Deficit Handling for Vote Unitarity PR methods

It’s possible for multiple candidates to win with less than a Hare Quota in a Vote Unitarity PR method (like Sequentially Subtracted Score), which is problematic because these methods are sequential, and so every candidate who won with a Hare Quota before the “deficit winners” took more points from their voters’ ballots than they needed to win; to keep things sequential, you’d have to retroactively lower the quota those candidates needed to win, which forces you to redo all the calculations every time someone wins with a deficit, making things very complex.

For an easier solution, you can restore this number of points to voter’s ballots when a winner gets less than the current quota (call them C1 or the “deficit winner”):
Number of points spent on non-C1 candidates * (1 - C1’s score/current quota)
Optionally: add (stored value from previous checks) to the value generated by the formula

The quota starts at a Hare Quota and drops to whatever the previous round winner’s point total was if no candidate got the current quota in that round and the current round.

Two modifications: if the formula would restore more total points to all ballots than the number of points C1 fell short of the quota, then divide the number of points restored to all ballots by a constant factor to ensure that the total number of points restored is equal to C1’s deficit (this preserves Vote Unitarity.)

And while this is an experimental step, check whether, in each round, restoring points can push any candidates to the current quota. If it can, all ballots that contribute to any such candidate get their points restored; all other ballots have the points they would’ve been restored “stored” away. We add this stored value to the number of restored points a future check would give to that ballot and keep checking, adding in each check’s to-be restored points into the stored value until that ballot can help elect someone at the current quota, at which point we restore all of the outstanding points to that ballot. The reasoning is that in non-Vote Management scenarios, the regular deficit handling formula somewhat over-prioritizes voters who already elected someone they like over voters who didn’t, because they have points to be restored that the unsatisfied voters never spent, so it favors larger parties.)

(For an example of the experimental step, if you have A 53 B 28 C 10 D 9 (5 winners, 100 votes) after 3 rounds this drops to A 13 B 8 C 10 D 9, and if we hadn’t run any deficit handling C would’ve won the final seat, since the final round would’ve been B 8 C 10 D 9. With no suspended restoration, B would win because they would have 8 points + 7/3 from the restoration = 10.333 points to C 10’s; this “suspended restoration” step first checks whether B can reach the current quota with the sum of all stored and to-be restored points; in this case, B can’t, because the current quota is 13 (from A3’s win), so it’s still B 8 C 10 D 9 and C wins.
I suspect suspended restoration is gameable somehow; if a voter strategically gives a 1/5 to a candidate that would’ve reached a quota without them, they can force point restoration that way. I don’t know if this is a serious issue, but it’s probably possible to significantly counteract it by sorting the ballots that contribute to each candidate that now meets the quota, and only restoring points to enough of the highest-contributing of those ballots to ensure the candidate reaches a quota, which is similar conceptually to Unitary Monroe, though it may slightly reduce performance under honest voting.
Perhaps a better solution is to only restore a fraction of the outstanding points, based on what fraction of your points you would be spending on the candidates that now meet the quota if you had your points restored. Actually, it might be necessary to run surplus handling here, to make sure voters don’t spend more than the current quota on the candidate once their points are restored.
You might be able to skip the delayed restoration if you always refuse to run deficit handling on the final 2 seats)

The idea behind deficit handling is that it’s similar in concept to surplus handling, relates to Vote Unitarity and the principle that the leading candidate in each round should win, and is probably one of the easiest methods to compute for when you have multiple deficit winners.

This happens with any race where there are more bullet voting groups than there are seats to be won. I do not agree that this is a problem. At each stage you elect the best winner based on current satisfaction. If equity of Utility is something you want to value above unitarity then I think @Marylander new system will do what you want.

This allows for Vote Management:

Oh, i understand the free riding issue. This is an issue but I doubt that more than a few percent of voters would play along with such a strategy even if a party could get word out without there being public fallout from a clear attempt to game the system. Keep in mind that STV has this flaw as well as Woodall free riding. Australia and Ireland dont seem to complain about either.

In the end it comes down to what you are willing to put free riding ahead of. I would put it ahead of vote unitarity but not clone immunity.

