To see why, read this term paper that my sister, a Republican sophomore at Dartmouth, wrote for a class in math in social science taught by Robert Norman. It is well written and addresses a very important issue that we as progressives share with the reasonable Right: fundamental voting reform. In the paper, she critically analyzes many voting methods, including pluarlity, IRV, coombs, condorcet, borda, and approval voting and concludes with an endorsement of approval voting.
After reading the paper, I am pretty much sold on approval voting and find myself strongly opposed to switching to IRV or other types of STV voting.
On a side note, my sister wanted me to note that she has not completely cited the paper (so please do not accuse her of plagerism just yet). If you would like a fully cited copy, I will post a link to the paper when it is finished.
Introduction
The 2000 United States Presidential election opened the floodgates to attack on the country's current method of electing public officials. Some jumped on the bandwagon to blame the Electoral College, ballot mishaps, and even racial discrimination. While attention is focused on these familiar irregularities and critiques, a much more serious and tangible problem is overwhelmingly ignored by the masses: the fundamental flaws of the voting procedure itself. The vast majority of political elections in the country utilize the system of plurality in which each voter selects a single candidate; the candidate with the highest number of votes is declared the winner. For centuries, voting theorists have argued that plurality voting is one of the worst of all possible choices compared to almost any other of the many voting system. However, despite their impressive and intermittent criticisms, little has been done to alleviate the problem.
Historical Legacy of Plurality
In elections with only two candidates, plurality voting functions well since the winner is guaranteed to have been the top choice of more than half the voters. But as soon as three or more candidates are on the ballot, the system can run into trouble. This problem has arisen many times in the past few centuries, with the most notable recent example the 2000 US presidential election. In the fiercely contested race, Republican George W. Bush won the state of Florida, and consequently the presidency, by merely a few hundred votes over the Democratic candidate Al Gore. Green Party candidate Ralph Nader garnered a significant 95,000 votes in Florida, and polls indicate that most Nader voters' second choice was Gore (Klarrreich, 2002.) Thus, if the race had been a head-to-head contest between Bush and Gore, Florida voters probably would have chosen Gore by a substantial margin. In races such as this one with two strong candidates, plurality voting is vulnerable to third-party spoilers: a weaker candidate splits a proportion of the electorate with one of the major candidates, thus causing the major candidate to lose even though he/she is more preferred by the voting population. As many Democrats have vocally asserted, should Nader have dropped out of the race? Mathematician Donald Saari of the University of California, Irvine does not think so, claiming, "We live in a democracy, and anyone should be able to run for any office. The problem was the bad design of the election" (Klarrreich, 2002.)
In races featuring a large number of candidates, plurality voting dilutes voter preferences. This creates the possibility of electing a leader whom the vast majority of voters despise. The French presidential election two years ago boasted sixteen candidates on the ballot; extreme right-wing candidate Jean-Marie le Pen, known for racist and anti-Semitic beliefs, managed to place second with only seventeen percent of the vote, and accordingly advanced to a runoff against the top candidate, incumbent President Jacques Chirac. Political analysts across the globe scrambled to explain le Pen's surprise success, ultimately accrediting it to voter disenchantment and a surge in right-wing support across Europe. Voting theorists paint a starkly contrasting picture, attributing the results to the fact that plurality distorts the preferences of the voters. Once again, Saari remarks, "The fact that le Pen was in the runoff had nothing to do with what the people wanted" (Klarrreich, 2002.) In the runoff election, Chirac trounced le Pen with 82 percent of the vote which suggests that while le Pen was at the top of a few voters' lists, he was near the bottom of many more. Another notable voting theorist, Steven Brams, astutely sums the election up, saying "There is no question that under almost any other system, le Pen would not have made it to the runoff" (Klarrreich, 2002.)
