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

VoteFair electionmethods at votefair.org
Tue May 4 09:48:53 PDT 2021

```Kristofer's example (below) helps me realize that I didn't clarify
something important:

Here in Oregon everyone votes by mail by marking a paper ballot.  This
means there is no way to enforce the Borda-count requirement that a
voter use each preference level for only one candidate.

Also the use of paper ballots means that if there are 10 candidates
there isn't enough space to offer 10 preference levels.

To avoid the complication that an election with 4 candidates would offer
4 preference levels on a Borda-count ballot and 6 preference levels on a
STAR ballot, let's suppose that the Borda-count (with runoff) ballot
always has 6 preference levels, regardless of the number of candidates.

And let's assume that ballots are not ignored when a voter marks a
Borda-top-two-runoff ballot with two or more candidates at the same
preference level.

With these paper-ballot-based constraints, Kristofer's example of Borda
ballots ...

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

... would change the "Borda" ballots into the same as the STAR ballots:

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

And to use Kristofer's wording here ...

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

...  the truncation issue, equal-rank issue, and skipping ranks issue
become the same for both methods.  Correct?

Which I believe means the first sentence here is valid:

> 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.

In other words the Borda-count method with the addition of a top-two
runoff, and always using 6 preference levels regardless of the number of
candidates, and not tossing out ballots unless the marks are not clear
(about whether the oval is marked or not marked), would yield the same
results as STAR voting.  Is this correct?  Or am I overlooking something
else?

Richard Fobes

On 5/3/2021 6:33 AM, Kristofer Munsterhjelm wrote:
> 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
>
```