A question about referendums

Isn’t there a worry that there might be Arrow-style cyclical preference rankings?

i.e. a majority would prefer A to B. A majority would prefer B to C and a majority would prefer C to A.

Hi Harry,

This is basically a question about social choice theory. Of course, a three-option (or four-option, or whatever) referendum is not “impossible”. The problem is that there are many different voting rules you could use for such a referendum, and there is no consensus which of these is the “best” or “correct” voting rule.

For a two-option referendum, one can make a fairly compelling argument for simple majority vote: under reasonable assumptions, and assuming there is no obvious asymmetry between the two options, simple majority vote is the unique best voting system. (There are theorems that state this formally.)

However, once you move to three or more options, there are an infinity of possible voting rules, each of which can claim to be “democratic” in some sense. All of these rules have good arguments in their favour, but all of them also have major weaknesses. For example, all of them can produce “perverse outcomes” (e.g. one option wins because the opposition was “split”; or a certain option that would have won becomes a loser after it gains more support; or some voters actually can get a better outcome for themselves by not voting, etc.). One major concern is that all of these rules are vulnerable to strategic voting.

What this means in practice is that there will always be some questions about the legitimacy of a three-outcome referendum. The losing side can always argue that the result is not legitimate —does not “really” reflect the Will of the People —because the wrong voting rule was used, some people voted strategically, etc.

You might think, “Why not just use Plurality Rule? It is simple, well-understood, widely used, and generally accepted.” But it turns out that, on closer inspection, Plurality Rule is actually one of the worst voting rules you could possibly use. (This, compounded with first-past-the-post regional representation, explains a lot of the problems with certain Western democracies, in my opinion.)

Social choice theorists have been studying this question for close to seventy years now (or seven hundred years, if you include the early work of Ramon Llull and Nicolaus Cusanus), and they still can’t come up with a really compelling argument that voting rule X is the best rule. (It’s like the old joke, “If you laid all the economists in the world end-to-end, they still wouldn’t reach a conclusion.”)

For Presidential elections in several countries (e.g. France), the voting rule is plurality rule with a run-off vote between the two front-runners. While this system has its problems, it also has much to recommend it: it is simple, well-understood, and guarantees that in the end, the winning option must be supported by a strict majority of the voters. If the UK was going to try for a three-outcome referendum, this might be the least-bad option.

Not an expert, but didn’t you people just recently have a referendum on what you call the Alternative Vote? (Answer: Not unless you call 2011 recent.) I don’t see any reason why it should be different for issues than candidates. Of course, it failed, so maybe people just don’t like this, but maybe whatever the opposite is of “Familiarity breeds contempt.”

No personal expertise, but I read the article below on The Conversation recently. In a nutshell, it says that deciding on the winner of a three-way referendum is a hard problem in social choice theory
https://theconversation.com/want-a-brexit-deal-referendum-a-major-voting-problem-is-being-overlooked-102678

Instant run-off referenda? That would be new to me. I think it wouldn’t be considered valid because referenda are already “complicated” enough. The idea of referenda is to get the will of the people on a question – it should be straightforward and simple.

In Canada we have the Quebec Reference Case and the Clarity Act to address our Referendum Issue. It mandates that referenda questions need to be clear.

That’s how it should go. If you are going to go full plebiscite, you need to be able to clearly state your question in a way that the least capable but still qualified voter can understand. I think that precludes having instant run-off or single transferable vote or any other second choice mechanism. And I don’t think a plurality among multiple options is compelling enough for referenda. So it has to be Yes or No.

With regards to the UK having a 3-way referendum on Brexit, there are several issues with this.

1. The referendum is being proposed to avoid implementing the result of a previous referendum. This is a never-ending series.

2. The three-way choice is being touted to split the Leave vote and make it impossible for Leave to win. Leavers may see this coming, say this is a biased poll and not vote, resulting in grievance being built into the UK political system.

3. The EU is a constantly changing thing. Do we have a referendum for every change? If not then very soon you are in “we didn’t vote for this” territory.

My view is that unless referenda are an integral part of your constitution, you should avoid them if at all possible.

