Different voting methods

Posted in

-by Steve Sisson-
The purpose of this post is to compare the current majority method of tabulating election results with two alternatives: Borda and Range.

To compare these methods, I will use a typical Presidential election with two strong candidates, the Republican #1 and the Democrat #2, and a weaker independent #3 as the available choices. I am assuming that the voter base is split roughly 50/50 between left and right wing supporters, and that the independent candidate has more support from the left than the right.

ELECTION #1 – This election tabulates votes using the standard majority vote counting –

The results are listed as:

Voter: Candidate for whom they voted. i.e. the first voter in this list voted for candidate #1, the second voter for candidate #2.

01: #1
02: #2
03: #3
04: #1
05: #1
06: #2
07: #2
08: #1
09: #2
10: #1

Results:
Candidate #1 - 5 votes
Candidate #2 - 4 votes
Candidate #3 - 1 vote

Candidate #1 wins.

The complaint with this system is that only half of the people voted for candidate #1, and so 50% of the voters are supposedly unhappy with the result. We don't know which candidate the voters liked second best, so we cannot be sure that these results show which candidate actually had the strongest support from all of the voters. If we assume that a person voting for candidate #1 hated the other two, and likewise for those voting for #'s 2 and 3, then the results would be fair. However, what if candidate #2 is still moderately liked by those who vote for candidate #1, while candidate #1 is strongly disliked by those who vote for #2? In this case, #2 would have wider support and would rate higher overall among all voters than #1, but #1 would still win the election. This doesn't seem fair.

A proposed alternative that addresses this issue is the Borda method, where the candidates are ranked in order of preference by each voter, allowing voters to select both a first choice as well as a second and third alternate and so on depending on how many choices there are total. In an election with three choices, the candidates get 3 points for a top ranking, 2 for second place and 1 for third place.

ELECTION #2 – Elction results this time will be tabulated using the Borda method. Rather than the voter simply indicating which candidate they like best, the Borda method allows the voter to rank all of the candidates in order of preference.

For this election we will assume that:
Candidate 1 has strong support from the right and moderate support from the left
Candidate 2 has strong appeal to the left and moderate support from those on the right
Candidate 3 has moderate support from the left and virtually no support from the right.

Listed as:

VOTER: 1st CHOICE, 2nd CHOICE, 3rd CHOICE

01: #1, #2, #3
02: #2, #1, #3
03: #3, #2, #1
04: #1, #2, #3
05: #1, #2, #3
06: #2, #1, #3
07: #2, #3, #1
08: #1, #2, #3
09: #2, #3, #1
10: #1, #2, #3

Results:
Candidate 1 - 22 points
Candidate 2 - 24 points
Candidate 3 - 14 points

Candidate #2 wins.

This appears to be better than the majority method because although more people placed candidate #1 in the top spot than did #2, every single voter got either their first or second pick elected, indicating that even if they didn't get the guy they wanted most, they will still be "OK" with the guy in office, because the candidate with the most overall support wins.

However, this scenario assumes that the Republican still has moderate support from those on the left and vice versa. Most of the time, the left and right are strongly opposed to the candidate from the other major party, and this is where problems with the Borda system become apparent. A more realistic scenario is that #1 has strong support from the right and virtually no support from the left, #2 has strong support from the left and virtually no support from the right, and #3 has moderate support from the left and virtually no support from the right. In strongly polarizing elections such as this, especially if the results are predicted to be very close, voters would exploit the ranking provided by the Borda method in an attempt to give the candidate from the opposing party the lowest possible score, resulting in the following scenario.

ELECTION #3 – Borda vote counting –
Candidate 1 has strong support from the right and virtually no support from the left
Candidate 2 has strong support from the left and virtually no support from the right
Candidate 3 has moderate support from the left and virtually no support from the right.

01: #1, #3, #2
02: #2, #3, #1
03: #3, #2, #1
04: #1, #3, #2
05: #1, #3, #2
06: #2, #3, #1
07: #2, #3, #1
08: #1, #3, #2
09: #2, #3, #1
10: #1, #3, #2

Results:

Candidate #1 - 20 points
Candidate #2 - 19 points
Candidate #3 - 21 points

Candidate #3 wins...

...despite having only moderate support from the left and no support from the right. Because it requires each candidate to be ranked in comparison to the others, the Borda method results in the candidate with the least support winning due to being lumped in the middle between the tactical votes from supporters of the two strong opposing candidates.

The Borda method only works when people vote their honest opinion, but the reality is that they will not do this if they fear their primary choice might lose due to the election being close. They will almost always give the guy whom they want to lose an artificially low score, and the failure of the Borda method is that it puts the "nobody" candidate in the middle by default.

