# Why do referendums use a binary choice rather than a numerical score?

In some democracies, citizens may be asked if they accept a law or the modification of a law in a "yes or no" referendum.

Most of the time, legal texts are complicated and specialized, so why does the government ask citizens to be absolutely sure in their vote of yes/no?

What difference would there be in a voting system where citizens were asked to give a proposal a numerical score between 0 and 100, where 0 would mean "no" and 100 would mean "yes"?

Has this kind of 'quantum voting' been used before, and if so, was it useful?

• WP calls this weighted preferential voting(en.wikipedia.org/wiki/Weighted_voting) Jun 19, 2021 at 8:03
• I am not sure I understand correctly what is proposed here. I assume this voting system is for referenda on specific matters, not to elect representatives, right? Then how exactly are the votes counted? When one votes "25", does that mean the "no" option gets 75 points and the "yes" option gets 25 points? And when "yes" gets more points than "no" from all voters, the proposal becomes law?
– Philipp
Jun 19, 2021 at 9:16
• If I understand correctly, "quantum voting" in this context does not mean weighted votes (each vote might have a different weight/coefficient), but the fact that each vote is fuzzy (from absolutely no to absolutely yes). I guess that when the average of votes is 50/100 or more, then the law is passed, right? Jun 19, 2021 at 10:17
• What does it mean to "accept 67% of a law?" Does that mean you are allowed to break it one day out of three, or you get to pick any two out of three clauses to obey and can break the rest, or what? A law either exists or it doesn't. Jun 20, 2021 at 4:11
• A better way to solve this is to split the proposal into parts and run individual votes on them. I.e. the UK referendum could’ve asked voters to decide not just Brexit but also choose from a list of options on how to handle the separation later on. Jun 20, 2021 at 20:45

Because the system highly incentivises strategic voting. The voting system you describe is more commonly known as score voting, although it has plenty of other names. It's been used before in the USA - the example given by Wikipedia is the election of officers in the Utah Green Party rather than on a referendum or legislative proposal - this used a ballot as shown below.

Image: Independent Political Report

However, in his paper entitled Strategic Evaluation in Majority Judgement, Professor Jack Nagel explains how the voting system is highly susceptible to strategic voting.

For example, using the ballot above, a voter might believe that Guymon is worth a score of 8, Twitchell a score of 6, and Styles a score of 3. To increase the chance of Guymon winning, the voter might instead score Guymon a 9, and to decrease the chance of Styles winning, score him a 0. This is an example of "pairwise polarisation" - the least-favoured candidate being scored 0, and the most-favoured candidate being given the maximum score. In "complete polarisation", all candidates are given either the maximum score or the minimum score - which, Nagel argues, degenerates the voting system to simple binary approval.

It's not hard to see how this logic would apply to a referendum-style vote - even if a voter would sincerely give a proposal 80/100, this implies that they would rather the proposal pass than not, and they are incentivised to score the proposal 100/100 in order to maximise its chances. The end result is that the ballot degenerates to a simple "yes/no" vote.

• And lest someone say that the worst case scenario is what we have now, a score voting system that degenerated in this way would give a big advantage to the side that most aggressively employs strategic voting, thus rewarding radicalism instead of compromise as intended Jun 19, 2021 at 15:54
• Also, simple approval voting (of multiple options) is in itself subject to tactical voting. i.e. if you prefer A over B over C, do you accept just A to increase the chance of getting your most favoured option; or approve A and B to increase the chance of not ending up with your least favoured option. Jun 19, 2021 at 20:31
• I don't have a source for this, but my assumption is that people are also very bad at assigning a numerical strength to their opinions (unless this is a skill they've specifically practiced, like those who make predictions with probabilities attached and calibrate their success periodically). If everyone assigns a different meaning to '6/10' then the high number of people who vote '6/10' doesn't give you much information, and may actively mislead. Jun 19, 2021 at 21:28
• A numerical score could also be for a question such as "What should be the speed limit on Main Street?" with an open answer, or "What % of childcare should be paid for by taxation (the rest by users)?". Do you know of any examples of referendums with such questions? Maybe those are less likely to lead to such polarisation? Jun 19, 2021 at 21:28
• @gerrit Depends on what you're doing with the votes. If taking the average, e.g., there is still strong incentive for polarisation - e.g. if about half of us believe the limit should be 30, and half believe it should be 50, then absent strategic voting, the average will be ~40. Which then gives voters incentive to pick e.g. 0 or 100 to pull the average closer to their desired outcome. OTOH, taking the median of votes would be less susceptible to this problem. Jun 19, 2021 at 23:49

