# Cases in which Approval Voting generates problems

In this topic I present some cases in which the Approval Voting seems to be causing problems, and I would like to know if I’m wrong.

1 winner

1. Bad representation of interests
These are the real interests of the voters:
Group 1 (98%): A[10], B[6], C[0]
Group 2 ( 2% ): A[0], B[6], C[10]
These are the respective marks in AV form
Group 1 (98%): A[X], B[X], C[]
Group 2 ( 2% ): A[], B[X], C[X]
With AV wins B even if he should win A, according to the real interests.
Here it seems clear to me that the winner should be A, but in your opinion how much should the percentage of the 2nd group increase to ensure that the victory of B is right?
In my opinion, exactly more than 40% (with group 1 less than 60%).

The groups are the same as in case 1), but with group 1 at 60% and the other at 40%.
The winner with AV is B but B isn’t the favorite of any group, therefore the following tactical votes are generated:
Group 1 (60%): A[X], B[], C[]
Group 2 (40%): A[], B[], C[X]
Group 1 manages to win A thanks to its tactical votes.
In this example we note that for a voter it’s better to give the cross only to 1 candidate, the best for him (if the best is on par with others, then he will give more crosses only to those).
This type of tactical voting makes AV similar to SNTV.

3. Bipartism
There are 4 candidates A,B,C,D among which A,B are the most supported while C,D minorities.
Pluto likes D a lot and everyone else sucks (even if A is slightly better than B); Pluto could vote like this:
Honest vote: A[], B[], C[], D[X]
In this case Pluto’s vote would be similar to a null vote being D a minority, and Pluto knows it before voting, so he decides to vote tactically:
Tactical vote: A[X], B[], C[], D[X]
Pluto’s vote can now influence the clash between A and B and that’s good for him, but the vote he created still supports A as much as D.
In conclusion, all voters who support minorities end up tactically supporting even 1 of the 2 most supported candidates.
This favors bipartism, even if it’s better than the single vote because at least here we see that minorities have support.

multiple winners
In addition to the problems seen in the case of only 1 winner, there is another.

1. Voters with wrong weight
By adding rules to the calculation of the results (which balance the weight of the votes) this problem is avoided.

I would like to know if these 4 cases are real AV problems or if I made some mistakes.
If you have other cases where the AV generates problems, indicate them in the reply.

In this example, C support from of 2% of the voters, so C is basically irrelevant; the winner obviously must be A or B. Therefore, it is silly for group 1 voters to vote A[X], B[X], C[], since when we ignore C, their vote is A[X], B[X]. It is more likely that group 1 would vote A[X], B[], C[].

You will have trouble making a voting method that does what you want, because systems that do not allow a majority that shares a first preference to force that first preference to win must have other problems. Either they allow a minority group to outvote a majority, which implies that either not all ballots are equal or not all candidates are equal (e.g. US Presidential elections), that they involve luck (e.g. random ballot), or are vulnerable to teaming (e.g. Borda, antiplurality).

Only in this example. There are other cases where approving multiple candidates makes the most tactical sense. If we add a third group that likes B the most 1/4 the size of the original electorate, we get:
48% Group 1, 20% Group B, 32% Group C. Then C voters have reason to approve B in addition to C, because B can win with their support, a preferable outcome. Strategic approval voting typically leads to the election of the Condorcet winner. This is not like SNTV.

That requires a system of proportional representation. There are several different ways to extend Score/Approval to a proportional representation system.

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Bad representation of interests
In the example they voted honestly and the result with the AV was incorrect. You have not denied this.
You just said that C is irrelevant (2%) but in the same example I expose that the problem could occur even if C (group 2) had 10%, or 20% or 30%.
In my opinion it’s worth up to a value less than 40%, always speaking of honest votes.
If group 1 had voted like this: A[X] B[] C[] seems to me a bit of a tactical vote, and I tried to use real interests here.

You are right that I will have trouble, but this also doesn’t deny the problem I have exposed.
The fact that a vote like this:
A[10], B[6], C[6], D[0] can tend to become like this:
A[X], B[], C[], D[] rather than like this:
A[X], B[X], C[X], D[] means that this type of tactical voting is similar to SNTV (not that the whole AV is similar to SNTV).
I agree that predictions of a certain kind on the results of the election can avoid (or generate) this tactical vote. I agree also that the presence of this tactical vote depends largely on the expected results of the elections.

