BC 2017: Non-Duvergerian watch

British Columbia votes for its provincial assembly on 9 May. Eric Grenier’s vote and seat projections at CBC suggest it could be a “non-Duvergerian” outcome, especially in the votes. The NDP looks likely to win a majority of seats, manufactured by the electoral system, as its votes are likeliest to stay below 45%. The Liberals are running well behind, but are also likely to be over-represented due to their more efficient votes distribution: Grenier’s average projection for the party would put them on just under 35% of votes but 39% of seats.

The non-Duvergerian aspect of the election is the Greens: average projection of 20% of the votes, which is on the high side for a third party under FPTP. The mechanical effect will crush them, however: average seats only 2 of 87. That, of course, is “typical” FPTP.

Where things get interesting is if the Greens rise even just a few percentage points. The “high” projection takes their votes up only to around 23% but their seats to 12 or more (around 14%)! Clearly, they are running second in many districts–particularly on Vancouver Island.

The reason this pattern is non-Duvergerian is that the so-called Duverger’s law says voters tend not to want to waste their votes on a third party. They should “coordinate” with the more-preferred of the two big parties. Further, this defection should be more likely after the third party has suffered from the mechanical effect of FPTP in past elections, as the Greens in BC have. Instead, their support is evidently growing.

But there is not much incentive to choose between the major parties when you have three-party competition throughout much of the province, and the third party could win a significant number of seats even if it stays well below 50% in the districts where it is most viable.

The BC NDP platform promises: “We’ll hold a referendum – and campaign for the “yes” side – on replacing our outdated voting system with a proportional one.”

Will they follow through if FPTP delivers them a majority?

Update: The tracker of where the party leaders are campaigning is interesting. Of course, there is much emphasis on swing districts, particularly for the NDP. However, this quotation from Richard Johnston (UBC political scientist) is spot on:
“You do have to campaign everywhere; you have to validate the existence of your candidates.” In Chapter 10 of Votes from Seats (forthcoming, Cambridge University Press), Rein Taagepera and I refer to this phenomenon as the “embeddedness” of districts in nationwide (here, provincewide) politics. We do not consider non-Duvergerian districts to be quite the puzzle that others do, because we expect voters to be more oriented towards the aggregate outcome than towards their own district, and parties to care about “showing the flag” even in districts they may not be very likely to win.

Uttar Pradesh, 2017

Election results have been released for the state assembly of Uttar Pradesh, India’s largest state. It was a big win for the federal ruling party, the BJP. The seat tally shows 312 for the BJP, with the second highest being the Samajwadi Party (SP) at 47. The SP, the ruling party since 2012, was in a pre-election coalition with the Indian National Congress (INC), which won just 7 seats. The Bahujan Samaj Party (BSP), which has been a significant party in the state in the past, won 19 seats.

Unlike 2012, when the SP majority in the assembly was achieved on not even 30% of the vote, this year’s BJP victory was a big win in votes, too. Not a majority, but a decisive plurality, at 39.7% of the vote. The SP-INC combine had 28.0% and the BSP 22.2%.

Note that the BJP managed a three-fourths majority (77.4%) of the 403 seats on not even 40% of the vote. The advantage ratio (%seats/%votes) was 1.95. That must be one of the biggest manufactured majorities under FPTP anywhere, at least in a large assembly.

Several other states have had recent elections as well. The news was better for the struggling INC in some, including Punjab, Goa, and Manipur, though its pluralities in these are short of majority status. The Aam Aadmi Party (which governs Delhi, but has had minimal success elsewhere) managed a distant second place in Punjab. See the results at the second link in this entry.

Republicans will likely keep their House majority – even if Clinton wins by a landslide – and it’s because of gerrymandering.

By Michael Latner

While the presidential race has tightened, the possibility of Donald Trump being defeated by a wide margin has some Republicans worried about their odds of retaining control of Congress. However, only a handful of Republican-controlled districts are vulnerable. Speaker Paul Ryan’s job security and continuing GOP control of the House is almost assured, even if Democrats win a majority of the national Congressional vote. How is it that the chamber supposedly responsive to “The People Alone” can be so insulated from popular sentiment? It is well known that the Republican Party has a competitive advantage in the House because they win more seats by narrow margins, and thus have more efficiently distributed voters. What is poorly understood is how the current level of observed bias favoring the GOP was the result of political choices made by those drawing district boundaries.

This is a controversial claim, one that is commonly challenged. However, in Gerrymandering in America: The House of Representatives, the Supreme Court, and the Future of Popular Sovereignty, a new book co-authored by Anthony J. McGann, Charles Anthony Smith, Alex Keena and myself, we test several alternative explanations of partisan bias and show that, contrary to much professional wisdom, the bias that insulates the GOP House Majority is not a “natural” result of demographic sorting or the creation of “majority-minority” districts in compliance with the Voting Rights Act of 1965. It is the result of unrestrained partisan gerrymandering that occurred after the 2010 Census, and in the wake of the Supreme Court’s 2004 decision in Vieth v Jubelirer, which removed legal disincentives for parties to maximize partisan advantage in the redistricting process.

Partisan bias tripled after congressional redistricting

We measure partisan bias using the symmetry standard, which asks: What if the two parties both received the same share of the vote under a given statewide districting plan? Would they get the same share of seats? If not, which party would have an advantage?

