Voting Issues: a Brief History of Preference Aggregation

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

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. may be voting. Individuals may vote not because they like their candidate but because they dislike another candidate more and STV is usually referred to as instant runoff voting (IRV) when it is wish to defeat them. That is certainly the case in an election with used to elect a single alternative. In this case, the lowest first-choice two relatively unpopular candidates, as in the 2016 U.S. presidential candidates are successively eliminated (and their votes redistrib- race, or in a field of multiple candidates, such as primaries. uted) until the field is reduced to two. The final round becomes an instant runoff because the top two go head-to-head. We can use RANKED VOTING SYSTEMS the order of the eliminated candidates to obtain an aggregated One way to counteract Arrow’s impossibility theorem is to for- choice. A similar but nonpreferential multiround system, called the mulate voting rules that relax at least one of his fairness criteria. exhaustive ballot, eliminates alternatives with the fewest number Usually, that’s the independence of irrelevant alternatives (IIA) of votes and asks for a revote if neither of the remaining alterna- axiom. IIA suggests that replacing an A > B > C vote with an A > tives achieves an absolute majority. In practice, both STV and IRV C > B cannot change the preference order between A and B. But are extremely difficult to manipulate from the outside, so they given that A > C > B suggests a stronger preference of A over B are considered to be a great improvement over majority rule.9 than A > B > C, preserving IIA is not always ideal. The world soccer federation, FIFA, and the International Olympic Committee use the exhaustive ballot to select host cities, while the The Borda count is one of the simplest ways to satisfy all of Arrow’s Academy Awards and Australian federal elections employ the IRV. required conditions apart from IIA. This method is named for an 18th-century French mathematician, physicist and mariner, Jean- The FairVote analysis discussed earlier, which concluded that the Charles de Borda, but was independently developed as long ago 2016 presidential election might have had a different outcome with

a different voting system, also applies the IRV method. Figure 1 One way to counteract Arrow’s summarizes the results of the simulation based on polls for the 2016 Republican primary election in Georgia. We can clearly see the impossibility theorem is to formulate benefits of using full rankings, which contain much more informa- voting rules that relax at least one of tion than a winner-take-all voting system. In this example, Trump led by far in the first round and thus won by majority rule. However, his fairness criteria. by taking into account all the preferences for the remaining candidates, consecutively eliminating the least-preferred options and Then the Kemeny aggregation is the most likely true ordering, recalculating the rankings on the reduced subsets, ranked-choice which produces the preferences. voting systems would have arrived at a different result because voters for Cruz and “all other candidates” would have preferred Despite its superior properties, Kemeny aggregation suffers from Rubio to Trump. a serious problem in practice. In computer science terms, it is NP-hard, even with only four voters;11 however, numerous fast WHAT IF THE GEORGIA GOP PRIMARY approximation algorithms have been developed that help this situa- USED RANKED-CHOICE VOTING? tion.12 Moreover, Kemeny’s time complexity is linear in the number of voters, hence it remains computationally feasible for up to ten 1st Round Transfers 2nd Round Transfers 3rd Round candidates. Another practical workaround is to use the so-called Trump 38.8% +3.8% 42.6% +6.3% 48.9% local Kemeny property — a ranking whose Kendall tau distance cannot be improved by a single swap of neighboring alternatives.13 Cruz 23.6% +3.2% 26.8% * –26.8%