I don’t think deficit handling hurts Vote Unitarity It does, but very marginally, and in a way that seems fair (see last paragraph.) In the first example Marylander provided,

If we change it to include four blocs of 5 voters for candidates E, F, G and H respectively, the example becomes:

Old candidates (A 53 B 28 C 10 D 9) / New candidates (E 5 F 5 G 5 H 5)

Now the Hare Quota is 24 instead of 20. Plain SSS elects A, A, B, C, and D, when A should’ve gotten a majority (they got 53% of the total points from ballots that supported the winners that were elected before the ineffective ballots were added.) This is largely because the A bloc had to spend more of its points on the inflated Hare Quota than the two deficit winners. Note that this still happens with only 2 blocs of 5 additional voters and 1 bloc of 1 additional voter (a more realistic estimate.)

To show how deficit handling fixes this, here is what SSS yields after the first candidate wins with a deficit ( C ):

Old candidates (A 5 B 4 D 9) / New candidates (E 5 F 5 G 5 H 5)

Deficit handling would check if A can be restored enough points to reach a Hare Quota after C wins with a deficit, and because nobody can, revises the effective quota down to what C got (10 points.) A can get that much if restored points running the formula with the old quota (Hare) (9.333 points to be restored and 14.333 points total), so they are restored points and can win the final seat outright with 5.333 points to spare.

In case you are thinking A should have 0 points to spare if they win the final seat, I believe this is merely an artifact of deficit handling, and isn’t actually an issue if we had added more seats. To fully solve it, though, we might want to restore only enough points so that A can get exactly the current quota, and “store” the excess for future rounds.

Another way of looking at why A should’ve won 3 seats rather than 2: if we look at each Hare Quota’s utility through a Sequential Monroe lens, we get 100% satisfaction in the 2 quotas for A and 1 for B, 41.67% in the one for C, and 37.5% for D. In terms of total utility, you have 53 x 2 + 28 x 1 + 10 x 1 + 9 x 1 = 153 points. Had we elected A instead of D, that final quota’s satisfaction goes up to %, and total utility goes up to 53 x 3 + 28 x 1 + 10 x 1 = 197 points.

However, you’re right that this hurts Vote Unitarity, and it even hurts the Sequential Monroe score; in plain SSS, 25 points are left unspent (A’s 5 + the 20 ineffective voters), while here, 29 points are left unspent (D’s 9 + the 20 voters.) I think it’s a worthy tradeoff overall, and it might even point to a logical contradiction in both unitarity and quota scores themselves. In terms of quota satisfaction, the quota with D goes down from 37.5% satisfaction to 20.83% satisfaction when A is elected instead.

We can look at the additional voters as essentially actively rejecting all of the possible winners; that’s why their inclusion shouldn’t affect who wins. I tested to see if A would still get 3 seats when the additional voters give 0.2 of a vote (1/5 in score) to all non-A candidates, which expresses that they don’t want A to win over any other candidate, and it turns out A only gets 2 seats; C wins the 4th round with 14 votes, and because A can’t match that (they can only get ~6.31 to-be restored votes and so a maximum possible total of ~11.31 votes), D wins the final seat with 13 votes. In this case, A got only 47.32% of all points given to possible winners, so it makes sense they didn’t get the majority.

Edit: ~~One way to skip deficit handling might be to give every candidate X number of points (I think the mathematical formula might be: number of voters * maximum score), but keep the Hare Quota the same. In the above example, we can give every candidate 100 points, in which case it’s (A 153 B 128 C 110 D 109). A wins 2 seats with 20 points each, then it’s (A 113 B 128 C 110 D 109). B wins 1 seat, then it’s (A 113 B 108 C 110 D 109). A wins 1 seat, then it’s (A 93 B 108 C 110 D 109). Now C wins the final seat, so the final result is still 3 A 1 B 1 C.

It turns out this doesn’t fix it. If you have (A 17 A 16 A 16 B 51), changing that to (A 117 A 116 A 116 B 151) with a Hare Quota of 20 still elects only 2 B and 3 A candidates, despite B having the majority of bullet votes.