We calculate seats/votes functions on the assumption of uniform partisan swing – if a party gains 5% nationally, it gains 5% in every district, give or take an allowance for local factors (simulated through random effects that reduce our estimates of bias). Linear regressions provide an estimate of what level of support the Democrats expect to win in each district if they win 50% of the national vote, given the national level of support for the party in actual elections, and we generate a thousand simulated elections with hypothetical vote swings and different random local effects for each district, to create our seats/votes functions.

Figure 1: Seats/Votes function for Congress 2002-2010

Figure 2: Seats/Votes function for Congress 2012

Figures 1 and 2 show the seats/votes functions under the 2002-2010 Congressional districts, and the 2012 post-redistricting districts, respectively. We observe a 3.4% asymmetry in favor of Republicans under the older districts. This is still statistically significant, but it is only about a third of the 9.39% asymmetry we observe in 2012. Graphically, the seats/votes function in Figure 1 comes far closer to the 50%votes/50%seats point. The bias at 50% of the vote is less than 2% under the older state districting plans, compared to 5% in 2012. That is, if the two parties win an equal number of votes, the Republicans will win 55% of House seats. Furthermore, the Democrats would have to win about 55% of the vote to have a 50/50 chance of winning control of the House in 2016. Thus it is not impossible that the Democrats will regain control of the House, but it would take a performance similar to or better than 2008, when multiple factors were favorably aligned for the Democrats.

Increased bias did not result from “The Big Sort” 

Perhaps the most popular explanation for increased partisan bias comes from the “Big Sort” hypothesis, which holds that liberals and conservatives have migrated to areas dominated by people with similar views. Specifically, because Democrats tend to be highly concentrated in urban areas, it is argued, Democratic candidates tend to win urban districts by large margins and “waste” their votes, leaving the Republicans to win more districts by lower margins.

The question we need to consider is whether the concentration of Democratic voters has changed relative to that of Republican voters since the previous districts were in place. In particular, if it is the case that urban concentration causes partisan bias, then we would expect to find relative Democratic concentration increasing in those states where partisan bias increases. In order to address this question, we measure the concentration of Democratic voters relative to that of Republicans with the Pearson moment coefficient of skewness, using county-level data from the 2004 and 2012 presidential elections. As shown in Table 1, in most states the level of skewness toward the Democrats actually decreased in 2012.


In twenty-seven out of the thirty-eight states with at least three districts, the relative concentration of Democratic voters compared to Republican voters declines. Moreover, in those states where partisan bias increased between the 2000 and 2010 districting rounds, those with an increase in skewness are outnumbered by those where there was no increase in skewness, by more than two to one. We also find that there was reduced skewness in most of the states where there was statistically significant partisan bias in 2012.

Of course, we should not conclude that geographical concentration does not make it easier to produce partisan bias. North Carolina was able to produce a highly biased plan without the benefit of a skewed distribution of counties, but to achieve this, the state Generally Assembly had to draw some extremely oddly shaped districts. While the urban concentration of Democratic voters makes producing districting plans biased toward the Republicans slightly easier, it makes producing pro-Democratic gerrymanders very hard. In Illinois, the Democratic-controlled state legislature drew some extremely non-compact districts but still only managed to produce a plan that was approximately unbiased between the parties.

Increasing racial diversity does not require partisan bias

Another “natural” explanation for partisan bias, one that is especially popular among Southern GOP legislators, is that that it is impossible to draw districts that are unbiased while at the same time providing minority representation in compliance with the Voting Rights Act of 1965. The need to draw more majority-minority districts, it is argued, disadvantages the Democrats because it forces the inefficient concentration of overwhelmingly Democratic minority voters.

There are four states with four or more majority-minority districts – California, Texas, Florida, and New York – and they account for more than 60% of the total number of majority-minority districts. Of these, Texas and Florida have statistically significant partisan bias, but California, Illinois, and New York do not, so the need to draw majority-minority districts does not make it impossible to draw unbiased districting plans. Yet many of the states that saw partisan bias increase do have majority-minority districts – or rather a single majority-minority district in most cases. It is possible that packing more minority voters into existing majority-minority districts creates partisan bias.

To test this possibility, we subtract the average percentage of African-Americans and Latinos in districts where those races made up a majority of the population in the 110th Congressional districts from the 113th Congress averages. Figures 3 and 4 display the results for these states. For majority-Latino districts, we find no evidence that states with increased Latino density have more biased redistricting plans.

Figure 3: Majority-Latino District Density Change and Symmetry Change

Figure 4: Majority-Black District Density Change and Symmetry Change

By contrast, states with increased majority-Black densities have clearly adopted more biased districting plans. Among the states with substantial reductions in partisan symmetry, only Louisiana (−1.9 %), Ohio (−1.0 %), and Pennsylvania (−0.9 %) had lower average percentages of African Americans in their majority-minority districts after redistricting. The three states with the largest increases in majority-Black district density, Tennessee (5.2 %), North Carolina (2.4 %), and Virginia (2.2 %), include some of the most biased plans in the country. This is not in any way required by the Voting Rights Act – indeed, it reduces the influence of African-American voters by using their votes inefficiently. However, it is consistent with a policy of state legislatures seeking partisan advantage by packing African-American voters, who overwhelmingly vote for the Democratic Party, into districts where the Democratic margin will be far higher than necessary.