Rubio 24.4% +6.0% 30.5% +20.5% 51.0% The task of creating an accurate consensus ranking is not only a All other 13.1% * –13.1% problem in politics. There are many situations where individual candidates “judges” focus on different evaluation criteria. In this context, the *Defeated Source: FairVote. aim of rank aggregation is to gather the knowledge and produce Figure 1 a best possible final ranking. The following examples illustrate the successful application of social choice theory in rank aggregation Still, neither majority rule nor preference ranking is perfect; neither problems in various other fields. consistently produces a Condorcet winner. However, this can be remedied by a more complex rank aggregation system called the APPLICATIONS BEYOND POLITICS Kemeny rule. Google is the most famous, and the most popular, search engine currently operating. However, there are dozens of other pub- RANK AGGREGATION WITH THE KEMENY RULE licly available specific and general-purpose query-based search John Kemeny developed a Condorcet-consistent voting method engines.14 Metasearch is the task of aggregating different search in 1959. Kemeny was a Hungarian-American mathematician and rankings. One of the field’s most important goals is to combat computer scientist and is most famous as a co-developer of the “spam.” The word comes from a skit from Monty Python’s Flying BASIC programming language; he also served as president of Circus in which a waitress reads off a menu that turns out to contain Dartmouth College. Kemeny’s rule selects (out of n! possible per- only Spam, a brand of canned cooked pork, to a customer trying mutations) the final ranking of preferences that minimizes the sum to avoid it. A web page is spam if it achieves a higher ranking in of the number of pairwise disagreements of individual preference search results than it deserves based on its content and link struc- orders, creating what’s known as a Kemeny consensus. Creating ture.15 Spam typically contains hidden text — out-of-scope links this consensus ranking involves calculating the number of steps specifically designed to achieve a top ranking for a wide range of a bubble sort — an algorithm that sweeps through rankings and possible queries. counts the number of swaps to make two lists equivalent — would require to transform each ranking to the aggregated one. This There is a way to effectively deal with spam. The extended Condorcet results in what is known as bubble-sort or Kendall tau distance criterion (ECC) states that if you can partition alternatives into two (after British statistician Maurice Kendall, who developed the sets (X,Y), so that for each x in X and y in Y the majority prefers x metric in 1938); the larger the distance, the more dissimilar the over y, then each x is ranked above each y in the aggregate.16 (When two rankings, and vice versa. There is another intuitive probabilistic X has only one element, we arrive at the Condorcet condition.) By interpretation that can be drawn from the Kemeny rule.10 Suppose definition, spam targets the index of specific search engines. If a voter preferences are noisy versions of an underlying true ordering few search engines give a page a high ranking, ECC will ensure obtained by swapping alternatives with a probability less than 0.5. that the page rank drops in the metasearch result as the majority

shows greater preference for honest pages. It turns out that meth- so-called microRNA targets. Andrew Fire and Craig Mello received ods with a Kendall tau distance that cannot be increased by a the Nobel Prize for medicine in 2006 for RNA inference.20 This single transposition of two neighboring elements (local Kemeny process silences gene expression: Small noncoding RNAs, called property) are ECC consistent.17 A local Kemeny optimization can microRNAs, have crucial importance in this process by block- be performed on any ranking — for example, a Borda count — by ing messenger RNA from conveying genetic information from starting from the top alternative and recursively adding the next the genome to the ribosome; microRNA appears to be part of an preferred candidate and bubble the newly added element up until immune response. In drug development, the long-term aim is to the majority of voters agree with that. cure diseases such as cancer and HIV by shutting down genes. Experimental verification of microRNA targets is not an easy task, Link prediction is another field where social choice theory has been as each microRNA may affect a large number of genes. Many dif- successfully applied. Link prediction observes a network and tries ferent target prediction algorithms have been published, greatly to predict future connections. Successful link prediction algorithms disagreeing on the target gene ranking. A reformulated Kemeny help build better recommendation systems for buying products or algorithm that tries to equally distribute the total number of dis- services; predict outbreaks of diseases and crowds in transpor- agreements on the predictors has been effective in aggregating tation networks; and suggest new friends, jobs and partners on target predictions.21 social media sites. The elementary predictors are the topological features, which express the properties and relationships among Finally, social choice theory naturally arises in the context of mar- the nodes. They usually include the number of common neighbors, kets. By design, prediction markets, such as horse racing, pay for the shortest path between nodes, the ratio of common neigh- successful wagers on the correct order of finish. Gamblers seeking bors to the union of neighbors, known as Jaccard’s coefficient, or an edge can combine ranked elementary predictors, such as a the PageRank of the nodes (PageRank is Google’s search engine horse’s track record, with voting functions. This method works best algorithm for ranking pages). Intuitively, if two nodes have many when the predicted order carries most of the information, because common neighbors, they will more likely link in the future. Better differences among the horses cannot be quantified without a lot results are achieved if the whole feature set is used simultaneously. of noise. In this case, information loss due to ordinal rankings is A major trend is to use supervised classification models on the recouped by the robustness of the ranked elementary predictors features to predict the linking of two nodes. This model can be and by the superior aggregation method. Voting rules will especially trained on the historical evolution of a partial or full graph. flourish when magnitude information is not available. Financial analyst recommendations are available from a number of services, An alternative approach is to create a rank, which represents the such as the Institutional Brokers’ Estimate System (IBES). The order of linking probability derived from a feature, and aggregate service archives recommendations on more than 40,000 companies the ranks in line with social choice theory. The predictive power by hundreds of analysts and makes social choice theory a useful of different features can vary a lot, so it’s preferable to have a tool for developing investment signals. voter-weighted version of the Borda count or Kemeny rule. Such voting rules have been found to more accurately predict future con- POLITICAL IMPLICATIONS nections in the DBLP computer science bibliography than classical Given their success in a wide range of areas, why haven’t these solutions such as k-NN, Bayesian methods and decision trees.18 preference aggregation methods been adopted more broadly in According to other studies, the approximated Kemeny rule that political elections? After all, majority rule has drawbacks. The aggregates features like degree centrality or PageRank also out- spoiler effect presents a scenario in which a less popular candi- performs standard supervised learning techniques when it is used date draws votes from a major candidate with similar positions, to detect outbreaks of viral influence on Twitter.19 allowing smaller parties to change the outcome. The system is also susceptible to gerrymandering, the practice of gaining political Rank aggregation also has proved useful in computational biol- advantage by manipulating the shape of congressional districts. As ogy, where a recent area of research has focused on deciphering Gibbard points out, strategic manipulation in most voting systems is unavoidable; however, majority voting is extremely susceptible to Rank aggregation also has proved reporting fake preferences to gain better outcomes. According to Duverger’s law, attributed to French sociologist Maurice Duverger, useful in computational biology, the first-past-the-post election rule favors the emergence of a where a recent area of research has two-party system.

