The "official" and "unofficial" definitions of "Condorcet" (and about range voting and red herrings)

[Adapted from a electorama web post by Warren D. Smith in August 2005.]

Most political science books define "Condorcet critierion" in a way that implicitly or explicitly assumes the voting method uses ranked ballots. That is because Condorcet himself had them, and only them, in mind. Our point is that there are two natural ways to generalize the definition of the "Condorcet criteron" to more general voting methods.

To try to say so hits a wall of opposition from traditionalists, which is a red herring (i.e. distraction from the main issue of which is better – range voting or Condorcet methods?). To avoid being distracted by that red herring, let us simply make the "classic" Condorcet definition be:

CLASSIC CONDORCET(CC for short): Elects a winner W, whom the voters based on the counts of X>W and W>X type votes, agree is pairwise-superior to all others X (if such a "Condorcet winner" exists – in the sad case where none exists, then no claim is made). Only applicable to: voting systems in which candidate rank orderings are deducible from the votes.

Then the "unofficial Warren D. Smith Condorcet property" of a voting system (which is related and also interesting) is

WDS-CONDORCET (WC for short): Elects a winner W, whom the voters would agree is superior to all others X (if such a "WDS-Condorcet winner" exists – if none exists, then no claim is made) in each paired-off 2-candidate election "W versus X", using the same voting system but with all but these two candidates erased from all votes. Only applicable to: voting systems in which "erasing all candidates from all votes, except for a preselected pair of candidates" has a meaning.

Another view: The "classic" CC definition assumes that the 2-candidate comparisons are based on plurality voting (PV). With this definition, there are circumstances under which the range voting (RV) winner is not the Condorcet Winner (CW). But I think it is more reasonable to do the 2-person comparisons on the basis of the same system used for the multi-candidate election being considered. [That is the WC definition].

If this is done, then RV always picks the CW (and, in fact, the RV winner is always a CW). Note that this is not the case for Instant Runoff Voting (IRV), where even considering 2-candidate IRV elections, IRV can fail to elect the CW. – Prof. Steven Unger (Columbia University, NY) in a 2008 web post independently rediscovering this same idea.

[That is, in a 3-candidate IRV election, IRV can fail to elect a candidate who would defeat every rival in 2-candidate IRV elections. Example: 2 votes B>A>C, 2 votes C>A>B, and 1 vote A>B>C. B wins this 3-candidate IRV election but A beats both B and C pairwise by 3:2 majority vote.]

Comparison of the two definitions CC and WC

  1. WC seems applicable to a strict superset of the voting systems to which CC is applicable, so in that sense it is a more useful, important, and valuable property.
  2. WC and CC happen to be equivalent on all "ranked-ballot voting systems which reduce in the 2-candidate case to majority vote." I.e. every system that CC has ever been tested on, in all preceding political science literature (apparently). (This justifies our claim that both are fully valid ways to generalize the original conception.) But they are not equivalent if applied to voting systems more general than that, e.g. range voting:   Example.
  3. Is CC a desirable property for a voting system to have? It sounds that way at first, but it is a known theorem that any voting system obeying CC must exhibit "add top" and/or "add bottom" failures (e.g. adding a new vote ranking the current-winner top, can cause him to lose). That makes CC sound less like a desirable property. And there is also the Fishburn counterexample and Condorcet self-contradiction example.
  4. Also, a simple instance where CC clearly is undesirable, is the "free the slaves" vote in the early USA, where, say, 60% of the voters would vote to keep slavery, and 40% (we assume the slaves are allowed to vote) would vote to ban slavery. However, the right decision is to free the slaves, and a voting system that were somehow able to weight the votes based on the utility to that person (slaves: freedom has high utility. Slaveowners: having a slave has utility, but a lot less) would produce a superior result utility-sum-wise. In this scenario range voting with honest-enough voters would be capable of freeing the slaves, although no classic-Concorcet system could do so.
  5. Is WC a desirable property for a voting system to have? Sounds that way, yes.
  6. Both Range voting, and all so-far-constructed "Condorcet voting methods", obey WC.
  7. Range voting elections (ignoring exact ties) always produce a WC-Condorcet winner.
  8. All other so-far-constructed "Condorcet voting methods" do not always produce either a WC-Condorcet winner or a CC-Condorcet winner.
  9. So in that sense, which is very highly related to the yardstick used by Condorcet proponents themselves, range voting is superior to every so-far-constructed Condorcet voting method!

More details – there is really also a third way ("SC") to generalize the definition of "Condorcet" and Range Voting obeys one out of three (two out of three with a bit of generous fudging) while "traditional Condorcet" methods also obey one out of three.

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