Why Range Voting is better than Condorcet methods

(Executive Summary)   (Return to main page)

Well, first of all, what is a "Condorcet method"?

There are two possible answers. With the first definition of "Condorcet method," range voting is Condorcet, in which case debate over – issue moot.

First definition: A "Condorcet method" is any voting method that obeys the "Condorcet property" that it always elects a "beats-all-winner" if one exists. A "beats-all-winner" is a candidate who would beat every other candidate in the two-choice head-to-head election got by erasing every other candidate from all votes.

Well, according to that definition, Range Voting is a Condorcet method, since if you erase all candidates and all numerical votes for them in all range votes – except for two candidates A and B – then A will beat B in the resulting 2-choice range election if and only if A beat B in the original election. Because erasing the votes for the others has no effect on A and B's individual totals.

Second (subtly different and more traditional) definition

Therefore we shall concentrate on this different second definition from now on: (quibbles about this)
A "Condorcet-winner" is a candidate who would beat every other candidate in a two-choice head-to-head election with every other candidate killed, and the voters now get to re-vote on just the two live candidates knowing that the other candidates are now dead (but remaining consistent in the sense they never reverse a previously-stated preference relation "A>B" if A and B are the two live candidates). Now a "Condorcet method" is any voting method that always elects a "Condorcet-winner" if one exists.

The two definitions actually are the same if we only are considering methods that take as input votes that are exactly rank-orderings of all the candidates such as "A>C>B>D>E" in a 5-candidate election among {A,B,C,D,E}. Condorcet had not considered methods with more expressive kinds of votes than just a rank ordering (such as range voting where you can express different intensities of preferences, not just the bare fact you prefer A over B) hence the distinction between these two definitions did not arise as an issue in his mind.

The reason this slight definitional change makes a difference is because a range voter who had originally voted A=56, B=57, could (and we assume would) now in the new shrunk 2-choice election change that vote to A=0, B=99. So with the new definition, range voting is not a Condorcet method.

"Free the slaves" example illustrating the fact range is not Condorcet: Consider a multi-option range voting election in which two of the choices were

  1. Free the slaves and ban slavery.
  2. Make no change to the status quo, i.e. keep having slavery.
  3. ...
Suppose the votes were: 40% say "ban slavery" rating it 99 points higher than keeping slavery; the other 60% say to "keep slavery" rating it 50 points higher than banning slavery. (These numbers are fairly realistic assuming the slaves are allowed to vote and the votes are honest.) In that case "free slaves" beats "keep them" by a lot. But now if the election were redone restricted to these two choices only, and every voter in the new election maxed out his votes to 99 and 0, then "keep slavery" would win over "ban it" by 60-40. (Another example.)

Remark: This example also illustrates the fact that obeying the traditional definition of the Condorcet property is not necessarily always good! This is an example of the "tyranny of the majority" where a majority of voters who prefer something by a little, cause immense harm to a minority of voters who prefer the opposite by a lot. Although range voting can still cause tyranny of the majority, it at least offers the hope – if enough voters choose to be honest about the situation – of escaping from it. With Condorcet methods, there simply is no hope for such escape.

Although range voting is not one of them, there are many known Condorcet methods that do obey this second definition. They vary from "more complicated to describe than range voting" to "way more complicated to describe than range voting." (If you don't believe me, try writing a computer program to do both. The range voting program will be shorter.) The first such method was invented by Condorcet himself in 1785. One of the latest and greatest such methods is Markus Schulze's beatpath method, invented in 1997, and my own "maxtree method" which still has not been published as of 2005.

The fact that all Condorcet methods are more complicated to describe than range voting, is a bad thing. But aside from the issue of complexity, we want to know –

Which method is better?

That depends how you measure "better." First, measured by the "Bayesian regret" yardstick, range voting is robustly better than every Condorcet method so far tested, both for honest voters and especially for strategically-exaggerating voters. (To me, that says it all and we need not go further. But some people, oddly, remain unconvinced, so we shall go further!)

