SODA voting (Simple Optionally-Delegated Approval)

From Electowiki
Revision as of 18:15, 17 February 2012 by Homunq (talk | contribs) (Full, step-by-step rules)

Jump to: navigation, search

Simple Optionally-Delegated Approval (SODA voting) is a single-winner voting system based on approval voting. As in approval voting, voters get a ballot which lets them vote for as many or as few candidates as they like. Unlike in approval, voters who don't want to decide yes or no for every candidate can choose to delegate their ballot, trusting their favorite candidate to "fill in the ballot" by adding further approvals. The voters' ballots are counted and preliminary totals are announced. Then, once everyone knows how much support and how many delegated ballots each candidate has, those who do not expect to win can add approvals to the delegated ballots they control, in order to help others who would be good compromise options.

Approvals added by a candidate to their delegated ballots are worth the same as direct approvals from voters. Both types of approval are added together and the winner is the candidate with the most total approvals.


Sample Ballot

Vote for as many candidates as you approve, or vote for one candidate to delegate your vote:
▢ Adams (Prefers Churchill, then Bachelet, then Deakin)
▢ Bachelet (Prefers Deakin and Churchill)
▢ Churchill (Prefers Adams)
▢ Deakin (No preferences declared)
▢ ____(write-in)____
▢ ____(write-in)____

If you only vote for one candidate, your vote is delegated to that candidate, unless you check the box below. A delegated vote means that if your favorite cannot win, they may use your vote to approve of a compromise who can, according to their preferences above.
▢ Do not delegate. Do not let the candidate I voted for add approvals to my vote.

Note: If you vote for more than one candidate, or if you vote for a write-in or other candidate with no declared preferences, your vote is not delegated. In that case, no-one can add approvals to your vote, and it does not matter whether you check the box above.

Full, step-by-step rules

Rules marked "(optional)" are suggested, but only the four steps in bold (1a, 2b, 3b, and 3b.i) are necessary for the system to function. Items in italics explain the justification behind each step. Terms are defined on the right.

1. Candidates publicly declare their rankings of the other candidates
a) Before the election, all candidates must rank the other official candidates in order of preference. Equal rankings are allowed. These pre-announced preferences allow voters to make informed decisions in step 2; and prevent corrupt vote-selling in step 3, where candidates may only approve others in a way that is consistent with their preferences.
b) (optional) Candidates may enforce any prior agreements of mutual preferences by changing their own preferences in response to those of others*. This enforcement mechanism is designed to deter betrayals, so that it should almost never actually need to be used. Candidates, realizing that changing preferences at this point will only make them look untrustworthy, will prefer agree on mutual approvals beforehand.

Enforcing mutual approval agreements: If A ranked B higher than B ranked A, then A may respond by lowering B's ranking. B may respond by raising A, and A may then respond by restoring B to the original ranking. If B raised A above C, then C may lower A, but A may not respond. Three rounds of response are all that are allowed.

2. Voters vote delegated plurality-style votes or non-delegated approval-style votes

a) (optional) Ballots list all candidates in order from candidates who submit the most complete* rankings of other candidates, to those who submit the least. Where this is a tie, ordering is random. This is a slight incentive for candidates to submit a complete ranking in order to appear high on the ballot.

Most complete: Completeness of ranking is scored by how many possible pairs of other candidates contain two candidates at different levels.

b) Voters may approve one or more candidates, and also have the option to mark "Do not delegate". Delegated votes* and total approvals* are tallied and announced for each candidate. Announcing full tallies makes step 3 predictable and fair.

Delegated votes:The delegated votes for candidate X is the number of ballots that approve only X and do not mark "Do not delegate". X will get a chance to effectively add additional approvals to these votes; or, if X has less than 5% approval, then additional approvals will be added automatically, in order to attain a better result by X's predeclared preferences.

Total approvals:The total approvals for candidate X is initially the number of ballots that approve X, including X's delegated ballots. This number increases through the process as other candidates use their delegated ballots to approve X.

