Voting Systems

Voting is a useful means of dispute resolution when large numbers of people have to choose among alternatives. Voters choose officials such as presidents, prime ministers, governors, mayors, legislators, city councils, and school board members.

What is required for an election to be “fair?” In 1785, Marquis de Condorcet suggested that fairness required the winner to be the candidate who could defeat the others one-on-one in pairwise elections. But, there is a possible paradox. Consider three voters choosing among three candidates for a single office:

Voter 1: Prefers Candidates in A B C order Voter 2: Prefers Candidates in B C A order Voter 3: Prefers Candidates in C A B order

Each candidate has one first, one second, and one third place result. Each candidate is preferred over one of the others by two voters. There is no clear winner, nor even a fair way to reduce the candidates by one for a runoff election. That is, Condorcet’s suggestion may not always produce a winner. Of course, the likelihood of such perfect symmetry is infinitesimally small in any election with a substantial number of voters, but it does provide a method for judging fairness, as no method of aggregating votes and determining winners should result in the defeat of a Condorcet winner.

Arrow (1950, 1951) demonstrated that election results can change based on the way elections are conducted. Specifically, his theorem [alternatively, paradox] states that when there are two or more voters and three or more candidates for a single position it is impossible to design a voting system that meets the following five criteria for rationality and fairness:

· The vote must rank all possible choices relative to one another, and all candidates must have the same rank no matter how votes are counted.

· Each voter should have equal weight.

· Every candidate should have a chance to win.

· Strategic voting should be impossible. A voter should not be able to hurt one candidate by promoting another.

· If every voter prefers a certain candidate, then so must the result. If this condition is met, the choice is Pareto optimal.

Arrow's theorem is often expressed somewhat loosely as “No voting method is perfectly fair” or “Every voting method is flawed.” One of these flaws is distortion of outcomes by spoilers or clones. Perot in 1992 probably drew more votes from Republicans and allowed Clinton to win; Nader in 2000 probably drew more votes from Democrats and allowed Bush to win.

Another flaw is strategic voting, in which the supporters of a candidate try to eliminate the strongest rival in the other party. It is possible, although difficult to coordinate in the real world except in special circumstances. In US states with open primaries, strategic voting becomes possible if one party has an uncontested, and the other a contested primary. One such case was the 2008 Michigan presidential primary. The Democrats had only one candidate on the ballot so independent-leaning Democrats were encouraged to vote for the weakest Republican candidate in hope of increasing the chances of the Democrats in the presidential election. If Ohio State University named the members of the University of Michigan football team and vice versa, alumni would have been in an uproar because it would ensure fielding the weakest teams for both schools in one of football’s greatest rivalries.

Many different systems exist for conducting many different kinds of elections (Brams & Fishburn 1991, Nurmi l987). Eight major ones, each with its own strengths, weaknesses, and variations, are described below.

Majority voting requires a winning candidate to have 50% + 1 of votes cast. Sometimes, particularly for tax measures or impeachment, super-majorities such as 60% or 66.33% are required. In other systems, double majorities (of turnout and approval) are required. In majority voting for offices, if no candidate receives the required majority, a runoff election occurs with a predetermined number of candidates (usually the top two). Majority voting is common in European national elections. Strategic voting is possible.

Plurality voting requires a winning candidate to have more votes than any other candidate. The system produces a winner in the first round every time, and is common in US elections for public office at all levels of government. Spoilers and clones can be significant factors in plurality systems. The US system of independent state primaries spread over several months to select presidential candidates is in effect a system of successive plurality elections in which the candidates themselves decide when to drop out. The Electoral College that elects US presidents equals the total number of senators (2 per state) plus congressmen (435) plus three from the District of Columbia. The intent was to prevent a few big states (originally New York, Pennsylvania, and Virginia) from dominating elections.

