Criteria for voting methods

I have never found an analysis and comparison of voting methods that uses the following principle:

In situations in which the vote in sufficiently close, I don't care what the result is, provided it is selected from the "reasonable set"

In other words, I don't care about all these pathologies/violations of properties if they only occur when the vote is really close, and as long as they don't select a "dark horse", i.e. someone who is not one of the popular candidates who should "reasonably" be the only ones who can win. If the vote is close, I consider the will of the group to be indeterminate, and I wouldn't mind flipping a coin.

This doesn't rule out all consideration of pathologies. I won't tolerate the spoiler effect, in which a compromise candidate loses (even if they are everyone's second choice), except when elections are close. And as stated above, I won't tolerate pathologies like <a href=""></a>, even when they are close. One might distinguish between minor and major pathologies, major ones (such as DH3) being intolerable even when elections are close.

The rest of this webpage won't apply the previous principle much because I haven't seen much research working along those lines, and I have not yet had time to research it myself.

In addition to the previous principle, there is another principle that I have seen used, but not universally:

The system must return good results when any proportion of the population is sincere/strategic (i.e. from 100% sincere to 100% strategic, and anywhere in the middle).

In other words, a system is no good if it requires sincerity or if it requires strategy. However, I am willing to bend this in the direction of allowing a system which requires some strategy, provided that strategy is not very complex. (note: according to , this is opposite to the preference of Saari, who apparently prefers to cater to sincere voters first)

"Good performance" means the following:

Finally, as for strategies, I am especially sensitive to the situation where each voter faction is strongly polarized and has trichotomous preferences about the candidates (i.e. each candidate is either a "good guy" or a "bad guy" or "unknown").

Which single-winner voting system do I currently prefer?

For single-winner elections, i am trying to decide between range voting and Condorcet. Currently i prefer range.

Range vs. Condorcet

i am still reading about it, but so far here are two arguments that i find persuasive in favor of range over Condorcet:

(1) Bd the DH3 pathology

When there are a number of competitive frontrunners, the strategic choice in Condorcet is to pretend that you prefer generally unpopular candidates over the opponent frontrunners. this can lead to the unpopular candidates winning.

<a href=""></a>

(DH3 = "dark horse 3")

(2) spoiler effect/wasted votes/favorite betrayal)

if you like a 3rd party (call them "C"), but you think they are not competitive, and you have a preference between the popular contenders (call them "A" and "B"; let's say you like "A"), then under condorcet you can have more influence in the result if you insincerely rank A over C. so 3rd parties are punished in Condorcet, similar to how it works in plurality.

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(3) More Condorcet than Condorcet

I don't yet understand the proofs of the arguments in this section as well, so I'm not as sure about this part.

(note that I use range voting and approval voting interchangably, since in some situations the optimal strategy for range voting may look a lot like approval voting)

The argument is that, paradoxically, although Condorcet methods are guaranteed to pick the "nominal" Condorcet winner, due to strategic incentives, Condorcet methods will actually pick the "real" Condorcet winner less often than range voting.

The "nominal" Condorcet winner is a candidate who fulfills the Condorcet criterion with respect to the preferences stated by the voters when they vote.

The "real" Condorcet winner is a candidate who fulfills the Condorcet criterion with respect to the sincere preferences of the voters.

The reason these might be different is that strategic voters might choose to misstate their preferences during voting.

Condorcet can fail to due to strategy

Because of strategic voting, it is possible for voters who are using a mixture of strategies with Condorcet to elect someone other than "real" Condorcet winner, even when a real winner exists: <a href=""></a>.

Under certain condition, approval voting always elects the Condorcet winner

When certain conditions hold, and when there is a Condorcet winner, approval voting always elects that Condorcet winner: <a href=""></a>. I am not very comforted by the following assumption, however: "they order the candidates from best-to-worst, then select a "threshold" T, and they approve the candidates above T." -- Wikipedia says, "An optimal approval vote will always vote for the most preferred candidate and not vote for the least preferred candidate. However, an optimal vote can require voting for a candidate and not voting for a more preferred candidate."

Wikipedia gives a stronger theorem: "... if all voters are rational and cast a strategically optimal vote based on a common knowledge of how all the other voters vote except for small-probability, statistically independent errors in recording the votes, then the winner will be the Condorcet winner, if one exists.", citing Laslier, J.-F. (2006) "Strategic approval voting in a large electorate," IDEP Working Papers No. 405 (Marseille, France: Institut D'Economie Publique). See also When Voters Strategize, Approval Voting Elects Condorcet Winners but Condorcet Methods can Elect Condorcet Losers.

Under certain assumptions, range is more likely to elect the Condorcet winner than Condorcet methods are

Under the previous assumption, as well as some others including the assumption that all voters have a good idea of the positions of other voters (perhaps this could be accomplished via iteration? I don't know of a theorem that shows convergence, though), it has been shown that, because of strategic voting, approval or range might be more likely to elect the real Condorcet winner than Condorcet methods themselves: <a href=""></a>.

Footnote 12 in this paper argues the same:

Cases based on foreknowledge

The following page argues that strategic range voting (under the same assumptions as before) will always elect the Condorcet winner if one exists and if they are thought to be one of the top two contenders beforehand: . Furthermore, the Condorcet winner MIGHT be elected if they are the third contender. This is not the case for IRV or plurality.

Finally, if knowedge of the other voter's dispositions is given by a pre-vote poll in which everyone votes sincerely and in which votes are tallied in the same way as the actual vote, then that page argues that there is more chance that range voting will end up with the Condorcet winner placing first or second than for IRV, plurality, or approval voting.

