What if Canada adopts MMP?

Among the electoral system types to be considered by Canada’s upcoming reform-proposal process is the mixed-member proportional system (MMP). What might we expect Canada’s effective number of seat-winning parties (Ns) to be under MMP?

As noted in the previous post on comparing the Alternative Vote to FPTP, Canada has had almost exactly the Ns, on average, that we should expect it to have, given its assembly size (around 2.6). Thus I will start from the premise here that Canada would continue to have about what we expect if it had MMP staring in 2019 (or whenever). That is, recent elections the country are neither surprisingly under- or over-fragmented, so there is no reason to think the country would over-shoot the expectation under a new system. (It might be more realistic that it would under-shoot, but that depends on how much we believe there is pent-up demand for new parties or for growth of existing smaller ones. I will leave that aside here.)

The answer to this “what if” depends on the precise MMP model. What Li and Shugart (2016) have shown is that a minor addition to the Seat Product Model (of Taagepera, 2007) captures two-tier compensatory systems well. MMP is a type of two-tier compensatory system, so let’s apply that model to a hypothetical Canadian MMP. The formula is:


In this formula, MSB is the “basic-tier seat product”, defined as the mean magnitude of the basic tier, times the total number of seats in that tier. For a typical MMP system, we retain M=1 in the basic tier, so the basic-tier seat product is simply the number of seats elected in that tier. The t in the formula stands for “tier ratio”, which is the share of the total assembly that is elected in the compensatory tier. This is an exponent on a constant term that is empirically determined to be 2.5. (Determined from the broader set of cases on which this is tested.*)

If a country’s electoral system is “simple”, meaning there is just a single tier, as with FPTP, then the above formula reduces to Taagepera’s (2007) seat product model:


(In a system with no upper tier, t=0, and MSB=MS.)

For purposes of estimating, I am going to assume the assembly size will remain the same, currently S=338. I will further assume that it would be politically difficult to reduce the number of districts (ridings) in the basic tier of an MMP system too much below the current number (which is, of course, 338). I will adopt two thirds of the current number as my estimate of “not too much”. In such a scenario, we get a basic tier of 225 seats, giving us t=.33:


I did not plan my scenario to get to three-and-a-third, but it has a nice ring to it. And seems pretty reasonable. For comparison purposes, this is not much different form what Canada had in 2006 (3.22) or Germany, an actual MMP system, had in 1998 (3.31).

Based on another formula in Taagepera (2007), which is empirically very accurate, we can also derive an expectation for the seat share of the largest party (s1):

s1 = Ns-.75.

For our hypothetical MMP system in Canada, this implies the largest party with just over 40% of the seats.

We can tinker with the scenarios. For instance, suppose the assembly size were increased to 400, with half the seats in the basic tier. Then we get:


As would be expected intuitively, the fragmentation of the House goes up due to the larger compensation tier, and in spite of the basic-tier seat product being correspondingly lower. This scenario would have an expected s1=.366.

Note that I have ignored thresholds here. My ongoing research with Taagepera suggests that thresholds matter, but unless the threshold is very high (more than 5%) or the seat product, MS, is extremely high, the value of the threshold has much less impact than the parameters discussed here.

In conclusion, under common Mixed-Member Proportional (MMP) designs, Canada could expect its House to have an effective number of parties ranging from 3.33 to 3.8, compared to a recent average of around 2.6 under FPTP. Its largest party could be expected to have around 36% to 40% of the seats, on average, compared to majorities or nearly so in recent elections.

* Note that this parameter’s empirical derivation raises the risk that it could be an artifact of the particular sample we have, and thus not reliable for determining an expectation value, as I am doing here. Fortunately, unless it is off by a wide amount, it does not make a large difference. For instance, 2.33 =1.26, whereas 3.33 =1.44. This variance in our prediction is much less than “normal” fluctuation from election to election in many countries.

Does AV mean higher or lower effective number of parties?

