Add-On Multi-Winner Proportional Representation


It is entirely possible to transform an election utilizing the score/range voting method (or the single selection method, or the approval method) into a multi-winner proportional representation election. That is, there is a method for implementing a proportional representation (multiple winner) election (e.g. for a legislature) without the involvement of parties, which could utilize strategic hedge simple score voting:

“Tranches” correspond to seats in a legislature, but also, approximately, to non-majority groups or interests. Here are the fundamental parameters and variables:

T = The Total Number Of Tranches (e.g. (one less than) the number of seats to be filled).
N = A Tranche Number (these tranche numbers run from 1 to T – a “strongest winner” is at the top, and the tranches form “layers,” with tranche #1 below the strongest winner, tranche #2 below tranche #1, and so on).
W = The Strongest Winner’s Total Number Of Votes.
B = The “Base”, or end point beneath each tranche. (for a given N (or tranche number), as determined by the equation below).

B = W * ( 1 - [(N /( T + 1 )]^2 )

Strongest Winner’s Total = 310
Total Number Of Tranches = 7
(For a total number of seats = 8)

For each tranche number (N) there is a base number (B):

B = 310 * ( 1 - ( N/ 8 )^2 )

Strongest Winner is --> 310 (This candidate has the Strongest Winner’s Total, and wins the “top” seat.)
Tranche Number 1 Winner is at or above --> 305.15625 – but is below the strongest winner (This is the “B”, that is, the base for this Tranche #1).
Tranche Number 2 Winner is the strongest one at or above --> 290.625 – but is below base #1.
Tranche Number 3 Winner is the strongest one at or above --> 266.40625 – but is below base #2
Tranche Number 4 Winner is the strongest one at or above --> 232.5 – but is below base #3
Tranche Number 5 Winner is the strongest one at or above --> 188.90625 – but is below base #4
Tranche Number 6 Winner is the strongest one at or above --> 135.625 – but is below base #5
Tranche Number 7 Winner is the strongest one at or above --> 72.65625 – but is below base #6
Tranche Number 8 Winner would be the strongest one at or above --> 0.0 (However, #8 cannot be a winner; the total number of tranches is only 7.)

The strongest winner is elected.

== Please amend this:
Then the candidate with the number of votes equal to or above the Tranche Number 1 base is the winner in that tranche.

Then the candidate with the number of votes equal equal to or above Tranche Number 2 base is the winner in that tranche.

And so on, down to Tranche Number 7.

== To read as:
Then the candidate with the largest number of votes equal to or above the Tranche Number 1 base, but below the stronger winner, is the winner in tranche 1.

Then the candidate with the largest number of votes equal equal to or above Tranche Number 2 base, but below base 1, is the winner in tranche 2.

And so on, down to Tranche Number 7.

Note that we could evaluate the winner of tranche 7 first, and the winner or tranche 6 next, and so on, and get the same result, since the order in which the evaluations are performed is irrelevant.

The scores of the winners cluster around W, the strongest winner, and “thin out” as the tranche number increases. This is because the equation describes a parabolic curve. Note that always-corruptible political parties are not involved in this proportional election method at all (it is a party-less method). It is not exactly “proportional” in the sense that the tranches do not necessarily reflect the interests of groups, yet it effectively provides a “voice” for such minority groups.

The “bases” distribute according to an inverted and translated graph of a parabolic curve, so the formula is very simple. You could visually examine this at:

Of course, as with all methods, a sufficient number of candidates are required to facilitate the procedure.

There is one very severe problem left to be solved if this solution is to be completely realistic. I will solve it after a little rest. Thanks for the patience!

(Note: I would have put this in the “Multi Member Voting Systems” thread, but I screwed that up because it’s so hard to “get everything right” with something new like this. Sorry.)


I’m struggling a bit to understand this method while I find it very interesting.

  1. how is this different from picking random winners?
  2. even in a election with many voters and many candidates is it guaranteed that each tranche consisting of an interval of (score-)totals has one (or more) candidates? I can imagine some score-intervals having no candidates in them. Perhaps better to first rank all candidates on their total and then make the intervals on the rank-number not on their total?


The effect of the tranches seems to be to essentially punish candidates for being popular. I fail to see why this is a desirable goal.


if I understand it right:

  • a more popular candidate gets in an earlier tranche.
  • there’s no way of knowing in advance if an earlier tranche has more or less candidates in it then a later tranche
  • there’s no way of knowing in advance where the precise tranche-bases will be
    conclusion: it’s a unusual way to elect random candidates
    I honestly wanna learn, anyone spot my reasoning-mistake?


I agree with your assessment, although I’d describe the choices of candidates as arbitrary rather than random.


There are two types of deterministic party agnostic voting methods: Sequential algorithms and optimization algorithems. Sequential algorithems elect one candidate at a time, and optimization algorithms optimize a ‘quality function’ that grades every possible set of n candidates based on how ‘optimal’ that set is. For example, sequential proportional approval voting (or as most people, CES included, refer to it as just proportional approval voting) is simply a sequential approximation of the approval ballot version of harmonic voting optimization algorithem (which, confusingly, is also called proportional approval voting by social choice scholars, as well as on Wikipedia).