Also, I believe deficit handling fails when you have A1 17 A2 16 A3 16 B 51 with 5 winners, because not enough points can be restored to B when it’s A 16 A 16 B 11 or A 16 B 11 for B to win. To fix this, I think you have to keep track of how many points are not being restored because of the multiplicative factor (the one that reduces total points restored to ensure they match the points that the previous winner fell in deficit of theirquota), and then store those in the stored value (I think these particular points should only be restored in the next round, even if other points are to be restored in the current round.) Using this in the example, we have B1 and B2 winning with Hare Quotas, then A1 winning with a deficit, so B is due to get 40 * (1 - 17/20) = 6 points back. Because A1’s deficit was only 3 points, however, we curtail this to 3 for the moment and store the remaining 3 away, and 11 + 3 = 14, which would be B’s total if all points were restored, doesn’t reach the current quota (20), so these points are stored away rather than restored. For the next seat, A2 wins, and now B is due to get 40 * (1 - 16/17) = 2.35 points back. Because A2 won with a deficit of 1 point (the current quota is 17 because A1 got 17 and nobody in the current round can reach that), this 2.35 is curtailed to 1 point. However, the 3 points that would’ve been restored in the previous round + the 3 points that were shaved off because of the multiplicative factor from that round + this 1 point from A2’s victory mean that B currently has 7 stored points. Add that to B’s current total of 11, and it’s 18, which lets B reach the current quota of 17, so the points are restored, and B wins the final seat with 18 points.

Edit: Deficit handling sometimes fails to stop Vote Management when there are an even number of seats to be elected. An example is (A 53 B 28 C 10 D 9), with 4 winners and a Hare Quota of 25. The first 3 winners are elected with full quotas, then it’s (A 3 B 3 C 10 D 9), and C wins, yielding a winner set of (A, A, B, C).

But if the A voters split up, that gives (A1 19 A2 17 A3 17 B 28 C 10 D 9), and after B wins with a full quota, A1 wins with a deficit of 6 points. Restoring those 6 to B voters doesn’t give B enough points to reach the current quota (25), so they’re stored away. A2 wins next, and total points to be restored are A1’s 19 * (1-17/19) = 2 and B’s 25 * (1-17/19) = 2.63. Even if you restored all of those points (deficit handling requires they be reduced proportionally to match the total number of points of deficit the previous winner had), B would have 11.63 points total, not enough to win the final seat against A3’s 17. So the final winner set is (B, A, A, A). Deficit handling does stop Vote Management if the majority and minority are united against each other, though, like if C and D voters here had also voted for B.

Anyways, deficit handling should probably only be used with an odd number of seats for maximal effect.

This is complex in motivation and explanation to the layperson but is dead easy programatically. I can add it to the simulation code in like 10 min. Is this better than the other ideas? I find it more justifiable. The result will be that voter blocks behave as if they did vote management perfectly. This gets rid of a lot of free riding. What about other issues? I think we are good for clones and monotonicity but is there something I am missing? Compared to SSS this shifts winners towards larger voting blocks. However, I think SSS has a bias towards small groups which is what made vote management useful.

To be clear the current subtracted score at each round for each voter is the score they gave that candidate times the current quota divided by the sum(score) for that winner accross all voters. I do not think we even need to start with a hare quota as the original quota. We can just take it all for the first winner since we will give it back anyway. This changes the order but on the set of you have more than three winners since this process only has a memory of the last round. Or is there an example otherwise…

This preserves vote unitarity since strictly we only know it is a linear function of score.

(“This” refers to Marylander’s method, right?) I believe so. It’s certainly better than my idea, which is featured in the OP here. I think it is actually the benchmark for how good a sequential Monroe method can be at resisting Vote Management.

That is fine for simulation, but in a public election, this would unnecessarily slow down the calculation of the results of the majority of the candidates who had won with full quotas. Not only that, but it impairs understanding when you have to give people simple examples of how the method works (it’s simple for the candidate with the most points at a given point to win, harder if that doesn’t happen when some other candidate gets more points and wins because of the deficit handling.) Basically, imagine if we had to have the yellow and green lines appear for every round:


“Capped Quota” is better.

@Marylander, could you help find such an example if it exists?

No, this is you just recalculate as a specified each round.

You start back at the beginning and elect all that have been elected. Except instead of subtracting the score you did before you do it based on a new quota. That quota is the quota used in the previouse round. @Marylander 's system splits the difference between this quota and the winning quota. Or something…

By fractions of seconds. This would not be slow. The currenty RRV implementation recalculates each round.

