[EM] Renaming SARA to "majority score"

Jameson Quinn jameson.quinn at gmail.com
Thu Oct 27 11:27:42 PDT 2016

I'm renaming the system I'd called SARA to "majority score voting
<http://wiki.electorama.com/wiki/Majority_score_voting>". The SARA acronym
is nice, but it relies on putting the rating categories out of their
natural order (which would be the much-uglier "SAAR").

I've also renamed some of the rating categories; they are now "support,
assist, accept, reject" with a default of "accept". I believe that these
words give a better intuitive sense of what each rating does.

I've done some significant work on the wiki page. In particular, I showed
criteria compliances. I also went through minor spoiler, chicken dilemma,
and center squeeze scenarios, showing both honest results and probable
tactics. One important note is that chicken dilemma truncation tactics will
definitely backfire for a subfaction that's smaller than 25% if there's any
subfaction that's larger than 25% (which must be the case in a two-way CD).
Since, given reasonable margins of error for polling, there will very
rarely be a smaller CD faction which can be confident that it's over 25%,
this is yet another reason that CD issues will never get off the ground in
this system.

For instance, let's say that the total error (both statistical and
structural) of polls is about 4%, and that a faction will not attempt CD
strategy unless it believes itself to be the smaller faction but over 25%.
That would mean that the smaller faction would have to be 29% and the
larger one 33%; leaving just 38% for the minority threat faction. That's an
awfully tight window for the scenario to work; given polling fluctuations,
it would be very unlikely that a scenario would be stably inside this
window for long enough for the faction to plan and execute a CD betrayal.

Here's the wiki page as it currently stands:

Majority score voting

Majority score voting chooses the candidate with highest score among the
serious candidates who aren't rejected by a majority. Step by step, it
works as follows:

   1. *Voters can support, assist, accept, or reject each candidate.
   Default is accept. Candidates get 2 points for each percent of "support"
   and 1 point for each percent of "assist", for a total of 0-200 points.*
      - *Obviously, you should support the best candidates (perhaps a
      quarter of them), and reject the worst (perhaps half of them). For the
      rest, the candidates who are between average and good, a simple rule of
      thumb is to assist when you're afraid of somebody worse, and accept when
      you are hoping for somebody better.*
   2. *Eliminate any candidates rejected by over 50%, unless that leaves no
   candidates with over 50 points.*
      - *If possible, the winner shouldn't be somebody opposed by a
      majority. But this shouldn't end up defaulting to a candidate
who couldn't
      at least get accepted by over 1/2 or supported by over 1/4.*
   3. *Highest points wins. In case of a tie, fewest rejections wins.*
      - *This finds the candidate with the widest and deepest support.*

Note: Majority score voting was originally called SARA voting, an acronym
for the 4 ratings voters can give. However, putting these ratings
out-of-order is confusing, even if it results in a nice-sounding acronym.
Contents [hide <http://wiki.electorama.com/wiki/Majority_score_voting#>]

   - 1Criteria compliance
   - 2An example
   - 3Other scenarios explored
      - 3.1Worst case?
   - 4As the first round of a two-round system ("majority score with
   - 5Relationship to NOTA

Criteria compliance[edit

Majority score passes the favorite betrayal criterion
<http://wiki.electorama.com/wiki/Favorite_betrayal_criterion>, the majority
criterion <http://wiki.electorama.com/wiki/Majority_criterion>, Independence
of irrelevant alternatives
independence of irrelevant alternatives
the assumption of fixed "honest" ratings for each voter for each
candidate), Independence of clone alternatives
Monotonicity <http://wiki.electorama.com/wiki/Monotonicity>, polytime
, resolvability
O(N) summability
and the later-no-help criterion

There are a few criteria for which it does not pass as such, but where it
passes related but weaker criteria. These include:

   - It fails the mutual majority criterion
   <http://wiki.electorama.com/wiki/Mutual_majority_criterion>, but passes
   if the mutual majority in question unanimously rejects all candidates
   outside their mutually-preferred set. It also passes if the number who give
   the strongest candidate in the set a rating below "support" is less than
   three times the amount by which the overall group exceeds 50%.

   - It fails the Condorcet criterion
   <http://wiki.electorama.com/wiki/Condorcet_criterion>, but for any set
   of voters such that an honest majority Condorcet winner exists, there
   always exists a strong equilibrium set of strictly semi-honest majority
   score ballots that elects that CW.

