| #voters | Their Vote |
|---|---|
| 8 | B>C>A |
| 6 | C>A>B |
| 5 | A>B>C |
In this 19-voter example (containing, for those who care, a "Condorcet cycle") the winner is B under a vast variety of election methods that employ rank-orderings as votes. (Namely, all these methods declare B the winner: Plurality, Borda=Dabagh, Plurality with Instant Runoff, Black, Schulze-Beatpath, IRV=Instant Runoff Voting, Loring, BTR-IRV, River, Maxtree, Tideman Ranked Pairs, Improved-Dodgson, Simpson-Kramer, Nanson, Rouse, Raynaud, Arrow-Raynaud, Condorcet Least Reversal, Woodall DAC, Sarvo-Plurality, Sinkhorn, and Keener-eigenvector.)
But if the six C>A>B voters insincerely switch to A>C>B ("betraying their favorite" C) then the "lesser evil" A becomes the winner under all these voting systems (and is the Condorcet-winner), which in their view is a better election result.
Lying paid.
This favorite-betrayal example is very important because, once voters understand that exaggerating their stances on the apparent-frontrunners can be necessary to prevent the "greater evil" frontrunner from winning, strategic voting is guaranteed, often causing "third parties" to tend to die out (since the strategic voters will not "waste their vote" on honest-favorite third-party candidates like C whom they perceive as having "no chance of winning").
In all of the voting systems we just mentioned, therefore, one might expect, or at least worry, that artificially enhanced self-reinforcing 2-party domination is still going to occur. Note that "Condorcet methods" are vulnerable to this problem: of the methods just mentioned, Black, Schulze-Beatpath, Loring, Nanson, Rouse, Raynaud, BTR-IRV, River, Maxtree, Tideman Ranked Pairs, and Condorcet Least Reversal all are Condorcet methods. Condorcet advocates might hope to escape from that trap by instead employing variant-Condorcet methods that permit "ranking equalities" and that are based on "winning votes" rather than "margins." That initially sounded like a possible way out, but it isn't. Indeed, Kevin Venzke showed that every Condorcet method (whether ranking equalities are allowed or not) suffers "favorite betrayal," completely wiping out all such hopes of wriggling out of this trap.
Note also that Instant Runoff Voting (IRV) also is vulnerable to this problem and in fact experimentally IRV does lead to 2-party domination. For this reason "third parties" are silly to advocate IRV. (Click here for more re this silliness. See also this other more realistic IRV example.)
This example perhaps may seem contrived and you might think it only occurs rarely. Not so. This kind of insincere-exaggerating voter-strategy is optimal asymptotically 100% of the time in a mathematical model of a "large random electorate" with any Condorcet voting system using full rank-orderings as votes. And ditto for IRV voting. Proof.
Range voting, however, is immune to this problem in the sense that in any 3-candidate election, it is never strategically desirable for any voter to range-vote as though X>Y when his true opinion is Y>X. So in the situation illustrated, the six C>A>B voters would not find it strategically desirable to cast a range vote in which A>C>B. And the word "never" above makes us believe that with range voting, 2-party domination is not inevitable.