Second, yes, obeying the "elects Condorcet-winner if exists" property, sounds good, at first. But unfortunately it is known to automatically cause the voting system to disobey a lot of properties which also sound good! Once you realize these non-obvious logical implications, then you realize that Condorcet's property isn't as "good" as it sounds. Four examples:

Four known theorems about Condorcet methods:

  1. Votes are orderings Theorem: Any voting method that always elects a Condorcet-winner if one exists, automatically uses votes that are exactly partial-orderings of all the candidates (with equalities optionally allowed). That is, your "vote" is a set of preference binary relations such as "A>B, B>C, A>D, D=E, F>G" which are mutually consistent (i.e. acyclic) and hence in which it is impossible for a voter to express the idea that he prefers A over B more strongly than he prefers B over C.
  2. Honest votes hurt Theorem: A voting method that always elects a Condorcet-winner if one exists, automatically disobeys the property that by casting an honest preference-ordering vote, you cannot make the election outcome get worse from your point of view. Specifically, in every Condorcet method there is either an election in which by adding a new vote ranking the current-winner top, you cause him to lose ("add-top failure") or in which by adding a new vote ranking the current-maximum-loser bottom, you cause him to rise in the rankings ("add-bottom failure"), or (usually) both. Meanwhile: range voting obeys this "honest votes can't hurt you" property and can never exhibit add-top failure or add-bottom failure. (There might be higher voter turnout with systems in which staying home and not voting, is never strategically superior to voting!) [Proof.]
  3. Strategic voters kill you Theorem: A voting method that always elects a Condorcet-winner if one exists, automatically elects the candidate whom all voters unanimously agree is the worst, in any scenario where (a) there are 3 kinds of voters: A-fans, B-fans, and C-fans, in roughly equal numbers; (b) there is at least one other candidate W they all agree to be worse and hence not in contention; (c) the voters strategically exaggerate in their votes to try to "make them have more impact" by claiming their favorite candidate is best and his 2 main rivals "worst". This is a very common scenario and the result it yields is worst possible. We call it the DH3 pathology.
  4. Subdistrict inconsistency Theorem: With range voting, in a "country" divided into two "districts," if candidate A wins in both districts, then he necessarily wins in the country as a whole. Any voting method that always elects a Condorcet-winner if one exists, automatically fails to obey this "subdistrict consistency property."
  5. Favorite Betrayal Theorem: With a voting method that always elects a Condorcet-winner if one exists, automatically there are situations in which voting your favorite candidate top, is strategically stupid. (With range voting, this is never strategically stupid.)
  1. Condorcet methods automatically sacrifice a lot of voter-expressivity compared to range voting.
  2. This means that Condorcet voters can feel strategically forced to "betray their favorite" because honestly ranking him top in their vote, can be strategically stupid and cause the "greater evil" to win. In range voting, honestly ranking your favorite top, is never strategically stupid and can never affect the race between the "greater evil" and the "lesser evil." (An appealing delusion about this.) This was a big part of the reason range voting was better than plurality voting: In plurality voting, third parties tend to get shut out because voters feel it is strategically stupid to vote for them, even if they honestly consider them best. Range voting cured that disease – why would you want to risk re-infection by going with Condorcet methods?
  3. Range voting does not behave very badly when voters decide to strategically exaggerate. In the scenario described in theorem 3, the most popular among {A,B,C} would win with range voting, not W, so everything would be fine. The point is that Condorcet methods often react extremely badly to strategic voters. And voters have lots of incentive to strategically exaggerate: In the theorem-3 scenario if the A- and B-fans decided to vote honestly while the C-fans exaggerated, then C certainly would win. So the A- and B-supporters would figure "we can't just stand idly by and let the C-supporters do that to us – we have to fight fire with fire." But then W gets elected and everybody suffers! That is because all Condorcet methods are extremely allergic to the kinds of dishonest exaggeration typically employed by voters, whereas range voting is comparatively immune. Range voting is designed to be good for both honest and strategic voters. (Think by allowing rank-equalities and basing everything on "winning votes" rather than "margins," Condorcet methods can be saved from this criticism? Wrong; in fact nothing works.)

Another problem with Condorcet methods – especially the more complicated ones in which your vote is allowed to be a partial ordering and/or is is allowed to express optional equalities (e.g. a vote in such a system might be "A>B=C>D=E>F, G>C") – is: you can't run them on most voting machines in use today. You'd need to design and build new kinds of voting machines. (And the Condorcet methods that allow equalities or partial orderings are even more complicated to describe than the ones that just accept ordinary full rank-orderings with equalities disallowed!)