  1. How does this method compare to other sequential algorithems, such as reweighted range voting, proportional score voting (proportional approval voting + the KP transformation), and Ciaran Dougherty’s mallocated score voting (with my modification so that the measure of how much a ballot contributes to a candidate getting elected is independent of the score that that ballot assigns to other candidates), etc.?

  2. Is there an optimization algorithm with a quality function that this method is trying to approximate (and by approximate, I mean for each round, does it elect the candidate that elects the candidate that will best increase the value the quality function would assign to the set of candidates that is that candidate plus the candidates already elected). If not, can you define a quality function method that this method sequentially optimizes?

  3. How does the optimization algorithm version of this method compare with other optimization algorithms, such as the score ballot version of harmonic voting, PSI voting, Elbert’s method, Monroe’s method, etc.?


Yes, this is a most interesting question! I will address this problem.

There is absolutely no such thing as an “earlier tranche” with this! Where did that come from?

It is not “sequential” in any sense meaningful to me.

We begin with a simple formula, plus we begin with knowing how many seats need to be filled (T + 1):

The election is held at one time, and there are no runoffs. This gives us the strongest winner’s total number of votes (W). From this, we can easily calculate the base numbers (B1, B2, B3,…). So everything is now set in stone. And we now also know where the candidate’s scores are located within the known tranches.

So nothing is sequential.

I’ll address other significant issues; probably tomorrow. Thanks for the responses, they help me to organize my perspectives.


I meant with “earlier tranche” for instance N=1 and with a “later tranche” for instance N=T (or many other examples). Sorry if my choice of words was confusing, I just wanted to refer to your own described procedure:


As reflected in the “amendment” I have made to the original article, no given tranche is to be considered to be “earlier” or “later” than another tranche. Rather, as I said, "the tranches form layers,” one below another. So instead, any given tranche should be thought of as being “above” or “below” another given tranche.

As I mentioned, there does exist a serious problem with this tranche-proportional system. As noted in the comments above, if a tranche winner is simply defined as the candidate with the largest number of votes equal to or above Tranche Number (N) base, but below the (N - 1) base (the highest tranches are the lowest-numbered), this result will be almost random, or arbitrary. Thus it would be highly likely that some new, non-incumbent would win in any given tranche, in any given new election. And this is dreadful. An untested new winner could not have been elected on the basis of his or her previous political performance. People who want term limits seem to ignore this major dilemma.

From time to time “paladins” appear, who actually do good things for their constituents, and these paladins therefor get re-elected many times. But as described above, this tranche-proportional system would nearly always continually remove such paladins, and this is really disastrous. Something must be altered, so here is a solution. If any candidate who is an incumbent within a given tranche manages to receive enough votes to qualify as a potential winner within that tranche, except that there are potential winners who received more votes, That incumbent will displace (or “bump”) any non-incumbent winning candidates.

This may initially displease folks who always want to through the “bums” out, but if we have strategic hedge simple score voting, the worst of the “bums” will not be in place to begin with.

One more detail! If a potential paladin winner actually becomes the normal winner in a tranche above the one he or she would otherwise become the paladin winner in, then he or she will become ensconced in the higher tranche, instead of becoming the paladin winner in the lower tranche.

Note that it could require a few election cycles before new methods begin to function smoothly.


But then the system no longer treats candidates impartially. Any election impacted by ‘impartiality failure’ (i.e. ‘relabeling’ the candidates would change the outcome) will be seen as illegitimate.
But the larger issue here is that since candidates’ overall scores determine their tranche, they don’t differentiate between different interests in society, just different levels of overall support. So the representatives won’t provide representation for different interests but for different levels of popularity - the lower numbered tranches will elect popular candidates, and the higher number tranches will elect unpopular ones. There’s no guarantee that these will be different interests; maybe the system will elect progressively more extreme versions of the same interest group.


I don’t see anything I would call an impartiality failure, nor anything that would be seen as illegitimate. The rules are the same for everyone, even though they favor certain incumbents for sensible reasons.

There are only two real interests in society: the “elite” overlords, and the everyday common people. All else is manufactured distinctions.


By impartiality, I mean that the system should not take into account the identities of the candidates, only the information provided by the votes. Specifically, if some of the candidates are switched on every ballot (e.g. A becomes B, B becomes C, C becomes A), then the set of candidates elected by the method should show the exact same switch. (So to continue with the previous example, if A and B are elected, then after switching the candidates, B and C should be elected.) Giving incumbents an advantage causes this to be failed. If a tranche has an incumbent and a non-incumbent, the incumbent will win, even after swapping their positions on every ballot.