There may be a conventional wisdom among people who study comparative electoral systems that the Alternative Vote (also known as Instant Runoff or Majority Preferential) tends to suppress the effective number of parties, compared to plurality (First Past the Post, or FPTP). Or maybe it is just me, but I will admit to having such a notion. After all, Australia is a pretty strict two-party system, isn’t it?

The correct way to approach the question of whether AV means a higher or lower effective number of parties (N) than FPTP is to ask: What we should expect N to be, given the country’s seat product?

As explained by Taagepera (2007) and further elaborated and tested by Li and Shugart (2016), the seat product is a country’s mean district magnitude (M), times its assembly size (S). The Seat Product Model says that the effective number of seat-winning parties (Ns) tends to be the sixth root of this product: Ns=(MS)1/6.

The model is logical, not a mere product of empirical regression work, although regression tests confirm it almost precisely (Li and Shugart, 2016).

When all districts elect just one member, thus M=1, the Seat Product is just the assembly size, S. Hence we take the sixth root of S to get an expectation for Ns. What if we do this for Australia’s House of Representatives? We get an expectation of 2.31.

The actual Ns for Australia’s elections since 1984, the year S was increased from 125 to 148 (subsequently it has increased to 150, a minor change) is… 2.53. However, I believe that figure (I am using Gallagher’s Election Indices) treats the Coalition parties as one in elections before 2010.

In the two most recent elections, Ns has been 2.92 and 3.23. The notes to Gallagher’s Election Indices indicate that for these elections the Liberal Party, the Nationals, and the Liberal National Party of Queensland are treated as separate parties. In my opinion they should be so treated, although I suppose one could have a debate about that.

The actual mean is thus above the expectation for a hypothetical FPTP of the same size assembly. If we use the figure of 2.53, it is obviously not much higher than 2.31 (the ratio is 1.10). However, if we consider the value, at least in recent elections, to be around 3.0, it is about 1.30 times the expectation value.

Contrast this with the UK, where elections of the same period (1987-2010) have a mean Ns=2.30. This is just what we expect for FPTP, right? Not much over 2.0. Not so fast! The UK has a huge assembly, and with S=650 (aprpox., as it varies over the period), we should expect Ns=2.94. The UK actually has one of the more under-fragmented assemblies, according to the Seat Product Model, with this recent-period average being only 78% of expectation.

So how about Canada, where AV is one of the potential reforms being considered? Over a similar period (1984-2011) we get Ns=2.63. With S around 300 during this time, we should get Ns=2.59. So Canada pretty much nails the expectation of the model.

So, should we expect Ns to go down if Canada were to adopt AV, as (what I characterized as) the conventional wisdom would have it? Or should we expect it to go up?

I would not be inclined to say ‘down’. I will just leave it at that for now.

‘Seat Product Model’–audio version

The audio-slides version of Li and Shugart (2016) is now available!

As previously announced, the publication details and abstract are as follows:

The Seat Product Model of the effective number of parties: A case for applied political science

Yuhui Li, Matthew S. Shugart

Electoral Studies 41, March 2016, pp. 23–34.


This paper extends Taagepera’s (2007) Seat Product Model and shows that the effective number of seat-wining parties and vote winning parties can both be predicted with institutional variables alone, namely district magnitude, assembly size, and upper-tier seat share. The expected coefficients are remarkably stable across different samples. Including the further information of ethnic diversity in the models hardly improves the estimate of the effective number of parties, and thus the institutions-only models are preferable on the grounds of parsimony and the applicability to electoral-system design or “engineering”.

The Greens, electoral reform, and the Canadian leaders’ debate

Last week, Canadian party leaders participated in a debate. It is currently the only one scheduled to include the Greens leader, Elizabeth May (the party’s only MP).

The debate included an entire segment devoted to, as moderator Paul Wells put it, “Canada’s democracy — how it works, why it doesn’t always work as well as we hoped.” (From my south of the border perspective, I can only admire a debate that actually asks such a question, rather than implicitly assuming that the debate and election themselves are proof of how great democracy is working, but enough of that digression for now.)

The first question within this segment of the debate went to May, and the exchange, which you can read in the transcript, is very interesting.