3. Candidates take turns to publicly use their delegated votes, starting from the candidate with most approvals

a) (optional) There is a one-week period for candidates to analyze and negotiate based on these preliminary results. Actually, the optimal strategies for all candidates and the resulting winner will already be obvious. Usually, all candidates except this winner would concede as soon as preliminary results are announced. However, for the occasional candidate inclined to act irrationally in a way that matters - say, by not delegating to an ally, even though the alternative is to see an enemy elected - this interim period would give them a chance to rethink things and come into reason.
b) As long as no candidate has approvals equal to over 50% of voters, each candidate in turn must use some or all of their delegated ballots* to approve zero or more other candidates. At each step, the candidate with the highest current approval total who has not yet used their ballots has their turn to do so. If, considering the votes, a candidate can win, they will probably prefer not to approve other candidates. If there are two allied candidates who need to cooperate to win, the one with more approvals has first chance not to approve the other. That way, when it's the weaker candidate's turn, they will no longer have the possibility of winning, so the weaker will approve the stronger.

Use their delegated ballots: The process by which a candidate effectively adds approvals to the votes delegated to him or her. When, for example, candidate A uses their delegated ballots to approve of B and C, then B and C's approval totals are each increased by A's delegated vote total. When it is a candidate's turn, they may choose to approve of nobody else, for instance if they believe they will win; but this counts as using their delegated votes, and they will not be given another chance to change that decision. When all candidates have used their votes, the election is over.

i. If the candidate using their delegated votes has more than 5% approval, they may choose which candidates approve on their delegated ballots, consistent with preferences, with equally preferred ties (optionally) broken by current approval totals*. This tie-breaking by approval totals resolves the "chicken dilemma", as explained below.

Consistent with preferences, with equally preferred ties broken by current approval totals: For example, candidate A may not approve B and not C unless either A predeclared a preference for B over C, or A predeclared B and C equal and B currently has more approvals than C.

ii. (optional) If the candidate using their delegated votes has 5% approvals or less, their delegated ballots are used automatically* to approve their preference between the frontrunners. Automatic approvals respect predeclared preferences but prevent minor candidates from having discretionary kingmaker power.

Automatic approvals: When approvals are assigned automatically for candidate X, they are set to approve as many of their pre-declared preferences as possible without approving both of the two candidates with the highest current approval totals that they prefer differently. (For example, say that X's preferences were for (A and B), then C, then (D and E), and the approval totals are 40, 30, 10, 20, and 15 alphabetically. A has the highest total; B has the next-highest, but does not count, because they are preferred equally to A; and D has the next-highest. So X's delegated ballots will be assigned to approve as many candidates as possible without approving both A and D; that is, they will approve A, B, and C.)

4. When step 3b ends, highest total wins

Write-ins: If write-ins are allowed, write-in candidates can not declare preferences or use delegated votes, but can be approved at will by official candidates, regardless of the official candidate's declared preference order.

Notes on rules

Finish resolving the "Chicken Dilemma"

Rationale for breaking equal-ranked ties using approvals in step 3.b)i.: One of the toughest situations for any voting system is the "Chicken Dilemma" between two near-clones. Say there are 60% "Blue" voters and 40% "Orange" voters; but the Blue voters are split between two candidates, say 35% for Navy and 25% for Sky. Any good voting system will allow one of the Blue candidates to win, if all the Blue voters support both of their candidates. But under many systems, there is a temptation for the Sky voters not to support Navy, so as to make sure that Sky, rather than Navy, is the winner. But if the Navy voters respond in kind, then Orange could defeat them both.,76A4FB,FF9900&chd=s:VPY&chp=4.71&chl=Navy%7CSky%7COrange

This is called the "Chicken Dilemma" because it resembles a game of chicken, in that both blue factions benefit from being the only one to deny support, but both lose if both deny support. In fact, since no single blue voter can reliably know exactly how other voters in the two blue factions will behave, it resembles a blindfolded game of chicken.

