(Skip to conclusions) (Executive summary)
1. What they are. Range voting: In an N-candidate election, each vote is an N-tuple of numbers each in the range 0 to 99. The Kth number in the tuple is a "score" for candidate K. You take the average of all the Kth entries to find the average score for candidate K. The candidate with the highest average score wins. (Voters are allowed to leave an entry blank to denote "don't know anything about that candidate." Blank entries not incorporated into average.)
Borda: Each vote is an ordering of the N candidates from best to worst. (Voters are not allowed to omit a candidate they know nothing about, and are not allowed to regard two candidates as equal.) The top ranked candidate gets a score of N-1, the bottom ranked 0, and more generally the Kth-ranked gets a score of N-K. All scores are summed and the candidate with the highest total score wins.
Range and Borda voting are actually very similar as far as their advantages are concerned, but differ a lot with respect to Borda's disadvantages. (One way to see it: the only difference between Borda and RV is that RV lets you choose what point assignments you'll give to the candidates; while Borda sometimes makes the choice for you. That makes RV more democratic, expressive, and responsive than Borda.)
2. Strategic Borda Voting. The biggest problem with Borda is that it reacts very badly to "strategic voting."
I remember one time when I worked for NEC Research Institute and we had to vote to decide who, among about a dozen candidates, to hire. There were several camps, each favoring a different candidate who excelled in one way or another. There were also many mediocre candidates – nonentities – whom nobody particularly wanted. Arguments grew impassioned.
So then our boss, who definitely was several watts shy of being the brightest bulb in the box, said "we have to be fair about this. Let's vote." And he right then (apparently) independently reinvented the Borda count system and told us all to provide our Borda votes (preference orderings) – by secret ballot of course, since he'd read in some managerial handbook that secret ballots were better. Then he'd add them up. There were one or two attendees who immediately objected that "strategic voting" would foul that up. My boss, of course, wasn't going to listen to a bunch of whining nerds about that.
Well, of course, since everybody there was an arrogant pushy scientist, everybody quickly figured out that the thing to do was to rank your favorite first, then artificially rank all of his perceived major rivals artificially last. Finally, by the rules of the Borda system, the non-entities had to be ranked somewhere in the middle. (Of course, you could honestly rank all the top candidates top, but that would be like having one-tenth of a strategic vote, unacceptably weak. Nobody could afford to be that stupid and ineffectual. Plus, there was no public embarrassment about submitting lies in your ballot, since everybody's ballot was secret.) Then, my boss made a great show of closing his eyes and shuffling all the ballots... then opened them and added them up... and golly... that was odd... the most-favored candidates all seemed to be ones that had been dismissed as nonentities before... they must have a good deal more support than he'd realized... hmm... the result-order really seems quite illogical and random... but after all, this clearly IS the most fair possible voting system, so we have to concede that Mr. Putz really is the truly most-favored candidate, unexpected though he may be... ok, meeting over, thanks for all your input, folks.
NEC Research Institute eventually collapsed and nearly all its scientists were fired. This particular meeting, in its small way, was one contribution to its downfall. (The parent company NEC still survives, thanks to a bailout by the Japanese government.)
AS A PICTURE: if the 3 good candidates are A,B,C then the strategic votes are:
A > nonentities > B > C (cast by about 1/3 of the voters)
B > nonentities > C > A (cast by about 1/3 of the voters)
C > nonentities > A > B (cast by about 1/3 of the voters)
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total: A,B, and C each get an average score of about N/3,
whereas the nonentities get, on average, a score of about N/2.
So a nonentity always wins and the 3 good candidates always
are ranked below average. In this kind of scenario Borda actually
performs worse than plurality voting.
Incidentally, this "DH3 pathology" is both very serious (near-pessimal candidate elected) and very common (occurs whenever there are 3 comparable rivals) and it happens not only under Borda voting, but under all the commonly-touted voting systems of "Condorcet type." This severe reaction to strategic voting seems a very good reason to discard all Condorcet-type voting methods!
Here's another related anecdote about Borda, related by Steve Eppley: (Salvador Barbera told this story during his course on strategy-proofness at Caltech several years ago, and insisted it was a true story.) Once upon a time at a major university in Europe, the economics department was hiring a new colleague. There were 4 applicants: One was a world-class macroeconomist, one was a world-class microeconomist. The other two were mediocre, one clearly better than the other. When it came time to select one by voting using Borda, about half the department preferred the macroeconomist, with the microeconomist as their second favorite. The rest of the department preferred the microeconomist, with the macroeconomist as their second favorite. They each had a pretty good understanding of each other's sincere preferences, and thus expected the election would be close. You can guess what happened: When they voted, each voter raised the two mediocre candidates over his/her second favorite, hoping to manipulate the outcome if the Borda count was close. As a result, one of the mediocre candidates was voted everyone's second choice and had the largest Borda count.
