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.

36 thoughts on “Does UK 2015 mean the death knell for Duverger’s Law?

  1. Note how, in the third graph, Canada’s long-term average is almost precisely what the Seat Product Model expects. This is striking, given how often in the political-science literature it is alleged that Canada is an exception to Duverger’s Law.

    I have a district-level graph for Canada like the one in this post for the UK. They are almost identical, although Canada’s district-level mean tends to be a little higher than the UK’s.

    In other words, both countries are about equal “exceptions” at the district level, whereas UK is a fairly egregious exception to the national-level seat expectation (in a low direction) but Canada nails the prediction.

  2. And yet, it is still being parroted as the reason there is no third party at all in the US (eg: whereas there are obviously other factors involved in making the US such a purely two-party system.

    • Yes, exactly. Someone should post a comment at that blog regarding the overly facile application of electoral-system theory to the US party system.

  3. NZ look like it was the middle of the pack during the FPTP era. How egregious is the U.S FPTP electoral system is? Is it disproportionate? It is too soon to say for the UK as the UKIP will most likely do very well and be the third largest party in votes.

    Could a FPTP jurisdiction embrace Australian style preferential voting? David Cameron will regret that he campaign against the Alternative Vote system as it is probably for the Conservatives the least worst alternative to PR.

    Could voters in a FPTP jurisdiction with a multi-party system voting strategically end up waking up shocked that the translation of votes is perverse?

    • New Zealand’s value of effective number of seat-winning parties (Ns) was on average over the post-WWII era just about where it should have been, for its size assembly. However, the effective number of vote-earning parties (Nv) had been rising steadily. This, along with the two consecutive spurious majorities, is a major part of the story as to why proportional representation was adopted.