focused on deciphering so-called To combat some of these disadvantages, a number of countries microRNA targets. have implemented instant runoff voting. For instance, Australia

uses such a system in general elections, Ireland for presidential Gamblers seeking an edge can elections, the U.K. and Canada to elect major party leaders and the state of Maine to elect U.S. senators and representatives. However, combine ranked elementary predictors, voters are often resistant to changing age-old practices, particu- such as a horse’s track record, with larly when it involves a more complex substitute. In 2011, the U.K. proposed the alternative vote referendum to replace its plurality voting functions. system with IRV in most elections. Two-thirds of the population rejected it, with the majority backing the referendum in only ten and Hillary Clinton, millions of Americans found themselves forced of 440 local voting areas: Cambridge, Oxford and eight others. The to vote for a major-party nominee they plainly couldn’t stand or to Kemeny rule has seen no practical use because of its computa- risk electing the candidate they hated even more by casting their tionally infeasible algorithm. ballot for a third-party contender.”22 Although Berman backed up his argument with public opinion data on candidates since 1980, In fact, we are yet to see a society that follows the metasearch the outcome of the 2016 race cannot be attributed solely to that. engines and uses local Kemenization applied to IRV results to As we’ve seen, majority voting can have such an effect if individual ensure that elections are Condorcet-consistent (with extremist voter preferences for all candidates are disregarded — a situation parties falling in rank like spam in a search engine). made more severe as the number of candidates grows.

CONCLUSION This is bad news for the 2020 Democratic Party primaries, given All voting systems have flaws that produce practical problems, yet the size of the field. However, multiple states plan to partially apply democracy remains the most accepted way to govern individuals IRV as a voting method in their early primaries and caucuses. Some around the globe. As Winston Churchill famously said, “Indeed, it have even dared to switch completely to ranked-choice voting: has been said that democracy is the worst form of government In 2020, Maine — the first state to introduce ranked preferences except all those other forms that have been tried from time to time.” in congressional elections — will become the first one to adopt ranked-choice voting in primaries.23 Still, a healthy democracy requires credible voting systems that are viewed as fair and reflect as much information as possible And that’s the good news. Election systems are improving slowly about individuals’ preferences. As a starting point, asking voters but steadily, which may lead to better social choices that reflect to submit an ordered list of preferences for candidates may be a preferences more accurately. It can’t come too quickly. As Abraham big step toward an optimal solution. Majority rule will continue to Lincoln said in 1856, in a quote that may be just as relevant today produce unexpected results, even outcomes not backed by a major- as it was then, “Do not mistake that the ballot is stronger than ity, but attempts to improve voting systems increase year by year. the bullet.” ■

The results of the 2016 U.S. presidential election highlight the flaws Marton Farkas is a Regional Research Director at WorldQuant and has a of majority voting and reveal a profound need for a change in the Master’s degree in mathematics from the University of Cambridge. system. According to Atlantic writer Russell Berman, the 2016 Dusan Timotity is a Senior Quantitative Researcher at WorldQuant and “election pitted the two most disliked candidates in the history of has a Ph.D. in financial modeling from Budapest University of Technology public polling against each other. In the race between Donald Trump and Economics.