This dilemma can be a problem for many voting systems. Approval voting, Range voting, Majority Judgment, and many Condorcet systems such as Schulze voting — all of them good systems — all can have some version of this problem. There are a few voting systems, such as IRV and some versions of Condorcet, which correctly elect Navy no matter what the Sky voters do. But these systems go too far in fixing the problem, and thus get bad results in some situations that superficially resemble the true "chicken" scenario in some way.

SODA already does better than most systems in resolving the chicken dilemma. The two blue candidates can each make sure that the other one includes them in the predeclared preference order. Then, if all votes are delegated, Navy, starting with more approvals, will add approvals to those delegated votes first (after Orange), and not approve Sky. Sky will be forced to approve Navy, or see the orange candidate win. Thus, Navy will win — the correct result.

But if the Sky voters are especially strategically-inclined, they could give Sky non-delegable approvals instead of delegable votes. If Sky did not have enough delegable votes to help Navy defeat Orange, then Navy would be forced to approve Sky, and Sky would win. Of course, the Navy voters could do the same thing; either "defensively" or "in retaliation", it doesn't matter. And so, while SODA resolved the game of chicken for the candidates, it is still a possibility if the voters are ruthless and engaged enough.

This rule would allow Orange to give the win to Navy (the correct winner) if Orange wished to; in the case where the Sky voters had attempted the strategy in the previous paragraph. Remember, if Orange had cared to, they could have declared a preference between Navy and Sky to start out with. This thus resolves the chicken dilemma completely, without causing other bad scenarios; something no other serious voting reform proposal can claim.

Unfortunately, while the rule itself is simple, it's taken 6 paragraphs to explain the rationale. This is probably too much complexity for most voters, and so this refinement is not recommended for an initial implementation of SODA. However, it is good to know that such a clear, simple fix exists, in case chicken-dilemma-style strategizing became a problem later on.

Prevent weak minor candidates from having kingmaker power

Step 3.b)ii. This rule helps make this system more attractive to major-party politicians. But it's a principled rule, not just a sop to the major parties. Consider the "kingmaker" case: in a basically 50/50 split, some tiny party has the balance of votes, and manages to extract concessions far bigger than their base of support justifies, just in order to delegate those votes or not. That's unjust, and this rule would prevent it.

5% is a good cutoff here; for instance, nationally in the US, that's tens of millions of voters, and enough to deserve a voice. It shouldn't be too high, because this rule is effectively taking power away from voters; that's only justified if the faction is so small that the power is not legitimate, and so it's better to err a bit on the small side if anything. But under 5% - that is, under 10% of the winning coalition - doesn't deserve kingmaker power.


Tennessee's four cities are spread throughout the state

Imagine that Tennessee is having an election on the location of its capital. The population of Tennessee is concentrated around its four major cities, which are spread throughout the state. For this example, suppose that the entire electorate lives in these four cities, and that everyone wants to live as near the capital as possible.

The candidates for the capital are:

  • Memphis on Wikipedia, the state's largest city, with 42% of the voters, but located far from the other cities
  • Nashville on Wikipedia, with 26% of the voters, near the center of Tennessee
  • Knoxville on Wikipedia, with 17% of the voters
  • Chattanooga on Wikipedia, with 15% of the voters

The preferences of the voters would be divided like this:

42% of voters
(close to Memphis)
26% of voters
(close to Nashville)
15% of voters
(close to Chattanooga)
17% of voters
(close to Knoxville)
  1. Memphis
  2. Nashville
  3. Chattanooga
  4. Knoxville
  1. Nashville
  2. Chattanooga
  3. Knoxville
  4. Memphis
  1. Chattanooga
  2. Knoxville
  3. Nashville
  4. Memphis
  1. Knoxville
  2. Chattanooga
  3. Nashville
  4. Memphis

In this simplified example, all the residents of each city agree on the rankings of all the other cities, so there would be no reason for anybody to do anything but bullet vote. Memphis has the first option to choose approvals for its delegated votes, and, as the leader, decides to approve noone but itself. Nashville goes second; it is the pairwise champion (Condorcet winner), so it also declines to approve any others. Chatanooga and Knoxville would approve each other and Nashville, to prevent Memphis from winning. Nashville would then be the winner, with 58% approval after delegation.

