An appealing delusion about Condorcet voting methods

The A>B>C "full support" myth

Mike Ossipoff once stated:

I advocate Range over Condorcet, as a public proposal. Assigning points from 1 to 10 is already familiar to people...

But if it's a question of which I'd prefer if I could enact whatever single-winner reform I wanted, I'd choose Ranked-Pairs(wv) or the wv version of BeatpathWinner/CSSD. [Editor's note: these are two Condorcet methods that Ossipoff particularly likes. Later, in 2007, Ossipoff said he preferred SSD for public elections and CSSD for organizations and committees with few voters. Both CSSD and SSD are the same as Schulze's beatpath method except for how they handle tie-breaking.]

Why would I choose Condorcet if the enactment decision were entirely up to me? Because, with Condorcet(wv), the voter who feels that it's necessary to fully support Gore against Bush will still be fully helping Nader beat Gore. (End quote.)

In contrast, Ossipoff feels that in range voting, you can fully support Nader versus Gore, or fully support Gore versus Bush, but not both at the same time.

Our Reply busting this myth

That was an appealing point by Ossipoff, although in fact, I suspect Ossipoff is wrong. I am not precisely sure what "fully supporting A versus B" means to him, but anyhow, it is a known theorem of Moulin & Perez that by casting a vote saying A>everybody, your vote can cause A to lose, or by casting a vote saying everybody>B, you can prevent B from being the bottom-loser (or, usually, both kinds of paradox are achievable) and this is true in every Condorcet method.

I do not see how you, by (say) casting the vote Nader>Gore>Bush, causing Nader to lose when he otherwise would have won, would be "fully supporting" Nader! (Incidentally, in range voting, this paradox never arises: your vote A>everybody can never cause A's defeat, and your vote saying everybody>B can never cause B to rise above bottom.) So I think Ossipoff's appealing point here, is actually an appealing delusion.

Disagree? Then consider the following "voting method":

  1. you supply rank-order ballots,
  2. I ignore them and choose whomever I want to be the winner.
As far as I can tell, you, by disagreeing, presumably feel that this method also has the property that a voter can "fully support A>B and B>C at the same time"??

In my view, it shouldn't be about some bogus perception of "full support." It should be about real support, which means, helping A win or at least not causing A's loss. If your vote containing A>X doesn't help A win (and in fact causes A to lose) then it was not "fully supporting" A in my book, sorry. With that definition: no Condorcet method always allows a voter to fully support A>B and B>C at the same time.

Still disagree? It seems to me if your goal is to "fully support A versus B" then, in either Condorcet or IRV voting, you should vote A>others>B. This should give you a strictly greater chance of electing A and not B, than any other vote (if, e.g, the other voters act randomly). [For example, consider this Condorcet election in which an A>C>B vote elects A but C>A>B fails to do so and elects B. Also you can see this discussion for how a vote X>Y>Z can elect X while X>Z>Y elects Z.] Similarly if your goal is to "fully support B versus C" you should vote B>others>C. And, obviously, you can't do both at the same time.


Indeed, no rank-order voting method whatever (whether rank-equalities are allowed in votes, or not), always allows voters to fully support A>B and B>C at the same time – in the sense that there always exists a 3-candidate election situation in which casting the honest vote A>B>C causes a worse election result (from that voter's point of view) than casting some other (hence dishonest) vote would have caused. That fact (for deterministic voting systems that are not "dictatorships") is the famous Gibbard-Satterthwaite theorem.

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.

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