That page contrasts this with Condorcet. If a sincere Condorcet pre-poll identifies the Condorcet winner, then strategic voters could make the actual Condorcet vote come out so that someone else wins.


here is's summary page for range vs condorcet:

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Possible disadvantages of range voting

Lessening the impact of moderates

The strategic choice for range voting is to give the highest possible rating to your favorites, and the lowest to your most feared opponents. "Moderate", non-strategic voters who don't give the top rating to their favorite candidate, nor the bottom rating to their favorite's major competitors, dilute the power of their vote. In effect, giving any score other than the highest or the lowest is saying, "I voluntarily give up some of my voting power because I don't care very strongly; I am willing to let the other, more opinionated voters have a greater say in the decision".

This could lead to a situation where people who are careful and measured in their opinion, and who sincerely let their votes reflect that, are underweighted, and people who are overly sure of themselves are overweighted. This would be bad because (if you assume that moderates are more intelligent than extremists) it underweights intelligent people.

However, I suspect that instead, what will happen is this. Intelligent people will discuss with other intelligent people and the fact that the best way to "get your way" is often to vote high-low will become widely known, at least among intelligent people, and probably among everybody. During a vote, intelligent people will vote high-low on those candidates that they have an opinion about. If they vote in the middle on other candidates (in a way that lessens the impact of their vote), it will only be when they make a conscious decision to let other people decide amongst their "middle choices", because they don't know enough or don't care enough about them. If they think that much of the voting public consists of people who are too ignorant or headstrong to be granted that power, then the intelligent people will not choose to cede their power that way.

So, in other words, intelligent people will not be "taken advantage of", unintentionally lowering the impact of their vote because they don't understand what's going on. Sooner or later, all of the intelligent people will learn the actual effects of voting various ways in range voting, and they will use this knowledge to wield range voting to their best advantage.

If anyone is "fooled" by range voting's ability to let you declare "I don't care as much" by voting in the middle, it will be less intelligent people. So the intelligent people will be overweighted, not underweighted.

Problems with range voting

Condorcet loser

Under some conditions, approval voting can elect the Condorcet loser! Here is the example from the wikipedia page on approval voting:

The voters all live in one of four cities. Depending on which city they live in, they like each of the other cities a certain amount. They want to vote on a city (i.e. the candidates are the 4 cities). Here is a table that tells you how many people live in each city, and how much the residents of each city like the other cities:

City Faction of Voters Memphis Nashville Chattanooga Knoxville
Memphis 42%10015100
Nashville 26%01002015
Chattanooga 15%01510035
Knoxville 17%01540100

Each voter votes for ("approves of") any candidate with an above-average likeability. In this case, each cities' voters vote for only their own city, except for Knoxville, whose voters approve of both Knoxville and Chattanooga. Memphis wins. But Memphis is the Condorcet loser! (Nashville is the Condorcet winner).

Note that in this example, the voters weren't strategically taking into account any information about what other voters were going to do. If, however, the voters believe that Nashville is the likely winner, and Memphis is the likely runner-up, and if they act on that belief by approving of cities which are preferred over the Nashville (and they approve of Nashville if they like it over Memphis), then Nashville wins. (if the voters think Memphis is favored to will win, followed by Chattanooga (the result of the "no-information" case that we discussed first), then there will be a three-way tie).

Perhaps there is a theorem similar to the "no-conflict theorem" in that shows that strategic approval voters with foreknowledge of everyone's preferences never elect a Condorcet loser? I would suppose that if it is thought that a Condorcet loser may be elected, everyone is going to vote against (not approve) them.


Given a population of sincere voters with fixed preferences, there can be many election outcomes in approval voting, depending on how the voters decide to set their thresholds, i.e. how many candidates they approve.

See Saari, D.G. and Van Newenhizen, J. (2004) "Is approval voting an ‘unmitigated evil?’ A response to Brams, Fishburn, and Merrill" Public Choice 59(2) pp. 133-147.

An example is given (I've changed it somewhat here). Let's say there are 100 voters. 99 have the same preference: they love A, they are neutral about B, and they hate C. So, they all approve both A and B. The 100th voter loves B, is neutral about C, and hates A. So he approves B and C. So B will win. As the paper says, that is because approval voting only allows the voter to partition the candidates into two groups. The first 99 voters had the power to separate A and B from C, but they couldn't also express a separation between A and B.

Note that strategy can prevent this outcome. If only a few of the 99-strong faction vote for just A, then A wins. Coordination is not needed; each voter in the faction could, with probability 90% vote A and B, with probability 10% vote just A. I note that this sort of strategy is easy to accomplish, because it's the sort of thing that might happen naturally anyway. Of course, in more contentious elections, there would be benefit to tightly coordinating the number of people doing one thing or another. Hopefully, by the time this becomes an issue, the election will have become so close that, under my first principle, I don't care if there are minor pathologies. But I'm not at all sure.

In response to implications that use of strategy can prevent bad outcomes, the article states that "With AV the voter needs strategies just to be sincere." This may be true, but if the choice is between requiring strategy or requiring sincerity (as noted above, Condorcet seems to require sincerity in order to elect the Condorcet winner in some cases), I would choose to require strategy.

In section "A positive feature of AV", the paper opines that approval voting works well when voters have dichotomous preferences, and badly if their preferences are more complex. I think that is the sort of situation that inspires strategic voting, however, so a method that is partially specialized to that situation is not so bad.

