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

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

VoteFair-2
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?

Both methods simply add numbers that are directly (although not
necessarily explicitly) specified on the ballot.

On a printed ballot the column titles for STAR voting are different from
a typical ranking ballot:

"5" instead of "First choice"
"4" instead of "Second choice"
"3" instead of "Third choice"
"2" instead of "Fourth choice"
"1" instead of "Fifth choice"
"0" instead of "Sixth/last choice"

Yet in this six-level case the Borda count would assign 5 points to the
first choice, 4 points to the second choice, and so on down to 0 points
to the sixth/last choice.  And those numbers are added, just as they are
in STAR voting.

If a top-two runoff (a pairwise comparison between the two candidates
who have the highest number of points) is added to the Borda count, then
the two methods are counted the same.

Of course STAR voting and the Borda count (either with or without
top-two runoff) have different ballot-marking instructions:

Specifically, the Borda count method (either with or without the top-two
runoff) requires that a voter not use the same preference level for more
than one candidate, and requires that a voter not skip (leave unmarked)
any preference levels.  Of course there is no obviously "best" way to
count a ballot that violates either of these requirements.

In contrast, STAR voting allows multiple candidates to be marked at the
same preference level.

In addition, voters marking a STAR ballot are encouraged to indicate the
strength of their preferences by rating their favorites much higher than
their strongly disliked choices.

In both cases the use of the top-two runoff discourages the tactic of
marking most candidates at the highest and lowest rankings (which would
become equivalent to approval voting if everyone used this tactic).

Otherwise, are there any other important differences?

As a related question, are there any academic references to using the
Borda count with a top-two runoff?  If so, such references would also
apply to STAR voting, right?

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."

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?

Also, to repeat my initial question, am I overlooking anything in
believing that the mathematics behind the Borda count with a top-two
runoff is equivalent to STAR voting?

In advance, thank you for any help.

Richard Fobes
The VoteFair guy
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Re: [EM] STAR voting equals Borda count with top two runoff?

Kristofer Munsterhjelm-3
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
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Re: [EM] STAR voting equals Borda count with top two runoff?

Toby Pereira
In reply to this post by VoteFair-2
They're not equivalent, and you even go on to explain the differences. STAR is score with run-off and Borda with run-off is, well, Borda with run-off.

Of course, by limiting it to 6 scores, if voters want to make sure that they would make a difference in a run-off between any pair of candidates, it does have the potential to make it a little bit Borda-y. However, at the other extreme, if a full continuum were allowed, it could become very approval-y.

Toby

On Monday, 3 May 2021, 00:33:59 BST, VoteFair <[hidden email]> 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?

Both methods simply add numbers that are directly (although not
necessarily explicitly) specified on the ballot.

On a printed ballot the column titles for STAR voting are different from
a typical ranking ballot:

"5" instead of "First choice"
"4" instead of "Second choice"
"3" instead of "Third choice"
"2" instead of "Fourth choice"
"1" instead of "Fifth choice"
"0" instead of "Sixth/last choice"

Yet in this six-level case the Borda count would assign 5 points to the
first choice, 4 points to the second choice, and so on down to 0 points
to the sixth/last choice.  And those numbers are added, just as they are
in STAR voting.

If a top-two runoff (a pairwise comparison between the two candidates
who have the highest number of points) is added to the Borda count, then
the two methods are counted the same.

Of course STAR voting and the Borda count (either with or without
top-two runoff) have different ballot-marking instructions:

Specifically, the Borda count method (either with or without the top-two
runoff) requires that a voter not use the same preference level for more
than one candidate, and requires that a voter not skip (leave unmarked)
any preference levels.  Of course there is no obviously "best" way to
count a ballot that violates either of these requirements.

In contrast, STAR voting allows multiple candidates to be marked at the
same preference level.

In addition, voters marking a STAR ballot are encouraged to indicate the
strength of their preferences by rating their favorites much higher than
their strongly disliked choices.

In both cases the use of the top-two runoff discourages the tactic of
marking most candidates at the highest and lowest rankings (which would
become equivalent to approval voting if everyone used this tactic).

Otherwise, are there any other important differences?

As a related question, are there any academic references to using the
Borda count with a top-two runoff?  If so, such references would also
apply to STAR voting, right?

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."

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?

Also, to repeat my initial question, am I overlooking anything in
believing that the mathematics behind the Borda count with a top-two
runoff is equivalent to STAR voting?

In advance, thank you for any help.

Richard Fobes
The VoteFair guy
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Re: [EM] STAR voting equals Borda count with top two runoff?

VoteFair-2
In reply to this post by Kristofer Munsterhjelm-3
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
>
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Re: [EM] STAR voting equals Borda count with top two runoff?

Kristofer Munsterhjelm-3
On 04/05/2021 18.48, VoteFair wrote:
> 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.

Presumably the current Plurality elections aren't really Approval
elections, so I would imagine they could enforce it the same way they'd
throw out Approval-style ballots under the current system?

>> 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?

Yes. However, I don't think it would be of much help, because the
"Borda" you end up with is not the Borda that's analyzed in academic
literature. So you may call it (a generalization of) Borda, but you
would have to show that results pertaining to Borda itself apply to this
generalization. (Borda is not bad because it has the name "Borda", but
because it behaves in a particular manner.)

For instance, the clone independence problem relies on the range of
points in Borda being a function of the number of candidates in the
election. Once you make the range fixed, that particular aspect of Borda
changes.

It would also alter instant runoff versions (Nanson and Baldwin) as some
of their behavior depends on candidates being given e.g. 1...5 points
when five candidates remain, then 1...4 points once someone has been
eliminated.

-km
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