Miscellaneous
Comments on Voting Systems

Jon Kirwan

Introduction

Voting lies at the heart of representative government and participatory democracy. In fact, it plays an essential role whenever control is shared and group choices about policy or action must be made. Party nominations, stockholder elections, U.S. economic goals, and community priorities are often decided through voting.

Voting is a solution to the problem of turning individual preferences for different outcomes into a single choice by the group as a whole. And although many procedures have been developed to reduce a list of alternatives to a desired option, it's surprising how few people realize that the method of voting they use can significantly impact the outcome of an election. Worse is just how likely it is to find that the person or issue with the highest tally doesn't actually hold a majority of the votes cast.

Every voting method occasionally gives paradoxical results, so no one method is always best. Methods such as majority rule, plurality wins, elimination and runoffs, sequential pair-wise comparisons, various weighted or scoring schemes, approval voting, and a host of other partitioning schemes that choose between successive subsets of potential outcomes; all of these have situations where they choose poorly, some much more than others. The problem lies in choosing one that minimizes the chance of counter-intuitive results.

Kenneth J. Arrow was able to prove, in 1952, that finding an absolutely fair and decisive voting system is impossible. He received the Nobel Prize in Economic Science for his work on the theory of general economic equilibrium and won the 1986 von Neumann Theory Prize for his fundamental contributions to the decision sciences.

Although searching for a perfect voting scheme is fruitless, it isn't a waste of time trying to improve existing methods. For example, we all know that engines cannot perform with 100% efficiency, but that fact shouldn't stop the search for better efficiency and utility.

Simple Isn't Always Good

Plurality voting is arguably the simplest method of selecting an outcome: everybody votes for their favorite choice and the choice with the most votes wins. But what about the case where there are 20 options to vote for and a winning plurality of only 6% succeeds. This is certainly a possible outcome for issues with factions of nearly equal size, but where one faction has a slight numerical advantage over the others. With 20 different options, and an average of 5% each, it's possible to see a 6% plurality win.

Now, of course this could be solved by successive runoffs where a few of the top choices are re-submitted to the voters. But consider another situation with only three possible outcomes: less taxes, the same taxes, and more taxes for everyone. Let's assume for the moment that the breakdown of support on these three outcomes is: 40%, 35%, and 25%, respectively.

If everyone votes their first choice, those wanting less taxes would win by plurality. But what if the voters already had a pretty good idea of the breakdown of support? Wouldn't it make sense for those wanting more taxes to compromise their vote and side with those wanting to keep taxes the same? The outcome would then be to leave taxes alone - which is probably preferable for those wanting more taxes than would be allowing the "less-tax" group to win. This is called strategic or insincere voting. Of course, you already know that this kind of coalition voting actually happens!

A pure plurality system of voting generally performs so poorly with multiple choice questions that it is rarely used. More often, the choices are broken down into smaller groups (such as party members) that are offered in preliminary voting rounds (such as primaries) before a small set of choices (the candidates) are taken from the winners of the preliminary rounds and finally offered.

Rather than spend too much time beating up on one voting system though I'd like to describe a hypothetical case that illustrates a fundamental problem of voting methods. It'll be easier to see some of the issues surrounding the selection of voting systems.

Take Your Pick

Here's an extreme case and it'll shed some light on just how important it is to select an effective voting system. Let's assume that there are 55 delegates to a national, political party convention, at which five party members have been nominated to be the presidential candidate. We'll call the nominees, Alpha, Beta, Gamma, Delta, and Eta.

Each delegate must rank all five nominees according to his or her preference. Although there are 120 conceivable distinct rankings, many fewer than that will appear in practice because electors tend to split up into blocks with similar rankings (and, of course, because we only have 55 delegates in this example.)

Let's assume that the 55 delegates submit their preferences and we find only 6 different schedules among them.

Here's the breakdown:

Preference Schedule

Delegate count:	18	12	10	9	4	2
1st choice	Alpha	Beta	Gamma	Delta	Eta	Eta
2nd choice	Delta	Eta	Beta	Gamma	Beta	Gamma
3rd choice	Eta	Delta	Eta	Eta	Delta	Delta
4th choice	Gamma	Gamma	Delta	Beta	Gamma	Beta
5th choice	Beta	Alpha	Alpha	Alpha	Alpha	Alpha
For example, 9 delegates indicated their preference for Delta, Gamma, Eta, Beta, and Alpha, in that descending order.

Now let's look at how different outcomes can result from different voting methods. It's interesting to see how "reasonable" methods can yield such diverse results.

Plurality

If the party elected its candidate by a simple plurality, nominee Alpha would win with 18 first-place votes, in spite of the fact that Alpha was favored by less than a third of the electorate and was ranked dead last by 37 delegates.

Runoff

If the party decided that a runoff election should be held between the top two contenders, who together received a majority of the first-place votes in the initial ballot, then nominee Beta outranks Alpha for 37 of the 55 delegates and is declared the winner in the runoff.

Eliminate the loser

If a sequence of ballots were used, eliminating the nominee with the fewest first-place votes at each stage, then the elected candidate would be Gamma.

In this example, the first ballot would eliminate Eta. Of the six delegates voting for Eta in this first ballot, 4 had listed Beta as their second choice, so their 4 votes would go to Beta in the second ballot. The other two "Eta-voters" had Gamma listed as their second choice, so Gamma would pick up their votes on the second ballot. This second ballot would eliminate Delta. Those voting for Delta in the second ballot would then select their runner-ups in the third ballot. On the final ballot, Gamma would win.

Borda count

Since each delegate provided a complete preference schedule, the party might have decided to use a straight Borda count to pick the winner.

Each first-place vote would get 5 points, each second-place vote would get 4, and so on. The highest total score, 191, is then achieved by Delta, who wins. Note that Alpha has the lowest score of 127 and Beta, the second worst, with 152.

Condorcet

Here, each nominee is matched head-to-head with every other.

Eta wins over Alpha by a vote of 37 to 18, over Beta by a vote of 33 to 22, over Gamma by a vote of 36 to 19, and over Delta by a vote of 28 to 27. Eta is the only nominee to beat every other nominee, when paired against only one other nominee at a time.

Good Can Be Simple

A better method is approval voting, where each voter is allowed to give one vote to each of the candidates or options on the slate. No limit is set on the number of candidates that an individual can vote for: they can approve of as many choices they like and withhold approval for those they don't. This system replaces the "one person, one vote" with "one candidate, one vote" and the winner is the candidate or option receiving the largest number of approval votes.

I like approval voting because it's a method that is robust and yields good, intuitive results with rare exceptions. It is particularly good in multi-candidate contests such as party primaries and it works well where more than one candidate or option may win; for example, in electing a limited number of new members to an academy or board from a larger group of candidates.

Approval voting is practical and simple to implement. It's quite easy to let voters punch out a few more holes, if they wish, or mark a few more boxes. The only change is to then permit and accumulate those votes, when tabulating them, and it's easy to explain, too. Finally, it gives voters greater freedom in expressing themselves without requiring complicated ranking schemes.

Approval voting is simple and versatile and it deserves widespread implementation as the preferred method for social choice.

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    Creation Date:  Mon 29-Mar-1999 15:22:27
    Last Modified:  Tue 25-May-1999 19:18:55
    Copyright (C) 1999 Jonathan Dale Kirwan