Chatanooga could, before the election, not include Nashville in its preference list, hoping to force Nashville to approve it. But in that case Memphis would approve Nashville to prevent Nashville from being forced to hand the election to Chatanooga, and so Nashville would win with an even larger majority. Therefore, Chatanooga will not attempt this.


SODA has advantages for many groups. In fact, most of the advantages would fit in more than one of the categories below, so the division is somewhat arbitrary. Also, on the talk page (click "discussion" above) there are also two "hard sell" SODA pitches for two different audiences, which restate these advantages in more-opinionated terms.

For voters

  1. SODA is extremely easy for the voters; in fact, no voting system is simpler to vote. Plurality, by restricting you to only one vote, also makes it possible to mistakenly "overvote", spoiling your ballot. There is no such way to accidentally invalidate your ballot under SODA. Also, both Plurality and Approval require a conscientious voter to consider strategy and polling status; SODA allows a simple bullet vote to still be strategically as strong as possible, regardless of the candidate standings.
  2. Under SODA, there is no need for dishonesty from individual voters. A voter can safely vote for any candidate that they honestly agree with, without fear of that vote being wasted; or safely vote an honest approval-style ballot, if they do not agree with any candidate's preference order. This is drastically different from plurality, where voters must dishonestly spurn "spoiler" candidates as a matter of course.
  3. SODA does not require you to trust any politician. Any vote delegation is both safe (you can see where your delegated vote may go) and entirely optional. There's no "smoke-filled room"; any voter who dislikes how their delegated vote might be used, can simply choose not to delegate.

For society (results)

  1. SODA is far more likely to arrive at a majority result than Plurality (or even IRV). Winners will thus have a clearer mandate. In fact, SODA may be more likely to elect the pairwise champion (aka Condorcet winner, the candidate who could beat all others one-on-one) than any other system (except SODA-DAC). See the technical discussion below for the assumptions that would make this true.
  2. SODA would increase voter turnout, leading to better, more-democratic results.
  3. However, unexpected, relatively unknown or unqualified winners will be as rare or rarer under SODA than under Approval or a Condorcet system. In a polarized society, Condorcet or Approval can have such a strong tendency to elect centrists that even unqualified, largely-unknown centrists have an advantage over better-known candidates; SODA will not have such a tendency unless the stronger candidates consciously choose this as a compromise.

For society (process)

  1. SODA balances voice for minority leaders with power for majority winners or coalitions. In fact, you could say that SODA combines the best of both worlds - the negotiated, everyone-gets-a-voice majority coalitions of parliamentary government, with the decisive, buck-stops-here clear winner of a US-style system.
  2. SODA would reduce negative campaigns. A negative personal attack against opponent A would often just shift votes to another opponent B who would end up sharing them back with A in the delegation round. (In fact, some parties might decide to run two candidates per race, to explicitly take advantage of this phenomenon). Meanwhile, the candidate carrying out the attacks could also suffer if voters saw them as a slimy mudslinger.
  3. Like many other voting reforms, SODA would reduce the influence of money in political campaigns. Plurality, with its overriding need to be a frontrunner, exaggerates the importance of money. Any candidate who's not one of the two best-funded, is almost certainly at best a spoiler, and is thus aggressively ignored by voters and media. SODA in particular, by encouraging meaningful campaigns and get-out-the-vote operations by minor candidates, while still ensuring that the extra turnout those generated would have an effective impact in deciding between the major candidates, would help substitute grassroots people-power for dollar-power in campaigns.
  4. SODA is easier to count and more fraud-resistant than most systems, including IRV. It also can be run on most voting machines, including even old and outdated systems.