In the section "Strategic manipulation and sensitivity", the paper says that the results of approval voting can often be changed by a small number of individuals.

In general, I found this paper lacking in that it constantly made broad analytical claims about how most profiles have instability or sensitivity, but I feel there is a prior probability distribution over profiles. Hence, I don't care if the raw number of all possible situations makes approval voting come out bad, I only want to know if it is likely to come out bad, given my prior distribution. For example, the paper notes that AV works better when voters have dichotomous preferences, and I think that situation will come up often.

Unfortunately, I'm not exactly sure what my prior distribution is. However, I'm hoping that given examples which are particularly likely or unlikely, I would be able to identify them. This state of ignorance makes the analytical method of the paper inapplicable. What is needed is a boatload of examples of all these times where approval voting fails. Then I could look at the examples and determine if they seem likely to me; if they do not, perhaps I will be able to pull out a set of general properties that they share that I think are unlikely to occur, and then a theorem could be proved about approval voting being good under restricted conditions. But this paper is strongly in favor of the analytical approach and doesn't provide many examples.

In other words, there is room to hope that most of the bad outcomes that are mentioned in the paper come from scenarios that I would consider unlikely.

multiple round idea

i'm sure this has been thought of, but how about having multiple rounds of range voting, adding the results at the end? following the wikipedia Tennessee example ( We assume that on the first round, voters assume that all candidates are equally likely, and approve of anyone over the average utility. On each subsequent round, voters assume that the winner and runner-up of the previous round (NOT the winner and runner-up of the total sum) are the likely winner and runner-up. In all subsequent round, each voter votes for anyone they like better than the presumed winner, and they also vote for the presumed winner if they like them better than the runner-up. When the winner is tied, people vote for the city they like best out of the ties. When the runner-up is tied, people vote for the winner only if they like them better than both runner-ups.

Round Winner Runner-up Memphis Nashville Chattanooga Knoxville
1Memphis Chattanooga 42263217
2Three-way tie 42585858
total after 2Chattanooga Memphis, Nashville 84849075
3Nashville Memphis 42681517
total after 3Nashville Memphis 12615210592

Nashville over Memphis is stable (a fixpoint), and is the Condorcet winner.

Now let's assume that, after the first round, voters think the presumed winner and runner-up is the winner and runner-up of the total sum.

Round Winner Runner-up Memphis Nashville Chattanooga Knoxville
1Memphis Chattanooga 42263217
2Three-way tie 42585858
total after 2Chattanooga Memphis, Nashville 84849075
3Nashville Memphis 42683217
total after 3Nashville Memphis 12615212292

Presumably people's actual estimate of the presumed winner and runner-up is some interpolation of the previous two cases, perhaps with some extrapolation to the "future" of these two cases, or some finding of fixpoints.

It remains to be proven that such an interpolation converges, and how fast it converges.

If Memphis, seeing that it was also a runner-up, withheld its votes from Nashville in round 3 of the second case, then Nashville still wins but it takes 3 more rounds:

3Nashville Memphis 42263217
total after 3Memphis Chattanooga 12611012292
4Three-way tie 42585858
total after 4Chattanooga Memphis 170168180150
5Nashville Memphis 42683217
total after 5Nashville Memphis, Chattanooga 212236212167
6Memphis Chattanooga, Knoxville 42263232
total after 6Nashville Memphis 254262244199

IRL we can expect faster convergence because people will be trying to accurately assess everyone's utilities with polls and stuff, and also because people who take advantage of the range voting will probably help things out.


here is the range voting advocacy site (advocates range for single-winner elections):

<a href=""></a>

Which multiseat (multiple winner) voting method do I prefer?


Notes on voting on proposals by grant review committees

If you allow any group of more than n reviewers to veto any proposal, where n is small compared to the size of the group, that's 'rough consensus' (or 'strict consensus' if n=0). This will select only the most 'safe' proposals (as opposed to controversial ones, which will probably tend to rule out novel/innovative/risky ones or ones which are very strong in some ways but weak in others). (related: 'very strong in some ways but weak in others'; in business you sometimes hear that you should 'hire for strength', eg hire the candidate who is very strong in an area that the team needs and is lacking, even if they are not as well-rounded as some other candidats).

Less safe is 'approval voting', where you select those proposals which have support from the largest number of reviewers. This will still select only the 'safe' proposals, however.

The opposite of this, which maximizes controversial/novel/innovative/risky proposals, is to divide the budget between the reviewers and let each reviewer decide which proposals to fund up to the limit of their sub-budget (eg each reviewer gets to fund their favorite proposal(s); said another way, any proposal which is one of someone's favorites gets funded). (you can't do this in most elections because generally the number of seats available is less than the number of voters; but that is usually not the case in grant review committees).

Something in the middle is score voting. Let each reviewer give each proposal a score (with a fixed upper bound, eg a score between 1 and 10), then the total score for each proposal is the sum of all reviewers' scores for it. If you wanted you could have the reviewers rank all the proposals and make their score be 1/rank, but that's more work. If you wanted you could also include how much money each proposal asks for by multiplying each score by 1/money. If you wanted you could make this method a proportional one by using proportional score voting (reweighted range voting) to select the winners (recommended). Note however that strategic score voting reduces to approval voting, so this may not give you much more 'innovative' results.