Demography is not destiny

The bias we observe is not the inevitable effect of factors such as the urban concentration of Democratic voters or the need to draw majority-minority districts. It is for the most part possible to draw unbiased districting plans in spite of these constraints. Thus if state districting authorities draw districts that give a strong advantage to one party, this is a choice they have made – it was not forced on them by geography. The high level of partisan bias protecting GOP House control can only be explained in political terms. As we show in our book, Pro-Republican bias increased almost exclusively in states where the GOP controlled the districting process.

One of the immediate consequences of unrestrained partisan gerrymandering is that, short of a landslide Democratic victory resembling 2008, the Republicans are very likely to retain control of the House. But one of the more profound consequences is that redistricting has upended one of the bedrock constitutional principles, popular sovereignty. Without an intervention by the courts, political parties are free to manipulate House elections to their advantage without consequence.


“Votes that count” in different electoral systems

Throughout the discussion on electoral-system reform in Canada, I have seen various social-media posts that purport to compare the percentage of votes that “count” in Canada vs. in existing PR systems. Typically, these posts will cite a figure of 50% of votes counting in Canada and 95% or more in PR countries.

The numbers seemed fishy to me. I do not doubt that votes are substantially more likely to “count” towards election of a representative under PR than under FPTP or other majoritarian systems. But the 95% figure seemed too high.

As I happen to have a dataset of district-level electoral results in many countries at my disposal, it was not too hard to subject the claim to a test. The harder part is knowing how to operationalize “count”. I chose two methods, based on my understanding of the complaint that reform advocates have against FPTP. The problem, it will be seen, as I suspect both methods are being used by these advocates, but different methods for different systems.

The first method is the one that I believe they are applying to Canada, which is: “did my vote contribute to the election of someone in my district?” Guess what–when your district elects one, for many voters the answer is “no”. (And this may be sufficient reason to want to ditch the system!) The second method is one that I suspect the authors of these posts apply when looking at PR systems: “Did the party I voted for win representation?” That is, I suspect a district-level standard is being applied to FPTP, but a national-level one to PR countries. My calculations seem to bear that out.

If we use the first method–district-level count of votes for parties (or candidates) that did not win in the district, we almost nail the 50% figure for the most recent Canadian election for which I have data. In 2011, I get 50.4% of votes across the country “counting” in the sense that they were cast for the winner in the voter’s district.

Across the 25 FPTP elections for which I have complete data, the average figure is 55.8%. The lowest figures are around 47% in several UK and Indian elections, with the highest being 62% or more in the US and Barbados. (What do these latter two countries have in common? Very few votes going to parties other than the top two.)

I then apply that same standard to PR systems. A pause is needed here. Canada is very highly unlikely to adopt nationwide PR, either with a single national district (which I think we can say is politically and constitutionally impossible) or with districts but also nationwide compensation (as in Germany or Denmark, among others). Thus I consider the relevant metric to be those PR systems that employ districts, plural, and no nationwide compensation.

Using the same standard–votes “count” when cast for a party that wins a seat in the district–the mean for (districted) PR systems is 87.2%. That figure is a lot higher than 50%, I will grant. It is also a good deal lower than 95%.

If we look at nationwide (single-district) PR, guess what? 96.5%! A few specific elections come in at 99%, such as the Netherlands, 2002, and Israel, 1951. (Most Israeli and Dutch elections are over 97%, but Netherlands, 1952, was at a paltry 94.7%.) That’s great! However, most PR advocates, unless they are real purists (and not at all realists) do not advocate the adoption of the Dutch or Israeli type of PR.

What about the second standard? Under this one, your vote “counts” if it was cast for a party that won representation somewhere (at least one seat), even if it did not win in your district. As I noted above, I suspect this is the standard being applied, at least implicitly, to the PR countries in the social-media posts I have seen.

By this standard, districted PR systems’ average percentage of votes that count rises to 93.7%. We are almost to 95%! For nationwide PR, it obviously does not change (there’s only one district). What about for FPTP? 97.1%. Canada, 2011, comes in at 99.1%.

I do not actually know if a Canadian voter feels “represented” if she voted NDP but the NDP candidate in her district lost. I suspect many do feel so represented, or else the NDP would not get more than trivial vote shares in districts where it has no chance of winning. Yet it does. Greens can also get votes near or above their nationwide percentage even in many districts that are totally hopeless for them. Perhaps they feel represented by Elizabeth May (the one Green MP) even if they reside in a different district or province. Not as well represented as if their own MP was Green, presumably, but my point is that voters probably tend to think of national party systems when voting, even in districted systems. Yes, even in systems in which districts have one seat apiece.

There may be many reasons to prefer PR over FPTP. I can think of quite a few myself. But the idea of a vote “counting” towards representation may not be one of the more meaningful criteria to use. Or, if it is used, it might be OK not to exaggerate. The difference between 50% (Canada, 2011) and 87% (mean for districted PR) is impressive enough, using the first (district-based) criterion. We don’t need to pretend that twice as many votes “count” under PR as under FPTP. 1.74 times as many is still a lot more!

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.

The new UK constituencies

For the next (expected 2020) UK election, the assembly size will be reduced from 650 to 600, and the balance in the number of constituencies across the UK’s component units (including English regions) will be shifted. Ron Johnston, at the LSE blog, has a rundown of the changes.