For activists

  1. SODA is arguably better than Plurality voting in every way (Pareto dominant). Although other systems may offer even-greater advantages over plurality, they also come with certain disadvantages. Those disadvantages may be minor; but they give an opportunity for lobbyists, corrupt politicians, and other reform opponents to attack the reform, and put reform advocates on the defensive. A SODA advocate can devote all their attention to its advantages.
  2. SODA would lead to less infighting. Because it solves the spoiler problem, there's no need for pointless debate between radicals who want to pressure their side and moderates who want to support it. Substantive debate would continue, but with less need for unhelpful acrimony.
  3. Because it allows more voices in the debate, SODA would support the passage of other reforms such as campaign finance reform. The current two-party system sees these issues, which should be bipartisan, through a partisan lens, crippling their progress. By breaking the two-party monopoly, SODA would open the potential for more cross-partisan alliances and independent support on these matters. (See the first process advantage above. One way that SODA would bring new ideas into the public debate is by giving a better voice to the leaders of smaller parties.)

For politicians

  1. SODA should be generally acceptable to honest officeholders, who are winners in a Plurality two-party system. Most of their familiar ways of thinking about the campaign would still work - except that it would reduce negative campaigning. Plurality-style voting still works just fine, and if most votes are for major parties, this system will cleanly allow a major party to win, in many cases without going to the delegation round (especially if the major-party candidates do not pre-announce delegation preferences, thus preventing an extorting minor party from demanding their delegated votes).
  2. SODA would make a politician's job more fun. Less time spent on fundraising (see process advantage #3), fewer attack ads from opponents (see process advantage #2); what's not to like? Sure, nobody would be a politician today without a high tolerance for these problems, but even for politicians, negative ads and the fundraising treadmill have gotten out of hand.
  3. SODA reduces the threat from unserious candidates. Under plurality, an even an unqualified candidate can be a spoiler. Under Condorcet or Approval, they can prosper simply because of second-choice support from two polarized camps. In fact, in the latter case, candidates can do better the less voters know about them! If there's anything that annoys a serious politician, it's losing to an unqualified cipher. SODA has no such problem, unless one of the candidates with more first-choice votes puts their reputation on the line by explicitly choosing the centrist twice (in order to avoid something worse). That's not going to happen for an unknown.

Criticism and responses

"There are other systems which are better in some ways."

This is true. Condorcet, Range Voting, and Median systems (MJ, MCA, or Bucklin) each have some claim to be the "best voting system". But SODA is the best system which has no downsides versus plurality. All those other systems require more-complicated ballots. All of them require more-complicated, or even dishonest, strategic decisions from the voter, to get the most effective vote.

So in the end, while any of those other systems would be, in my opinion, a clear net benefit versus plurality, with SODA you don't need any qualifications like "net benefit" or "in my opinion". It is simply better, in every way.

"Delegation is undemocratic"

Simple response: if you don't want to delegate, don't delegate. You can still vote for whomever you want.

Disliking delegation is no more a reason to oppose SODA, than disliking chocolate is a reason to oppose candy stores. SODA allows delegation, it doesn't force it.

Also, since candidate's additional approvals must accord with their pre-declared preferences, there is no opportunity for strategy as long as those preferences were honestly-declared. And the preferences do not represent back-room wheeling and dealing; they are public positions. The various risks of dishonestly declaring one's preference clearly outweigh the unlikely chance that doing so would give some strategic benefit.

Simple response to a politician who makes this argument: "You just want the only smoke-filled room to be the one inside your skull." That is, minority factions should have a seat at the table, as long as everything is done transparently. The two-party system stifles some of the valid concerns of the members of the party coalitions. In SODA, all vote totals, preference orders, and final approvals added to delegated ballots are known; in the end, that's not a smoke-filled room, it's a transparent seat at the table, with a just degree of power which is derived from the people.