Another way to interpolate between approval voting and a system where favorite proposals are funded is to fund only those proposals which are in the favorites list of at least b different reviewers, where b is a parameter; or, to allow each reviewer to give each proposal a thumbs up or a thumbs down, and then to also select favorites, and fund those proposals which are on at least b people's favorites list, and which also got at least 'a' thumbs-ups, where a is another parameter (generally a > b).

In general, the requirements that each winner have a certain level of 'support' (thumbs-up/approval) and also appear on a certain number of favorites lists can be used as a preprocessing technique for other election methods.

This suggests a 'mixed score voting' system in which some fraction of the 'seats' (winning proposals) are selected via approval voting, some fraction by score voting, and some by favorites lists. A single score voting ballot with three score options, 0,1,2, can be used, where 0 means disapprove, 1 means approve, and 2 means approve and favorite.

Two differences relevant to voting method choice between the grant review committee scenario and the mass election scenario are:

other (conflicting) ideas for grant reviews:


what is the range in score voting (range voting)

In score voting, another argumjent for 0,1,2; this is the nonnegative portion of -2, -1,0,1,2, which maps onto the common "terrible, poor, no comment, good, great".

Multiseat equivalent of a runoff?

what is the multiseat equivalent of a runoff? mb instant runoff (rcv)?: i bet not though. google doesnt give much:

for proportional voting, mb just have a second round of proportional score voting after first using it in round 1 to eliminate most candidates. How big should round 1 be? maybe m^2 (where m is the number of seats available)? that seems too large. Maybe (maxscore+1)*m? Mb 2*maxscore*m or mb e*maxscore*m? for maxscore=2 those factors are between 2 and 3

hmm mb the last 'runoff' round for score voting should always be approval voting (0,1), bc this reduces strategy.


arrow's endorsement

"Well, I’m a little inclined to think that score systems where you categorize in maybe three or four classes probably (in spite of what I said about manipulation) is probably the best." [1]



interesting document describing a workshop at which voting method experts voted on voting methods using approval voting. The winner was approval voting, by a landslide, but Alternative vote and Copeland also were top candidates. Notably, plurality voting got zero votes.

(what are Alternative vote and Copeland? Here they are:)

Alternative vote [Alt] Each voter submits a ranking (possibly incomplete) of the candidates. One first counts the number of times each candidate appears as top-ranked (his plurality score). The candidate with the lowest plurality score is eliminated. In a second count, the votes for this candidate are transferred to the second-ranked candidate (if any) on these ballots. The process is then repeated again and again until one can- didate is ranked first by an absolute majority of the votes (original or transferred) is elected. See Farrell (2001), Farrell and McAllister? (2006). Other names for this pro- cedure or its variants: “Hare” system, “Single Transferable Vote”, “Instant runoff".

Copeland [Cop] Each voter submits a ranking of the candidates. For each candidate one computes his pairwise comparison score, that is the number of challengers this candidate beats under pair-wise majority rule. The candidates with the largest score are chosen. This Condorcet-consistent rule does not specify how ties (which are common when there is no Condorcet winner) are broken. See Laslier (1997). Other name: Tournament score.


some random technical arguments:


score voting with runoff is called 'star voting' sometimes:

by 'runoff' is meant an instant runoff that counts the number of people that voted one candidate higher than the other (ignoring by how much)

this method does mech better than plain score voting, according to [2]


3-2-1 voting is also interesting. It does very well according to [3]

"voters give each candidate one of 3 ratings — good, acceptable, or bad — and you find the winner in 3 steps. First, the 3 semifinalists are the candidates rated “good” the most; second, the 2 finalists are the semifinalists rated “bad” the least; and third, the winner is the finalist who’s rated higher on more ballots (ie, the winner of a virtual runoff)."


[4] recomends:


Approval voting is a voting method where you can “approve” (support) as many candidates as you want, and the candidate approved by the most voters wins. Its VSE is around 89-95% for most levels of voter strategy. That’s not the best of the methods I tested, but it certainly is the best “bang for the buck”; a simple reform, with basically no downsides, which improves outcomes hugely.

3-2-1 voting is a voting method where voters give each candidate one of 3 ratings — good, acceptable, or bad — and you find the winner in 3 steps. First, the 3 semifinalists are the candidates rated “good” the most; second, the 2 finalists are the semifinalists rated “bad” the least; and third, the winner is the finalist who’s rated higher on more ballots (ie, the winner of a virtual runoff). This method’s VSE runs from 92-95%, even with strategic voters. Also, because it’s one of the methods which best avoids giving an unfair advantage to those strategic voters, and because its simple ballot format is approachable for all voters, it encourages honest (non-strategic) voting. In my opinion, this is the best single-winner election method for large-scale political elections with diverse voters.

Score runoff voting (SRV), also with different possible score levels. This is like score voting (explained below), except that you choose the top 2 candidates based on scores, and then find the one of them who’s rated higher on more ballots (ie, the winner of a virtual runoff). With enough possible score levels, this has a VSE of 91% all the way up to 98% —— better than even 3-2-1 voting. The only reasons I chose to highlight 3-2-1 voting above this method is that 3-2-1 has a simpler ballot and resists strategy slightly better. But SRV is undeniably a top-shelf election method, and arguably the best out of all the ones I tested. "


more evidence for plurality being especially bad:

"Plurality voting, also known as choose-one plurality or first past the post: This is the most common election method in the English-speaking world. It’s also in most situations the worst out of all the methods I’ve tested, with a VSE of only around 75%. It often gets “spoiled” results, where a weaker candidate wins due to vote-splitting; it encourages strategy; and it leads to uncompetitive politics, dominated by big parties (and their big donors) who get their votes as much through fear as through hope." -- [5]