You are making me dizzy. The method I described is not yet “perfected” and could still use a bit of fine tuning, but…

People anonymously mark paper ballots, which go right into a box. It’s score voting. The votes are then counted and added up. The number of votes determines who wins. That is it. There is no “switching of candidates” or anything of that sort. The candidates are not ranked.


The problem is that you are using outside information to determine the winner. If the only information I have about the election is the ballots, that should be enough to figure out who won- I shouldn’t need to know anything about the candidates. With your method, I need to know their incumbency status.


Well yeah we need to know about their incumbency status, which of course is common knowledge, in order to obtain paladin preservation. This is a very good thing. And I don’t see how it could lead to any problems.


How can you be sure that the rule will only protect paladins and not crooks?

Another major problem is ‘unanimity failure’.
Suppose we are using this method to fill 5 seats. The range voting winner has an average of 78.574, so they are elected, and four tranches are established.
Base 1: 75.43
Base 2: 66.001
Base 3: 50.29
Base 4: 28.29
Suppose everybody scores Candidate A 66. Because A is below Base 2, A is in the third tranche. Since A is so close to the maximum value for that tranche, A will very likely be elected. Suppose everybody scores Candidate B 70. B is in the second tranche, but has an average more than 5 points less than the maximum for that tranche. It’s possible that someone has a higher score in that tranche and defeats B. In that case, A is elected and B is defeated, even though the voters unanimously preferred B to A. No one’s interests are served by electing A instead of B.


Paladins by definition have acquired a track record as a result of their previous activity in the legislature (or whatever). That in itself is the best protection from “crooks.”

The whole idea of proportional representation is based on the concept that some relatively small number of higher vote getters will lose to some lower vote getters so as to obtain proportional representation for minorities – this is really an iron rule.

Since the behavior of other voters can never realistically be predicted, any strategy to coordinate votes to elect some specific candidate is wildly unrealistic. Even if it could be done it would not be a disastrous outcome.

Remember that I always advocate strategic hedge simple score voting, whereby voters can (hopefully strategically) cast from (1) to (10) votes, or better, from (5) to (10) votes to each of as many candidates as they desire. This allows for hedge strategy voting, and the ballots are easy to hand-count.


But the minority whose interests are protected by electing A over B in the example I gave is a minority of zero.


I always begin with the essential criterion that any election method must thoroughly disrupt the “elite” party capture effect, which is usually conceived as “two-party domination.” And the only method that meets it is strategic hedge simple score voting (SHSSV), whereby voters can (hopefully strategically) cast from 1 to 10 votes, or better, from 5 to 10 votes to each of as many candidates as they desire. This allows for hedge strategy voting, and the ballots are easy to hand-count.

This is qualitatively different in its effects from approval voting, and from the system proposed above, which does not prioritize strategy, allows the number of votes per candidate to be from 0 to 99, and appears to require “averaging.” (This averaging introduces excessive complication, yet provides very little benefit.)

Attempts to create winners by voting just below tranche bases seem very far fetched. The relevant candidate could easily lose by coming in just above the base, or else lose simply by not obtaining the largest number of votes in the “target” tranche. Especially if there are only 1 to 10 votes for electors to work with.

== Please amend this:
It is most important to observe that even if a method causes an actual majority vote winner to lose an election, that is not necessarily a major problem just so long as he or she is not an “elite” overlord sponsored/controlled candidate.

== To read as:
It is most important to observe that even if a method causes an actual majority vote winner to lose an election, that is not necessarily a major problem just so long the winner is not an “elite” overlord sponsored/controlled candidate.


Any truly proportional (i.e. 1 quota of voters should be able to force the election of one candidate regardless of how everyone else votes) method can avoid two party domination provided there are enough seats per constituency.

The point I was trying to make with the unanimity example was that tranches do not necessarily correspond to minority interests. Some candidates in the tranches with lower bases may be there because nobody thinks they are very good, but still be elected because they happened to be close to the base line, while someone whom everyone scores higher loses because they were in a higher base tranche. Yes, my example had a 0 to 99 scale, but such cases are still possible with a 0 to 9 scale. Also, if there are no ‘blank’ scores (something summed score methods do not allow), then using averages rather than sums does not change the winner.

Yes, it would be a risky strategy, but for candidates who are not contenders for the range voting winner seat, it’s not obvious that their best option is to always going to be to push to score as highly as possible. Also, factions could minimize the risk by running many candidates, thus allowing them to take multiple shots at the target.

Ultimately, the problem is that score sum (and average, for that matter) only measures a candidate’s overall level of support; it cannot tell where it came from. It cannot discern between candidates with strong support from a small group versus candidates universally perceived as rather weak. It also cannot discern between which groups have already received representation and which haven’t. (That is, the lower scoring winners’ supporters may be a subset of the higher scoring winners’ supporters, even when the higher scoring winners’ supporters are a slim majority.) The only way to make these distinctions are by examining the individual ballots (as PR methods based on reweighting do.) Your method just doesn’t use enough information to accomplish its goals.