Bipartism
Fail NESD * -> bipartism.
NESD pass * -> no bipartism.
NESD * not applicable on AV -> another method must be used to evaluate bipartism.
2BC = 2 most supported candidates
In my example I clearly point out that Pluto has an interest in voting 1 out of 2BC to make sure that he can decide who out of 2BC wins. This means that by hypothesis all minority voters give an X to 1 of the 2BC, so the NESD* rule is used.
In my example we note that although minority candidates are supported (no bipartism), at least 1 of the 2 dominant candidates is supported in the same way so you can know in advance that the minor candidate cannot win (bipartism).
However, if Pippo’s favorite candidate is expected to be very close to 2BC, then it makes sense for Pluto not to vote for the 2BC, falling back into the NESD and therefore canceling bipartism for that 3rd highly supported candidate (but not for the other minorities). In reality, this particular situation can also occur in SNTV in which a new highly supported 3rd party that beats the others appears, receiving the vote of Pluto…

Voters with wrong weight
In short, decrease the power/score of the voters every time 1 of their voted candidates wins.
It seems complex and very speculative to me, but it should work.
Okay, I’ll take this point away to avoid distractions, since you listed it here.

Define “honestly”. I charge that the approval threshold is relative rather than absolute, and so it’s just as honest for a group 1 voter to vote A[X] B[] C[] as A[X] B[X] C[]. The only way an approval vote can be dishonest is if it misorders a voter’s preference. For example, a group 1 voter who voted A[X] B[] C[X] would be dishonest because they said C>B. While not all honest votes (by the “does not order a threshold” definition) are equally powerful (voting for no one or everyone would be honest yet have no voting power), Approval has the advantage that in any 3 candidate election, the most powerful vote is an honest one in the sense that it does not misorder a preference. This is not true of ranked choice systems (or of your proposal from the other thread). There are specific scenarios where they differ for 4 or more candidates, but such scenarios aren’t closely connected to problems like favorite betrayal that reinforce today’s party duopoly.

Sure, I agree that Approval is not always guaranteed to return optimal results. But we have to have some method of deciding elections. I would prefer a score system with a greater range than approval’s {0,1}, but I still prefer Approval to any form of RCV. Any method has to be able to convince the public to enact it, and for this purpose Approval has the advantage of simplicity over pretty much any other reform proposal. No reform at all means we stick with the awful FPTP system, so while flaws in Approval are certainly of theoretical interest, they can only justify practical action if they are coupled with a case that an alternative does a better job.

The reason I supplied the NESD link was because NESD describes the same scenario you bring up. In your scenario, you argue that Pluto has an incentive to approve one of the two major candidates. (It also makes sense for Pluto to vote for any minor candidates that Pluto prefers to both major candidates.) The fact that approval passes NESD means that if every voter is like Pluto and approves one of the two major candidates (as well as some minor candidates), it is still possible for a minor candidate to win.

Honest vote
If my real (honest) interests are: “A is twice as good as B, while C is the worst of all” the form of the honest (and basic) vote is the following (proportional form):
A[2], B[1], C[0]
Once this is done, all changes to this proportional representation of interests can have repercussions as errors on the result.
These changes may depend on the form of the vote or on various tactical votes.
AV: A[X], B[X], C[]
In the AV the proportion between A and B (A is double the value of B) is not respected therefore this can already generate (mandatory) errors on the result, regardless of the method used then to “add” the marks between them.
Score: A[10], B[5], C[0]
DV: A[64], B[33], C[0]
Both Score and DV give you the opportunity to adequately represent your interests (they don’t force you to represent them badly like AV, SNTV, RCV).
Then we need to evaluate how much tactical votes influence the real form of interests (the proportional form).
Another eg: A[8], B[2], C[1], D[0] (proportional form) means “A 4 times better than B, B 2 times better than C, D the worst of all / hated”.

It’s certainly very easy to write and introduce AV, but I prefer to evaluate criteria of theoretical correctness rather than practical simplicity.
I’m also very much against RCV methods, AV is better than those but as you already know I prefer DV.