Cumulative voting gives each voter a fixed number of votes, usually equal to the number of seats at stake, to distribute among candidates in any way desired, including giving all of one’s votes to a single candidate. Such a system ensures that minorities (however defined) can elect representatives approximately equal to their proportion in the community without gerrymandering. Consider a city population divided between minority Greens and majority Blues. Assume an election to select a city council, with each voter having as many votes as there are contested seats on the city council. The optimal strategy for the minority Greens is to run candidates approximately equal to their proportion of the electorate, and to concentrate all of their votes on those candidates. This will insure that the majority Blues will not have enough votes to prevent the election of those candidates. If the Greens dilute their vote across a larger number of candidates, they risk losing all seats. Conversely, despite their advantage in numbers, if the Blues try to shut out the Greens, they risk losing control of the council. For both sides, putting up a full slate of candidates for all seats risks losing all races.

Ordinal voting requires voters to rank candidates from highest to lowest. Some systems require ranking every candidate while some do not. Some systems allow for tied rankings. Calculating a winner can be complicated and ties are possible. Borda devised one such system in 1781. He proposed having each voter assign 0 points to the candidate he ranks lowest, 1 to the second lowest and so on up to C-1 points for the top-ranked candidate, where C is the number of candidates. Each candidate’s points are summed across all voters, and the candidate with the highest total wins. Unfortunately, voters are likely to rank candidates strategically to give their preferred candidate the best chance of winning by giving the lowest ranking to the most powerful rival regardless of actual preference. The effect is even greater if voters do not rate some of the candidates (Nurmi 1987). This strategy is easy for every voter to carry out, making a Condorcet winner unlikely.

Single Transferable Vote or Instant Runoff is a variation on ordinal voting devised in the 1850s by Thomas Hare and used in Australia, Fiji, Malta, Nauru, Northern Ireland, and the Republic of Ireland. It is applicable to at-large elections of multi-seat bodies and requires voters to rank candidates in preference order. Initially, the first preference of each voter is used to order the candidates. The number of votes needed to win is determined after all votes have been cast using a preselected formula. The most common, the Droop quota, is 1+ v/(s+1). An election run under the Droop quota in which 1000 votes were cast for 9 seats would require 1 + 1000/(9+1) = 101 votes to win a seat.

Approval voting allows voters to rate each candidate simply as “acceptable” or “unacceptable.” The candidate with the most “acceptable” votes wins. The most prominent use is the selection of the United Nations Secretary General. It eliminates the problem of clones and spoilers. It is more likely than plurality voting to select the Condorcet winner. It is simple for people to understand and use, and works on existing voting machines. Compared with majority voting with runoffs, it is less likely that the candidates eliminated in the first round would not have been able to win in the second. This is particularly the case when similar candidates split the vote in the first round or in primaries conducted to select party candidates, where it might be a particularly useful system. It also eliminates the dilemma faced by members of minor parties between wasting their vote to prevent election of their least preferred candidate and expressing their true opinion. However, ties are more likely in this approach than in most. Claims that it will reduce negative campaigning or increase voter turnout are unproven, the opposite seeming equally plausible. It also is likely to produce a profusion of candidates, such as the California 2003 gubernatorial recall with over 200 candidates, most running only in hope of gaining a little publicity and ego-gratification.

Range voting permits voters to express their opinion about each candidate by assigning each a score within a predetermined range (usually 1-5 or 1-9). Olympic diving and gymnastic competitions use range voting. Voters may give the same rating to several candidates. Ties are unlikely. Range voting has potential for reducing two-party domination, eliminates cloning, and is simple enough to run on existing voting machines, but fraud is possible if voters leave some candidates unranked. Some argue the ranking will increase voter turnout; some that the complexity will reduce it. It increases the advantage of strategic voting and becomes confusing to voters as the number of candidates and range of allowable points increase. In some versions, voters may express no opinion about a candidate, in which case it is necessary to establish quorums to prevent a candidate with a few high scores and a large number of abstentions from winning.

Weighted voting gives each voter a different numbers of votes based on some predetermined qualification. Corporate elections in which individuals get one vote for each share of stock held is a form of weighted voting. Another is the UN Security Council, in which five permanent members have veto power.