… and the referendum we are being asked to consider now is one that requires us to approve a legal agreement that runs to over 500 pages. Seriously … that is why we elect people to represent us isn’t it?

3-option referendums are nothing special. An example is the practice of putting a proposal and counter-proposal to referendum. For example a Popular Initiative in Switzerland, after gathering the required number of signatures, first is debated by the government and then by Parliament. Parliament can approve it, reject it, or offer a counter-proposal. In the latter case, the initiators may withdraw the initiative in favor of the counter-proposal or both proposals are put to referendum. Voters then have three votes: yes or no to proposal A, yes or no to proposal B, and they can state a preference in case both proposals reach a majority. So there are three clear options: Status quo, proposal A, proposal B. (https://de.wikipedia.org/wiki/Volksinitiative_(Schweiz))

Some other jurisdictions allow this practice. There is absolutely nothing problematic about it.

The above explained procedure in my view is way clearer and makes democratically more sense than a three way ranked alternative. The assumption is that the proposals go in the same direction but one is more radical while the other is more of a compromise so radical voters can support the radical proposal but also support the compromise in order to get at least moderate change. Normally the compromise is more likely to win. However if some radical voters really don’t like the compromise, they may split the “change” vote and thus help the status quo to win.

In the Brexit case, the status quo would be Remain and the proposals would be “Leave with Deal” or “Leave no deal”. Ardent Brexiteers would vote for both proposals and state a preference for either. Remainers would vote Remain and probably state a preference for Deal over No Deal. Suppose there still is a majority for Leave (probably not the case), almost all of them would have to be prepared to vote Yes twice in order to ensure that at least one (probably Deal) wins. But that seems unlikely since at least some Brexiteers would prefer the status quo over No Deal but some others passionately hate the Deal.

Generally I agree that it is problematic to use direct democratic instruments when they are not part of the constitutional tradition. Britain should either never have gone down that path, or should have adopted a constitutional framework that specifies when and how referenda are to be used. Switzerland has a long tradition in that respect but they too continue to learn. The procedure explained above was adopted (at the federal level) only 30 years ago, before that voters could only make one choice. This usually favored the status quo.

My view is that unless referenda are an integral part of your constitution, you should avoid them if at all possible.

Thus spake Dipper

@TM 17

“In the Brexit case, the status quo would be Remain and the proposals would be “Leave with Deal” or “Leave no deal”.”

My understanding is that the status quo would be no deal, since A50 has already been triggered.
I say this because while this doesn’t change the logic of your argument, it shows that the situation is complex legally, in such a way that makes a referendum difficult.

This doesn’t mean that it is impossible, but it means that whomever wins the losers will never fully accept the result (as is already happening with the brexit referendum, anyway).

Why would you hold a referendum in the first place? Presumably it’s to legitimise a decision. There is no “will of the public” to be discovered, it’s just answers to poll questions. And the subtext here is finding a way to block Brexit – unlike Dipper I don’t find that illegitimate, but the Commons already has all the power it needs to do so. However, MPs sense – rightly! – that they’d be hammered at the polls for their betrayal. So any referendum that might undo Brexit would have to have at least as much political legitimacy as the original referendum, or it would fail to accomplish its task.

Suppose the voters were to break down as follows, which is pretty plausible:
45% No deal, second choice May’s deal.
15% May’s deal, second choice Remain.
40% Remain, second choice May’s deal.

No Deal is the plurality winner. Remain is the STV winner. May’s deal is the Condorcet winner – i.e. it beats every other option in a head-to-head. All have pretty good arguments. So who should win? Now, maybe this wouldn’t be so bad if we had a long history of such referenda, but we don’t. We would be making up the rules on the fly. Worse, we would have polling data, so the rules would be being made up on the fly not by disinterested parties based on abstract principles, but by partisans based on how much advantage said rules would give them. And even if we could somehow agree in such a partisan atmosphere (as I think we should) that Condorcet is the fairest and most legitimising procedure, we still have to ask why these are the options. Why isn’t “renegotiate the deal” an option, for instance? I strongly suspect that this would be more popular than any of your three options…

This will not achieve any popular legitimacy, unless somehow one option manages to win an absolute majority. And if one option could win an absolute majority, we don’t need 3 options to begin with.