"Sounds too much like a parliamentary system, which has caused problems in [Italy / Israel / wherever]"

SODA does involve some parliamentary-style negotiation for about a week after each election. But then it's done, and the winner is in power until the next election. Unstable parliamentary countries, or ones where minority parties have inordinate power, are that way because minority parties can leave the governing coalition at any time. SODA does not have that problem; in many ways, it's really the best of both worlds between a 3-branch system and a parliamentary system.

"Why go to the trouble of pre-announced rankings and a second round? Why not just have candidates pre-announce the approvals they will add to their delegated votes?"

This sounds appealing, but would not work if two similar candidates were in a close race to see which had more first-choice votes, while an opponent stood ready to take advantage if they split the vote (that is, in the Chicken Dilemma, also discussed above). The system as it stands allows the similar candidates to see, after the votes are counted, which of them deserves to win: the one with the higher approval total. That one will go first and not delegate their votes, and then other one (of necessity) will.

In general, this system, because it provides perfect information on voting totals at the time when delegation is happening, will make strategy obvious. This has the paradoxical result that, as long as most voters agree with their favored candidate's chosen ranking order, this system will in practice be more Condorcet compliant than a Condorcet method (because strategy could confound a true Condorcet method more often than SODA).

Criteria Compliance

SODA satisfies monotonicity, the favorite betrayal criterion, the majority criterion, and the mutual majority criterion. Depending on assumptions and definitions, it can pass the Condorcet loser criterion.

It does not satisfy the independence of irrelevant alternatives criterion in general, but it does if the "irrelevant alternative" is assumed to delegate votes in the same way as any candidate whose delegable votes they supplant. Similarly, it can only pass the participation and consistency criteria if it is assumed that candidate delegations do not change. These assumptions are not realistic, but they do show that the method is in some sense "close to" passing these criteria.

Condorcet criterion (includes highly technical discussion)

SODA fails the Condorcet criterion, although the majority Condorcet winner over the ranking-augmented ballots is the unique strong, subgame-perfect equilibrium winner. That is to say that, the method would in fact pass the majority Condorcet winner criterion, assuming the following:

  • Candidates are honest in their pre-election rankings. This could be because they are innately unwilling to be dishonest, because they are unable to calculate a useful dishonest strategy, or, most likely, because they fear dishonesty would lose them delegated votes. That is, voters who disagreed with the dishonest rankings might vote approval-style instead of delegating, and voters who perceived the rankings as dishonest might thereby value the candidate less.
  • Candidates are rationally strategic in sharing their delegated vote. Since this process is sequential, game theory states that there is always a subgame-perfect Nash equilibrium, which is always unique except in some cases of tied preferences.
  • Voters are able to use the system to express all relevant preferences. That is to say, all voters fall into one of two groups: those who agree with their favored candidate's declared preference order and thus can fully express that by delegating their vote; or those who disagree with their favored candidate's preferences, but are aware of who the Condorcet winner is, and are able to use the approval-style ballot to express their preference between the CW and all second-place candidates. "Second place" means the Smith set if the Condorcet winner were removed from the election; thus, for this assumption to hold, each voter must prefer the CW to all members of this second-place Smith set or vice versa. That's obviously always true if there is a single second-place CW.
  • Delegated votes are honest. That is, voters do not delegate their vote to a candidate whom they disagree with. Though such a strategy might work in limited circumstances (when a candidate with exactly the right preferences was available to use to create a Condorcet cycle which didn't include themself), it would be both more trusting and risky than the average strategic betrayal, and also more cold-bloodedly dishonest. To exaggerate the case: how often are psychopaths good at trusting people?

The assumptions above would probably not strictly hold true in a real-life election, but they usually would be close enough to ensure that the system does elect the CW.

SODA does even better than this if there are only 3 candidates, or if the Condorcet winner goes first in sharing delegated votes, or if there are 4 candidates and the CW goes second. In any of those circumstances, under the first three assumptions above (but without needing the last one), it passes the Condorcet criterion, not just the majority Condorcet criterion. The important difference between the Condorcet criterion (anyone who beats all others pairwise must win) and the majority Condorcet criterion (anyone who beats all others pairwise by a strict majority must win) is that the former is clone-proof while the latter is not. Thus, with few enough strong candidates, SODA also passes the independence of clones criterion.