If it were not for the plurality system, Abraham Lincoln might never have become the sixteenth president of the United States. In the four-candidate 1860 election, Lincoln was a polarizing figure, popular with many Northerners but abhorred by an overwhelming number of Southerners. Stephen Douglas, Lincoln's closest competitor, was more broadly popular, and although he did not receive as many first-place rankings as Lincoln, he was nearly everyone's second choice. With Lincoln holding a prominent place in the pantheon of American presidents, the result of that election might seem desirable in retrospect. However, that is a completely different question than if the voters actually preferred Lincoln on Election Day of 1860. Furthermore, not all controversial outcomes from plurality can be seen in such a good light. One only has to look at Salvadore Allende's 1970 rise to power in Chile via a plurality victory of merely 36.3% that did not reflect the majority views of the country. Within three years, the military dictatorship of Pinochet overthrew the populist leader due to his inability to enact reforms as he lacked support in Congress due to his unpopular left-wing agenda.
The Need for Change in Light of Arrow's Theorem
In light of the above historical examples, there is a pressing need to implement a new voting system. With the many proposals currently floating around academic and political circles, it is important to determine which system is most representative of people's preferences. The term voter preference is a vague way of defining how well a voting system captures and aggregates voter and societal inclinations when selecting the winning candidate. This begs the question, is there a best voting procedure? In 1952, Kenneth Arrow, a professor emeritus of Economics at Stanford, proved that no voting system is completely free from counterintuitive outcomes in a work that later won him a Nobel Prize. Arrow looked at voting systems that satisfy two properties considered highly preferable. First, if everyone prefers candidate A to candidate B, then A should be ranked higher than B. Second, voters' opinions about candidate C should not affect whether A beats B: after all, if you prefer cake to pie, finding out that brownies are also available should not cause you to suddenly prefer pie to cake. These sound like reasonable restrictions that any voting system should encompass, yet Arrow ingeniously proved that the only voting system that always satisfies both of them is a dictatorship where a single person's preferences determine the outcome.
Arrow's Theorem is a very important theoretical result and an engaging proof, but the criteria he uses to depict an ideal election method may be unnecessarily rigorous. In other words, a less restrictive set of criteria that can be satisfied through practical voting systems may be perfectly acceptable to reasonable people. In any case, Arrow's Theorem certainly does not imply that any method is equally effective, or perhaps in this case equally defective, as some electoral reform activists have suggested. Thus, while Arrow's Theorem shows that no system is flawless, many voting procedures reflect voter preferences more effectively than plurality voting. This paper seeks to identify and examine voting criteria, namely monotonicity, Condorcet, and Independence of Irrelevant Alternatives, to evaluate the most popular voting procedures. The concluding remarks will also delve into current legislative efforts to replace plurality.
Voting Criteria
The literature detailing voting criteria is quite extensive and there seems to be an endless number of criteria advocated by various scholars. A voting criterion is a statement of value, a quality one hopes voting methods will meet. The criteria themselves are not methods; rather each criterion gives a condition from which a certain outcome should follow. Ranging from the independence of clones criterion to the consistency criterion to the Schwartz criterion, there is considerable debate on what qualities are vital in a voting system. Much emphasis is also placed of the degree of manipulation a system encourages or allows for. Each criterion provides valuable insight into the paradoxes of voting, however, it is both impractical and beyond the scope of this paper to delve into each and every criterion. Instead, this paper focuses on voting criteria that are most highly desirable for a fair election: monotonicity, Condorcet, and Independence of Irrelevant Alternative. If this paper were to be expanded, it would also look into the degree each system encourage strategic voting.
Voting Procedures
Just like a wide variety and large number of voting criteria exist and are constantly invented, the same can be said when it comes to voting procedures. Employing different voting methods on the same profile can, and often do, elect varying winners. As the historical examples highlighted, many past controversial elections that used plurality produced a winner that did not necessarily reflect societal preferences. The 2000 US Presidential election, in particular, has breathed life into search for a more representative voting method. The most widely discussed and probably most viable options are Single Transferable Vote (STV), Coombs', Approval voting, and Borda count, with the majority of academia spread among these main camps. As it is most likely that if a change is made to deviate away from plurality voting one of these systems would be implemented, this paper will focus on applying the above criteria to these methods, as well as plurality. In other words, by examining these proposed methods in light of the selected criteria, we will have a better idea of which of the methods will best represent societal preferences.