3. Contrast this with Strategic Range Voting. Strategic voting also hurts us in the range system, but not nearly as much. For example, in the scenario above, nobody would feel strategically forced to middle-rank the nonentities, and in fact, they would, quite honestly, rank them last. Hence, all the votes would consist of 99s and 0s for the good candidates, and 0s for the nonentities. (The 0s for the good candidates would be dishonest strategic attempts to hurt rivals.) Despite this strategic dishonesty, one of the 3 good candidates would still necessarily win.
4. Voter Expressivity – and What if the voters actually are honest? Well, they can be more honest and more expressive with range voting than with Borda voting, so we would expect better results.
In range voting, voters can express the idea that they think 2 candidates are equal. In Borda, they cannot.
Range voters can express the idea they are ignorant about a candidate and want to leave the task of rating him to other, hopefully more knowledgeable voters. In Borda, they can't choose to do that.
Borda voters who decide, in a 3-candidate election, to rank A top and B bottom, then have no choice about C – they have to middle-rank him and can in no way express their opinion of C. In range voting, they can.
If you think A>B>C>D>E, undoubtably some of your preferences are more intense than others. Range voters can express that. Borda voters cannot.
Surveys show that a lot of voters want to just vote for one candidate, plurality-style. In range voting they can do that by voting (99, 0, 0, 0, 0, 0). With Borda, they can't do it. [Asterisk: Borda with ballot "truncation" would allow plurality-style voting, which also goes some way to allowing expression of ignorance about some candidates, albeit only a downward-biased sort of ignorance. But permitting this kind of Borda voting gives up various beautiful mathematical properties discussed in Donald G. Saari's books...]
5. Strategy and 2-Party Domination. Most will, in an election like Bush v Gore v Nader 2000, exaggerate their good and bad opinions of Bush and Gore by artificially ranking them first and last, even if they truly feel Nader is best or worst. They will do this in order to give their vote the "maximum possible impact" so it is not "wasted". (If they voted Nader top, then their vote would only have half the impact they get by exaggerating Gore versus Bush. They can't afford to waste half their vote like that because their opponents aren't going to be that stupid.) Once they make this decision, with Borda, Nader automatically has to go in the middle slot, they have no choice about him. If all voters behave this way, then automatically the winner will be either Bush or Gore. Nader can never win a 3-candidate Borda election with this kind of strategic voters. (Unless it is an exact 3-way tie and the tie-break goes Nader's way, which'll never happen in reality. Note: Nader can win this kind of election if there are more than 3 candidates.)
In view of this, third parties would be silly to push Borda. They should advocate range.
Analogously, in range voting, if the voters exaggerate and give Gore=99 and Bush=0 (or the reverse) in order to get maximum impact and not waste their vote, then they are still free to give Nader 99 or 0 or anything in between. Consequently, it would still be entirely possible for Nader to win with range, and without need of any kind of tie. (For example, if 51% of the voters rated Bush=99 and Gore=0, and 49% rated Bush=0 and Gore=99, and 60% of the voters rated Nader>90, then Nader would win easily with range voting.)
Think this kind of strategic thinking won't matter much? Wrong. The "National Election Study" showed that in 2000, among voters who honestly liked Nader better than every other candidate, fewer than 1 in 10 actually voted for Nader. That was because of precisely this sort of strategic ploy – these voters did not wish to "waste their vote" and wanted "maximum impact" so they pretended either Bush or Gore was their favorite. (Same thing happened with voters whose true favorite was Buchanan.) In short, strategy has an enormous impact in the real world, and over 90% of real voters act strategically and not honestly, given the chance. [Another, different sort, of example where strategic voting is known to have had a huge impact, was described on p.65 of Lakeman & Lambert's book: a 1950 Gallup poll showed 38% of British voters wanted to vote Liberal but only 9% did.] That is exactly why third parties always die out and the USA is stuck with 2-party domination.
6. And what about with more than 3 candidates? Chaos. In plausible many-candidate situations, with optimal voter strategy, the Borda winner becomes almost random. (Not "2-party domination," rather, "utter chaos.") Range is "Borda done right" by giving the voter flexibility. Borda's bizarre reactions to strategic voting can actually arguably make it a worse voting system even than plurality (e.g. plurality handles the DH3 pathology without difficulty).
7. Think this kind of chaos wouldn't happen in real governmental elections? Wrong. In fact, it is even worse than we said: A lessened form of the DH3 pathology can occur with only two strong candidates – if they are close to being tied under strategic voting, then the most strategically-above-average among the mediocrities will win. (See Barbera's anecdote above for an example.) The Pacific Island Republic of Kiribati (pop. 60,000; formerly the Gilbert Islands) is, as far as we know, the only government which has adopted Borda voting. [Nauru uses its own, different, weighted positional voting system, which seems to work better than Borda.] It immediately suffered from exactly this kind of problem as the two "most popular" candidates were eliminated and two dark horses who did not campaign and were not recognized as serious contenders were nearly elected [See p.368 of Benjamin Reilly: Social Choice in the South Seas: Electoral Innovation and the Borda Count in the Pacific Island Countries, Int'l Political Science Review 23,4 (2002) 355-372. Steve Eppley says it is important to point out here that the Kiribati Borda election in this paper was not a public election; it was a nomination vote (something like a "primary" in the USA) in which the voters were the national parliament.] It appears that Kiribati then abandoned the Borda system and went back to the (often superior for strategic voters) plurality system and now has 2-party domination.