ENDNOTES

1. Steve Lohr and Natasha Singer. “How Data Failed Us in Calling an Election.” 6. Kenneth J. Arrow. “General Possibility Theorem for Social Welfare Functions.” New York Times, November 10, 2016. Social Choice and Individual Values. New Haven, CT: Yale University Press, 2012. 2. Lizzie Dearden. “President Donald Trump: Four Hours That Shook the World — 7. Allan Gibbard. “Manipulation of Schemes That Mix Voting with Chance.” How the Polls Swung and the Election Results Came In.” Independent, November Econometrica 45, no. 3 (1977): 665–681. 9, 2016. 8. Jon Fraenkel and Bernard Grofman. “The Borda Count and Its Real-World 3. Andrew Douglas, Rob Richie and Elliot Louthen. “Simulating Instant Runoff Flips Alternatives: Comparing Scoring Rules in Nauru and Slovenia.” Australian Most Donald Trump Primary Victories.” FairVote, March 4, 2016. Journal of Political Science 49, no. 2 (2014): 186–205. 4. Marquis de Condorcet. “Essay on the Application of Analysis to the Probability 9. John J. Bartholdi III and James B. Orlin. “Single Transferable Vote Resists of Majority Decisions.” Paris: Imprimerie Royale, 1785. Strategic Voting.” Social Choice and Welfare 8 (1991): 341–354. 5. H.P. Young. “Condorcet’s Theory of Voting.” American Political Science Review 10. Tyler Lu and Craig Boutilier. “Learning Mallows Models with Pairwise 82, no. 4 (1988): 1231–1244. Preferences.” Proceedings of the 28th International Conference on International Conference on Machine Learning (2011): 145–152.

11. William W. Cohen, Robert E. Schapire and Yoram Singer. “Learning to Order 18. Manisha Pujari and Rushed Kanawati. “Supervised Rank Aggregation Approach Things.” Journal of Artificial Intelligence Research10 (1999): 243–270. for Link Prediction in Complex Networks.” Proceedings of the 21st International World Wide Web Conference (2012): 1189–1196. 12. Alnur Ali and Marina Meila. “Experiments with Kemeny Ranking: What Works When?” Mathematical Social Sciences 64, no. 1 (2012): 28–40. 19. Karthik Subbian and Prem Melville. “Supervised Rank Aggregation for Predicting Influencers in Twitter.” PASSAT/SocialCom 2011: 661–665. 13. Cynthia Dwork, Ravi Kumar, Moni Naor and D. Sivakumar. “Rank Aggregation Revisited.” March 2003. 20. Andrew Fire, SiQun Xu, Mary K. Montgomery, Steven A. Kostas, Samuel E. Driver and Craig C. Mello. “Potent and Specific Genetic Interference by Double- 14. Christopher Ratcliff. “Say Good-bye to Google: 14 Alternative Search Engines.” Stranded RNA in Caenorhabditis elegans.” Nature 391 (1998): 806–811. Search Engine Watch, February 25, 2016. 21. Debarka Sengupta, Aroonalok Pyne, Ujjwal Maulik and Sanghamitra 15. Nathan Francis. “Voting as a Method for Rank Aggregation and Spam Reduction Bandyopadhyay. “Reformulated Kemeny Optimal Aggregation with Application in on the Web.” Department of Computer Science, Yale University (May 9, 2005). Consensus Ranking of microRNA Targets.” IEEE/ACM Transactions on 16. Cynthia Dwork, Ravi Kumar, Moni Naor and D. Sivakumar. “Rank Aggregation Computational Biology and Bioinformatics 10, no. 3 (2013). Methods for the Web.” Proceedings of the 10th International World Wide Web 22. Russell Berman. “A Step Toward Blowing Up the Presidential-Voting System.” (2001): 613–622. Conference Atlantic, September 20, 2019. 17. Michel Truchon. “An Extension of the Condorcet Criterion and Kemeny Orders.” 23. Maggie Astor. “Maine Voters Will Rank Their Top Presidential Candidates in (1988). Cahier 98-15 du Centre de Recherche en Economie et Finance Appliquées 2020.” New York Times, September 6, 2019.

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