STV, also known as Instant Runoff Voting, has voters rank the candidates. The candidate with the fewest first-place votes is dropped, and that candidate is erased from the voters' preference lists. The ballots of voters who had placed him/her first are converted into votes for their second choice. From the remaining candidates, once again the one with the fewest first-place votes is dropped. The process is repeated until only two candidates remain; the candidate with more top votes wins. There is no need to hold repeated elections because the voters provide their complete ranking of all the candidates when they vote. STV is used to elect the Australian House of Representatives, the president of Ireland and the Parliament of Nauru. Coombs' method is also an elimination procedure where each voter ranks all the candidates. However, it works in reverse compared to STV, eliminating the candidate with the most last place votes from all the voters' ballots and repeating this procedure until once again, only two candidates are left. Like STV, the winner is ultimately indentified by determing which candidates has the most top place votes. Because of their similarities, both of these methods will be examined together throughout the paper. This is not to say that there are not substantial differences between the two, which will be noted when appropriate, but that they share enough common ground that it makes the most sense to lump them together.
In Approval voting, each voter simply votes for, or "approves," as many of the candidates as desired without ranking them. As in plurality voting, the votes are counted, and the candidate with the most votes wins. No new voting equipment is needed, and the change to the current voting rules is trivial: "vote for one" simply becomes "vote for one or more." Although trivial to implement, Approval voting goes a long way toward overcoming the "lesser of two evils" problem inherent in our current voting system artificially entrenching the political system into a two-party duopoly without effective competition from other parties. Voting for a third party would no longer be seen as a "waste of a vote," leading to a great diversification of parties and political ideas. Approval voting is currently used by a number of scientific societies.
Borda count, like STV and Coombs', requires the voter to rank the candidates. Each candidate receives 1 point for each last place vote received, 2 points for each next-to-last point vote, etc., all the way up to N points for each first place vote, where N is the total number of candidates. The candidate with the largest point total wins the election. The voting system was devised by Jean-Charles de Bordo in 1770, and was used by the French Academy of Sciences to elect members for several decades until Napoleon rose to power in 1800. The Borda count is often employed for determing sports awards, including the Most Valuable Player in Major League Baseball. It is also used for parliamentary elections in the Eastern European country of Slovenia, several south Pacific islands, and was one of the voting methods of the Roman Senate.
Monotonicity
The monotonicity criterion is one of the most intuitive and highly desirable of all voting criteria. It states that if a candidate X wins and one or more voters change their preferences in a manner that favours X, then X should still be the winner. The criterion is one of voting theory's most bewildering paradoxes; if a candidate is in the lead before the actual election date, performing well in a debate or giving a persuading speech that attracts more supporters to his/her cause should not make him/her lose. A monotonic violation can work in the other direction, too; not voting for a candidate can actually cause them to win. Suppose, for example, that 35 percent of voters prefer ABC; 33 percent prefer BCA; and 32 percent prefer CAB. In STV, C will be eliminated first, leaving A and B. A picks up C's first-place votes, winning handily 67 percent to 33 percent to B. But now suppose A makes such an inspiring speech that some voters who liked B best move A into first place, so now 37 percent rank the candidates as ABC, 31 percent as BCA, and 32 percent as CAB. Now, A faces C in the runoff instead of B. The votes that ranked B first become votes for C, and C beats A, 63 percent to 37 percent. Should A be penalized for attracting more supporters? This paradox can occur in any voting procedure featuring more than one round. Thus STV and Coombs both violate monotonicity, while plurality, Approval voting and Borda count do not.
