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Perfect Examples:
Flaws with Voting Systems

July 2002

Stanford Economist Ken Arrow received the Nobel Prize in 1972 for proving (in 1951) that there is no such thing as a perfect voting system.  This means that for any system one can come up with an example that makes that system look bad.  The question, therefore, is not whether one can come up with an example, but rather how realistic that example is.  A second question is whether one has any real examples of absurd outcomes, instead of invented ones.

This page lists examples that illustrate flaws with a few commonly discussed voting systems:

  • Plurality voting
  • Approval voting
  • Pairwise (Condorcet) voting
  • Instant runoff voting
  • Borda count (point system:  4 for a 1st choice, 3 for a 2nd and so on)

We leave it up to you to judge how realistic these scenarios are and whether you can find any real-world examples of actual elections that involved such scenarios.  We've got some more information about this topic on our website.


Plurality voting has a flaw that occurs frequently:  when a small number of people vote for a third party candidate, they can swing a close race to the major party candidate opposed by a majority of voters. This happened in the 2000 presidential with Democratic candidate Al Gore and Green Party candidate Ralph Nader both in the state of Florida and nationally.  It also occurred in the states of New Mexico and Iowa, where Pat Buchanan supporters swung the states to Gore and in a US Senate and a state senate race in Washington State, where Libertarian candidates swung the race to Democrats, which ended giving Democrats control of both the state senate and the US senate, after Jim Jeffords defection from the Republican Party.

Approval voting

In approval voting, voters can vote for or approve of as many candidates as they like, and the candidate with the most votes wins the election.

Imagine the following scenario:

     Percent of   Candidate (w/grade)
     Electorate   in order of preference
Group 1  40%   Jones (A), White (C+), Smith (C-)
Group 2  35%   Smith (A+), Jones (C), White (F)
Group 3  25%   White (A), Smith (B+), Jones (F)

Under a two-round or instant runoff, no one has a majority after the first round, so White is eliminated, her supporters prefer Smith, and Smith is elected 60% to 40%.  This makes sense. White had little enthusiastic support, and when the race went to Jones versus Smith, most of the voters preferred Smith.

But what happens under Approval Voting?  If each of the factions does not bullet vote, but rather approves of their top two choices, Jones gets 75% of the vote, White gets 65% of the vote, and Smith gets only 60% of the vote!  Even though most voters prefer Smith to Jones, Jones wins!

Well, when the uproar dies down, what does Smith do in the next election?  He asks his voters to bullet vote, of course.  After all, if his enthusiastic supporters had simply bullet voted the first time, he would have had a better chance of winning; his voters caused Jones to win.  (Obviously, if only Smith's voters bullet vote, and everyone else votes the same way in the second election as in the first, White will win.  This will cause Jones to also tell her supporters to bullet vote, either when she gets wind of Smith's strategy, or in the third election.)

Thus we see that this sort of thing is contagious, as anyone who has been around politics (as opposed to academia) knows, and before long, everyone asks their supporters to bullet vote, causing Approval Voting
to degenerate into simple plurality.  If this didn't happen in the first election, it would happen in the second and subsequent elections.

Pairwise (Condorcet) voting

In the above example, there is no pairwise winner. 

Smith defeats Jones, 60%-40%
Jones defeats White, 75%-25%
White defeats Smith, 65%-35%

So in this situation, you have to use another method to eliminate one of the candidates to determine the winner.

In addition to the flaws revealed by specific examples, approval and Condorcet voting suffer from another real world problem:  they encourage candidates to avoid taking public stands on controversial issues, since the least offensive candidate can emerge victorious.  Thus, compared to instant runoff voting, they create a less-well informed electorate.

Instant runoff voting

A commonly cited example is one in which two extreme candidates have strong core support, neither can appeal to a majority, and a compromise candidate has weak core support but is preferred by a majority over the other two candidates.  For example:

Candidate    Support
Jones            45%
Marvin Moderate  15%
Smith            40%

If the supporters of the extreme candidates prefer the moderate to the other extremist, then the moderate candidate would win head-to-head races against both of the other candidates.  In an instant runoff or a two-round runoff with this example, the compromise candidate is eliminated, and one of the extremists is elected.

As an astute visitor pointed out, instant runoff voting generally does a better job of finding the true compromise candidate than either plurality or two-round runoff elections.

Borda Count

In an election with n candidates using a Borda Count, a candidate receives n-1 points for each 1st choice, n-2 for each 2nd choice and so on.  The candidate with the most points wins the election.

The chief flaw in this system is that your vote for your 2nd choice can end up defeating your first choice.

Acknowledgements:  Big thanks to Steve Chessin, Jim Lindsay and others who contributed text and examples to this page.

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Copyright 2002 The Center for Voting and Democracy
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