Imagine the research-design opportunities for analyzing personal-vote behavior:

Some current MPs will see their current seat dismembered, and may worry whether they will be selected for another; David Cameron has promised all current Conservative MPs that they will have a seat to fight in 2020, but it may well be very different from the one they currently represent. And so much change will break the bonds between MPs and both their constituents and their party organisations – some of them of long standing – that will have to be rebuilt before the 2020 contest. Many MPs may spend a lot of time building support in their new constituencies rather than serving their existing ones – let alone debating and decision-making in Westminster.

The partisan effects also could be substantial: “The Conservative lead over Labour will probably be widened with the new seats,” says Johnston. However, the extent of this impact is unclear as, given the LibDem collapse and the rise of UKIP and Greens in 2015, “there are fewer marginal seats than at any time since 1945.” Conservatives, especially, have many very safe seats. Still their path to a majority in the House of Commons looks better for 2020 under the constituency revisions than was the case in 2015, when the manufacturing of their majority by the FPTP system was a close call.



Alberta election follow-up

The ThreeHundredEight blog has a follow-up on the recent Alberta election. Key point of interest:

The idea that the PCs and Wildrose share the same voter pool is simply wrong. The right wasn’t divided. Rather, the anti-PC vote was divided between the New Democrats and Wildrose.

I suspect this is a more general phenomenon: when it appears that some party won due to divisions in its main opponent’s base, things are quite likely not as they appear.

UK 2015 and Duverger’s Law

Before the election, I said that it was premature to declare Duverger’s Law dead. With an apparent late swing, relative to what opinion polls were showing, in English votes from Labour and especially from Liberal Democrat to Conservative, I am going to say that my “prediction” was not the worst one on this election!

The swing from Labour might be interpreted as nationally focused voting–“which government would I prefer?”. English voters certainly seem to have recoiled from the idea of a Labour government dependent upon keeping the Scottish National Party content. The swing from Liberal Democrats was far greater than had been anticipated, and looks like the district-level “psychological effect” working as anticipated. The party generally benefits more than others from incumbency, given its incumbents’ reputations as good constituency MPs. In this sense, the Lib Dems could be thought of as the party that made FPTP “work”–it is supposed to be a system in which local representation matters, after all. In this election, however, it seems the LibDems suffered major desertion even in districts where they were up against Conservatives. (Defection to Labour from Lib Dem was widely expected, ever since they entered the coalition in 2010.)

In making the case for Duverger’s Law not being dead yet–even if it has been on life support in the UK for some time–I suggested that this election would be more “top two” than the last one, at least in England. That looks like a good call. The following table shows the percentage of votes and seats for Conservative + Labour in 2010 and 2015 in the UK as a whole, and in England.

UK UK England England
votes seats votes seats
2010 65.1 86.8 67.7 91.7
2015 67.3 86.7 72.6 98.5

Even with Scotland included, there was a slight increase in the top-two percentage, back up over two thirds (which is still low for a “classic” FPTP system!). In England alone, it was more dramatic, and the collapse of the Lib Dems, plus the failure of either UKIP or the Greens to win more than a seat apiece despite major growth in votes, sure is noticeable in the top-two seat percentage. The UKIP vote in England was 14.1% in this election, compared to 3.5% in 2010. The Greens won 4.2%, up from 1.0%. The mechanical effect is especially alive and kicking!

The increase of these two small parties’ votes obviously cuts against the notion of a Duverger’s Law rebound. Yet in spite of their increases, we still see an almost five percentage-point increase in the Labour + Conservative percentage, thanks to the major third party having collapsed to fourth place: the Lib Dems, in England, fell from 24.2% to 8.2%! Their seats–still England only here–fell from 43 to 6 (57 to 8 in the UK as a whole).

So, yes, overall a more Duvergerian result. But let’s not overstate it. UK-wide, the effective number of vote-earning parties (NV) went to 3.93. That is by far the highest it has been in the entire post-WWII era. The next highest were 3.71 (2010) and 3.60 (2005). So the trend as of 2015 remains upward, and fairly substantially so. The effective number of seat-winning parties (NS), on the other hand, did decline, although only a little bit: to 2.53 from 2.57. As I pointed out in the previous post, based on Taagepera’s Seat Product, the UK’s large assembly should lead us to expect NS=2.9. So it is still much more a “two-and-a-half” party system than expected–even with the massive SNP win (56 of Scotland’s 59 seats).

Speaking of the Scottish result, that near sweep certainly is a gift from the electoral system. The SNP’s 94.9% of the seats in Scotland comes on almost exactly half the votes. By contrast, the formerly dominant party in Scotland, Labour, in 2010 had 69.4% of the seats on 42.0% of the votes. In 2010, the SNP was a distant second in votes (19.9%) and an even more distant third in seats (6), compared to the Lib Dems (11 seats on 18.9%). What a difference a 30 percentage point swing can make under FPTP!

It is also noteworthy that the SNP received 1,454,436 votes in the parliamentary election. The YES side in the independence referendum last September obtained 1,617,989. Given the turnout differences, the YES vote was 44.7%, so just over five percentages point less than the SNP obtained in this week’s election. (In the last Scottish Parliament elections the SNP had 902,915 votes, which was 45.4%.)