Bipartism
2BC = 2 most supported candidates
No, if all the voters (of minor candidates and those who already vote for the 2BC) give at least 1 vote to the 2BC candidates, it means that 100% of the voters vote for at least 1 of the 2BC and therefore at least 1 of the 2BC wins by force (if it isn’t the absurd case of parity 50%-50%).
This case falls into the NESD * that the AV doesn’t pass.
Changing hypotheses, that is Pluto votes only the minority candidate he likes, then he falls back into the NESD context and the bipartism falls.
The question: is it more likely that Pluto will vote only the minority candidate he likes (becoming useless in the fight between the 2BC, however disadvantaging bipartism) or will Pluto also give 1 vote to 1 of the 2BC (participating in the clash between the 2BC, but favoring bipartism)?

Keep in mind with Approval, a 3rd party can win with the approval of voters who also voted for one of the 2 major parties. For example:

50 A>B
50 C>B

Assuming everyone votes for their 1st choice, A and C have 50 approvals. Now, supposing that >50 voters also approve their 2nd choice, then B wins.

It’s also realistically possible for a 3rd party to win with a few strategic voters. With something like

25 A>B
30 B>A
45 C

it’s more than possible that a few A>B voters bullet vote (only support their 1st choice) while most of the B>A voters approve both their 1st and 2nd choices such that A ends up winning. For example, A could end up with 55 approvals and B only 50, with C at 45.

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Sounds to me like you are arguing for a multi-member Cardinal system. Welcome to the club. There are a ton and the real question is which is best. Your system would work as one but it fails clone immunity and I think monotonicity. This would put it below a lot of other systems right away. There is a list of existing options here

I’ll explain better:
2BC = 2 (expected) most supported candidates.
Vm = voter who likes a (expected) poorly supported candidate.
VM = voter who likes 1 candidate out of 2BC.

1. Vm also votes for 1 of the 2BC, and VM doesn’t vote for minority candidates.
2. Vm votes only preferred poorly supported candidates, and VM also votes for minority candidates.

The more case n.1 occurs, the more bipartism occurs (vice versa with case n.2).

Your examples fall into case n.2 (no bipartism).
In reality, however, it is very likely that Vm, to avoid that his vote is null in the clash between the 2BC, often ends up voting even 1 of the 2BC. The VM, knowing this behavior of the VM, to avoid that some minority can overtake his favorite candidate, doesn’t vote poorly supported candidates.
All of this often leads to case n.1 (bipartism) rather than case n.2.
The VM’s reasoning takes into account the similarity between parties, for this reason it makes sense.

It’s not a certain bipartism like SNTV, but it’s a probable bipartism (if you have valid predictions on what the 2BC will be, and often you have).

about 50 A> B
about 50 C> B
(I said “about” because those are predictions of the results, not certainties).
A and C are the 2BC so the voters are all VM and tactically prefer not to win B. To win B they should all agree on accepting a second choice as a winner (which does not happen in reality, this is the problem for which tactical votes are born).

25 A> B
30 B> A
45 C.
The 25 are Vm and have an interest in supporting B as well, in the hypothetical final struggle between B and C (the 2BC).
The 30, knowing the frequent reasoning of the Vm, have no interest in supporting A (less supported candidate).
In the end he would probably win B and not A.

I repeat, it’s not a certainty, I just think that it ends up more in case n.1 rather than in case n.2.

I’m only describing the negative sides of the AV.
When you say “your system” do you mean Distributed Voting? Because DV isn’t a cardinal system but a cumulative one.

If you refer to the DV then:

• don’t fails clone immunity due to redistribution of points.
If there is only one winner and I vote like this: A[0], B[25], C[25], D[25], E[25]
while another votes like this: A[100], BCDE[0]
it seems that candidate A is much more advantaged but from time to time the worst is eliminated; e.g. they are eliminated in order D, E, B and in the end my vote will be: A[0], C[100] because the points move respecting my interests expressed in the vote and in the end it will be worth as the one that immediately gave all 100 points at A (distributing the points a lot didn’t make him less worth in the end).
• in DV if there is a way to disadvantage A’s victory by giving him more points with a tactical vote (monotony fails), then there is also a way to disadvantage A’s victory by distributing the points in other candidates differently, in particular by increasing those given to another candidate (satisfied monotony).
Ex:
G1-33: A|99 B|1 C|0
G2-30: A|1 B|99 C|0
G3-37: A|0 B|1 C|99
A wins (I eliminate worse B, I redistribute the points and A is better than C).
If 8 G3 voters (which I will call G4) vote like this:
G4-8: A|99 B|1 C|0
they are favoring A (favoring the previous winner) and this then leads to B’s victory in the end (making A lose).
However, G4 voters could have voted more intelligently as follows:
G4-8: A|0 B|99 C|1
respecting their interests (they avoid giving 99 points to the hated candidate A) still obtaining the victory of B, without canceling the monotony.
In summary: by using intelligent tactical vows, monotony is respected by the DV.