In the specific case of a one-law referendum, which is what the question was asking about, what would be the purpose?

Let's pick a subject that will be polarizing and motivate people strongly, abortion for example. Now we have a referendum to allow it (Ireland did that recently, IIRC). Could be a referendum to forbid it, doesn't matter.

What does it mean to be 67% for abortion rights? Your brain is saying "well, I am 33% against too"? If you were voting on two different laws or two different political offices, then yes, you might "spend" 67% of your voting power to have your way on your most important subject, and 33% on your second important choice.

This logic might even work for 1 office and 5 candidates. 67% to your favorite candidate, 33% on your second-best choice. That makes sense.

But, again, for voting on one law and one law only, with a binary yes-no, what is the purpose of this quantified vote system? (please don't use quantum, it means quantified in discrete steps, not just quantified and poor word is overused ;-). You either want something or you do not. If you don't care - don't vote at all.

p.s. One valid reason for a quantified vote would be places that have laws forcing you to vote, even if you didn't care: voting 50/50 would essentially spoil your ballot, assuming there was no other way (such as voting blank or checking off both Yes and No). But that's a special case at best.

• "You either want something or you do not." - I'm having a bit of a hard time wrapping my head around this concept. Is it ever the case that you want or do not want something 100%? Isn't the much more realistic situation that there are pros and cons, and while you would strongly prefer one outcome (therefore allocating most of your support to that side), you also see some points of the opposing side are not without merit? Jun 20, 2021 at 21:11
• @O.R.Mapper without a specific policy deployed as part of the referendum to enable it, such a distinction is meaningless. You either get it the new law/policy or you don't. What policy would being 64% in favour of it even work for? To take this answer's example, would a higher pro-choice percentage move the late term abortion deadline? Who would even want to write a law like that in the first place? Lets vote on having a social security system, and the % support maps to the dollar figure of public spending? Jun 21, 2021 at 12:31
• @Jontia: No, no mapping, no modification of the proposed policy. Simply a representation of the fact that I see many points (w.l.o.g.) in favour of a policy, but also some against. Hence, I support both sides. One more, one less, but decidedly not just one, while expressing absolutely no support to the other side. Jun 21, 2021 at 12:42
• It seems reasonable someone could rank where they stand on something like brexit on a 0-10 scale. One assumes most non voters are 4 to 6s on this scale, which is why they didn't vote on such a crucial issue. Of course tactical voting still is the problem.
– eps
Jun 21, 2021 at 13:59

This answer is in the same vein as the one by CDJB, but explained a little differently. Consider the following voting system: There is a referendum, and everyone who votes gets to choose two things.

1. Whether they vote "for" or "against."
2. How much their vote is worth, or how much "weight" it has, on a scale of 1 to 100.

Then, the side that gets the most points wins. In other words, if there are only two voters, and one votes "against" with a weight of 20, while the other votes "for" with a weight of 70, the "for" side wins.

My voting system here actually turns out to be equivalent to your voting system, in the following sense: For every vote in your voting system, there's an equivalent vote in my voting system, such that the end result when totaling the votes as described by my voting system is identical to what it would have been if totaled in your voting system. For example, someone who votes 75-25 in favor of "for" in your voting system would vote "for" with weight 50 in my voting system.

