NOVEMBER 26, 2019 Voting Issues: A Brief History of Preference Aggregation Given the surprising results of recent elections, voting methods have drawn lots of attention. Research in social choice theory reveals the underlying complexity — and flaws — of different methods of expressing preferences. By Marton Farkas and Dusan Timotity WorldQuant, LLC 1700 East Putnam Ave. Third Floor Old Greenwich, CT 06870 www.weareworldquant.com 11.26.19 VOTING ISSUES: A BRIEF HISTORY OF PREFERENCE AGGREGATION PERSPECTIVES DONALD TRUMP’S VICTORY IN THE 2016 PRESIDENTIAL ELEC- major successes in areas as seemingly different as search engine tion produced one of the biggest shocks in U.S. electoral and ratings methodologies and molecular biology and genomics. history. The sharp swing in Election Day forecasting at the New York Times reflected just how surprising an outcome it But let’s begin with elections, which lie at the heart of democratic was. As the newspaper’s Steven Lohr and Natasha Singer political systems and led to the birth of social choice theory in the wrote, “virtually all the major vote forecasters... put Mrs. first place. [Hillary] Clinton’s chances of winning in the 70% to 99% range,” until the actual results began coming in.1 As the day THE THEORY OF AGGREGATING turned to night, Trump unexpectedly took the lead in a number INDIVIDUAL CHOICE of reliably Democratic states, including such bellwethers as Social choice theory dates to the mid-18th century, when the Michigan, Pennsylvania and Wisconsin, which resulted in his Marquis de Condorcet, a French philosopher and mathematician, Electoral College victory despite trailing Clinton by several presented his ideas on the pitfalls of making collective decisions million votes nationally.2 The outcome not only highlighted based on individual preferences, and supported them mathemat- polling failures, but it raised questions about everything from ically. Condorcet was a central figure of the Enlightenment, well primary contests involving large numbers of candidates to known and controversial for his forward-looking views on slavery the Electoral College and its proportional voting system. and freedom; he died in prison in 1794, in the midst of the French Revolution. One of his pioneering scientific works, the “Essay on According to Andrew Douglas, Rob Richie and Elliot Louthen, the Application of Analysis to the Probability of Majority Decisions,” analysts at FairVote, a nonprofit organization that advocates for provided the basis for social choice theory.4 The essay defines a way electoral reform in the U.S., the 2016 election would have had a of voting in which “an alternative defeats every other by a simple very different ending if the voting had been based on preference majority.” A so-called Condorcet winner is defined as a candidate rankings.3 The authors argue for a change of voting methods in or issue that defeats or ties every other alternative in pairwise the U.S. to capture more of the electorate’s complex individual majority contests.5 preferences for candidates, particularly in campaigns with crowded fields. For example, their analysis reveals that on Super Tuesday If an election process consistently finds the Condorcet winner when 2016 — March 1, when the largest number of states held primary it uniquely exists, then it has what’s known as the Condorcet prop- elections — Trump would have lost nine of 11 states instead of erty. However, in many cases no such collective decision emerges; picking up seven if voters had submitted a ranking of candidate no single candidate wins a majority of the pairwise contests. This preferences rather than picking just one individual, as in the usual is known as the Condorcet paradox. For example, consider three majority voting process. If that had occurred, Texas Senator Ted candidates — A, B and C — and three voters: x, y and z. If x pre- Cruz, not Trump, might well have been the leading Republican fers A over B, y prefers B over C and z prefers C over A, there is candidate for president in the subsequent election. no Condorcet winner. The paradox arises from the fact that while individual preferences may be “transitive” (that is, if a voter pre- Of course, that was not the case, and majority rule — also known fers x over y and y over z, then we can assume x is preferred over as a plurality, first-past-the-post or winner-take-all voting system z), the collective preference may end up as “intransitive” (x is not — remains the predominant voting methodology, although the preferred over z). This paradox often blocks the creation of an number of political systems that use preference voting in some optimal, transitive order of candidates. Another way to say this form are growing in the U.S. and around the world. The study of is that while individual preferences are rational, or transitive, the voting, or, more technically, preference aggregation, is part of a collective decision may be irrational, or intransitive. discipline