Cascading Vote and Delegative Transfer Proportional Systems

Two families of new proportional voting methods

13 min readSep 28, 2023

An interesting fact about Single Transferable Vote (STV): It yields proportional representation (specifically, proportionality for solid coalitions) regardless of how candidates are eliminated. Instead of eliminating the candidate with the fewest votes, you could eliminate the candidate with the lowest Borda score. Or the Condorcet loser. If you wanted to be especially perverse, you could even eliminate the candidate with the most votes (while still electing candidates with more than a full quota of support). Regardless of what you do, you still get proportional representation.

This suggests that you can take your favorite single-winner voting method, use it for eliminations in the STV algorithm, and get a proportional voting method. There’s just one problem with this: A lot of single-winner voting methods allow equal rankings (or scores), and we want to accommodate them. But STV is based on voters having a single vote that gets transferred; if voters voted for multiple candidates who were elected simultaneously, this could result in these voters effectively having multiple votes, violating proportionality.

Cascading Vote

The solution is to only elect one candidate at a time. Instead of votes giving their highest ranked/scored remaining candidate one vote, we say that voters offer a vote to each of the highest ranked/scored candidates on their ballots. When a candidate is elected, they accept the offered votes of a full quota of the voters who offered them a vote; these accepted votes cannot be offered to other candidates in future rounds. And if multiple candidates are being offered enough votes to be elected, use the single-winner method you’re using for eliminations to choose the best one of these candidates to be elected; those not chosen may still be elected in later rounds.

This yields the “cascading vote”: When a candidate is eliminated, voters who ranked/scored that candidate strictly above all others have the votes they are offering “cascade” down to the highest tier of support on their ballots where there is at least one remaining candidate. For example, if you’re using 5-star ballots, all the candidates you’ve given four stars to have already been eliminated and the only candidate you gave five stars to has just now been eliminated, your (offered) vote cascades down to all the candidates whom you’ve given three stars.

If there aren’t any candidates left to whom you’ve offered any support, your vote doesn’t cascade down to all remaining candidates, however. As with STV, your ballot is exhausted in this case. And if the number of remaining candidates equals the number of seats that still need to be filled, all of the remaining candidates are elected. This makes the single-winner method used as the basis of the cascading vote method especially important.

(I’ve elided the question of how to choose which votes are accepted. Practically speaking, you want to use some weighted form of surplus handling such that, if you offered a vote to a candidate who was offered half again as many votes as she needed to be elected, your vote stays active at 1/3 of its original weight. Cascading Vote methods aren’t meaningfully different from STV with regard to surplus handling.)

One notable difference between cascading vote methods and STV (along with various cardinal proportional voting methods like Allocated Score) is that only the tallies from the original ballots are used to determine the elimination order. That is, if a ballot’s vote is accepted by a winner, that ballot remains just as influential as any other for determining which candidates are eliminated. This approach has a number of advantages:

It’s simpler; there’s no need to recalculate every candidate’s support each round for the purpose of eliminations.
It’s more equitable. Voters whose votes go to the first elected candidate have just as much influence over eliminations as voters who don’t use their votes until the final round. (Contrast this with Allocated Score, where voters forfeit all of their influence once they’re put in a candidate’s quota, giving voters who don’t get put into a quota before the final round more influence than everyone else.)
It gives less of an incentive for free-riding, as a consequence of being more equitable in this manner.

Before we describe a concrete voting method, let’s flesh out what we mean by “using your favorite single-winner voting method for eliminations”. After all, single-winner voting methods pick out a single winner; they don’t necessarily pick out a loser in an obvious way. Here’s the general solution:

Reverse the ballots. A 5-star ballot with the scores {Alice: 5, Bob: 0: Carol: 1, Dmitri: 3} gets converted to {Alice: 0, Bob: 5: Carol: 4, Dmitri: 2}; a ranked ballot with Alice>Bob>Carol>Dmitri gets converted to Dmitri>Carol>Bob>Alice.
Use the chosen single-winner voting method with the reversed ballots, with all the eliminated and elected candidates ignored.
Eliminate the winner of the election with the reversed ballots.