8. Clones. Suppose several near-identical "clone" candidates run. Under plurality voting: they split the vote and all lose. Under Borda voting (basically): with enough clones, one is assured victory!
Just to be sure you follow that, let's examine this in detail.
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Cloning the Bores gives them a huge advantage! The Boring party can just arrange to have a lot of Bores enter the race, and game over! Totally Boring! Of course, the Mushites could try to defeat that by intentionally lying in their votes by ranking the Bores in the opposite of their true order, to try to cancel out the Borites and make Mush win. But this strategy causes them to be massively dishonest in their votes and risk not only a Bore-victory, but in fact a victory by the worst of the bores! (Which would in fact happen if the Borons counterstrategized by also being dishonest in their Bore-orderings!) This is all crazy!
And it gets crazier. The Mushites, knowing this, would probably try to fight back by also nominating their own clones, say Mush1, Mush2, Mush3, and Mush4, so with more clones that the Bores, they'd win. But then the Bores could sponsor some more clones Bore4, Bore5, Bore6. And so on. The war of the clone armies. It would make no sense whatever. It would be all about which side had the gall to generate the most ridiculously huge set of clones, and have virtually nothing to do with what the voters actually wanted.
Now compare the situation with Range voting. Aaaah. Much better: clones don't matter, election result not affected. Can't manipulate the election by creating or abolishing clones.
Which would you prefer?
9. Removing an "irrelevant loser" candidate. In case that was too abstract for you, consider this related simple Borda example by Paul Johnson, involving 4 candidates and 7 voters. Start looking at the situation on the left.
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Now suppose it is discovered that the far-last-place loser D – top-ranked by nobody – was a criminal and a non-citizen and hence was not eligible to be a candidate. (Actually, felons are not allowed to vote in, e.g, 2005 Florida, but it is legal for them to hold office. But never mind that.) Fine. Eliminate D from the election, do everything over again with the same orderings of A,B,C in all votes. Surely we will get the same winner as before, right? Wrong! The result of the Borda election now is completely reversed!
(In contrast, with range voting, eliminating some non-winner candidate from all votes never changes the election result. Also note in Johnson's example that if the winner C drops out, that reverses A-versus-B.)
10. One-party domination: One interesting consequence of Borda's "teaming" effect above (that a party, by running many clones, increases its chances of victory) is that Borda can lead over time to one-party domination. (Sort of like two-party domination, but even less fun.) Here's an observation of that by Tim Hull.
I'm [a student] at the University of Michigan, and we use a variant of the Borda count for our elections where you get as many votes as open seats. Slates of candidates typically contest elections as "parties", and most discussion of elections revolves around these parties.11. Bayesian regret (For Statistics Nerds): Extensive computer simulations of millions of artificial "elections" by W.D.Smith show that range voting is the best single-winner voting system, among a large number compared by him (including Borda, Plurality, IRV, Condorcet, Eigenvector, etc.) in terms of a statistical yardstick called Bayesian regret. This is true regardless of whether the voters act honestly or strategically, whether the number of candidates is 3,4, or 5, whether the number of voters is 5 or 200, whether various levels of "voter ignorance" are introduced, and finally regardless of which of several randomized "utility generators" are used to generate election scenarios.
Anyway, the system as-is works better than at-large plurality, but it still leaves much to be desired. The biggest problem with the current system is that the largest party slate always wins a disproportionately high number of seats – so large, in fact, that competition has generally withered away.
Smith's papers on voting systems are available here as #56, 59, 76, 77, 78, 79, 80...
12. Voting machines. You can't run Borda elections on plurality voting machines. (No way for plurality machines to detect illegal Borda ballots, for example.) You can run range elections on any plurality voting machine, without modification, right now.
13. "Write-in" candidates. There seems no way, or no pleasant way, to allow voting for "write-in" candidates in Borda elections.
14. Summary. Isn't the purpose of voting to provide information about your opinions? Why would you want to have a system (Borda) that forces you to express less information, when you can have one (Range) that permits you to express more?
If you think 2-party domination is a bad thing and would like to see a greater diversity of parties and more voter choice, then why would you want Borda (in which, with strategically-exaggerating voters in a 3-candidate election, 3rd parties have no chance) when you could have Range?
And why would you want a system (Borda) that we know reacts very badly to strategic voters, often giving a below-average candidate the victory and often exhibiting "chaotic" behavior – including in the only actual government (Kiribati) where it has been used?
(Duh!) You want range voting. Forget Borda voting.