Paul Wells: Our first question on this to Elizabeth May. Ms. May, you’ve called the government we have now an elected dictatorship and you’ve called for electoral reform, but this election will be won and lost under the current electoral system. Do you worry that Green candidates will take support away from other parties that could defeat this government? Might the Green Party help reelect this government?

Elizabeth May: When I refer to the government as an elected dictatorship, it’s not personal in any way to the Prime Minister nor to his party…

The only job description for a member of parliament is that found in the Constitution, which is to represent your constituencies.

So we need to actually revisit parliamentary democracy, understand that this election isn’t about electing a prime minister — we don’t do that in this country; we elect members of parliament. And their job is to find the government that will hold the confidence of the House, so we can work for Canadians…

As far as Greens being concerned about this, not at all. We have had success and we’ve now had election – my election in Saanich–Gulf Islands, but across provinces — in British Columbia Andrew Weaver, in New Brunswick David Coon, in Prince Edward Island Peter Bevan-Baker. All of us got elected by driving voter turnout.

So instead of fixating on this splitting the vote non-problem, vote-splitting, we need to focus on the real problem, which is 40 percent of Canadians in the last number of elections haven’t voted. And vote abandoning, in my view, is a much bigger problem than vote-splitting…

Paul Wells: You’ve said we don’t elect a prime minister, and that’s true, but we saw quite a mess of a coalition crisis in 2008. Are we headed towards that sort of arbitrage among parties after the next election if there’s no majority?

Elizabeth May: I can’t tell you how committed Green MPs as a caucus will be to working with other parties, working across party lines to ensure that we go from a precarious, perhaps two-year minority parliament to a stable, productive, effective parliament, because you look at really great parliaments in this country, and I refer viewers back to Lester B. Pearson where the small group of NDPers under David Lewis and Diefenbaker in the Conservatives and Lester B. Pearson delivered our social safety net.

I find the exchange interesting for the effort to drive the discussion way from vote-splitting and choosing a prime minister–two common perceptions of elections in parliamentary systems using first-past-the-post electoral systems. These are perceptions that are, of course, harmful to small parties. So May attempts to emphasize local viability of Greens, and the advantages of cross-party cooperation in a non-majority parliament.

Then things returned pretty quickly to business as more-or-less usual, with Liberal leader Justin Trudeau engaging NDP leader Thomas Mulcair in a debate over the Clarity Act (regarding another potential Quebec secession referendum).

A bit later, Wells raises the issue of electoral reform directly, referencing the proposal of the Liberal Party (see p. 8 of the linked PDF). I will just quote a few snippets here. PM Stephen Harper (Conservative):

Well, I think it’s a very fundamental change to the way our political system would work in this country. We have a Westminster system. Voters are able to elect governments. They don’t elect coalitions that make up the government later. And you know, Canadians – Paul, this has come up before. It was subject of a referendum and plebiscite in Ontario and Prince Edward Island and British Columbia. I have not found Canadians who want to make this fundamental change. In fact, whenever Canadians are asked, they reject it. We know the rules. Let’s play under the rules that Canadians support.

Mulcair did not use the immediate opportunity to talk up his party’s stated commitment to introducing proportional representation (MMP, specifically). Instead he talked about the current government’s “Unfair Elections Act” (it is, actually, of course, the Fair Elections Act). Later, however, Mulcair did say, “We think that there are three main things we can do with regard to our institutions. The first is to make sure that every vote counts with a proportional representation system.” (The others were “open up parliament” to more public scrutiny and abolish the Senate.)

Unless I missed it, Trudeau himself never mentioned his own party’s commitment to electoral reform. Perhaps he thought it was enough that Wells invoked it and gave Harper a chance to denounce it.

El Salvador joins the panachage ranks, president’s party holds steady

El Salvador held its legislative election on 1 March, using a modified electoral system. The country had already left behind the closed list in 2012, replacing it with an open list. This year the country moved to panachage, the variant of open list in which voters may vote for candidates on different lists (sometimes called “free list”).