[6] says that the bullet voting argument against approval isn't really a problem: " The issue is that in approval voting, it’s not entirely clear what constitutes an “honest vote”; how many candidates should a voter approve? This has led some people to criticize the method, suggesting that it leads to too many “bullet votes” for just one candidate. However, when I tested the method with realistic clusters of voters and issues, I found that including up to 70% of bullet voters actually improved the outcome by a tiny amount, making it slightly more robust against strategy (whether or not the strategic voters bullet voted). Thus, I find this criticism to be without merit. "


Proportional Approval Voting (Multi-Winner):

"Each voter chooses (no ranking) as many candidates as desired. Voters may not vote more than once for any one candidate. Add all the votes. Elect the candidate with the most votes in the first round. After each round, use the following formula to reweight the ballots: (1/ (1 + Total Approvals Given to Elected Candidates)). Continue to elect candidates and reweight the ballots each time until all the seats are filled." [7]


personally i still think "two hands" (my name) score voting (score/range voting with a range of 0,1,2) is probably superior to approval because i feel that psychologically, people will feel that they should 'bullet vote' a lot with approval just because they will feel 'disloyal' to their favorite candidate if they don't give that candidate a higher score than evryone else.

and if there is almost all bullet voting, this will delegitimize approval and cause people to want to switch back to first-past-the-post.

adding a runoff to score voting is a good idea imo.

i like 'two hands' voting better than wider ranges b/c:


MMP (mixed member proportional), GOLD, and AV+ also get some buzz.

And STV (but i think it's too complicated) and IRV (but i think it has poor theoretical results)


interesting that approval/score tend to form a 'family' that theoretically does pretty well even with strategic voters (in score voting's case, runoffs may be needed), and that 3-2-1 and median-based methods also do well, by seemingly different means. Perhaps these 3 can be combined?

Especially 'two hands' score voting seems like it could be easily combined with 3-2-1, at least in the single-winner context: maybe just use two-hands score voting (non-proportional) to select the 3, then drop the candidate with the most 0 votes, then have an instant runoff between the last two. Like two-hands, this could pretty easily be done in physical assembly, just with 3 rounds instead of 2.

however, that's a little more complicated than just two-hands, and it encourages negative campaigning.

"majority judgement' seems to be the best and most well-known 'median' method, and here it is: "Voters freely grade each candidate in one of several named ranks, for instance from "excellent" to "bad", and the candidate with the highest median grade is the winner." [8].

The range voting guy dislikes median methods: [9]

Again, this could easily be combined:

Computing the median isn't very quick in a physical assembly, because you'd need to count the number of 0s, 1s and 2s instead of just the total number of hands (assuming that you can't just tell people to sort themselves by moving to sides of the room, then peeling off an outlier from each side one by one...). But there is some connection that i don't understand yet between medians and methods (such as majority approval voting) that first consider only the highest score, and then consider lower scores (is the connection just that after the initial round, now you are lumping together the highest score and the next-highest score? probably.). mentions the following:

" In particular, Chris Benham published his voting method "Majority Choice Approval" (MCA) on the internet in 2004 (or before; I am not sure when Benham first did it). MCA is:

In a physical assembly, this would, again, require either writing not just the total for each candidate but also the number of 0s 1s and 2s, or it would require having multiple rounds. So how would the multiple rounds work? Maybe:

Can we modify this to make it a range of 0,1,2, yet still reasonably quick in a physical assembly? I don't really see how. But.. the impediment to doing more complex stuff with 0,1,2 is just that the count has to be made 3 times instead of once. But having 3 rounds takes just as much, and probably a little bit more, time. So maybe:

sounds complicated. Can slightly simplify one step by realizing that once we have the 0s, 1s, 2s written down we can just say c'compute the median':

This is already getting pretty complicated, and it's not clear that it would capture the performance of Majority Judgement; the median part is close to Benham's Majority Choice Approval, but lacks the 'remove one ballot of median grade and recurse' part of Majority Judgement; i'm not sure how important that part is. Adding that part in would make it more complicated.

There may be some redundancy between the median component here and the 3-2-1 component; namely, if we look at score voting and ask what the median component here fixes, it's probably that, relative to score voting, it penalizes candidates who have a high score by virtue of having lots of +2 but at the cost of lots of +0s (eg polarizing candidates); or, said another way, it rewards candidates who have fewer +0s even if that means they have fewer +2s. But the elimination in step 3 of a candidate with lots of 0s penalizes the same thing.

So, maybe we can dispense with the median component, since i dislike complexity.

Also, maybe move the elimination of the top-3 candidate with the most 0s to after the second round, because otherwise in Round 1, many voters may give 0 because they don't know a candidate rather than because they don't like them. This also removes most of the incentive to do any negative campaigning until in between Round 1 and Round 2. However, if ballots are available, then Round 2 could be 'instant', eg there need only be one round in that case.

However, this isn't really a great method for physical assemblies, because asking to count the number of 0s, 1s, and 2s is too much to ask; in a large assembly, there is a sea of hands, and identifying which pairs of hands are coming from one person, as opposed to coming from two separate people, and counting people not raising any hands is asking too much.


so, a proposal:

Each step here can have ties and i'm not quite sure what to do about them -- i'd like to keep it simple.

This is a combination of SRV (with a range of 0,1,2) and 3-2-1 voting.

A modified version can be used in a physical assembly without ballots:

A flaw here is that allowing two hands in the first vote but not in the subsequent votes will confuse people.