Finally, below is a table of NS, NV, and the gap between them since 1945. The final column is an “expected NV” derived from NS, based on another (as yet unpublished) Taagepera formula that I will put below the table. The noteworthy thing is that we could expect the NV– NS gap to be around .4, given the actual  NS, values in UK elections. In most elections since the resurgence of the Liberals (in votes) in 1974 it has been above that. The gap has been more than 1 in each election since 1997, but surged all the way to 1.4 in 2015. This is extraordinarily high; in fact, I record a gap that high in only 31 of 517 parliamentary elections worldwide (details in the first comment below).

The 2015 result thus appears to confirm that there is demand for more party representation than the electoral system can deliver, but due to what we might call the “Duvergerian rebound”, I have to agree with Alan Renwick that the probability of electoral-system reform has gone down, rather than up, as appeared at least somewhat likely (despite many obstacles) if the result had been as anticipated in pre-election forecasts.

year Ns Nv Nv-Ns ‘Expected Nv’ from Ns
1945 2.12 2.58 0.46 2.56
1950 2.08 2.44 0.36 2.52
1951 2.05 2.13 0.08 2.49
1955 2.02 2.16 0.14 2.47
1959 1.99 2.28 0.29 2.44
1964 2.06 2.52 0.46 2.50
1966 2.02 2.42 0.4 2.47
1970 2.07 2.46 0.39 2.51
1974a 2.26 3.15 0.89 2.68
1974b 2.25 3.13 0.88 2.67
1979 2.15 2.87 0.72 2.58
1983 2.09 2.83 0.74 2.53
1987 2.17 2.85 0.68 2.60
1992 2.27 3.03 0.76 2.69
1997 2.12 3.23 1.11 2.56
2001 2.17 3.33 1.16 2.60
2005 2.46 3.6 1.14 2.87
2010 2.57 3.71 1.14 2.97
2015 2.53 3.93 1.4 2.93
mean 2.18 2.88 0.69 2.61
mean, 1983- 2.30 3.31 1.02 2.72
mean, 2005- 2.52 3.75 1.23 2.92

‘Expected Nv’ here is NV=(NS3/2 +1)2/3.

Trust me, it works amazingly well across hundreds of elections under different electoral systems. To see the derivation, you will have to wait for some forthcoming Shugart-Taagepera work!


PEI 2015

Thanks to Wilf Day for the reminder that there is also a provincial election in Prince Edward Island this week–today, in fact.

It will be another large manufactured majority, with Liberals currently on 18 seats with a bit over 41% of the votes. Conservatives 7 seats on 37%, and a Green MLA elected (11% province wide). The NDP looks to be shut out in seats, despite currently being just slightly ahead of the Greens in votes.

As for the one Green win, in Kellys Cross-Cumberland, it is not close. The Green candidate, Peter Bevan-Baker has 54.8%, and the Liberal trails far behind on 27.6%. I know nothing about Kellys Cross-Cumberland or Bevan-Baker, but I am always intrigued by constituencies where Greens win, given how rare they are.

Alberta 2015 election

5 May is provincial election day in Alberta. It really is hard to exaggerate just how tectonic a shift the result of the election result will be–if there is not some major polling error or late swing.

The Conservatives have ruled since the 1971, sometimes with massive majorities. They are now clearly in third place, according to the ThreeHundredEight projection. The NDP–yes, Canada’s most leftwing major party–is poised to win. The farther-right (almost tea-partyish for you south-of-the-border folks) Wildrose is in a clear second.

Yes, an NDP government in Alberta! One of the biggest oil-producing jurisdictions of North America, and the political base of the Canadian Conservative PM, Stephen Harper.

Wildrose led polls late in the 2012 campaign, but not by as much as NDP leads now. On the late collapse and the tea-party parallel, see a Weekly Standard article from just after the 2012 election. I like how that article says, “It’s as if the Tea Party movement here in the U.S. had been more of a real revolt, with tired taxpayers and fiscal conservatives fed up with overspending by Republicans and Democrats alike forming their own party.” Yes, just like that–and then the voters saying, forget both the old and this ‘new’ right, and while we are at it, forget the old center-left, too; let’s give social democracy a try!

There has been a discussion in the comment thread to the post about the 2012 election that has been going on for a while now. But it seemed about time to start a new one related to this election.

Does UK 2015 mean the death knell for Duverger’s Law?

Patrick Dunleavy, writing at the LSE blog, says that the “UK’s current general election looks certain to put another nail in the coffin of the most famous proposition in political science – Duverger’s Law.” This “law” is, of course, that the plurality rule in single-seat constituencies (FPTP) tends to result in a two-party system. With projections generally indicating that the largest UK party will not win much more than 35% of the vote and 45% of the seats, it does indeed look bad for our famous law.

On closer inspection, however, the UK has been drifting away from the law’s expectations for some time now, and yet even this election does not look like it will mean a break with the “law”. Assessing the performance of the law as a set of expectations for election results depends on just how we define what the “Duvergerian” norm should be, so let’s look at a few different standards and how the UK stacks up, both over time and in this current contest.

1. It is not exactly news that the UK voters and party elites have been less than conformist to the law. Consider the following graph.


The graph plots the effective number of parties, which is the standard way that electoral-systems specialists measure the fragmentation of a party system–it gives us a size-weighted measure of the party system, where each party’s weight is its own vote or seat share.* We can calculate the effective number on either votes (NV, the red line) or seats (NS, in blue).