El Salvador is one of the few pure presidential systems still using an electoral cycle (a long-term interest of mine) consisting of all non-concurrent elections, with presidential terms of five years and legislative terms of three years. This election is in the first year of the incumbent president’s term, and offered neither “surge” nor decline in his party’s legislative support.

I was first tipped off to the change to panachage by a remark in a Tico Times article just before the election that said:

For the first time, voters will be able to select individual candidates from any party rather than being forced to vote for a single party with an established list of candidates. Voters can still opt to simply choose a party.

This is definitely far more detailed than your average journalistic note about an electoral system. But to be sure, I checked with the Tribunal Supremo Electoral, which has a very useful gallery of ballot images and instruction cards. Below I post an image of the instructions for the district (department) of Cabañas.

ElSal 2015 guia

It offers the voter five options:

      1. Vote solely for the list of a party or coalition;
      2. Vote for a list and mark the photo of “one, various, or all the candidates” on that list;
      3. Mark the photo of “one, various, or all the candidates” in one list (without also indicating a list vote);
      4. “Mark candidates of distinct political parties or coalition” or candidates of distinct parties and a non-party candidate”, not exceeding the total number of deputies elected from the district.
      5. Mark the photo of a non-party candidate.**

It is option 4 that clearly establishes panachage.

By contrast, in 2012, the options excluded any mention of marking candidates across different lists.

El Salvador open list 2012

See also images of a portion of the ballot from 2015 or 2012.

For a long time the only panachage systems in use at the national level (to my knowledge) were in Luxembourg and Switzerland. In 2005, Honduras adopted such a system, and now one of its immediate neighbors has followed suit. (I believe Ecuador still uses a panchage system adopted several years ago; Venezuela used one at least once, but only at the municipal level.)

As for the election itself, there was a considerable delay in reporting the results. It does not seem that the panachage system had anything to do with the delay.

Preliminary results suggest that the opposition ARENA will have a plurality of seats, with 35 (of 84). That would represent a gain of two seats from the 2012 legislative election. The governing FMLN is likely to have 31 (no change) and its ally GANA 10 (-1). The PCN (the pre-1979 ruling party, still just hanging on) is expected to have 6 (-1) and the PDC would have 2 (a doubling of seats for this party that was the main alternative to the right before the civil war ended).

El Salvador hence essentially maintains its long-term relative stasis and close left-right division. The incumbent president, Salvador Sánchez Cerén of the FMLN, was elected by a very narrow margin a year ago. This election barely changes the balance of power, even though it occurs within the first year of a president’s five-year term and hence could qualify as a “honeymoon election”. In Salvadoran politics, there really isn’t much of a honeymoon, or any near-term prospect for the much-anticipated realignment. The big swing to the FMLN in the 2009 legislative election looks, in retrospect, like a blip within what is otherwise ongoing stasis.

It will be interesting to see if the move to an electoral system allowing cross-party voting for the first time begins to break down El Salvador’s remarkably rigid partisan lines.***

* Elections could be concurrent every 15 years, as they were in 1994. However, in 2009, the legislative election was shifted to the “counter-honeymoon” and held in January, with the presidential election, as usual, in March.

** In both options 4 and 5, the reference to independents is singular; logically if voting for an independent is an alternative to voting for a party, one could vote for only one independent just as one can vote for only one party. On the other hand, in a panachage world, one actually can vote for more than  one party (assuming a vote for any candidate also counts for the party on whose list the candidate was nominated), so why would there need to be a restriction on the number of votes that a voter may give for independents? (Note that it is probably quite difficult for an independent candidate to be elected in any case.) [I edited this footnote shortly after posting.]

*** Whether the partisan lines over El Salvador can ever be broken down in the United States is another matter.

Visualizing the impact of two-seat districts

In response to my post about the two-seat districts in Burkina Faso, David Altman (a leading scholar on Chilean and comparative politics) contacted me to say I was not quite right in my contention that the Chilean experience showed that two-seat districts systematically advantage the second party or bloc. Fair enough. I made my point utterly without nuance–as if the second list was always advantaged more than the first–and I elided the distinction between national and district-level effects.