Is there a proportional multiwinner analog (assuming written ballots)? Maybe not quite, but:

?? what is the multiwinner proportional analog of a runoff??

('collapse' is not quite a precise multiwinner analog of runoff because a voter could vote a 1 for candidate A and a 2 for candidate B, and this preference will be lost in a collapse operation, whereas in an instant runoff it would be preserved)

I don't really like the 'collapse' step. It's not really doing anything different than the 'reverse and collapse' step; both are penalizing candidates who only won the first step at the cost of many 0s. So maybe just:

is the second step really any better than the other 'reweighted range voting with support' kind of stuff? probably, yes, because the factions get an proportional chance to eliminate their enemies.

hmm how about instead of the previous third step, we do an 'intelligent expand-collapse':

although if we're willing to introduce heavy-duty computation at this step, maybe just compute preference rankings then use STV?


hmm, this explanation of 3-2-1 voting [10] (which has a great reminder of a great Simpson's cartoon about the problems with the spoiler effect) has the following complexities:

" A couple of extra rules cover some unusual situations. First, no two semifinalists can be from the same party; that stops one party from sweeping the election by running three identical candidates. Second, it takes at least 15% “good” ratings to become a semifinalist; so when there are only two serious candidates, a third unknown can’t win by mistake. "

the idea of identifying who is not from the same party is too easily abused. so how about we use proportional score voting for step 1?

1. Proportional/reweighted score voting on -1,0,1 to select 3 candidates 2. Eliminate the candidate with the most -1s (in case of a tie, the candidate with the most -1s and the smallest (non-re-weighted) score) 3. Runoff (possibly instant, if there were written ballots) between the final two candidates

This has a fairly obvious generalization to multiseat:

1. Proportional/reweighted score voting on -1,0,1 to select 3*SEATS semifinalist candidates 2. Eliminate the SEATS candidates with the most -1s (in case of a tie, the candidate with the most -1s and the smallest (non-re-weighted) score), leaving 2*SEATS finalist candidates 3. Proportional/reweighted score voting to select SEATS winners out of the 2*SEATS finalist candidates

Even this multiseat generalization can be instant if the ballots were written. If the ballots are not written, have 3 separate rounds of voting, requiring a show of hands n+4 times, where n is the number of candidates (if you could -1s, 0s, and 1s are counted separately for each candidate, then you only need one initial vote and one runoff vote, but this would require 3n+1 times which is probably slower).

otoh maybe for single-seat we should WANT 'one party sweeping the election by running three identical candidates' --- after all in a single-seat election, a majority party is going to win anyways, so shouldn't a runoff give voters a choice between two of that party's favorite candidates? but then that doesn't generalize to multiseat as well.

hmmm.... maybe we need more than 3 semifinalists... eg imagine that there are 3 main parties, each with 2 main candidates... if one party is bigger than the others, then one of its candidates is probably gonna win, so we want the finalists to both be from that party... so we'd need at least 4 semifinalists in this scenario, and then eliminate 2 of them? hmmm not sure if that's good either...

let's try it:



recursive halving candidate pool using 321 criteria (instead of reducing to fixed number 3)

that is:

--- makes a case for GOLD for multiwinner, even against reweighted range:

(it likes STV pretty well too, but i think STV is too complicated)

for single winner, it likes:

the author says:

"Here's a comparison table. Tab 1 has multi-winner, tab 2 has single winner. I like GOLD and 3-2-1, but star/SRV is also good."


random forum thread:


great overview:


i guess the analog of a runoff in multiwinner is a multiwinner extension of a concorcet method. CPO-STV is such a method. So you could do CPO-STV in the last round to make an analog of 3-2-1. But that's computationally complex; even for a 3-seat election, that's 6 choose 3 = 20 scenarios to compute. I prefer methods that could be done by hand, if necessary.

but i guess you could do 3 rounds of score voting:

0) have voters assign -1,0, or 1 to each candidate, with 0 being the default 1) select 3*#seats via reweighted score voting 2) change votes of '1' to '0', then eliminate 1/3 (that is, #seats, since we had 3*#seats) via reweighted ANTI-score voting (that is, reweighted approval voting after combining +1 and 0 votes) 3) go back to the original votes, then change votes of '-1' to '0', then out of the remaining 2*#seats, select 1/2 (that is, #seats) via reweighted score voting (that is, reweighted approval voting after combining 0 and -1 votes)



combining the -1 and 0 in the last round reminds me of 'majority choice approval', a median-like method. Recall that above i said:

" interesting that approval/score tend to form a 'family' that theoretically does pretty well even with strategic voters (in score voting's case, runoffs may be needed), and that 3-2-1 and median-based methods also do well, by seemingly different means. Perhaps these 3 can be combined?


"majority judgement' seems to be the best and most well-known 'median' method, and here it is: "Voters freely grade each candidate in one of several named ranks, for instance from "excellent" to "bad", and the candidate with the highest median grade is the winner." [11].

The range voting guy dislikes median methods: [12]


" In particular, Chris Benham published his voting method "Majority Choice Approval" (MCA) on the internet in 2004 (or before; I am not sure when Benham first did it). MCA is:



so i guess combining the -1 and 0 in the last round is a good place to start. Another nice thing about that is that it blunts the impact of negative campaigning; -1 (as opposed to the default 0) is only useful in round 2, and because round 2 is proportional, your faction only gets to remove a quantity of candidates proportional to its strength; after removing your allocated quantity of most hated candidates, further -1s don't help your faction. So, in a two-party scenario, if everyone in your faction just marks everyone in the other party with -1s that doesn't help you much; you should rather identify only the most hated candidates in the other party and mark only them down.

