# [EM] Are tied rankings valid in IRV?

Rob Lanphier robla at robla.net
Sun Apr 20 14:27:19 PDT 2003

```Alex Small wrote:

>Let's look at a scenario where IRV encourages insincere rankings, and see
>if generalized IRV would encourage equal rankings rather than reversing
>the order of 2 candidates.
>
>Say that your preference is A>B>C, and no candidate has first place votes
>from a majority of the voters but C has the lead in first place votes.
>Say that B beats C pairwise, and C beats A pairwise.
>
>You don't want B to be eliminated, so you definitely rank him first.
>However, you do want A to be eliminated.  Although there will be cases
>where ranking A and B equal will suffice to push B ahead of A, there will
>still be cases where you also want to rank A second, to ensure that A is
>eliminated rather than B.
>
>
>A second case:  Your preference is still A>B>C.  A has enough first place
>votes to not be eliminated, but B and C are in a close race.  A beats C
>pairwise but B beats A pairwise.  You want B to be eliminated, so you
>don't rank him equal to A, but you want C to advance, so you rank him
>equal to A.  There's no disincentive to rank A in first place, so your
>ranking should be A=C>B.
>

I'd like to spell this scenario out to clarify how it would work.

Let's say that the sincere rankings would be

40:A>B>C
16:B>A>C
15:B>C>A
29:C>B>A

In this particular race, the Condorcet winner would be B, and the IRV
winner would also be B (just barely, thanks to a first round elimination
of C).  So, some A voters would be motivated to engage in order reversal
under standard IRV, but that would be a risky strategy.  Under tied-pref
IRV, a bunch of A>B>C voters could have tied rankings in first place:

37:A>B>C
3:A=C>B
16:B>A>C
15:B>C>A
29:C>B>A

This would give C the edge over B in the first round (A: 40, B: 31,
C:32).  The final election would be (A: 56, C:47).  Not too many folks
could do this, though, without risking an election of C.

Also not that other voters could engage in a defensive (even offensive)
strategy similar to Approval.  The most obvious example would be the
folks who prefer C>B>A:
37:A>B>C
3:A=C>B
16:B>A>C
15:B>C>A
3:C=B>A
26:C>B>A

Now the first round is (A: 40, B:34, C: 32).  C is eliminated.  In fact,
the correct voting strategy is exactly the same as Approval (vote for
candidate if preferred to the second place option).  If the (B and C)>A
voters use Approval strategy, they are pretty unassailable from
tied-pref attacks from A:

37:A>B>C
3:A=C>B
16:B>A>C
15:B=C>A
29:C=B>A

Here we see that the result is (A: 56, B: 60, C: 47).  C is eliminated
in the first round, and B beats A.  There's nothing that A voters can do
to affect this, since the only thing that greater A=C>B votes does is
pump up C to the point that it eliminates A from the race.

So, I'm becoming convinced that tied-pref IRV behaves like Approval.
However, I'm also becoming convinced that IRV advocates would never go
for it, because it looks *too much* like Approval.

This points out something that is superior about Condorcet voting.  It
pretty gracefully handles tied rankings, in a way that doesn't
profoundly impact strategy.   This seems like a good talking point to
those that would argue in favor of IRV over Condorcet.

Rob

```