[EM] Comparing CW with "write-in pairwise" CW

Kevin Venzke stepjak at yahoo.fr
Sun Oct 27 11:26:17 PDT 2013

```Hello,

Recently I wrote about the concept of allowing write-in votes in individual pairwise contests. I prefer the "approval" formulation which depicts each pairwise contest as an Approval race. In effect the meaning is that an ordinary pairwise win of X over Y is replaced by a tie, when more voters prefer some other candidate Z to both X and Y, than the number of voters who prefer X to Y.

The motivation behind this concept is to try to come up with a pairwise relationship with some intuitive relevance to the election as a whole. If candidate X has no possibility of winning the election, then I want to recognize as "noise" his pairwise wins, because they don't (or should not!) aid him, and any effect they do have will be arbitrary.

We could define a "write-in pairwise Condorcet winner" (let's call it "WICW") who suffers no "write-in pairwise" losses, and correspondingly a WICW criterion which says that a WICW must be elected if there is one.

In practice, in the three-candidate case at least, this is very similar to standard Condorcet or Smith. Even when throwing out cases with majority favorites, a quick simulation suggests they only disagree 6.5% of the time. I wanted to note the types of disagreements, though, because there are only two types. One is illustrative of the motivation behind the concept, and the other is an unexpected result.

The first type of disagreement makes up 70% of them: Here WICW demands a single winner while Condorcet/Smith is indecisive among the three candidates. A random example:
29 A
18 A>B
26 B>C
27 C>A

There is an A>B>C>A cycle, but WICW doesn't recognize the B>C win, making C undefeated. According to WICW, write-ins for A>{B,C} nullify the B>C win. (Note that SDSC and Plurality also disqualify B from winning. Additionally SFC disqualifies A.)

To take a step back: Why regard pairwise contests at all? To my mind, what they can simulate is a test of whether supporters of one candidate feel cheated when the other candidate is elected. In the case of B vs. C we want to know whether we can elect one of those two without irritating the supporters of the other. The answer is: the election of one or the other gives neither side a strong reason to complain.

The second type of disagreement, the other 30%, is the case where Condorcet picks a single winner while WICW picks *two* possible winners:
42 A
30 B>C
28 C>B

Here we have a mutual majority and a CW, but WICW is indecisive. The B>C pairwise win isn't counted, because once the majority splits up between B and C camps, neither can contend with the A>{B,C} voters. This is quite an oddity and I'm not sure what to make of it.

Does it suggest any actual insights? I kind of think it does: While B is unambiguously the superior candidate, it isn't by much. It's quite possible that if we forcibly extracted second preferences from the A voters, that they would prefer C. It's unknown, it may matter, and WICW says as much.

Furthermore, there should be no shame in being indecisive, considering that Condorcet is often indecisive as well.

On the other hand, I could easily believe that this result is indicative of a need for a "second draft" of the concept, that wouldn't suffer from this problem. One thing you could easily do is remove non-winners and then refigure WICW's result. But that takes it further from the original idea and the justification becomes cloudy.

I should say, my point isn't really to define a new criterion (and less still, a method). It's more that I want to define a model that matches the logic or aesthetic that I tend to use when evaluating scenarios.

Kevin Venzke

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