Although all voting systems are vulnerable to strategic voting, systems which are not monotonic suffer from a greater degree of manipulability than systems that are monotonic. Strategic voting occurs when a voter's ballot is insincere if his/her reported preference order differs from his/her true preference order in hopes of improving the final outcome. People can, and likely would, vote tactically to elect their preferred candidate by voting against that candidate, and thus changing the order of elimination in nonmonotonic voting systems. Proponents of STV and Coombs' claim that violations of monotonicity, while not desirable, do not occur often enough to draw serious attention to the problem. Mathematician Robert Norman of Dartmouth College, however, has recently proved otherwise. Norman looks at close elections and proves that violations of monotonicity occur much more frequently than proponents of STV would like to admit. Norman showed that the proportion of all profiles in close elections that violate monotonicity increases, eventualling leveling off at just over 10% (Norman, 2004.) To the best of my knowledge, there has been no work completed documenting the frequency of monotonic violations under Coombs', although intituitively, the level of occurence should be similar. Such a high occurrence of monotonicy violations in both of these voting procedures is a serious blemish. Further research on the nature of these violations is needed to judge the liklihood of the problem arising in realistic situations; some have claimed that the violations are more frequent when cyclic preferences exist, a phenomena not often found in empirical data on previous elections.
Condorcet
Perhaps the most widely discussed voting criterion is the Condorcet criterion. The name comes from the eighteenth century mathematician and philosopher Marquis de Condorcet who developed the criterion to critique Borda count. In the Condorcet criterion, the winner, known as the Condorcet winner, must beat all other candidates in pairwise comparisions when such a winner exists in a given profile. Condorcet's paradox occurs when there are cyclic preferences, and thus no candidate can defeat all others pairwise. None of the discussed voting procedures meet this criterion. Instead, one must look at a voting system's Condorcet effeciency to estimate the percentage of a class of elections where a Condorcet winner exists in which the voting system system elected the Condorcet winner. This number is a voting system's "Condorcet efficiency". To estimate the efficiency of each voting system, several political scientists have used computers to simulate groups of voters. Merrill estimates that plurality, STV, Coombs, approval, and Borda count have Condorcet effeciencys of 60, 88, 90, 67, and 85 respectively (Merill, 1988.) Merrill then further explores Condorcet efficiencies in more complex situations to find that STV's chance of electing the Condorcet winner drops in a polarized society. Its efficiency rises with rising voter uncertainty about candidates' positions on issues but still remains lower than most other rule's. Finally, the efficiency of STV drops as the number of candidates increases, seeming paradoxical to STV proponents' claims that the system promotes viable third parties (Merrill, 1988.)
A similar criterion is the Condorcet loser criterion which states that if a Condorcet loser exists, it must not win. A Condorcet loser is an alternative that loses to all other candidates in pairwise elections. Of the discussed voting systems, only plurality and approval voting fail to meet this criterion. However, it is important to note that neither of the procedures violate the standard excessively. In fact, approval voting will elect a Condorcet loser when one exists roughly 1.8% of the time, while plurality does the same 3.1% of the time (Nurmi and og Uusi-Heikkila, 1985).
There are times, however, when electing a Condorcet loser does not look like too bad of an option when using approval voting. One example is the below profile:
1 voter likes C best, B next best, and A last;
2 voters like B best and A almost as well, but rank C very low;
2 voters like C best and A almost as well, but rank B very low.
Under approval voting, A receives four votes while neither of the other candidates gets more than three, and thus A wins. Yet, A is the Condorcet loser. However in this case, considering the size of the preference different, it does not seem unreasonable that A should be victorious. Even if one does not consider the size of preferences, there are instances where the Condorcet loser does not look too shabby, such as next profile:
3 voter rank the candidates A, B, C, D, E
1 voter ranks them B, C, D, E, A
1 voter ranks them C, D, E, B, A
1 voter ranks them D, E, B, C, A
1 voter ranks them E, B, C, D, A.