David showed me a graph he had generated in which it is clear that, on average, there is little difference between the vote-seat gap for the main center-left alliance and that of the right-wing alliance; in fact, the gap tended to be slightly higher for the center-left. It is the right that has been consistently second in nationwide votes, so the data seemed to challenge my “systematic” claim.

In the Burkina Faso post, I went over a number of examples from the district level in that country where there was a large advantage in terms of the translation of votes into seats for the second party. I do not think David and I disagree about the district-level effects. But what about the national?

For reasons Taagepera and Laakso* first emphasized in 1980, the best way to visualize how electoral systems treat parties (or alliances) is not by using the difference between seat and vote shares for each party, but by the advantage ratio, A:

    A=(percent seats)/(percent votes).

This is the approach I will use here. The first graph simply plots the distribution of A values in the Chilean elections from 1989 to 2013 (Chamber of Deputies only), using a kernel density plot. It also marks the arithmetic means with vertical dashed lines.

Chile Adv ratios kdens

Of course, with only seven elections, we should be cautious in interpreting how “systematic” the effect is. Yet the trend is clear: the right-wing alliance (referred to in the graph as “Alianza” and shown in blue) has a notable tendency to have higher advantage ratios than has the center-left (“Concertación”).** Yet in every election the Alianza has been second in nationwide votes, albeit by widely varying margins.

The second graph shows the advantage ratio against the nationwide gap in votes between the two parties (Concertación vote share minus Alianza vote share), with each election labelled.

Chile A vs vote gap

Note that as the gap grows, there is a tendency of the Alianza’s advantage ratio to increase. There is actually a small such tendency for the Concertación, too, although it is much closer to a flat line. More importantly, we would normally expect that as the top two parties/alliances grow farther apart in votes, the first one would gain more in seat share. Yet what the Chilean pattern shows us is that the second one gains more, relative to its vote share, as the gap grows. This is not something we would expect under any other district magnitude (except maybe M=4) under any proportional (or semi-proportional) allocation rule. It is this “relative to its vote share” point that I meant to convey. Two-seat districts are a good way to get the second force more over-representation than that of the first force.

If we turn to individual elections, we see that Alianza data point is higher than that for the Concertación in 1989, 1993, and 2005, all years in which the Alianza was more than ten percentage points behind the Concertación, plus in 2009 when the two alliances were almost even in votes. On the other hand, the large vote gap for the Concertación in two elections, 1997 and 2013, resulted in a higher advantage ratio for the Concertación. In 2013 I assume that has to do with the center-left having expanded its electoral reach by incorporating new parties and rebranding its list as Nueva Mayoría. I am not sure what the explanation is in 1997; the gap was large that year, but as the data plot shows, not unusually so.

I find it interesting that the worst year for the Concertación, by this standard, was 2005, which I had noted at the time as being unusual in having a bloc of parties compete against each other in the first round of the presidential elections while remaining united on legislative lists–the Alianza’s two main component parties had separate presidential candidates. The result in the legislative races was their best advantage ratio since the first democratic election of 1989.

It looks to me like two-seat districts in Chile have indeed generated a systematic greater advantage to the second alliance over the first, even if the effect is tempered by other features of specific elections.

* Rein Taagepera and Markku Laakso, “Proportionality Profiles of West European Electoral Systems,” European Journal of Political Research 8 (1980):423-46; see also Rein Taagepera and Matthew S. Shugart, Seats and Votes: The Effects and Determinants of Electoral Systems (Yale University Press, 1989), in particular chapter 7.

** The right has run under different names in different elections. I am using “Alianza”; the center-left changed names from Concertación to Nueva Mayoría in 2013.


The data, by year (pardon the plain text formatting):

year Conc_votes Alianz_votes A_Conc A_Alian
1989 .515 .342 1.116505 1.169591
1993 .554 .367 1.052948 1.135331
1997 .505 .363 1.138614 1.078972
2001 .479 .443 1.078636 1.072235
2005 .518 .387 1.045689 1.162791
2009 .444 .435 1.06982 1.111111
2013 .477 .362 1.17051 1.127993
Mean .4988571 .3855714 1.096103 1.122575