One final detail: cascading vote methods need to use the Droop quota, rather than the Hare quota, to ensure proportionality. (If you’re unfamiliar with these quotas you can ignore this section; it’s not important for anything later in this post.) For most single-winner voting methods that a cascading vote method can be based on, a slim majority can have total control over the elimination order. This means that if the Hare quota was used for a 2-winner election, a faction comprising 51% of the electorate could ensure that they won both seats. Using the Droop quota avoids this problem; in a two-winner election, the Droop quota ensures that any faction with the support of over a third of the electorate can win a seat.

All of this is probably easier to understand in more concrete terms, so let’s look at a particular cascading voting method.

Score Cascading Vote

Score Cascading Vote (SCV) uses Score as the chosen single-winner voting method. So voters score the candidates from (say) 0 to 5. Each voter offers a vote to each candidate who is given the highest score on their ballot (usually a 5, though if a voter casts a ballot on which no candidate is given a 5 then the highest-scored candidates are still offered a vote). Tabulation goes in rounds, where in each round:

If the number of remaining candidates equals the number of seats that still need to be filled, elect all the remaining candidates immediately.
If no candidate is offered a Droop quota of votes, the candidate with the lowest score is eliminated. Ballots that scored this candidate strictly higher than all other remaining candidates have their (offered) votes cascade down to the remaining candidates who were scored at least as highly as all other remaining candidates.
If exactly one candidate is offered a full Droop quota of votes, that candidate accepts a Droop quota of these votes and is elected. These votes can no longer be offered to other candidates. Ballots that scored this candidate strictly higher than all other remaining candidates and didn’t have their votes accepted by the winner have their (offered) votes cascade down.
If multiple candidates are offered a Droop quota of votes, whichever of these candidates has the highest score (across all ballots) is elected. A quota of votes is accepted by this winner; votes that aren’t accepted can cascade down.

I consider SCV to be significantly better for electing a legislative body than all previously proposed voting methods. I prefer it to single-winner voting methods because it's proportional, and proportional voting methods are the best way to end the two-party system and make gerrymandering irrelevant. I prefer it to STV and party list systems because SCV incentivizes candidates to care about the opinions of all voters, not just their core base of support. This is a key point because incentivizing candidates to care about the opinions of opposing voters is a promising path to depolarization, and depolarization is critical for averting a civil war or democratic backsliding.

Finally, I prefer SCV to Allocated Score and other proportional cardinal voting methods because of voter strategy. Under Allocated Score, if you give a relatively tolerable candidate in an opposing party a 2 it can put you in their quota, which discourages voters from giving such candidates anything above 0. SCV won’t put you in a candidate’s quota until there is no remaining candidate who you’ve scored higher; it makes more sense to give a lot of candidates nonzero scores under SCV.

STAR Cascading Vote

STAR-CV is very similar to Score Cascading Vote. The first difference is that you have STAR-style runoffs for every elimination. Instead of the candidate with the lowest score getting eliminated, the two candidates with the lowest scores will face off; whichever of them is scored higher than the other on fewer ballots will be eliminated. The second difference is that STAR Voting, instead of Score Voting, is used to elect candidates when multiple candidates are offered a quota of votes. When there are exactly two such candidates, this simply means that whichever of them is scored higher on more ballots is elected.

The downside of STAR Cascading Vote over Score Cascading Vote is the added complexity. But this complexity is unproblematic with computerized tabulation; all that’s required is a tally of the preference matrix.