Again, if the voters prefer their first choice enough to vote for only one candidate, A, the Condorcet loser, will win. It would take a high degree of collusion and manipulation to elect anyone other than A. Once again, in this case, the Condorcet loser does not necessarily look like a bad choice. Thus, it is questionable how necessary and highly desirable the Condorcet Loser criterion is, as the above examples present cases where electing a Condorcet loser does not seem like a bad outcome.
Independence from Irrelevant Alternatives
The final criterion discussed in this paper is Independence from Irrelevant Alternatives Criterion (IIAC). According to IIAC, adding another candidate into an election should not change the winner, unless the inserted candidate actually wins; nor should eliminating a candidate change the winner, unless the eliminated candidate was previously the winner. In other words, IIAC is the technical terminology for the classic "spoiler" problem. Arrow uses IIAC as one of his axioms to prove that there is no perfect voting system. However, Arrow's definition of IIAC assumes that every candidate except the winner is irrelevant, a proposition that many would undoubtedly find surprising and unrealistic. A more reasonable definition of IIAC is needed that captures the key aspects of the criterion while still allowing viable voting procedures to meet it.
One slightly less stringent definition of an "irrelevant" candidate is based on the Smith set, the smallest set of candidates such that each candidate in the set beats, in the pairwise sense, each candidate not in the set. If one candidate beats each of the other candidates in their one-on-one race, then that candidate is the clear winner and is the only member of the Smith set. If no candidate beats every other candidate in their pairwise competitions, then a cyclic ambiguity exists where A beats B, B beats C, and C beats A, and the Smith set contains multiple candidates. Such an occurrence would once again fail Arrow's definition of IIAC. Note, however, that each member of the Smith set must have defeated at least one of the other members; hence none of them are "irrelevant" in the normal sense of the word. Only the candidates not in the Smith set can be considered irrelevant. This is sometimes referred to as Local Independence of Irrelevant Alternatives. Even with this understanding, none of the voting procedures examined in this paper would satisfy the relaxed "Independence from Irrelevant Alternatives Criteria." Furthermore, the definition of IIAC is well established in academic circles and cannot simply be redefined.
Politicians can often eliminate outcomes by introducing irrelevant alternatives to change who wins under almost all voting systems. That means politicians can make the winner become a loser by introducing a candidate who is less popular than the former winner. Introducing irrelevant alternatives includes the strategy by which parties financially help emerging candidates on the opposite end of the political spectrum to divide the potency of the opposition. Such a political trick is fairly simple and common. However, violations of this criterion are not uniform across the board. Merrill used a spatial model of 200 voters and 5 candidates repeated in 1,000 elections to determine the frequencies of such violations across the most common voting procedures. As would be expected, plurality topped the chart, with violations occurring 19% of the time. Approval voting, Borda count, STV, and Coombs violated the criterion 9%, 7%, 6%, and 1% respectively (Merrill, 1998.) More research is needed to collect empirical data on the likelihood of such violations in actual voting scenarios.
Concluding Remarks
As more and more people begin to recognize that the shortcomings of American elections are more serious than shoddy voting equipment and insufficient election administration, the need for change has crystallized. It is a problem that many of America's top leaders win with less than a simple majority of the popular vote. On the state level, the trend is even more apparent.
The last two California governors, including Arnold Schwarzenegger, won their elections without support from a popular majority. In Massachusetts, the Democratic primary for governor was won by a candidate with a mere third of the vote. Since 1990, most states have had governors who won elections with less than 50 percent of the popular vote (Hill and Richie, 2005.) And then there's the problem of filling vacancies through special elections. Such elections are notorious for low voter turnout, and by law if no candidate wins a majority in a congressional special election, a runoff election is required to fill the vacancy, an additional financial burden for taxpayers. Candidates also had to raise more money for the runoff, and independent expenditures tended to soar (Hill and Richie, 2005.)