So the question is: do you consider all these disadvantages to outweigh the advantage of obeying Condorcet's property? If you do, then Condorcet methods are not for you.

If we, striving for simplicity, demand voters produce full rank orderings, and disallow partial orderings, then all Condorcet methods have the severe disadvantage that they do not allow a voter to express ignorance. In a large election like the 2003 California Governor Recall election with 135 candidates, a Condorcet voter would be forced to provide a full rank ordering of all 135 candidates. Meanwhile, a range voter could just rate the candidates he understands, and then conveniently say "leave the rest blank" or "make the rest all have score S, where S=32 (or whatever other common value that voter prefers)."

Strategy and 2-party domination in Condorcet methods with orderings as votes

Most people, 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 do this in order to give their vote the "maximum possible impact" so it is not "wasted." Once they make this decision, with any Condorcet method based on rank-orderings as votes (with equalities disallowed) Nader automatically must 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-way Condorcet 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.)

In every Condorcet method based on full rank-orderings as votes, in this scenario this kind of exaggeration is the only strategically-effective vote. There was once some hope that by going to Condorcet methods not demanding full rank orderings – i.e. permitting rank-equalities – and based on "winning votes" rather than "margins," such exaggeration would no longer be strategically useful. However, this example nixed that hope.

Meanwhile, 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 the third-party candidate Nader 99 or 0 or anything in between. Consequently, it would still be entirely possible for a third-party candidate such as Nader to win with range, and without need of any kind of tie. In range voting, exaggerating the major-party candidates A and B to pretend A>C>B (when you really think C>A>B in a 3-candidate election) is never best strategy.

In view of this, third parties would be silly to push any Condorcet method that uses candidate orderings (with equalities disallowed) as votes. They should advocate range.

Think this kind of strategic thinking won't matter much? Wrong:

  1. 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 plurality voters act strategically and not honestly, given the chance. That is exactly why third parties always die out and the USA is stuck with 2-party domination.
  2. Click here for a proof that this kind of insincere-exaggerating voter-strategy can be strategically-optimal asymptotically 100% of the time in a mathematical model of a "large random electorate" with Condorcet voting. This suggests those Condoret methods will lead to 2-party domination.
  3. Another (admittedly weak) piece of evidence for the conjecture that Condorcet methods will lead to 2-party domination: The National Election Study has collected data about supposedly-honest voter feelings about various candidates (and about other things and people) on a 0-100 subjective scale during every presidential election year since the 1940s. Based on that data, it has been inferred that in every presidential election since NES started in the 1940s, the plurality winner has been a Condorcet winner. (Asterisk: in 2000, Gore was the plurality and Condorcet winner!) If Condorcet always yields the same result as plurality then Condorcet certainly would lead to 2-party domination! [However, warning: just because it has happened 14 times or so in a row is not a huge amount of evidence it always happens, and also the story conceivably would have differed with strategic Condorcet voting.]

Could it be that Approval Voting is, in practice, more likely to produce Condorcet Winners than "official" Condorcet methods?!

Counterintuitively, we can prove that under reasonable assumptions Approval and Condorcet voting actually are not in conflict (no-conflict theorem) and it is plausible that approval voting will actually be more likely in practice to elect honest-voter Condorcet winners, than "official" Condorcet methods!

And because strategic range voters generally vote approval-style, the same would be true of range voting elections with strategic voters. In other words:

Even if you don't quite buy all that, we think you still will agree that in practice, one should expect no great advantage for Condorcet methods over the simpler range voting system.

Further reading

Herve Moulin: Condorcet's Principle Implies the No Show Paradox, J. Economic Theory 45 (1988) 53-64.

Joaquin Perez: The Strong No Show Paradoxes are a common flaw in Condorcet voting correspondences, Social Choice & Welfare 18 (2001) 601-616.

Markus Schulze: A New Monotonic and Clone-Independent Single-Winner Election Method, Voting Matters 17 (Oct. 2003) 9-19. (Later expanded version available from Schulze.)

Warren D. Smith: The voting impossibilities of Arrow, Gibbard & Satterthwaite, and Young, (survey) paper #79 here.

H.P.Young: Condorcet's theory of voting, American Political Science Review 82 (1988) 1231-1244.

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