With this equivalence in place, it becomes easier to see why the incentives for voting are a little bit weird; although it seems like it, the number assigned to your vote in your voting system is not actually an expression of how much the voter agrees with the proposal. Rather, its an expression of the voter's confidence in their own vote. Obviously, a strategic voter who is looking to maximize the chance that their preferred outcome is chosen would always assign their vote a weight of 100.

But even if we accept for now that some voters would assign their vote a weight less than 100 because their vote "should" be worth less than that of others (maybe they believe they are less informed than the average voter), it's still not clear that this would actually produce better results. In particular, people who are likely to be confident and self-assured about their opinion would have a disproportionate impact on the political decision.

Referendums typically use binary choice because statistics, psychology and logic.

• How strongly do I even agree with some referendum?

Trying to put that on a scale of 0 to 100 is just going to be very prone to inaccuracy.

• What does some percentage agreement even mean?

Let's say the question is whether dogs should be allowed inside. Someone might 95% agree that dogs should be allowed inside at night or during bad weather, but 40% during the average day, and maybe 5% that they should be allowed to sleep in our beds and 20% that they can be on furniture. If they were to try to average that all out, they might say that's about 70% agreement. But all those numbers could also be very different and they could still reach the same 70%. Not to mention that each of the above percentages can be broken down further.

One percentage just doesn't give that much information.

• Different people may give different scores for the same level of agreement.

If you ask people to rate things on a 1-5 star scale, some will give 3 stars for something perfectly acceptable (to keep 4 stars open for above average and 5 for excellent), while that would be 5 stars for others.

The same might apply to voting in a referendum, so this could introduce a lot of noise.

• How do you combine all those percentages?

Maybe you say 2 75%'s, 1 100% or 50 51%'s are all equivalent, but some might argue that very strong agreement should be given an even higher weight, while others might argue that all agreement should be weighted evenly.

With yes/no there's no problem, but with percentages you're opening the floodgates for disagreement about how to actually add up the results.

Not to mention needing to present this to the public. "60% of people agree" is a nice and easy summary, while it would be much less elegant and potentially more objectionable to say "60% of people agree, but most of them don't agree so strongly, while those who disagree disagree more strongly, so ultimately we decided to go with those who disagree".

• Only 0% or 100% make sense / people may "game" the system.

If I want the "yes" decision to go through, voting 100% is most likely to achieve that, so that's the optimal vote, even if my agreement is only at 60%. There's little logical reason from a game theory perspective to vote anything other than 0% or 100%.

Well, 50% might make sense to express indifference about the result, but most who would go for that option presumably wouldn't care to go and vote at all. There may be some exceptions, but that still wouldn't justify a percentage score, but rather a "maybe" or "I don't care" option (although for the most part that probably wouldn't be useful).

• It should all average out either way.

Given enough data, the end result should generally be the same regardless of whether they give binary choices or scores.

On both sides you're going to have some people who feel strongly and some people who don't care that much. And most of the people who don't care that much aren't going to vote one way or the other.

• Allowing people to assign weight to their opinions could have value if people had a total number of "vote points" that they had to divide among the issues, with a proviso that people would only have to spend vote points on issue where they won, and vote points allocated to issues where someone got outvoted would be transferred to other issues. Jun 21, 2021 at 22:37
• @supercat First notice that most issues would get decided by fanatics, as normal folks won't spend all of their points on stuff. Sometimes you might even opt to vote against your wishes just to keep your points for something more important (assuming you have to vote - otherwise nobody would). Besides, even if you believe that change still makes sense, now try to implement that in a way that doesn't break any of privacy guarantees. You can't - if vote buyer demands number of your vote points before and after voting, he knows the way you voted (and, if your side won, how many points you spent). Jun 22, 2021 at 8:40
• @supercat Letting people split/prioritise their vote among different issues (whether yes/no per issue or degrees of agreement) sounds rather distinct from letting people express degrees of agreement on just one particular issue. But I wouldn't want to try to argue that the former is better than just letting them vote on as many issues as they want. That would be some next level game theory, where people try to only vote on issues where their vote is most likely to change the result. If the choices are mutually exclusive (e.g. candidates in an election), that just sounds like ranked voting. Jun 22, 2021 at 10:12

Because the outcome is binary

The proposition either passes or it doesn't. As others have said, the only rational numbers to use would be 0% and 100%, and that's because you can only prefer one outcome over the other (or you don't care either way and then abstain or void your ballot).