Grassroots organizations have sprung up across the country supporting a variety of voting methods. It appears that the STV camp is the best funded and has had the most success in reform efforts. In fact, their biggest domestic victory to date occurred in San Francisco, California. In last November's election, San Francisco used instant runoff voting for local races. Two exit polls showed that city voters liked their new system and found it easy to use, including the city's many non-English speaking minorities. Previously, San Francisco decided majority winners in December runoffs. Citywide runoffs cost on average about $3 million, and voter turnout plummeted by as much as 40 percent in recent years (Hill and Richie, 2005.) Proponents claim that with IRV, San Francisco taxpayers are saving millions of dollars. Furthermore, the city also is electing winners when voter turnout is at its peak in November, and reducing the costs of campaigns. This phenomena, however, is not limited to San Francisco. Legislative bills for implementing STV were introduced in 22 states in the past two years, with states poised for real action in 2005. Ballot measures supporting STV passed by 2-to-1 margins in all three cities where it was on the ballot in 2004: Ferndale, MI.; Burlington, VT.; and Berkeley, CA (Hill and Richie, 2005.) Dartmouth College's own Student Assembly recently passed a controversial bill that will implement STV for student elections beginning this spring. Places like Australia and Ireland already have been using instant runoffs for decades to elect their highest offices.
The topic has drawn bipartisan support from Republicans such as Sen. John McCain of Arizona and Democrats such as Howard Dean, the new chairman of the Democratic National Party and former Vermont governor (Hill and Richie, 2005.) However, it must be noted that Dean, when informed of the voting paradoxes that arise when using STV claimed that he did not support STV, but rather the need to examine alternatives to plurality voting.
Although STV seems to currently have the most momentum, this does not mean that is the most ideal system. In fact, as the above analysis indicates, STV suffers some rather severe setbacks when viewed in light of voting criteria. The frequency of monotonicity violations is uncomfortably high, especially considering the nature of the criterion. The problem is even more glaring considering there are viable monotonic systems that capture many of the same advantages STV proponents champion. Approval voting does not violate monotonicity, but still encourages the development of workable third parties and elects the majority candidates.
This is not to say that Approval voting overcomes all voting paradoxes. There are instances when the procedure will elect a Condorcet loser. However, the chance of such an event is relatively low at less than 2%. Furthermore, this paper provided examples where electing a Condorcet loser did not necessarily look like a bad choice considering the profile. Making the change from plurality to approval voting is also the easiest and cheapest to implement. No new voting equipment would be needed, and the only rule change required would be to vote for "one or more" of the candidates.
Borda count also looked advantageous under the lens of the discussed voting criteria. However, it was not within this paper's scope to examine the degree to which each system could be manipulated. Borda count, because it gives candidates differential points based off of their relative ranking, encourages strategic voting. Voters will rank a candidate other than their first choice lower if it is more likely to win in hopes of helping their most preferred candidate by muting the competition.
Approval voting is much like Winston Churchill's famous quote on democracy: "Democracy is the worst form of government except for all the others tried." Unfortunately, as confirmed by Arrow's Theorem, there is no perfect voting procedure to aggregate the preferences of the public. Instead, one must weigh the advantages and disadvantages of each voting system via voting criteria. In the end, no system will leave unscathed. And while approval voting has its faults, it is the least faulty and thus the best choice to replace the status quo system of plurality.
Works Cited
Hill, S. and R. Richie. 2005. San Francisco's Innovation in Democracy- Instant Runoff. Christian Science Monitor.
Klarrreich, E. 2002. Are We Using the Worst Voting Procedure? Science News. 162(8): 280.
Nurmi, H. and Y. Uusi-Heikkilä. 1985. Computer simulations of Approval and Plurality voting: The frequency of weak pareto violations and Condorcet loser choices in impartial cultures. European Journal of Political Economy 2(1): 47-59.
Merrill, S. 1988. Making Multicandidate Elections More Democratic. Princeton, Princeton University Press.
Norman, R. 2004. Some properties of Single Transferable Voting.