Given the subtext, the referendum would presumably be set up with rules designed to favour the Remain outcome, so we would end up right where we are now – a House of Commons with a majority of members who would like to Remain, but who will lose their seats to justified popular anger if they vote that way. Why not just have a referendum between May’s deal and Remain and be done with it?

22: One can invent interesting voting procedures and have mathematical fun with them but I would really start with getting informed about those that have already been tried in the real world and have stood the test of time. Many political problems that Angloamericans are wrestling with have already been solved elsewhere.

Take this seriously, CT folks. Get to know the rest of the world at least a bit before daydreaming all those castles in the air.

Hi James,

I’m a big fan of “Cardinal Voting” (also called “Range Voting”, “Evaluative Voting”, or “Score Voting”). There is a little bit of research on this voting rule in the social choice theory theory literature, but not very much. As you say, it seems to be “obviously” a good idea. But it runs into two serious problems.

First, it is not clear how to interpret the scores that the voters assign to alternatives. For example, suppose scores are on a scale from 0 to 10, and I assign alternative A a score of 2, alternative B a score of 3, and alternative C a score of 9. What exactly does this mean? Clearly, it means that I like C more than B, and B more than A. But does it mean that I like C “three times as much” as B? It’s not clear that this even makes any sense. Does it mean that my preference for C over B is “three times as strong” as my preference for B over A? Again, it’s not clear how to interpret such a statement.

Since the meaning of the scores is ambiguous, it is difficult to mathematically justify one particular way of aggregating them over another way. For example, the “obvious” thing to is to add the scores, as you propose. But why add them? Why not add the squares of the scores instead? Why not multiply them?

Furthermore, since the meaning of the scores is ambiguous, there is a concern that different voters will interpret them differently. For example, suppose that I vote as described above. Meanwhile, you assign alternative C a score of 2, alternative B a score of 5, and alternative A a score of 8. Clearly, I feel that C > B > A, whereas you think that C < B < A. But should we argue that your preference for B over C is “three times as strong” as my preference for B over A? Perhaps you just interpret the numbers in a different way than I do, and actually the intensity of your preferences for C > B > A is exactly the same as the intensity of my preferences for C < B < A.

The fact that there is no obvious way to answer these questions makes social choice theorists nervous, and has caused many of them to just avoid this voting rule altogether.

(One way to interpret these scores is as expressions of voters’ cardinal utility functions, so that Cardinal Voting is a sort of “utilitarianism”. But there is no guarantee that the voters themselves will interpret their scores in this way. And even if they did, this raises questions about the correct way to make “interpersonal comparisons” of cardinal utility. So it does not really solve the problem.)

A second problem is that if voters vote strategically, then they will not use the full expressive power of the scores anyways. Suppose there are six alternatives, A, B, C, D, E, and F, and suppose that my preferences are A< B< C< D< E< F . Suppose that I believe (based on public opinion polls, etc.) that the two alternatives most likely to win are B and D. Then it is not hard to see that my optimal “strategic” vote is as follows: assign A and B the minimum possible score (i.e. 0), and assign D, E and F the maximum possible score (i.e. 10). (Strategically speaking, I can assign C any score I want, since it is unlikely to win anyways. So I lose nothing by being honest in my assessment of C).

If everyone votes like this, then “Cardinal Voting” essentially devolves into the “Approval Voting” rule, where voters simply assign a score of either 1 or 0 (i.e. “yes” or “no”) independently to each alternative. There is nothing wrong with Approval Voting (indeed, it has lots of good features). But it seems a bit redundant to introduce a more complicated “Cardinal” rule if in practice it ends up being equivalent to Approval Voting.

“how little Anglophones seem to know about the constitutional and political traditions of other countries”

Australia is Anglophone, and we have tried just about every voting system under the sun: single and multi-member plurality, instant runoff (with or without full ranking), Hare-Clark (with and without Robson rotation), list systems and many more.