Yes. Also, if you want to claim inventing it then you should put up a page on electowiki

I would consider cumulative voting a cardinal system because it has cardinal ballots. I would not consider it a score system though.

While technically true there might be a case where this has weird consequences. I can’t think of one right now but clones do cause the voter to change votes for others.

Monotonicity does not really have to do with strategic voting. IRV is nonmonotonic even for honest voters. This is somewhat the cardinal version of IRV. I would check some examples where IRV fails monotonicity in your system.

Thanks for the advice of electowiki.

• For the clones I can assure you that if the points are used only on the preferred candidates, then the weight of the vote will always end up entirely against the hated candidates (precisely because the 100 points remain constant thanks to the redistribution).
However, the way in which the points are distributed among the preferred candidates can change and tactical votes develop on this change.
• If I’m not mistaken, the monotony has a definition like “If the (honest) social choice is A, and one or more voters increase A in their preference scale, then the social choice is no longer A”.
I consider the increase of A in one’s preferences, a tactical vote (because it’s no longer the honest starting vote in which A would win).
Having said that, in DV the monotony fails in theory, but in practice the change of the vote that would make the monotony fail, is a stupid change, in the sense that this change can be done without making the monotony fail, but still losing A (all better respecting the real interests of the tactical voter, that’s why he’s smart in not making the monotony fail).

Eg, an honest DV vote like this: A [40], B [30], C [20], D [10], E [0]
will probably be converted to the following tactical vote: A [94], B [3], C [2], D [1], E [0].
The important thing however is that the tactical vote concerns only A so if A is eliminated, both votes (honest e tactical) become equal to: B [50], C [33], D [17], E [0] so the tactical vote is completely gone.
By eliminating the other candidates first, the tactical vote disappears but slowly.

Overall, I didn’t find a voting method that deviates less than the DV from a person’s real interests (note that the deviation also depends on the way the vote is written, not only on the tactical votes; eg. in the previous case, the AV would put erroneously all candidates on the same level using X, even if in reality they have different proportions).

Monotonicity isn’t significant just because of tactical voting potential. It matters because given two outcomes that generate nonmonotonic phenomena, logically at least one of them must be wrong.

I see two problems that you haven’t addressed: favorite betrayal, and expecting voters to learn the system.

Favorite Betrayal
It has similar problems with this to IRV, in ways that could promote two party domination. For example, suppose a major candidate A is preferred to their opponent B by 54 to 46. Another candidate, C runs, and draws support from a subset of A voters. However, not all of the A voters prefer C to B. (Suppose there are 10 of them, some of whom may start to give points to B as a hedge). If C can convince enough A voters to give points to C, and particularly to score C ahead of A, C can surpass A on the first count, leading to A falling into last place. But C will score only 4400 points in the final round, and B will score at least 4600, possibly more if some A voters hedge.

Voters Understanding the System
Casting a cumulative-style vote requires voters to distribute points in a manner that sums to 100 points. While most voters probably can do this correctly, when they are in the polling booth, they might not want to have to. Overvotes would be more common just because voters may make a mistake. The 100 point scale is often intimidating to voters. STAR voting initiatives have focused on a 0-5 point scale, because of research suggesting that it is what voters are most comfortable with. Warren Smith’s website once advocated using a 0-99 score voting scale, but later he changed his preference to a 0-9 scale for similar reasons. A large scale for distributed voting is necessary for it to function well in large elections, however, so it is not so easy to cut back the scale, unlike with score voting. Furthermore, complicated tabulation algorithms are often unpopular, which has been an obstacle for IRV in some places. DV’s back end is complicated in an IRV-like manner. Contrast with Score Voting’s back end of “most points wins”. All of this will probably cause voters to see distributed voting ballots as a large hassle.

Edit: reposted to the Sequential Elimination systems thread

I split this into a new thread with a reformulation to a proper score system. Please answer there as this thread is more about approval voting