Even if an elector can have a nuanced opinion on the subject, the options are only to either keep the status quo or to have this predefined package of changes to it. If you want to be fancy, you can have a complex "value function", but it is only evaluated at two points.

More complex voting systems do exist, but they are only useful where there are more possible outcomes, such as an election among more than two candidates or the composition of a board.

• If you could magic away the tactical voting issues, I don't see why a binary outcome is an issue in the abstract. Let's say you have a small group of 10 people debating a policy. If 5 are weakly opposed and 5 extremely strongly in favor it seems reasonable to apply the weighting and let the yeahs have it from a utilitarian standpoint.
– eps
Jun 21, 2021 at 14:09
• @eps It makes sense in your example, but what if you have 20 people voting, 15 of who are weakly opposed and 5 are extremely strongly in favor. Does it pass or not? Three times as many people oppose it as are in favor, but none of them have strong opinions. Even without the problem of tactical voting, do you side with the majority opinion or the strongly opinionated minority? Jun 21, 2021 at 14:58
• @Bobson: Suppose 2/3 of the people weakly prefer choice A to choice B, but 1/3 strongly prefer B. Suppose further that 2/3 weakly prefer B to C but 1/3 strongly prefer C. And 2/3 weakly prefer C to A while 1/3 strongly prefer A. No matter which policy is chosen, there will be one particular alternative policy which 2/3 of the people would prefer to the chosen one. Jun 21, 2021 at 17:12
• The point was that a binary choice is only necessary for the global/final outcome over all the voters; why should every citizen be binary when it is not necessary ? I suppose it is due to technical difficulties and demanding a binary choice is already a burden, so if one ask more it would even induce more abstensions I suppose ? Jun 21, 2021 at 18:06
• The second point is that numerical seems not to be involved in the choice. But as seen from the opposite way, the Parliament does not really care about the population neither, since the laws we have to vote about rarely have to do practically with the chooser, but on the other hand stuff the power wants us to do is not prone to discussion it seems. They say the population is consulted. Jun 21, 2021 at 18:22

The Australian Constitution can only be changed through a referendum, and it is a requirement that the change be first passed by the parliament and then approved by the people through a referendum.

One benefit of this mechanism is to prevent the government proposing a vague change to get approval, and then sneaking in something different once the approval has been given.

As a specific example, in 1999 there was a referendum to change Australia to a republic from a constitutional monarchy. This was rejected, in part because a significant number of voters did not agree with the proposed President appointed by parliament and wanted a directly elected president.

A vaguer 'Do you want to be a republic?' question, without a defined mechanism covering the powers and method of appointment of the president, may be considered and passed in a plebiscite but a referendum would still be required under to constitution to implement a change. This prevents a government using the vague question to justify appointing an all-powerful unelected president for life.

• This is all very interesting, but how does it address the question how score voting would affect the results of referenda?
– Philipp
Jun 21, 2021 at 8:30
• @Phillipp The question in the title is "Why do referendums use a binary choice rather than numerical score" The answer is because a numerical score is open to interpretation while a yes/no approval is not. Everyone casts exactly one vote of equal value. Really a specific example along the lines of Rad80's answer. It addresses the question in the title and first two paragraphs, but not the question about whether the alternative has been used. Jun 22, 2021 at 6:17
• What interpretation, the calculation of the balance were the same but instead of adding 1s one should add other numbers. But for this we would need to know arithmetics. But in fact this is useless since democracy is letting the people believe they could choose their dictator. Then they focus on him/her but this absolutely gives no information about where the coercitive power were. Jun 29, 2021 at 14:11