My favorite was the modified D’Hondt system – it came with a 15-page explanatory booklet which neither a PhD in economic theory, nor an Honours degree in pure maths enabled me to understand. It produced some bizarre results the only time it was tried (admittedly a bizarre election anyway) and was never used again.

It always amuses me to see fellow-Anglophones (some of whom aren’t too kind about Aussie intellectual ability) explain that instant-runoff is too hard for ordinary voters to understand.

The tactical voting problem in all cardinal voting systems comes from the lower preferences counting against higher preferences. This creates a perverse incentive to vote dishonestly.

An example using the modified borda count used in Eurovision. Say your national jury vote is honest and puts Serbia first (twelve points) and Sweden second (ten points). Add all the other jury and popular votes and Sweden wins by two points, the jury’s second-best outcome. If they had voted tactically given Sweden no points and still given Serbia twelve then Serbia wins by eight points. Which gives the jury’s preferred outcome.

While the exact details vary depending on which of the many cardinal voting systems you use they all have this problem.

@Tzimiskes 6

You don’t live in America, I suppose, where it appears to be Team Pepsi vs Team Coke all the time everywhere?

Okay, it’s taken me overnight, but I understand the force of one of the objections now!

Because it’s one-round and winner-takes-all, as Marcus says there is an incentive for people to not only express their preferences but to bake their predictions of other people’s votes into their own, by systematically overstating their preference for options that they believe are likely to be widely popular. This doesn’t necessarily come at any direct cost to their ‘real’ favourite, which they can also rate ’10’ (the equivalent of a ‘hopeless cause’ STV first preference). But it does distort the information they provide, which the French run off model, and the Swiss model both are at pains to account for. The French run-off model with its famous “first round to say what you want, second round to do what you must”, the Swiss by establishing completely independent votes for each discrete outcome. I think that this is a very serious weakness of the system I proposed as I proposed it.

29: “Australia is Anglophone, and we have tried just about every voting system under the sun” Point taken.

“My favorite was the modified D’Hondt system – it came with a 15-page explanatory booklet which neither a PhD in economic theory, nor an Honours degree in pure maths enabled me to understand.”

Interesting, the original D’Hondt system was used (? at least it was named after) Jefferson more than 200 years ago. It was used in Germany up to 1985 and replaced by Hare-Niemayer (for Anglos: Hamilton), which is more complicated but fairer to small parties. I have heard people (well, Americans) complain about the complexity of these algorithms but they really don’t matter to the voter (at least as long as there is a reasonable number of seats to assign). What voters need to know is that seats are assigned approximately proportionally (and in the German case that small parties under 5% vote share get nothing).

32: Two problems:
– The constitutional status of the 2016 referendum is entirely undefined
– The question put to the voters in 2016 doesn’t meet the requirement of clarity. They voted (by a tiny margin) against the status quo but not for any specified alternative.

I’d just like to thank Marcus again for taking the time to patiently share his expertise. I certainly feel that he’s clarified a lot for me.

To return to Harry’s opening question, then, it seems to me that if you have an uncontroversial ‘status quo’, the Swiss system is the gold standard. All options are discrete votes requiring their own majorities (no tactical voting), with a preference mechanism built in to adjudicate if multiple proposals get over the line. (The detail on the preference mechanism would be interesting – again, simple plurality?)

Unfortunately, the first referendum probably scuppered the ‘uncontroversial status quo’ criterion – we already voted to leave, so in the absence of a majority, would we stay in, or crash out? Making either a default would now seem like gerrymandering to a substantial part of the electorate.

The next alternative, then, would be a run-off system. Any one-round system would be susceptible to the following: a single poll, perhaps due to mere noise, shows one of the ‘remain-flavoured’ options and one of the ‘leave-flavoured’ options ahead of the others. Having only one shot at getting the most popular option, the two camps then begin to cluster around these positions, in an endless feedback loop, like a run on the banks in reverse. Run-off elections take the pressure off this. One of each kind of flavour will certainly make it through, so there’s space in the first round to be candid about what specifically you prefer.