There are two upsides of STAR Cascading Vote over Score Cascading Vote. First, STAR-CV further incentivizes candidates toward having broad appeal; the difference between a 1 and a 0 is bigger under STAR Cascading Vote than under Score Cascading Vote. Second, STAR-CV encourages the use of lower nonzero scores for more candidates in order to have a voice in more possible runoffs. Unlike single-winner Score, Score Cascading Vote rewards voters for using some intermediate scores; the use of 3s and 4s is necessary in order to have one’s vote cascade in a desirable manner. However, Score Cascading Vote does not provide much of an incentive to use the score of ‘1’; STAR-CV does.

Minimax Cascading Vote

The cascading vote formula also works with a wide range of ranked voting methods. In particular, we can use it with Condorcet methods. Here’s how it works with Minimax:

If the number of remaining candidates equals the number of seats that still need to be filled, elect all the remaining candidates immediately.
If no candidate is offered a Droop quota of votes and there is a Condorcet loser among the remaining candidates, the Condorcet loser is eliminated. If there is no Condorcet loser, the candidate whose greatest margin of victory against any other remaining candidate is the smallest is eliminated. Ballots that ranked the eliminated candidate strictly higher than all other remaining candidates have their (offered) votes cascade down.
If at least one candidate is offered a Droop quota of votes, whichever of these candidates would win a Minimax election against the other candidates who are being offered enough votes is elected. (If only one candidate is offered enough votes, this Minimax election is very boring.) Surplus votes cascade down.

For the most part, I prefer Score Cascading Vote and STAR Cascading Vote to the options that use ranked ballots because I think scoring ballots are better, in terms of voter psychology, at encouraging voters to give e.g. the 8th best candidate nonzero support; with a ranked ballot, even when equal rankings are allowed, I worry that voters will rank their favorite few candidates, giving each candidate a unique rank, and then stop ranking either when they’ve run out of room on their ballots or once they feel like they’ve already ranked “enough” candidates. Admittedly, this is just my intuition; I can’t point to any empirical evidence here.

One interesting option is to require voters to rank all the candidates with unique rankings like Australia does for many of its elections. (This would eliminate the need to talk about “offered votes”; every candidate with a quota of votes could simply be elected simultaneously.) This would ensure that voters indicate which disliked candidates are more tolerable than others. However, it would place a major burden on voters when there are a large number of candidates; having to rank more than 20 people in order to have your vote count would be a headache for many voters. Likely such an option would need to be combined with some sort of a primary to limit the field to a more manageable number of candidates (say, 10).

Delegative Transfer Proportional Methods

One issue with cascading vote methods, STV, and candidate-based proportional voting methods more broadly is that they’re a pain to tabulate. Specifically, they aren’t precinct summable; all the ballot data has to be sent off (either physically or electronically) for centralized tabulation.

However, looking at the SCV algorithm, we find that the eliminations don’t pose problems for precinct summability; they’re determined solely by each candidate’s total scores, which are trivial to determine in a precinct-summable manner. STAR Cascading Vote and Minimax Cascading Vote are similar; you just need the pairwise comparison matrix, which can also be determined in a precinct-summable manner. It’s only the part where one’s offered votes cascade down that is incompatible with precinct summability.

(Note for readers who are familiar with the technicalities of precinct summability (everyone should skip this paragraph): Making PR consistent with precinct summability requires that you greatly limit expressiveness in some fashion. In order to get true proportionality in a way that isn’t heavily reliant on vote management systems like Single Nontransferable Vote, you need to keep track of ballot weight (or at a minimum, whether a ballot has already been used to elect someone). Allowing ballot weight to be a function of both the full content of the ballot and the candidates elected so far means you need O((n-1)!/(n-w)!) separate tallies, where n is the number of candidates and w is the number of winners, in order to have a separate tally for each way ballot weight can change based on the winners so far.)

The only way I know of to get PR while maintaining precinct summability is to deny voters the freedom to have their votes get transferred however they want, and instead make them choose from a relatively small list of options. All party list systems can be thought of as special cases of this, where voters select a party and that party chooses how their votes will get transferred. (In the case of open list systems, the electorate determines how each party’s votes get transferred as part of the election, but voters still lack the option to decide everything for themselves.) The main alternative approach is to have each candidate publish a preference order of all the other candidates ahead of the election, have voters choose a single candidate, and have those candidates’ votes get transferred according to their preference orderings. This is what is done in voting methods such as PLACE and what we will use here.