known as social choice theory, which focuses on how people attempt to make optimal choices collectively. Voting turns Social choice research has revealed deeper difficulties in preference out to be far more complex than it may seem to a citizen pulling a aggregation. One of the most important insights is attributed to lever or filling out a ballot. economist and Nobel laureate Kenneth Arrow. In what he initially called his general possibility theorem for social welfare functions, This article examines a number of those systems and the ideas published in 1949 (later commonly known as Arrow’s impossibility and mathematics that support them. We will explore practices that theorem), he demonstrated that no ranked voting system, in which many people assume are straightforward and uncontroversial but voters rank candidates by preference, can meet criteria of fairness are, in fact, complex and often flawed. More broadly, the activity if voters have three or more distinct alternatives.6 The proper- known as preference aggregation has a thriving existence beyond ties required to define fair voting include unrestricted domain (all traditional voting in political contests. It has been used across preferences of all voters are taken into consideration), nondictator- many disciplines, from economics to philosophy, and has achieved ship (voting cannot mirror any single voter’s preferences without Copyright © 2019 WorldQuant, LLC November 2019 2 11.26.19 VOTING ISSUES: A BRIEF HISTORY OF PREFERENCE AGGREGATION PERSPECTIVES as 1435 by German philosopher Nicholas of Cusa. In this method, Social choice theory dates to the voters submit complete preference orders over n number of alter- th natives. For each voter, the top choice receives n points, the second mid-18 century, when the Marquis de gets n-1, and so on (the last alternative is assigned 1 point).The Condorcet presented his ideas on the final ranking is the order of the total sum of the scores. The point allocation is arbitrary; the top choice can receive n-1 points, or we pitfalls of making collective decisions can apply any monotonic function to allocate scores. An example is based on individual preferences. the Downdall system, in which the nth choice receives 1/n points. Due to the simple scoring, which does not consider pairwise com- considering other individuals), Pareto efficiency (no individual can parisons, the Borda count fails the Condorcet property. be better off without making someone else worse off) and the independence of irrelevant alternatives (combined preferences During Borda’s life, the French Academy of Sciences used his for A and B depend only on individual preferences between A and method to elect its members. However, after Borda’s death in 1799, B, and not on any third factor — say, C). Practically speaking, Napoleon Bonaparte became president of the academy and replaced the independence of irrelevant alternatives crops up when a new the Borda count with his own method. Nevertheless, it is still used candidate, such as a third-party candidate, joins a race. in academic institutions and political jurisdictions (for example, the Slovenian Parliament) to distribute minority seats, while the Arrow’s theorem directly questioned the ultimate fairness of dem- Downdall system is used in the Pacific island nation of Nauru.8 ocratic elections. Another increasingly popular preferential voting method is the University of Michigan philosopher Allan Gibbard went on to gen- single transferable vote (STV), which is designed to achieve pro- eralize Arrow’s ranked model to include cardinal preferences, portional representation in a multiseat contest. Voters list their meaning that voters can not only assign a ranking of preferences preferences from a slate of candidates. Votes are totaled, and a but can quantify differences among their choices by assigning quota is derived for the number of first choices needed to win a grades to candidates. Gibbard also includes nondeterministic pref- seat. The most common quota requires 50 percent-plus-one votes erence aggregation functions that introduce chance in determining and is known as the Droop quota: |(valid votes)/(seats to win+1) |+1. social choice (in practice, some votes are excluded randomly).7 Candidates who hit the limit are elected, and their surplus votes over Under such conditions, Gibbard’s theorem states that any process what was required to win are distributed to voters’ second choices, of collective decision making either ends up being dictatorial, limits pushing more candidates past the quota. If more candidates than possible outcomes to two options or encourages agents to act seats remain, the candidate with the lowest number of top votes is strategically — that is, submit preferences that don’t reflect their eliminated and their top votes are distributed to the second choices. true opinion but are made based on expectations of how others The process continues until every vacant seat is filled.