   - It fails the participation criterion
   <http://wiki.electorama.com/wiki/Participation_criterion> but
passes the semi-honest
   participation criterion

It fails the consistency criterion
<http://wiki.electorama.com/wiki/Consistency_criterion>, the Condorcet
loser criterion <http://wiki.electorama.com/wiki/Condorcet_loser_criterion>
, reversibility <http://wiki.electorama.com/wiki/Reversibility>, the majority
loser criterion <http://wiki.electorama.com/wiki/Majority_loser_criterion>,
and the later-no-harm criterion
An example[edit
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Imagine that Tennessee is having an election on the location of its
capital. The population of Tennessee is concentrated around its four major
cities, which are spread throughout the state. For this example, suppose
that the entire electorate <http://wiki.electorama.com/wiki/Electorate> lives
in these four cities, and that everyone wants to live as near the capital
as possible.

The candidates for the capital are:

   - Memphis <http://en.wikipedia.org/wiki/Memphis> on Wikipedia, the
   state's largest city, with 42% of the voters, but located far from the
   other cities
   - Nashville <http://en.wikipedia.org/wiki/Nashville> on Wikipedia, with
   26% of the voters, near the center of Tennessee
   - Knoxville <http://en.wikipedia.org/wiki/Knoxville> on Wikipedia, with
   17% of the voters
   - Chattanooga <http://en.wikipedia.org/wiki/Chattanooga> on Wikipedia,
   with 15% of the voters

The preferences of the voters would be divided like this:
42% of voters
(close to Memphis)26% of voters
(close to Nashville)15% of voters
(close to Chattanooga)17% of voters
(close to Knoxville)

   1. *Memphis*
   2. Nashville
   3. Chattanooga
   4. Knoxville

   1. *Nashville*
   2. Chattanooga
   3. Knoxville
   4. Memphis

   1. *Chattanooga*
   2. Knoxville
   3. Nashville
   4. Memphis

   1. *Knoxville*
   2. Chattanooga
   3. Nashville
   4. Memphis

Assume voters in each city support their own city; assist any city within
100 miles; accept any city between 100 and 200 miles; and reject any city
that is over 200 miles away or is the farthest city. (These assumptions can
be varied substantially without changing the result, but they seem
reasonable to start with.)
(must be) <50 >50
Memphis 42 0 0 58 (84)
Nashville 26 0 74 0 52
Chattanooga 15 17 26 42 47
Knoxville 17 15 26 42 49

Chattanooga and Knoxville both get under 50 points, but Nashville is above
the threshold. Thus Memphis, explicitly rejected by a majority, is
eliminated. Nashville wins.

If Memphis voters tried to strategize by rejecting Nashville in the above
scenario, it would have no effect.

If Chattanooga and Knoxville tried to strategize by supporting each other,
this has a chance of working, but Memphis could safely defend Nashville by
assisting it. Since Memphis is essentially guaranteed to be solidly
rejected, the Memphis voters have nothing to lose by defensively assisting
Nashville like this. A mere 13% of "assist" from Memphis's 42% — that is,
under a quarter of the Memphis population — would give Nashville 65 points,
more than double the combined size of Chattanooga and Knoxville, safely
defending it.
Other scenarios explored[edit

In all of the scenarios that follow, plurality voting
<http://wiki.electorama.com/wiki/Plurality_voting> would get the wrong
answer if voters vote honestly, and Majority Score would get the right

Say there are two major and several minor candidates (where "major" and
"minor" are defined by true support, not simply by media coverage). Voters
should reject one major candidate and any minor candidates who are worse;
support their favorite, whether they are major or minor; and accept
everyone else. In that case, one of the two majors will be rejected by a
majority; the other will win; and all minor candidates will show their true

Now suppose that there are two overall ideological "sides", and the
majority "side" is split into two subfactions, each smaller than the full
minority "side". This is called the "chicken dilemma
<http://wiki.electorama.com/wiki/Chicken_dilemma>", because many voting
systems encourage a "game of chicken" between the two majority subfactions;
whichever subfaction cooperates less will win, unless cooperation breaks
down entirely and the minority wins. A very closely-balanced chicken
dilemma would have 27% for subfaction A, 26% for subfaction B, and 47% for
the minority Z; whereas a more typical example might have 31%, 24%, and
45%. In Majority Score, each subfaction should support their candidate,
accept the other subfaction's, and reject the minority; while minority
voters should support their candidate and reject both subfactions. In this
case, the bigger subfaction will win.

Note that A, the bigger of the two subfactions, must necessarily be above
the 50-point threshold in step 2, so Z will be safely eliminated. In many
cases, B, the smaller subfaction, will fall short of 50 points. This helps
explain the rationale for putting that threshold at 50 points.

This is not to say that majority score solves the chicken dilemma 100%. If
subfaction B (the smaller one) is larger than 25% of the total, it is still
possible for them to win if they largely reject A while the A voters
largely accept B. And if both factions largely reject each other, Z can
win. But the key word there is "largely"; unlike the case with approval
voting <http://wiki.electorama.com/wiki/Approval_voting> or score voting
<http://wiki.electorama.com/wiki/Score_voting>, this strategy will not work
if it's done by only a few individual voters, but only if one subfaction
uses it significantly more often than the other. Also, the smaller
subfaction must be larger than 25%, which is rare. Given that organizing
such a coordinated betrayal in secret would be hard; that doing so openly
would invite mutually-destructive retaliation; and that even if it could be
done secretly and unilaterally, it would risk electing Z if the betraying
subfaction did not reach 25%; it seems that majority score voting has a
good chance of avoiding this problem.

