Marcus Pivato was a professor of mathematics at Trent University (Peterborough Ontario Canada) 2002-2014 and now is a professor in the Department of Economics at the Université de Cergy-Pontoise in France. He's studied voting a good deal and I respect him. (He may, however, not be in Canada right now, he emailed this from Belgium.) The below is advice he emailed about what Canada ought to do re voting. (I'm just re-posting his email here.) Although Pivato wrote this totally independently of me – acting in complete ignorance of my own thoughts – he came up with advice remarkably similar to my own views.
[I've also separately gotten some advice from politics professor Stephen J. Brams at NYU, albeit Brams was pretty aware of what I was thinking, so it was not independent. Brams seems to agree with me on the main points of my thinking.]
This is a very good question. Canada is a representative democracy with a Parliament, but no President. The official "head of state" is the Governor General, but he/she is an unelected figurehead with no real power (except in very rare situations). The real leader of the government (in effect, the "president") is the Prime Minister, who is the leader of the majority party in the Parliament. (So, to make a somewhat misleading analogy to the US system, the "Speaker" and the "President" are effectively the same person.)
This means that, although you officially vote for your local representative (your "Member of Parliament", or "MP"), you are not really voting for that person – you are in effect indirectly voting for the party which you want to form the government.
This is significant, because almost all the voting rules considered in the mainstream social choice literature assume that you are voting for a single entity (e.g. a president). But voting for one representative in a parliament is very different thing from electing a president. So almost all of the mainstream literature is inapplicable, and most of the standard rules are not appropriate.
I'm sure you are already familiar with all the many serious defects of single-member district plurality (a.k.a. "first past the post") voting systems, so I won't review them here. One reaction is to go entirely to a "proportional representation" (PR) model, but then you lose whatever small amount of "local representation" we have under the current system. (Although, as I mentioned above, that is mostly a fiction.) Another major problem with purely proportional systems is that they give a lot of power to entrenched political parties (who assemble the "lists" from which representatives are elected). Indeed, proportional systems don't even make sense without pre-existing political parties. [WDS comment: it is possible to design PR systems in the absence of parties, STV usd in Ireland does that and I have some better recommended ways to do that, but Pivato is correct that many PR systems, in particular "party list" systems, make no sense without parties. There also are "open" party list schemes in which the list-orders come not from party bosses but rather from voters, and those can hope to be more defensible about "assembling the lists."] Personally, I think political parties are an extremely pernicious element of government. Ideally, we could dispense with them altogether. Probably, they are inevitable and a necessary evil. But in any case, their power should be minimized.
Ideally, I would like a system where the distribution of seats in Parliament is "proportional" (i.e. a party which gets 8% of the popular vote gets 8% of the seats), and yet still provides people with local representatives. One system which comes very close to achieving this ideal is the Germany model, sometimes called "Mixed Member Proportional" system. In this system, each voter casts two ballots: one for their preferred local representative, and another one for the political party they prefer overall. (It is understood that these are not necessarily the same thing.) A local representative for each district is chosen by applying single-member district plurality to the first ballot. But the overall distribution of seats in the Parliament is decided using the vote distribution on the second ballot. The discrepancy between the "desired" seat-distribution (given by the second ballot) and the actual seat distribution arising from the first ballot is made up by supplementing the Parliament with extra members drawn from the party's lists. (These members do not represent any particular district.)
Personally, I think this model has a lot going for it. But I've never done a serious formal analysis (and neither has anyone else, as far as I know).
There is another system which I think is very interesting. In this system, each district elects several representatives (possibly quite a few), but they don't all have the same voting weight in Parliament. The voting rule within the Parliament is a weighted voting rule, where the weight of each member of Parliament is directly proportional to the number of people who actually voted for her.
So, for example, imagine a district with 130,000 voters, where there are ten candidates, labelled, A, B, C, ...., I, J. Suppose they receive the following proportion of the votes in this district:
A: 30% B: 25% C: 20% D: 10% E: 5% F: 3% G: 3% H: 2% I: 1% J: 1%
Suppose we impose a threshhold that a candidate must receive at least 5% of the vote to be elected. Then this district will elect A, B, C, D, and E as its representatives. For simplicity, suppose that only 100,000 of the 130,000 voters in this district actually voted. Then these five representatives would have the following voting weights in the Parliament:
A: 30,000 B: 25,000 C: 20,000 D: 10,000 E: 5,000
(Note that such "voting weights" could be incorporated into many different voting rules within the parliament – not just the usual majority voting rule.)
This rule has several advantages:
I have some vague conjectures that this voting rule has some very nice properties, but I haven't had a chance to do a formal analysis yet.
It is very important to recognize that you can have the most amazing electoral system in the world, and it is totally worthless without proper regulation of electoral spending and campaign financing. In my opinion, the system in the U.S. is completely out of control; basically the government is for sale to the highest bidder. The system in Canada and most European countries is not quite as bad, but rich individuals, corporations, and unions can still purchase far too much political influence.
I personally like a system proposed by Bruce Ackerman. At the beginning of the election (or perhaps, at the end of the previous election), each citizen is given a voucher of, say, $100. They can allocate this money as contributions to one or more candidates in their district, in any manner they see fit. They can even choose to withhold some or all of the money (in which case, no one gets it, including the voter themselves). The money from these vouchers is collected by the candidates and used to fund their campaigns. This is the only source of campaign funding in the election. All other campaign contributions, either by individuals or groups, are totally prohibited.
Of course, this doesn't stop wealthy and powerful organizations from simply purchasing their own advertising, in support of their preferred candidate or policy. However, one partial solution is to require every paid political advertisement to completely disclose all sources of funding larger than some threshold amount (e.g. $100). And "completely disclose" means disclosing the original source of the funding, not whatever "PAC" or other shell organization it was laundered through. This would not completely eliminate the influence of plutocrats like the Koch brothers. But it would help.
Postscript. One thing I should have said above: if I were going to adopt something like the German Mixed Member Proportional system, I would change one thing. On the "first ballot", the local representative for each district should not be elected using plurality rule. Because as we both know, plurality rule is awful. There are a lot of alternatives that would be better. Personally, I would favour range voting. Of course, when people vote strategically, range voting more or less collapses into approval voting; that would be okay too. I also wouldn't be opposed to Borda rule. Of course, the "second ballot" (used to decide the distribution of seats in the house) would not be any of these rules; it would simply say, "What party do you think is the best?" (A more complicated rule might ask each voter to specify the precise distribution of seats in the Parliament she thinks would be optimal, but I think this is way too complicated a question to ask most voters.)
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