We see that since 1945, and especially since 1974, there has been a mostly steady upward trend in the effective number of vote-winning parties (NV). Only in two elections in the 1950s was NV under 2.25, and in every election since 1992, NV has been greater than three. If Duverger’s Law means that the effective number of vote-winning parties should not be three or higher, then the law was broken in the UK some time ago.

Of course, maybe Duverger’s Law does not require the effective number of vote-winning parties to be less than three. After all, Duverger did say in his original presentation of what he said was closest to a true sociological law of all the propositions in his book Political Parties, that the national effect meant a two-party system in the legislature. He also said that the impact on the voter (which he called the “psychological effect”) was most felt at the district level.

The graph above shows that the effective number of parties in the legislature (NS) has indeed lagged well behind Nv. In fact, it had never been much above 2.25 until 2005 or above 2.5 until 2010. So we could say Duverger’s “mechanical effect”–the electoral system punishing parties other than the major ones–has continued to work even in the face of fragmentation of the vote. I would point out, however, in response to Dunleavy’s piece, that if NS creeping up above 2.5 signals an end to Duverger’s Law, that is nothing new in 2015.

What about the district level? The next graph compares NV in the average district to the national value of NV. (Note that NS is always one at the district level in such a system!)

It is not surprising that NV is higher nationality than in an average constituency. After all, some parties are more present in some parts of the country than others, and some seats are safe for one or the other (which results in a sharply lower Nv for such districts than for the nation as a whole). Still, district-level mean Nv has been above 2.5 in every election since 1983, and reached almost 3.0 in 2005, before actually tapering back just a bit in 2010 (despite that being an election with no majority party in seats at the national level). So, either Duverger’s Law has been on something like life support for decades, even at the district level, or at any rate, 2015 is not newly non-conformist to the law.

2. This election will actually be somewhat more “Duvergerian” than 2010 was. I mean in a more qualitative sense, as in “how top-two is the election?”, as opposed to the perhaps overly blunt instrument of a single indicator like the effective number of parties.

The big threat to Duverger’s Law is not the presence of regional parties like the Scottish Nationalists (SNP), who will do smashingly well in this election. Duverger explicitly recognized that regional parties were favored by the FPTP system. What is a threat to the claims of the law being in effect for a given FPTP country is a persistent third national party. In 2010, as in 2005 (and some earlier elections), there was such a party in the form of the Liberal Democrats (and their predecessors). However, with the LibDems’ impending collapse in this election, the top two parties–Labour and Conservative–are going to be much more dominant over the rest than they were in 2010. We should see each of these parties on 40-45% of the seats, and no UK-wide party with even 5% of seats. (Two non-regional parties could have 8-15% each of the votes, but the mechanical effect will squash them down in seats, as expected.)

What about the district level? I will venture that here, too, the mean district effective number of vote-earning parties (NV) will drop, compared to 2010. This is mostly due to the collapse of the LibDems, but also due to strategic/tactical voting in English constituencies where voters concerned about the close national contest between Labour and Conservatives lend their vote to whichever party is best placed in their constituency to defeat their least-liked national party. (Such trends are evident in some of Lord Ashcroft’s polls of marginal seats.) As for Scotland, most districts should be much more two-way fights between SNP and Labour, and a few LibDem-SNP, than was the case in 2010.

The 2015 election thus will have more of a top-two flavor than 2010 in both the UK as a whole and in Scotland.

3. The UK is actually under-fragmented, according to the most fully developed quantitative model of the “Duvergerian Agenda”, by which I mean Taagepera’s (2007) Seat Product Model (SPM). This says that the effective number of seat-winnng parties (NS) at the national level tends to be about the sixth root of the Seat Product, defined as mean district magnitude (M) times assembly size (S):

NS = (MS)1/6.

In both Taagepera’s own work and follow-up work I am doing, we find that the SPM is an excellent predictor of seat fragmentation in countries with various “simple” electoral systems (a category that includes variable-magnitude districted PR, single-district PR–and FPTP). For the UK, with M=1 and S=650, the models says that we should expect NS=2.9 in the UK. Note: 2.9, as in almost THREE, not TWO.

The graph below shows how well the SPM works on just those countries that use FPTP. Each country’s long-term average is shown with its abbreviation at the plotted position. The red line is the expectation of the Seat Product Model.


Apparently it works pretty well, overall.** However, the UK really stands out as one of the big outliers! Its average NS is much too low for its very large assembly, despite the single-member district system. In fact, it is more of an outlier than India, on the high side! (It is not, however, as egregious a low-outlier as the US.)

What about 2015’s likely result? Using the Election Forecast projections, the effective number of seat-winning parties would be around 2.6 in this election. So it is creeping upwards towards its expected 2.9, but it still won’t be there. In other words, the UK’s likely 2015 result still remains substantially more top-two dominated than we should expect, given its electoral system–when we include the size of parliament in our definition of “electoral system”, as we should.

In summary, it is hard to sustain the claim that the 2015 election is some sort of final nail in the coffin for Duverger’s Law. We looked here at several ways of defining what a “conforming” election result might look like. In some ways, the UK has been deviating for some time: the effective number of vote-earning parties has long been too high to really be a “two-party system”. While the effective number of seat-winning parties has been lower (as expected, due to the “mechanical effect”), it has crept up in recent elections, meaning 2015 is not really new. Even at the district level, it is not as if we have had consistent two-party dominance in recent times. Moreover, this election result will be more focused around the top two nationally than were the last two elections, and this pattern should be true of the district level as well as nationally. Finally, the UK’s electoral system actually should be able to accommodate an effective number of parties a good deal higher than it will in 2015 (or has in preceding elections), according to the most detailed quantitative model yet of the Duvergerian effect of electoral systems.