Here’s how a delegative transfer proportional (DTP) method works. As with cascading vote methods, each DTP method is based on a single-winner method that is used for eliminations in exactly the same way as in the corresponding cascading vote method. However, the way that votes move about is determined by the preference orders published by the candidates.

To start with, let’s suppose that each voter selects a single candidate as their favorite — there may be tied scores or rankings, but not for first place. Let’s also require all of the preference orders to include all of the candidates and not contain any ties, so instead of talking about “offered votes” we can just say “votes” and elect every candidate whose vote total surpasses the Droop quota. Now the transfer rule is extremely simple: whenever a candidate is eliminated, transfer their votes in accordance with the preference order of those voters’ first-choice candidates. (If your favorite is eliminated, and then the top candidate on your favorite’s preference order also gets eliminated, your vote is still transferred from there in accordance with your favorite’s preference order.) Transfers of surplus votes upon candidates being elected work the same way.

And that’s it! We’ve shown how to take a fairly arbitrary precinct-summable single-winner voting method and turn it into a precinct-summable proportional method. But there’s one last detail: We’d still like to give voters the option of indicating multiple first choices. The solution is to use equal-and-even Cumulative Voting for allocating votes in the first round: If you mark N candidates equally high and above all others, you provide 1/N votes to each of them, and each of them transfers their share of your vote in accordance with their preference order. And with this addendum, we have the delegative transfer proportional (DTP) formula for converting single-winner methods into proportional methods.

One way of thinking about transfers in DTP: For the purpose of transfers, there is one ballot for every candidate. Each of these ballots is initially weighted according to how many votes the candidate receives in the first round, and the reweighting works as it does under STV.

The “delegative transfer” part of DTP is not especially novel. What is novel is delegative transfers combined with using the full ballot data for eliminations, such that voters’ later preferences are still highly influential.

DTP-STAR

We started off by putting Score Voting into the cascading vote formula. What if we do this with the DTP formula? This isn’t exactly great because the only scores that it makes sense to use are 0, 4, and 5; there’s no downside to strategic exaggeration unless you're exaggerating all the way to a 5 and causing your vote to be split.

So let’s use STAR as our base single-winner method instead of Score. Here’s what DTP-STAR looks like:

Before the election, each candidate publishes a complete preference order of all other candidates.
Voters vote on 5-star ballots.
Each candidate receives a number of votes equal to the number of voters who scored them uniquely above all other candidates, plus half the number of voters who had them tied for first with exactly one other candidate, etc.
Whenever no candidate has a Droop quota of votes, whichever of the two lowest-scoring remaining candidates is preferred over the other on fewer ballots is eliminated. (Candidates’ preference orders have no effect here.) The eliminated candidate’s votes are transferred in accordance with the preference orders of the voters’ first-choice candidates.
Whenever a candidate has a Droop quota of votes, that candidate is elected. Excess votes are transferred as above.

Ultimately, I prefer STAR Cascading Vote to DTP-STAR. As a rule of thumb, I trust politicians less in determining who should be in office than in determining anything else, and the delegative aspect has a significant “ick” factor to me. But I don’t claim that this argument is remotely rigorous, and my feeling of “ick” is of negligible importance compared to the issues of democratic backsliding or a possible civil war. Since they use the same process for eliminations and the elimination algorithm is what yields the incentives for having broad appeal (and not being polarizing), DTP-STAR and STAR Cascading Vote should be similarly effective when it comes to depolarization. DTP-STAR delivers nearly all of the benefits of STAR Cascading Vote, and it delivers them while being precinct summable. I’m not convinced that precinct summability is actually all that important, but if it is, I think DTP-STAR is an excellent solution.