And that would seem to serve as an incentive to candidates to try to compete with others within their faction to be the most palatable to the opposing faction.

Also, for a pair of competing candidates within a faction, if they think they will both survive round 2, there's no benefit to them to negative campaign against each other; and otherwise the negative campaigning is aimed at voters in the opposing faction.


like 3-2-1 voting, we can also allow one level of delegation; each candidate may pre-publish a list of their recommended vote, and instead of marking anything down, you can write in the name of a candidate, in which case your ballot is treated as if it had voted their list. This has the added benefit that, for the reasons given above, these candidate-recommended lists are incentivized to not just vote -1 for all opposing party candidates, and, if the candidate given the list thinks themself extreme, to vote +1 for at least some moderates -- forcing candidates to admit which of the politicians outside of their faction are the least bad may promote social cohesion.


Another nice thing about this system is that, although there is only one positive preference level above the default (+1), bullet voting is still disincentivized at least for some voters because if you bullet vote and then your preferred candidate gets eliminated in round 2, you have no impact in round 3.

So if you suspect that you might have political views that seem extreme or obnoxious to a lot of other voters, you are incentivized to also choose some moderates to give +1s to rather than only bullet vote for your favorite candidate.


this doesn't quite reduce to 3-2-1 voting in the single-seat case. my hypothesis is that it will do nearly as well, however.


(later: note: triscore voting was posted at on Wed Sep 12 17:24:46 PDT 2018 )

so to reiterate, the current proposal, which i am tentatively calling 'triscore voting', is:

0) have voters assign -1,0, or 1 to each candidate, with 0 being the default 1) select 3*#seats via reweighted score voting 2) change votes of '1' to '0', then eliminate 1/3 (that is, #seats, since we had 3*#seats) via reweighted ANTI-score voting (that is, use reweighted approval voting after combining +1 and 0 votes, then flipping polarity, to select 1/3 of the candidates, then eliminate those selected) 3) go back to the original votes, then change votes of '-1' to '0', then out of the remaining 2*#seats, select 1/2 (that is, #seats) via reweighted score voting (that is, reweighted approval voting after combining 0 and -1 votes)

this is only about 3x as much computation as score voting with 3 choices, so it's still feasible to do by hand.

we also allow one level of delegation; anyone may pre-submit and pre-publish a list of their recommended vote, and instead of marking anything down, you can write in the name of a candidate, in which case your ballot is treated as if it had voted their list.


random forum discussion on drawing district boundaries while avoiding gerrymandering:

freen 2 days ago [-]

There is a simple, game theoretically optimal solution to gerrymandering: an iterative process of I cut, you choose.


oconnor663 2 days ago [-]

Is there a way to cut a cake between 3 people with this sort of approach? If there's not a way in general, could there be a way with the added assumption that cooperation to screw the 3rd person would be detectable and publicly shamed enough to disincentivize it?


ZeroGravitas? 2 days ago [-]

Apparently there is a way, but it's complicated and probably doesn't translate from cake to maps (you don't end up with neat slices, but rather a collection of different sized pieces):

And it's not clear if one of the cake cutters could sacrifice their own slice to benefit one of the others (in return for a bribe).


mc32 1 day ago [+1]

dragonwriter 2 days ago [-]

> Is there a way to cut a cake between 3 people with this sort of approach?

Cake? Maybe. Equal-population single-winner firdt-past-the-post political districts, no. But that's okay, because single-winner districts don't practically support more than two competitive parties, especially when you have separately elected statewide and national executives rather than a parliamentary system.

Of course, get rid of single-winner FPTP districts, and you not only can support more parties, you make gerrymandering mostly pointless, even if you do still have districting.


koheripbal 2 days ago [+2]

simonsarris 2 days ago [-]

You could do it the Settlers of Catan way.

(Suppose four players, P1 to P4. They each get to place two settlements, and there's an advantage to getting first pick. You place settlements in this order: P1, P2, P3, P4, P4, P3, P2, P1)

Now, there is a cake (map) to cut, and 4 parties. P1 draws one line, then P2, then P3. Then, each party picks one of the 4 slices, starting with P4.


jeffwass 2 days ago [-]

Yes, “envy-free” cake cutting.


oconnor663 1 day ago [+1]

pqh 2 days ago [+1]

hinkley 2 days ago [-]

IIRC the three slice solution is: I cut a piece. You decide if I get the piece or half of what’s left. The third guy cuts and whoever else doesn’t have a piece now, choses.



Combining multiseat Proportional Score Voting with 3-2-1

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.

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. 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 candidates. 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 winners.

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 candidates).

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)


" specialist 9 hours ago [-]

Agora's ballot privacy (secrecy) is apparently based on Neff shuffling.

A verifiable secret shuffle and its application to e-voting [2001]

All prior crypto-based voting systems I've studied rely on hash collisions, algorithmically simulating the secure one-way hash of physically dropping a ballot into a box, for an individual's ballot to get lost in the herd. But for this to work, ballots have to be simple and elections have to be large (enough).

In the USA, ballots are complicated and precincts are small. Appropriate for elections administration based on the Australian Ballot, bad for crypto-based balloting system.


I'm very surprised this is the first I've heard about Neff shuffling.