I believe, though have not proved, that the prior paragraph would always hold regardless of the number of candidates, if the candidates were to add approvals to delegated votes in the DAC order instead of simply in order of approval score. This is not part of the definition of the method for simplicity's sake.

Note that, although the circumstances where SODA passes the Condorcet criterion are hemmed in by assumptions, when it does pass, it does so in a perfectly strategy-proof sense. That is not true of any actual Condorcet system (that is, any system which universally passes the Condorcet criterion). Therefore, for rationally-strategic voters who believe and are correct that the above assumptions are likely to practically hold, SODA may in practice pass the Condorcet criterion more often than a Condorcet system.

SODA-PR (obsolete; see PAL representation)

(Section obsolete; see instead PAL representation.)

SODA-PR is the proportional representation version of the above.

First, to simplify the ballots, the population is separated into a "district" for each seat, and "districts" are grouped into sets of 2 or 3 "co-districts". The ballot for each district lists the candidates from that district in a larger font, and the candidates from its co-districts in a smaller font. Write-ins may be used to vote for candidates from other districts not listed on the ballot, so the districts only matter for ballot simplicity (Voters do not want to have a ballot with many dozens of candidates on it, but write-ins allow full freedom for those voters who want it). Larger parties will usually run one candidate per district; smaller parties may just run one candidate per co-district set.

As with SODA, voters can either vote an approval-style vote or delegate by bullet voting. Instead of a pre-announced ranking which allows them to choose a cutoff after the election, candidates have a pre-announced list of approvals that will be added to any votes delegated to them (beyond the votes they need to be elected). Candidates may make their approval of other candidates conditional on that approval being mutual, but they may not make it conditional on anything else about the other candidate's approvals. Thus, parties cannot protect corrupt members by forcing other candidates in the party to approve of them.

The counting process is as follows:

  • While there are more uneliminated candidates than empty seats:
    • Divide each ballot by the number of uneliminated candidates it approves
    • If there are any candidates with more than a Droop quota:
      • Elect the one with the highest score (previously "unique ballots")
      • Reweight ballots for the elected candidate to reduce the total weight by a Droop quota, starting with the ones delegated to that candidate
      • Add the elected candidate's pre-declared approvals to any remaining delegated ballots for that candidate
    • Otherwise:
      • Eliminate the candidate with the lowest score
      • Add the eliminated candidate's pre-declared approvals to any delegated ballots for that candidate
  • Finally, elect all remaining candidates to fill the remaining seats.

Note: If all votes are for one of the two main-party candidates in the voter's district, and if all candidates approve everyone from their party, and if the districts are divided fairly so that plurality would give a proportional result, SODA-PR gives the same results as plurality. These assumptions will not generally be perfectly true, but they will generally be close to true, so SODA-PR will give results that are recognizably similar to those of single-member districts. It is hoped that this would make it a more acceptable system to politicians who have won under single-winner rules.

Compatibility with the US electoral college

A state could adopt SODA for assigning its electors in the electoral college. If the state did not wish to dilute its voting power, it could adopt SODA, conditional on a certain number of total electoral votes being assigned in a non-winner-take-all fashion.

The way to do this would be to use a divisor-based Proportional Representation process to assign electors between delegated votes for each of the candidates and "undelegated approval electors".

Each candidate's electors would be sworn to vote as directed by that candidate (not necessarily for that candidate), if consistent with that candidate's pre-approvals.

Each "undelegated approval elector" would know the statewide non-delegated approval total, and the total of number of affiliated electors throughout the EC, for each candidate. Other "approval electors" would count as fractionally affiliated based on their known approval totals. These electors would be sworn to vote for whichever of the two candidates with the most nationwide electors had the highest statewide undelegated approval total. They would be chosen by the candidate for whom they were sworn to vote.