An example of a range voting election in which CC (classic Condorcet) is disobeyed but WC (modified Condorcet property) is obeyed

The three voters are Amy, Bob, and Cal, and the 4 candidates are A,B,C,D, and we use 0-9 range voting:

Voter score for A for B for C for D
Amy 9 1 0 0
Bob 0 0 3 9
Cal 3 9 5 0
Totals 12 10 8 9

A is the winner. Now if we omit candidates B and D, then sure enough A is still the range voting winner, confirming the WC property. But the traditional-condorcet property CC fails since C would be the winner then (using < and > counting), not A.

The two definitions CC and WC are the same if the voting method only takes notice of the < and > counts and reduces to majority vote in the 2-candidate case – but range voting is not such a method.

But what if I think the range voters would have rescaled their votes for A and C if B & D dropped out? Wouldn't that be more realistic, thus proving that CC is a more important property than WC?

Indeed. If that is the way you feel – you want voters to be able to change their votes to make them "more powerful" when candidates get removed – then what you really want is yet another (a third!) kind of "Condorcet" definition, which we shall call "SC." (It is inequivalent to the other two.)

STRATEGIC CONDORCET (SC for short): Start with some voters and candidates, voters supplying the votes they consider maximally powerfully beneficial for themselves in view of whatever (zero, partial, or complete) knowledge they have about the other voters. An "SC-winner" is a candidate X, who when
  1. all candidates besides X and Y are removed, and
  2. the voters then, in the new X versus Y two-candidate election, get to re-choose their votes in a way they would in these new circumstances consider maximally powerfully beneficial for themselves,
  3. using these new two-candidate votes and the same voting system, X always wins versus Y no matter who Y is.
A voting method is SC (or "S-Condorcet") if it always elects an SC-winner (in the original starting circumstances) if and when one exists.

Now. Does range voting obey SC? Actually, yes it does, provided we agree to regard a "maximally powerfully beneficial" range vote as one with all scores maxxed-out or minned-out to 9 or 0 (in approval style) and provided we demand all such votes in 2-candidate pairwise (reduced) elections must agree with those in the full election. However, no it does not, if the second proviso (more realistically) is removed.

What about other voting methods like Schulze Beatpaths – i.e. voting methods that the traditional-Condorcet gang regard as traditional Condorcet methods? Do they obey SC? No!!

Theorem: None of them satisfy SC even in 3-candidate elections!

Proof sketch: That is because, elsewhere in the CRV pages (e.g. see here, here, and here), we have proven that every Condorcet method based on ballots that are rank orderings of the candidates (with or without permitting equalities), exhibits "favorite betrayal" scenarios in 3-candidate elections, in which the strategically best vote is to lie by dishonestly ranking your favorite below top, causing some "lesser evil" candidate L to become the Condorcet winner. In such a scenario, L would lose versus some other candidate Y in a head-to-head election, because L would not really have been a Condorcet winner with honest voting (and with voters allowed to change their votes in the reduced L versus Y election, they'd now vote honestly so L would lose to some Y).

Remark: Incidentally, it has often been falsely claimed that Condorcet methods, while vulnerable to strategy in cyclic situations in which no Condorcet winner existed, are not vulnerable when one does exist. Here's a counterexample.


So, traditional Condorcetists – be careful if you try to claim your favorite method obeys Condorcet's property, but range voting does not! Because it isn't that simple!

The truth is there are three ways to generalize the definition of "Condorcet" to apply to voting systems that include range voting: WC, CC, and SC. Range Voting obeys WC but not CC. (Also not SC unless you are very generous about it and cheat a bit.) Traditional Condorcet systems obey CC, WC in cases where a WC-winner exists (but it does not always exist, whereas with range voting a WC-winner always does exist), and not SC. This fuller comparison, therefore, does not favor traditional Condorcet over Range Voting.

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