Duverger’s Law may be on life support, but it actually has been for some time, and it is not dead yet.

* For those unfamiliar with the index, a quick summary of what it does: It offers a summary of how many hypothetical equal-sized parties would be just as fragmented as the actual constellation of unequally sized parties. Thus if there are three parties, each with a third of the vote, we get NS=3.00. But if one of them is bigger than the others, the index will trend downwards towards two. If one of them splits in half (so the party shares become 1/3, 1/3, 1/6, 1/6) we now have four parties, but as they are unequal, the effective number will be less than four (but more than three–the actual value being around 3.6). It is computed by squaring each party vote or seat share, summing the squares, and taking the reciprocal of the sum.

** Sri Lanka and New Zealand are shown here only in their FPTP eras.

UK 2015 forecasts

With the UK general election three weeks away, Chris Hanretty of the electionforecast.co.uk team offers a comparison of their forecast with those of two other academic teams, ElectionsEtc.com and Polling Observatory.

The forecasts must estimate a nationwide vote share for each party, and then devise a means of projecting these figures on to the 650 individual plurality (FPTP) contests that make up a UK House of Commons election. Given these two stages for any forecast, Chris’s comparison includes running their own seat calculator on other projections’ vote shares.

One thing all the projections agree on is that no party will be close to half the seats, although the confidence interval on the ElectionsEtc and ElectionForecast figures for the Tories include the majority mark (barely). All agree that Labour has almost no chance of winning a majority.  Another point of agreement is that the leading party in votes will be the Conservatives, but on only 34%, with Labour just a percentage point or two behind. In each projection, the confidence intervals on vote shares overlap. So, yes, the race remains more or less tied.

For those of us who enjoy anomaly watches, the PollingObservatory forecast (as of 1 April) has Labour ahead in seats, 276-271, despite being just behind in votes. The other two put Conservatives ahead on seats as well as votes, although again with overlapping confidence intervals.

On the eve of the 2010 elections, I ran some numbers from FPTP elections around the world up to that time, asking how uncommon it was for the largest party have less than 36% and the third party have more than 25%. The answer, in my sample of 210 FPTP elections, was one (Nova Scotia 1998). There was at least one other that was missing in my data, Quebec 2007, where all three leading parties were within a range of 33.1 to 28.3. The question arose, because it was the consensus of the final polls that the UK was going to have such an election. However, it did not quite get there. The Conservatives made it just over 36% and the third-place Liberal Democrats slipped to 23.0. In other words, there may have been some last minute tactical (strategic) voting by just enough voters to make the result just a little more “Duvergerian”.

If we think of Duverger’s Law as a baseline expectation for an electoral system like the UK, the projections for this election might be said to conform, if all one cares about is the relative dominance of the top two over the rest. Thanks to the fall of the Liberal Democrats since entering the coalition in 2010, all projections agree that there will be a big gap between second and third. Of course, the top two on around 33% each is not exactly what we normally think of as a Duvergerian outcome, regardless of the gap between second and third. The interesting thing to watch is if there is sufficient late desertion of the UKIP (on 10-13%) and Greens (less consequentially, as they are under 5%) to push the leading party up over 36% again. If that is the Conservatives, there might be an outside chance of a majority government, depending on the constituencies where such tactical voting takes place. UKIP leader, Nigel Farage, recently called on supporters to vote Conservative in seats UKIP can’t win, and there is some evidence of UKIP slippage in recent Ashcroft polls of marginal constituencies.

For now, let’s suppose that the seat projections are about right. Are there any combinations of two (other than Labour-Tory) that would control a majority? The House contains 650 members, but Sinn Fein (of Northern Ireland) does not take the seats they win, which were five in 2010. That makes the majority threshold probably at 323. A combine of Labour and Scottish National Party (SNP) gets there, according to Polling Observatory–barely: 325 (or 324 if using ElectionForecast’s seat model).

How likely is Labour-SNP cooperation? Nicola Sturgeon, SNP leader, has been appealing for it–not for a formal coalition, but for blocking a Tory government. An exchange in the final debate (opposition parties only; video) was telling. Labour leader Ed Miliband forcefully refused, saying he had fundamental disagreements with the SNP, mainly over the possibility of a second Scottish independence referendum. He said no to a “coalition”. However, he pointedly did not directly say there could be no cooperation short of a governing coalition, although he shook his head dramatically when Sturgeon appealed for him to join her to block the Tories. He is in a bind, for sure. Sturgeon is probably right when she says left-leaning voters would never forgive him if he refused cooperation and allowed a Tory minority government to form. On the other hand, he certainly has to be careful not to signal intention of working with a party that would break up the UK.

A path to a stable Conservative-led minority or coalition government is hard to see, on these projected numbers. However, two of the projections have Conservatives and Liberal Democrats at 311. I find it hard to believe the Liberal Democrats would enter a coalition again with the Tories, but less hard to believe they could enter a looser arrangement. However, such a combination would still be a around a dozen seats short, and even the Northern Ireland Democratic Unionists (not in the projection models, but 8 seats in 2010) would not quite get them there.