But I know a lot about VoteHere?. Even though they are a proven bad actor in this space, I'll suspend disbelief and see if something good came out of their efforts. The Agora people appear smart, earnest. So maybe there's something here.

If Neff shuffling (or something similar) actually works for this application, it'd be remarkable. Least importantly, I'd have to update my world view. Specifically: no fully digital voting system can both protect the secret ballot and ensure a public vote count. (In practice, electronic voting systems do neither.)


PS- Scanning the other comments, feel compelled to point out:

Design the whole system. Understand election administration. Protecting the ballot is not enough. Information also leaks from poll books, voting history, etc., which then deanonymizes the secret ballot.

For Sierra Leone, Agora might be a great idea. Maybe the benefit of extending the franchise (reduced costs, increasing access) outweighs the loss of individual privacy.

reply "

iokevins 15 hours ago [-]

They used Agora:

(Aside: forgot to turn off my extension and so it appeared as, 'Sierra Leone just ran the first "Multiple copies of a giant Excel spreadsheet"-based election')



specialist 53 minutes ago [-]

Canada's elections are quite amenable to manual tabulation. Color me envious.

Years ago, a group of us activists in the USA tried to understand manual counting better. We timed ourselves on mock elections using the "sort and stack" system. It goes about as fast as you'd expect (and reminded me of being a junior bookkeeper reconciling accounts at the end of period, ahhh nostalgia).

We also self-funded some official manual recounts, just to get better metrics. It wasn't pretty.

USA's elections are generally over complicated, with many races/issues per ballot. To enable efficient manual counting, we'd have to redesign our ballots so that they could be physically separately into federal, state, local sub-ballots, for separate processing.

Replacing FPTP with Approval Voting and Proportional Representation would also help.

reply "


specialist 1 hour ago [-]

"... electronic ballots recorded in a proprietary system by a private company..."

This is the second biggest threat to elections. And probably the most timely.


My tour of duty as an election integrity activist radically changed my worldview on these things. I previously thought the gear was the biggest problem.

Now I know that the biggest threat is disruption. While well intentioned, HAVA caused a lot of disruption. Resulting in no one knows what the rules are. Ditto the continuous ongoing "reforms". Like changes in voter ID laws, rules, procedures. Moving poll sites. Etc. Any changes that must be made should be done incrementally, methodically.

The second biggest threat is the privatization of our election administration. Like you observe. No private entity any where should be responsible for verifying eligibility, issuing ballots, counting votes. Election administration is the most fundamental function a democratic government performs, its prime responsibility. It must be performed by citizens working for the government to have any legitimacy whatsoever.

The third biggest threat to our elections is our form of voting. The USA's FPTP (winner takes all) elections are very brittle, intolerant of the inevitable margin of error. Much better, for both democracy and election administration, would be to use Approval Voting and Proportional Representation.

Yes, I still believe the gear we continue to use remains a big open untreated wound.

reply "


a crypto voting (and treasury funding) method (based on -1,0,1 score voting)

"the proposed treasurysystem supports liquid democracy/delegative voting for better collabo-rative intelligence. Namely, the stake holders can either vote directly onthe proposed projects or delegate their votes to experts. Its core compo-nent is a distributed universally composable secure end-to-end verifiablevoting protocol. The integrity of the treasury voting decisions is guar-anteed even when all the voting committee members are corrupted. Tofurther improve efficiency, we proposed the world’s first honest verifierzero-knowledge proof for unit vector encryption with logarithmic sizecommunication. This partial result may be of independent interest toother cryptographic protocols. A pilot system is implemented in Scalaover the Scorex 2.0 framework, and its benchmark results indicate thatthe proposed system can support tens of thousands of treasury partici-pants with high efficiency."


another crypto voting method -- i'm not sure but it looks like it allows the voter to prove to a third party how they voted, so unless i misunderstand, i don't like it




2019 Python programming language governance system


good read:


tool "Multiple voting systems - Single choice, Approval voting, Quadratic voting, and more" Single choice voting Approval voting Quadratic voting Ranked choice voting (IRV) Weighted voting



mb relevant to something someday

the many faces of strategic voting


" For instance, the legislative branch was designed to require compromise, yet Congress, social media, and partisan cable news channels have co-evolved such that any legislator who reaches across the aisle may face outrage within hours from the extreme wing of her party, damaging her fundraising prospects and raising her risk of being primaried in the next election cycle.

Reforms should reduce the outsize influence of angry extremists and make legislators more responsive to the average voter in their district. One example of such a reform is to end closed party primaries, replacing them with a single, nonpartisan, open primary from which the top several candidates advance to a general election that also uses ranked-choice voting. A version of this voting system has already been implemented in Alaska, and it seems to have given Senator Lisa Murkowski more latitude to oppose former President Trump, whose favored candidate would be a threat to Murkowski in a closed Republican primary but is not in an open one.

A second way to harden democratic institutions is to reduce the power of either political party to game the system in its favor, for example by drawing its preferred electoral districts or selecting the officials who will supervise elections. These jobs should all be done in a nonpartisan way. Research on procedural justice shows that when people perceive that a process is fair, they are more likely to accept the legitimacy of a decision that goes against their interests. Just think of the damage already done to the Supreme Court’s legitimacy by the Senate’s Republican leadership when it blocked consideration of Merrick Garland for a seat that opened up nine months before the 2016 election, and then rushed through the appointment of Amy Coney Barrett in 2020. A widely discussed reform would end this political gamesmanship by having justices serve staggered 18-year terms so that each president makes one appointment every two years. " -- by Jonathan Haidt