Yes, this is much more complicated. I am not convinced it would be worth it.

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Oh, never mind then.

If I understood you correctly, this is still subject to Vote Management. Take the example of (A 53 B 28 C 10 D 9) with 5 winners. A wins, and the quota is 53. That yields (B 28 C 10 D 9), B wins, and the quota is 28. That yields (A 25 C 10 D 9), A wins, and the quota is 25. (A 3 B 3 C 10 D 9), C wins, and the quota is 10. Then it’s (A 33 B 18 D 9), A wins, and the final winner set is (A, B, A, C, A).

But if the 28 B voters had split into two blocs of 14 B1 and 14 B2 voters, after A wins first it would’ve instead been (B1 14 B2 14 C 10 D 9), B1 wins and the quota is 14. That yields (A 39 B2 14 C 10 D 9), A wins, and the quota is 39. That yields (B2 14 C 10 D 9), B2 wins, and B’s Vote Management strategy has managed to win them 2 seats rather than just 1.

Marylander’s system is unusual in that it is actually deciding who wins based on the quota itself; it is who can win with the highest quota if the previous candidates had to pay that quota that wins, rather than who has the most points. It’s fundamentally fairer, but this is why a simpler approximation which can ward off or at least make Vote Management risky is desirable.

Please keep Marylander’s method in mind if anyone is unwilling to go with SSS, as it might convince some Wolf Committee members or others.

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Yep, you are right. It did work for the examples I was playing with. I was looking at examples where the large group (A) was overspending and the middle group (B) was using vote management to steal from them. This is the case where they steal from the small group. Perhaps it only solves some of the vote management issues. Some might be enough though. This is a pretty simple system.

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In the end, it may even incentivize C to reach out to D voters to get enough points to beat B for that seat. So in that sense, some Vote Management countering can do just enough to let the voters figure the rest out.

Only if C and D are compatible. If one is communists and the other is Birchers, no such union is happening.

It’s possible, especially because the final seat in a utilitarian PR method is the one that has the least consensus (almost by definition under honest voting), but on the other hand, we are already putting a lot of faith into the idea that utilitarianism can unite most or all of the voters in a single-winner election, so it’s hard to say. On top of that, A voters may prefer a C over a B in the final seat. But you’re right, it might still be an issue.

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True, and capping makes it easier for A voters to give C a low level of support without losing weight for their own candidates.

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@Marylander Do you think this “dynamic retroactive quota” method is a net gain over SSS. It reduces vote management and removes Hare Quotas.

If the first candidate is unanimously approved, then the second candidate is picked randomly, which seems like a problem. (Although in practice, no one will have unanimous approval, so maybe not.)

I would put this in the same category as IIB. Not a problem in practice.

I meant, do you think it is good enough at reducing vote management to justify it?

Conjecture about the party list case: a party that receives at least as many seats as it would under D’Hondt cannot gain from vote management. A party that receives fewer than its share under D’Hondt can gain seats through vote management up to its D’Hondt share.

Examples:
A:24 B:23 C:22 D:21 E:10 (5 seats)
Each party would get one seat under this system. (Under D’Hondt, it’s 2-1-1-1-0). A can divide itself into 2 parties of 12 to win 2 seats, taking E’s. Any attempt by the other parties to take the 5th seat can only give it to A.

A:24 B:23 C:10 (4 seats)
Here, A wins 2 seats; B and C win one. B can take a seat from C by splitting into 12 and 11.

On the whole, it seems Keith’s idea makes Vote Management very risky, because a) the margins are too close to be confident that you’re not going to lose seats by trying Vote Management (you could have one bloc of voters be much larger than the other to guarantee winning one seat, but that only makes it a lot harder to have a chance at winning the other seat) and b) it only really fails when there’s a significant deficit for the final seat, yet this is when it most incentivizes the party that could potentially lose a seat to Vote Management to build consensus to try to muscle out the Vote Managing party.

Also, do deficit handling methods get better or worse when there is less party list voting? SSS should produce somewhat less of that after all, and if it gets better, than we can be even more confident in this idea.

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We’d need simulations for that.