Absent a late surge for the Tories, it still looks like a government led by the party with the second largest number of seats, and needing support from Scottish separatists, is the most likely result. Let me close with an understatement: This will be interesting.

Delhi assembly: Massive AAP win

This past Saturday voters in Dehli went to the polls to elect a replacement assembly for the one that was dissolved following the resignation of the 49-day minority government of Chief Minister Arvind Kejriwal.

While polls had generally foreseen a win by Kejriwal’s AAP (Common Man Party), none saw the massive win that the party obtained: 67 of the 70 seats, with the BJP (ruling party at the federal level since 2014) winning just 3. The Congress Party was shut out. The top three parties vote percentages were 54.2, 32.2, 9.7.

This is a stunning surge for the AAP. Its first-ever contest was for the Delhi election of December, 2013, in which it won 28 seats, second to the BJP’s 31 (Congress had 8). AAP formed a minority government, but then resigned when it could not get its signature anti-corruption bill through the assembly. In the federal election of May, 2014, the AAP flopped miserably, winning a third of the votes in Delhi but no seats. (It ran over 400 candidates across several sates, but managed just 4 seats, all in Punjab.) And now it has a solid majority of the vote and a near-sweep of the Delhi assembly’s seats.

As might be expected, some of the AAP’s changing fortunes comes down to the dynamics of first-past-the-post voting with multiple parties. The AAP actually won a slightly higher percentage of the vote in Delhi for the federal parliament in 2014 than it had won in the assembly election of 2013. The biggest source of new AAP votes clearly comes from the collapse of Congress from 24.6% in 2013 to less than ten percent this time. This swing is largely due to the Muslim community deserting the sinking ship that is Congress to block the Hindu-nationalist BJP. By contrast, when the BJP won a plurality of Delhi’s assembly constituencies in 2013, it did so with around a third of the vote, or roughly the same as in this latest assembly election. Evidently, it was the BJP’s sweep of Delhi’s seven federal constituencies in 2014 that was the aberration (46.4% of the vote to AAP’s 32.9 and Congress’s 15.1). The “Modi wave” did not carry over into the sub-national contest, and a third of the vote looks a whole lot worse when support for the the third party has collapsed.

Either Kejriwal is incredibly lucky, or he is the canniest politician in India, and had this all planned out from the start.

Botswana election, 2014

I’ve long been skeptical of Botswana’s classification as democratic/free by standard datasets we political scientists rely on. Based on a pre-election report by Amy Poteete for The Monkey Cage, skepticism seems justified. Mysterious accidents and other odd events involving political rivals, increasing partisan use of state assets…

The Botswana Democratic Party (BDP), the only ruling party Botswana has ever known, is feeling the heat, and is experiencing internal tension, including cases of losers in candidate selection running against the party’s official choice.

In the 2009 election, the BDP won over three fourths of the seats* on just 53% of votes (FPTP). That’s some serious disproportionality, but a result like that reveals electoral precariousness.

Using the dataset of the Constituency Level Electoral Archive, I took a look at just how many seats the BDP won only narrowly in 2009. In that election, its median margin over the runner-up was .188 of the district’s valid vote. It would need to lose at least 15 seats to fail to retain its majority. The 15th most marginal seat was won by .116. (Five were under .05.) There were 17 seats that it won with vote shares below that of its nationwide share of .533, including seven won with under half the vote. Thus the party looks somewhat vulnerable if there is a modest swing against it or if the defecting candidates are in close districts and take some chunk of the BDP vote with them.

Nonetheless, the BDP in 2009 did not face a single opponent in many districts; of the 17 districts where its vote was below the nationwide share, it won with a median margin of .081. Thus unless its opposition is more coordinated in 2014, the BDP will probably hang on even in the face of an adverse swing.

The election is on 24 October.

* The data I am working with show 44 seats, which would be 77.2%, while Poteete says 79%.

New Brunswick election 2014

The Canadian province of New Brunswick held a general election on 22 September. Notwithstanding some problems with the vote-tabulation system, and several lead changes during the night, the Liberals won by a good margin: 42.7% of the votes against 34.7% for the Progressive Conservatives (PC). The seats split 27-21, giving the Liberals 55.1% of the seats and an advantage ratio (%seats/%votes) of 1.29. This is not particularly remarkable by the standards of plurality (FPTP) systems, and is nothing compared to some significant anomalies the province has experienced in the past (see previous posts).

The Green Party won a seat. It had 6.6% of the vote province-wide. Its one victory was in the riding (district) of Fredericton South, where it won 31% of the vote, against 26% for the PC, 22% for the Liberal, and 20% for the NDP. Not even a third of the vote–sometimes plurality is good for small parties. But not as good, obviously, as the proportional system (of the MMP type) that was formally proposed but for which the planned referendum was cancelled–by a party that had just won on a plurality reversal. (I said NB elections had been anomalous!)

A “populist” party known as the Peoples Alliance elected no one on its 2.14% of the vote, but it did miss in one riding by a mere 27 votes. The NDP won no seats despite 13% of the vote. I am not sure how closely it missed in key ridings. Of course, it is hardly unheard of for a third party to win no seats under plurality rules despite such a substantial vote total. Nor is it unusual for a fourth party to win a seat despite having half the votes such a third party. It is a disproportional system, especially given the small assembly size, and regional distributions of support are critical. It is hard to argue against the proposition that New Brunswick should dust off that old proposal for a new electoral system.