As a final tricky scenario, consider what happens in the above case if the
minority Z prefers the smaller subfaction B. That results in a center
squeeze <http://wiki.electorama.com/wiki/Center_squeeze> scenario: one
where candidate B is a Condorcet winner
<http://wiki.electorama.com/wiki/Condorcet_winner> (able to beat either
rival in a one-on-one race) but an honest plurality loser (with the
smallest faction of direct supporters). This is more common than one might
think; candidate B is fighting an ideological battle on two fronts, while
candidates A and C are free to triangulate towards the middle without
losing supporters, so the fact that B has the smallest direct support may
not reflect a lack of quality. Thus, generally speaking, as long as each
faction considers their second choice to be more than half as good as their
first choice (taking the worst choice as 0), it's pretty clear that the
Condorcet-winner candidate B is the one who democratically "should" win
this election.

In this scenario, there are two ways that B could win. B voters could
reject both A and C; or A and/or C voters could assist B. Either of these
makes strategic sense for the voters in question, and either or both could
lead to a strong equilibrium
<http://wiki.electorama.com/wiki/Strong_equilibrium> result.

One interesting aspect in the above three scenarios: "assist" is only
strategically favored in the case of a center squeeze scenario, when the A
and C voters are helping B, whom they see as the the "lesser evil", beat
the other extreme. If there isn't a center squeeze, voters need only use
"support", "accept", or "reject". In fact, even in center squeeze, voters
could get by without "assist", as long as enough A and C voters were
willing to support B outright. So eliminating the "assist" option would not
substantially change the strategic outcome; but it would reduce the
expressive power available to the A and C voters in a center squeeze
scenario. Thus majority score sacrifices some possible extra simplicity in
return for this increased expressive power.

It is very rare to have a voting system which can deal with both chicken
dilemma and center squeeze. The two situations are very similar, even when
voted honestly, and yet the "correct" outcome is different. And under
strategic voting in many voting systems, it is very easy for the two
different scenarios to lead to identical ballots. Delegated voting systems
such as SODA voting <http://wiki.electorama.com/wiki/SODA_voting> can deal
with both; but without that kind of explicit participation in the voting
process from the candidates, it is very hard to find a system which deals
with both types of scenario better than majority score does.
Worst case?[edit

Majority score could still get a "wrong" answer in cases of a multi-way
Chicken dilemma, such that none of the subfactions reached 25%. This
scenario is avoidable if the subfactions compensate by "assisting" each
other's candidates, but that cooperation is subject to slippery-slope
chicken dynamics.

If this highly-specific scenario is the most-plausible worst case, it would
seem that majority score is a pretty good system.
As the first round of a two-round system ("majority score with runoff")[edit

If this system is used as the first round of a two-round runoff, then you
want to use it to elect at two finalists in the first round. Thus, run the
system twice. The first time, replace "50%" in step 2 with "2/3".

Then, to find the second winner, if the first-time winner got 1/3 or more
support, first downweight those ballots as if you'd eliminated enough of
them to make up 1/3 of the electorate. Otherwise, discard all of the
ballots which supported first-time winner. After downweighting or
discarding, re-tally the points and run Majority Score again.

If all the candidates in the first round got a majority of 0's, then you
can still find two finalists as explained above. But the voters have sent a
message that none of the candidates are good, so one way to deal with the
situation would be to have a rule to allow candidates to transfer their
2-votes to new candidates who were not running in the first round, and if
those transfers would have made the new candidates finalists, then add them
to the second round along with the two finalists who did best in the first
round. In that case, since there would be more than two candidates in the
second round, it would be important to use majority score for the second
round too.

Note: this "proportional two-winner majority score" system for the first
round is "matrix-summable", that is, summable with O(n²) information per
ballot for n candidates. This contrasts with the base majority score
method, which is summable with only O(n) information per ballot, and thus
can be counted voting equipment designed for counting plurality elections.
Relationship to NOTA[edit

As discussed in the above section, if all the candidates in the first round
got a majority "reject", then the voters have sent a message that none of
the candidates are good, akin to a result of "none of the above
<http://wiki.electorama.com/wiki/None_of_the_above>" (NOTA). Majority score
still gives a winner, but it might be good to have a rule to limit the
chance that such a winner would remain in office for multiple terms. This
could either be a hard term limit, so that such a winner could only legally
serve one term; or perhaps a softer rule that if they run for the same
office again, the information of what percent of voters had rejected them
should be next to their name on the ballot.
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