# [EM] STAR voting equals Borda count with top two runoff?

Kristofer Munsterhjelm km_elmet at t-online.de
Mon May 3 06:33:14 PDT 2021

```On 03/05/2021 01.33, VoteFair wrote:
> Recently I realized that, if I'm not mistaken, STAR voting -- "Score
> Then Automatic Runoff" -- is equivalent to the Borda count with a
> top-two runoff.  Is this belief correct?

Not really, because the scale is truncated and it's possible to both
equal-rank and skip ranks.

Consider the Borda clone scenario: you have two candidates (A and B):

Borda:
60: A>B
40: B>A

STAR:
60: A (5), B (0)
40: B (5), A (0)

A wins.

Now clone B four times (to overwhelm top two runoff):

Borda:
60: A > B1 > B2 > B3 > B4
40: B2 > B1 > B3 > B4 > A

A gets 60*4 = 240 points. B1 gets 100*3 = 300 points. B2 gets 60*2 +
40*4 points = 280 points. So B1 and B2 advance to the runoff, and then
B1 wins.

STAR:
60: A (5), B1 (0), B2 (0)
40: B1 (5), B2 (5), A(0)

A and one of the Bs advance to the runoff and then A wins.

And this happens even if the voters fail to equal-rank:

60: A (5), B1 (1), B2 (0)
40: B1 (4), B2 (5), A (0)

A gets 300 points. B1 gets 220 and B2 gets 200. So A and B1 advance, and
then A beats B1 pairwise.

So, in a sense you're right. But the distinctions make all the
difference: that the scale is fixed means it's harder to win by fielding
a thousand candidates, and that voters can skip ranks make it even harder.

> For those who don't know, fans of STAR voting heavily promote their
> method within the state of Oregon, and so far within Oregon it's used in
> a few places (party nomination and, as I recall, a couple of local
> elections).  A couple of state legislators have sponsored bills for
> adopting STAR voting more widely within Oregon.
>
> I've communicated with leaders of the organization that promotes STAR
> voting.  When I point out that tactical voting can undermine the
> fairness of STAR voting, they basically respond with what I would
> characterize as saying "I can't imagine any voting tactic that would
> cause any unfair results."  When I point out a specific tactic that
> would produce an unfair outcome they respond either by claiming the
> outcome is not unfair, or by saying something equivalent to "I can't
> imagine that kind of situation happening."

It is, unfortunately, a common response. Something similar can be seen
in IRV proponents.

Step 1: It's impossible.
Step 2: It's possible in theory, but it will never happen.
Step 3: It may happen, but not in the kind of elections we have now.
Step 4: It may happen in elections we have now, but momentum is more
important.

FWIW, I think that STAR (Range+runoff) is better than IRV and Range. I
still prefer advanced Condorcet methods to all of them :-)

> I think it would be helpful to point them in the direction of any
> meaningful mathematical analysis that relates to their method.  So does
> anyone here know of any academic articles that apply to using the Borda
> count with a top-two runoff?

I don't know of many: the reason is probably that, from an algorithms
perspective, if you have a top-two runoff you can just as easily make it
an exhaustive runoff. There's quite a bit written about
loser-elimination Borda methods (Nanson and Baldwin).

James Green-Armytage and Nicolaus Tideman wrote a paper about selecting
candidates for a manual runoff:
http://jamesgreenarmytage.com/runoff.pdf. Page 15 shows that Range is
more susceptible to strategy than Borda, and that choosing candidates
for a runoff by a Condorcet-IRV hybrid makes the method almost impervious.

It also shows the tradeoff between representativeness and utility:
picking the two candidates by Borda or Range will tend to provide runoff
candidates that are both quite good but that are very similar to one
another. On the other hand, some other methods given in the paper will
pick a second candidate that is representative of voters who don't feel
represented by the first, but may not be considered as good a candidate
as the first in general.

I'm not sure how they determined Borda to be more robust than Range;
from my arguments above, I would've expected the opposite to be true.

-km
```