One voting method that contains the advantages of Borda without the danger of artificially high rankings due to tactical voting is the "range" method. Range voting uses a points system to rate candidates. Unlike Borda where a voter arranges candidates first, second and third in order of preference, this system requires voters to rate each candidate individually using a scale from 0 to 9 (or some other high number), a higher rating indicating more preference toward that candidate. Like Borda, the candidate with the highest score is the winner.

ELECTION #4 – range method -
Candidate 1 has strong support from the right and moderate support from the left
Candidate 2 has strong support from the left and moderate support from the right
Candidate 3 has moderate support from the left and virtually no support from the right.

LISTED AS:

VOTER: CANDIDATE #1 SCORE, #2 SCORE, #3 SCORE
01: 9, 5, 1
02: 5, 9, 1
03: 0, 5, 9
04: 9, 3, 2
05: 9, 5, 5
06: 4, 5, 3
07: 4, 9, 5
08: 9, 5, 3
09: 1, 9, 8
10: 9, 7, 2

Results:
Candidate #1 - 59 points
Candidate #2 - 62 points
Candidate #3 - 39 points

Candidate #2 wins

Just as with the Borda method, the candidate with the highest overall support between all voters wins.

If we now use the range method in a scenario where voters are strongly polarized, the voting goes as follows:

ELECTION #5 – range method –
Candidate 1 has strong support from the right and virtually no support from the left
Candidate 2 has strong support from the left and virtually no support from the right
Candidate 3 has moderate support from the left and virtually no support from the right.

01: 9, 0, 0
02: 0, 9, 5
03: 0, 5, 9
04: 9, 0, 0
05: 9, 0, 0
06: 0, 9, 3
07: 0, 9, 5
08: 9, 0, 0
09: 0, 9, 8
10: 9, 0, 2

Results:
Candidate #1 - 45 points
Candidate #2 - 41 points
Candidate #3 - 32 points

Candidate #1 wins.

Note that the 3rd party candidate does not get placed artificially high because the voter does not have to put them in the middle by default. Also, there is no tactical advantage in voting dishonestly because each candidate is rated individually regardless of what votes his opponents receive. In highly polarized elections, the results of range voting appear to resemble the simple majority method.

Both Borda and Range were tested with two differing scenarios. The majority method was only shown once because the results would have been the same regardless of which candidate a voter liked second best. Only the candidate with the most "#1" votes wins in the majority method, which I believe does not produce fair results in cases with two candidates that share moderate support from the opposing party. The Borda method fails in ELECTION #3 where tactical voting skews the results, so it should be ruled out as a fair voting system.

So now the question is: Does the range method produce fair results in ELECTION #5? Just as with the majority vote counting, with the range method we still end up with 50% of the voters who did not see their first choice candidate win. Is this a problem?

If we analyze the data from ELECTION #5, we see the following indications:

Candidate #1:
50% of voters strongly approve
50% of voters strongly disapprove

Candidate #2
50% of voters strongly disapprove
40% of voters strongly approve
10% of voters moderately approve

Candidate #3
40% of voters moderately approve
40% of voters strongly disapprove
20% of voters strongly approve

Even though half of the voters gave the winner a score of 0, from the election data you can also see that the same was true for candidate #2. Therefore, both candidates were disliked equally. However, 50% of voters gave #1 a "9" score while only 40% of voters did the same for candidate #2, showing that candidate #1was liked more than candidate #2. One way to put it is that while the vote was equally negative for both candidates, it was more positive for #1 than it was for #2. In the range voting method, the vote totals are basically an "approval score." In this example, a score of 90 would be a 100% positive approval by all voters. Candidate #1 has a 50% positive approval rating, while candidate #2 has a rating of 45% and candidate #3 gets 29%. The candidate with the strongest approval rating among all voters is the winner.

If we take the same approach in ELECTION #4, we see that the winning candidate has a positive approval rating of 66%, while candidate #1 comes in at 64% followed by #3 with 41%. Therefore, I believe the range voting method does produce fair results in both experiments.

CONCLUSION:

The majority system works in highly polarized elections, but fails when there are two candidates that share support between the parties.

The Borda method addresses the shortcomings of the majority method by allowing voters to rank candidates in order of preference. However, because it requires candidates to be ranked in comparison with each other, it allows the use of tactical voting to produce false results.

Range voting is essentially the same as the Borda method, except each candidate is rated individually. Because the candidates are not compared to each other, it eliminates the flaw that breaks the Borda method.

To the best of my knowledge, no public elections are counted using the range method. Why not?

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