All this having been said, if you’re considering a one-round, multiple choice election, consider cardinal systems! Provided that you’re willing to make the assumption that the intervals individuals leave between options reflect something, and that this is true between as within individual voters, cardinal systems are probably superior to ranked systems if we’re aiming to adequately reflect preferences, though they suffer some of the same weaknesses to tactical voting as the rest of their family.

Yeah?

Ordinal voting state seem to have fewer perverse incentives than other systems. The biggest in practice in STV is placing a weak candidate you like first so if your real first preference are above quota before eliminations begin then your vote comes into play at a later stage at full value rather than as a fractional vote. Or you help get the weak candidate elected unexpectedly (this applies mostly if there are a relatively large number of available seats). Mostly however it is optimal to vote your real preferences as you would need a detailed knowledge of how other people are likely to vote.

Borda is a cardinal voting system, which means it has several perverse incentives that ordinal systems lack. The lower preferences counting against higher preferences is why they are rarely used for anything important.

Approval voting has the problem that it doesn’t account for relative preferences at all. It might benefit a disliked party to have a credible extremist party as no reasonable person would want the extremist elected they might approve all the moderate parties. Even ones they strongly dislike.

Re ccc and the Lessig piece:

Isn’t one of the reasons Trump won the Republican presidential nomination from a large field of candidates that the others expected him to fade and behaved nicely to him because they hoped to pick up his supporters afterward? In this case some more nastiness in the process would have been preferable.

A/K/A #32
——————-
I have no idea of the “constitutional” status of a particular referendum or of any of the weird-&-wonderful voting/lottery systems under consideration here. The question was asked in the first referendum by the government, using governmentcreated text, using a polling system run by the government. The answer was “Brexit!”.

The governmentcould have delt with that issue in Parliament but took the time and expense to ask the populace directly. The government didn’t get the answer that it wished. Any “Brexit” was going to have costs to the economy and to the nation. The duty of the government in this, and most other issues, was to get as good of a solution as possible. All choices have “costs”.

It is fraudulent to now claim that the populace didn’t understand the issue, when you mean that they didn’t appreciate your preference. Actually, you don’t seem to understand the issue(s) that are important to the populace.

I have no idea about the details of “constitutionality”. I am reluctant to use the term such as “honor” in a political discussion. The government asked a question directly to the populace, presumably to get past party and donor issues. The government seems bound to go for the best Brexit deal that it can get in light of the answer that it receivedor to get out of the way and let someone else take care of the problem(s). It is clear that the government has refused to do anything, so far, to reduce the expected adverse impacts of a Brexit.

I suspect that the original referendum was a ploy to relieve pressure on the government that didn’t go as planned. Too bad! Since the government made an “unusual” move to get feedback directly from the populace, it would seem that the government is obliged to follow through with the response that they received. Otherwise, the question becomes, “Who is the supposedly representative government actually representing?”

Most of the conversation here seems to be about the lottery system most likely to get the result that the pro-EU faction wants. In my untutored opinion, that train has already left the station. A significant portion of the UK populace doesn’t want what you are selling.

Most of the discussion has been in the abstract, so let’s look at the practicalities.

Assuming that the system is alternative vote/IRV, and that the alternatives are those set out by May: Her deal, No deal, No Brexit (ie revoke Article 50 declaration)*, it seems clear that No Brexit will get at least the 34 per cent needed to avoid elimination. If No Brexit gets a first round majority, there are no problems, so let’s assume that doesn’t happen.

Suppose everyone agrees on a left-right ranking from No Deal to No Brexit, with May’s deal in the centre. That is, any No Deal voter would prefer May to No Brexit and vice versa. That implies that, if No Brexit doesn’t win a majority in the first round, it will lose to May’s deal in the second, but might beat No Deal. I’ll assume that, since No Deal is, I think, too scary for most.

In this case, it makes sense for No Dealers to vote tactically for May’s deal in the first round, ensuring that May’s deal wins in the second round.

I don’t think this is the case, however. I think a lot of No Dealers would prefer staying in the EU to the “vassal state” offered by May. In that case, No Brexit beats either of the alternatives – it is what is called the Condorcet winner.

* How any of these choices actually play out depends, of course, on the response of the EU