[EM] Introducing Triscore Voting

Bayle Shanks bshanks.list at gmail.com
Wed Sep 12 17:24:46 PDT 2018

This is an attempt to combine the strengths of multiseat proportional score
voting (also called reweighted score voting, or reweighted range voting)
with 3-2-1. Apologies if this method is already known under another name.

This voting method can be feasibly counted by hand (but does require
written ballots). It reduces to something reasonable-seeming (and similar
but not identical to 3-2-1, due mainly to the proportionality in round 1)
in the single-seat special case. I have not done any formal or empirical
evaluation of this proposal. I conjecture that it will be superior to
vanilla Proportional Score Voting due to the elimination of the most
divisive candidates.

It is complicated to write out completely (see appendix) but can be
summarized and motivated simply:

0) Each voter gives each candidate a vote of -1, 0, or +1. 0 is the default.
1) The summed score of each candidate is computed. Proportional Score
Voting is used to select the best 3*#seats candidates (where #seats is the
number of seats to be filled in this election).
2) In this step, instead of the summed score, we only count -1s; 0 and +1
are treated the same. Proportional Score Voting is used to (proportionally)
eliminate the #seats most hated candidates, leaving 2*#seats remaining
3) In this step, instead of the summed score, we only count +1s; 0 and -1
are treated the same. Proportional Score Voting is used to select the

Round 1's function is a 'preprocessing' step to narrow the field to a small
number of contenders who have a good shot at winning (also, this is
important to limit the amount of computation needed when done by hand).
Round 2's function is to eliminate the most divisive candidates. Round 3's
function is to re-establish rough proportionality after round 2 (note that
Round 3 is just Proportional Approval Voting on the reduced set of

Round 3 ignores -1s in order to reduce the importance of negative
campaigning; -1s only matter in eliminating the most divisive candidates,
but they don't affect the choice of final winners after that (except to
break ties). Since Round 3 is proportional, between large factions in
manyseat elections, -1s have little effect on the proportion of seats won
by each faction, but they do have an effect on which candidates in each
faction win; this encourages candidates to compete with others in their
faction to be the least-hated by the opposition, and perhaps more
importantly encourages voters not to indiscriminately vote -1 for every
candidate in 'enemy' factions, but rather to choose some 'enemy' candidates
who are 'the best of a bad bunch' and give them 0s.

This is a semiproportional method; i conjecture that it is 'mostly
proportional', but a small, widely hated faction might not achieve
proportional representation if too many of their candidates are eliminated
in round 2.

I imagine that this method, which i might call 'triscore voting' (or does
it already have another name?) would combine the benefits of
proportionality with the elimination of very divisive candidates, and cause
elected officials to be somewhat responsive to the concerns of voters in
factions other than their own.


This is a voting procedure to select one or more winners out of a number of
candidates. The number of people to be selected is called the number of
'seats' and denoted here by '#SEATS'.

Each voter receives a ballot. The ballot contains a list of candidates, as
well as space for write-ins. For each candidate, the voter has three
choices: -1, 0, +1 (or equivalently: disapprove, neutral, approve; or
equivalently: thumbs-down, no opinion, thumbs-up). The voter may make a
choice for each candidate; for example, they may choose +1 for many
different candidates if they want. A ballot indicating no choice for a
candidate is the same as a ballot indicating 0 for that candidate.

There are then three rounds of calculation to progressively narrow down the
candidate pool until the winners are determined.

In the first round, for each candidate, a first-round score is calculated
by summing the values given to that candidate across all of the ballots. (3
* #SEATS) candidates are then selected by applying the Proportional Score
Subprocedure (see below) to the first-round scores to produce 3*#SEATS
winners. In case of a tie for last place, all tied candidates are selected.
All candidate that were not selected are eliminated.

In the second round, for each remaining candidate, a second-round score is
calculated by grouping +1 and 0 together and only considering whether or
not a ballot gave a -1 to a candidate. The Proportional Score Subprocedure
(see below) is then applied to the second-round scores to select (# of
remaining candidates - 2*#SEATS) 'winners', but this time those candidates
that 'win' are eliminated. In case of a tie for 'last' place, none of the
tied candidates are eliminated.

In the third round, for each remaining candidate, a third-round score  is
calculated by grouping -1 and 0 together and only considering whether or
not a ballot gave a -1 to a candidate. The Proportional Score Subprocedure
(see below) is then applied to the third-round scores to select #SEATS
winners. In case of a tie for 'last' place, the tie is broken by
eliminating all those except the one(s) with the lowest second-round score;
in case of a further tie, the tie is broken by choosing the one with the
highest first-round score.

=== Proportional Score Subprocedure ===
Each ballot is given an initial "weight" of 1.

Repeat the following P times, where P is the number of winners to be chosen:

1. The weighted scores on the ballots are summed for each candidate, thus
obtaining that candidate's total score.

2. The candidate with the highest total score (who has not already won), is
declared a winner.

3. When a voter "gets her way" in the sense that a candidate she rated
highly wins, her ballot weight is reduced so that she has less influence on
later choices of winners. To accomplish that, each ballot is given a new
weight = 1/(1+SUM/2), where SUM is the sum of the scores that ballot gives
to the winners-so-far ('winners-so-far' refers only to winners within the
current round of the Proportional Score Subprocedure)
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