructs a social ordering from this collection of individual orderings (called a profile). As is well-known, this type of voting is beset with paradoxes (counterintuitive outcomes) whenever there are more than two alternatives. NEWLINENEWLINENEWLINEChapter 1 surveys some of the problems associated to voting methods, amplifying the author's thesis that `our basic voting procedures can generate problems so worrisome that it is reasonable to worry about the legitimacy of most election outcomes'. The author gives examples of most common voting procedures and some of the problems arising from them, leading up to Arrow's theorem. He also gives a brief discussion of measuring power among unequal voters, using as an example the U.S. Electoral College. NEWLINENEWLINENEWLINEChapter 2 considers positional systems, that is, those where a voter assigns points to each candidate. These methods include plurality, antiplurality and Borda count. The author shows that there are profiles of \(n\geq 2\) candidates with up to \(n!-(n-1)!\) rankings as the positional system varies and that for three candidates almost 70\ under different positional systems. Focussing on elections with three candidates, the author introduces his `Saari triangle' method of encoding profile information in geometric form and shows how the different positional systems lead to a `procedure line' from which the range of possible results for a given profile can be read off. The author also compares approval voting to positional systems, showing that any result possible under some positional system is possible under approval voting. NEWLINENEWLINENEWLINEIn Chapter 3, the author turns to pairwise comparison methods and the central issue of what happens to the ranking of the remaining candidates if one of them withdraws. One of the main results here is that there exist profiles allowing essentially arbitrary re-rankings of the remaining candidates as the candidates are withdrawn in turn. However, not all positional methods allow arbitrary re-rankings -- the Borda count method does not and here the author begins his championing of the system. NEWLINENEWLINENEWLINEChapter 4 turns to the issue of strategic voting and the celebrated Gibbard-Satterthwaite theorem. The author gives a good analysis of strategic opportunities with several examples of agendas and runoffs. NEWLINENEWLINENEWLINEChapter 5 begins the analysis of how the various paradoxes arise. This is the theoretical core of the book. The author introduces the idea of a `ranking wheel' for Condorcet cycles and shows that positional systems are unaffected by the addition of equal numbers of voters with rankings forming a complete rotation of the wheel, for example \(A>B>C\), \(B>C>A\) and \(C>A>B\). Of course, pairwise comparison systems are affected. The surprising conclusion is that all problems with pairwise methods arise from this ranking wheel Condorcet effect. If the tied Condorcet cycles are removed, the remaining profile is additively transitive, that is, the differences in tallies add in a linear fashion. NEWLINENEWLINENEWLINENext, the author argues that voters with completely opposite rankings should cancel. He shows that this reversal effect explains the differences between positional systems and that reversal symmetry is respected only by the Borda count. Thus, the author views profiles in three parts: the basic profile; the Condorcet profile, and the reversal profile. By beginning with a basic profile and adding suitable Condorcet and reversal components, any desired paradoxical situation can be created. The Borda count is the unique method satisfying all the symmetries. NEWLINENEWLINENEWLINEChapter 6 looks briefly at applications of these insights into voting theory in different areas where statistics must be aggregated. The book is intended for the non-specialist reader. There are numerous examples throughout the text and results are often presented not in complete generality for clarity. The mathematical requirements are essentially simple geometry and algebra, although the reader will need a certain level of mathematical stamina. Technical proofs are not given; the reader is referred to the original publications. The book presents a very clear picture of how the author views the central issues of voting theory and provides an excellent entrée into his work.