  4. 1. “Duverger explicitly recognized that regional parties were favoured by the FPTP system” (MSS)
    True, which is why George Wallace not Henry Wallace was a potential Electoral College kingmaker. However:
    (a) “Regional” is a question of degree. In the Sixties and early Seventies, before the rise (or rebirth) of the SNP, a number of British political scientists considered the (then) Liberals as something of a “regional” party, being confined to the Celtic fringes.
    To some extent, any third or fourth party that manages to win seats under FPTP has to be “regional,” so the qualifier could be tautologous to the point of uselessness. One could devise a more precise metric, eg, how contiguous are the districts won by the “regional” party? If the SNP wins a large bloc in Scotland and none anywhere else in the UK, or the Bloc Quebecois sweeps most seats in Quebec while also winning zero in the ROC, this is more clearly “regional” than the Australian Greens winning one district in central Sydney, one in central Melbourne, one in Newcastle, one in Hobart, and one in far northern New South Wales.
    (b) It’s not FPTP per se but using separate districts (either single-member or small multimember) that favours regional parties. (Political scientists often conflate “FPTP” and “single-member districts,” despite the existence of the French National Assembly, the Australian House of Representatives, and the US Electoral College. Please, people: “winner-take-all,” “semi-proportional,” “proportional” and “supra-proportional” are the correct taxonomy. [*]
    [* “Semi-proportional” systems allow for the possibility that a party, other than that with the most votes on the final count, might win seats. “Proportional” systems allocate each party over a certain minimal threshold one seat per whole quota ¬- however defined ¬- or remainder over a specified percentage of a quota. “Supra-proportional” systems are even more generous, eg the Afrikaners’ proposal for a South African Senate in which every party polling over 5% of the votes would be allocated an equal number of seats: I don’t know of any legislatures that use a supra-proportional formula among parties ¬- among federal Member States is, of course, common ¬- but it is sometimes used for legislative committees and administrative commissions].
    This usually restricts the effect of the “regional parties” exception to legislative elections with districts and/or a non-winner-take-all voting system (ie, most national and state legislative bodies other than the Philippines Senate), although the exception can apply to direct executive elections if some kind of districting ¬- whether live electors or an automatic “unit rule” ¬- is used. (The US Presidency is the last remaining example I know of, but previous analogies were used in the past for the Presidency of Argentina and the Governorships of various US States.)
    2. Respect to M Duverger but I don’t think FPTP always and everywhere deters small parties. It does so only where the “top two” in the district are far away ahead of the third, fourth and lower contenders. (And relatively close, too, I suppose. If you were a Conservative supporter in a constituency where the SNP is sitting on 60% and Labour on 35%, you may as well vote your conscience for the Tory candidate on 5% ¬- you aren’t “throwing your vote away” because tactical voting wouldn’t have produced any result higher-up your scale of preference).
    I prefer my analogy that FPTP’s threshold is like a brittle glass window while more majoritarian systems (runoff, AV, Approval) are like a thick padded sponge. Once the window is smashed, anyone can get through it. Examples are US primary elections and Papua New Guinea ¬- both, interestingly, without strong party or factional mechanisms for making binding preselections. If there are four or five strong candidates already in the ring, the winner might get in with only 25%-30%. Suddenly your likely 15% support, which would be derisory if you were facing two contenders on 40-45% each, looks like a possible steppingstone to vault you to a plurality.
    3. I may be missing something, and doubtless Alan will correct me if that’s so, but surely “the number of effective parties = number of seats plus one” only applies when a single vote (or more accurately, a fully-cumulated vote) system ¬- PR-List, PR-STV, SNTV, even (if we’re talking single-seat districts) FPTP ¬- is used.
    Where a winner-take-all system is used (ie, multiple non-cumulable votes), with strongly partisan voting, Duverger’s Law should still apply, even in multiple-seat districts. The US Electoral College has nearly ten seats per district but certainly doesn’t have eleven effective competing parties. The Australian Senate pre-PR (1901-46) was usually divided (18-18, 21-15, 24-12, and once even 33-3) between Labor and the conservative coalition. There were not four effective parties during the era of three-seat MNTV.
    4. Another factor muddying Duverger’s Law is the strength of ideology. If supporters of the party running a (distant) third or lower are pragmatic, then yes, they should switch votes tactically to help the lesser evil defeat the greater evil. But if they’re very ideologically zealous, or extremely angry (eg Perotistas in 1992), they won’t “hold their noses” and vote tactically. Ironically, this may occur either because they are on the extreme Left or the extreme Right (“the Big Two are both equally tools of capitalism/ socialism”) ¬- or because they are virtuous centrists who don’t want to pick sides. (The Australian Democrats used to issue two-sided How-to-Vote tickets, so they wouldn’t be seen as favouring the ALP over the Coalition or vice versa ¬- ironically this meant that, after two decades of Australians expecting DLP preferences to help defeat the ALP despite the latter consistently polling a plurality, circa 1980 opinion polls again meant that the party with more first preferences was probably likely to win. Likewise, some anecdotal evidence from British observers of 1983 and 1987 suggested that, while Alliance voters detested Thatcher, they also disliked Michael Foot-era Labour nearly as intensely, and may well have exhausted their ballots ¬- or even preferenced the wetter Tory candidates ¬- had Britain used a preferential system of voting.
    Related to intensely-held ideology is ethnicity or religion where these are politically salient factors. If you vote for XYZ Party because everyone in your tribe does so, then voting for a different party becomes a betrayal of your family and your ancestors, even if XYZ Party is running a distant third in the opinion polls. (See: Ireland, Northern: India).
    5. Finally, the effectiveness of FPTP in scaring away voters from “wasting” their votes on distant-third candidates depends a lot on how accurate the opinion-polling methods are. By all accounts, many people who voted or campaigned for the British Lib/ SDP in 1983, or the Perot ticket in 1992, seriously thought their favourites might break through to first or second place, or at least win enough seats in the executive-choosing body to play kingmakers. It is possible they were fooled by looking at popular votes only, because 27% and 18% are very respectable totals. The Alliance very nearly came second UK-wide, and Perot did run second in Maine. Remember David Owen telling SDP activists in 1982 “go back to your constituencies and prepare for government”? If two or more parties on the same side of the spectrum think they’ve got a chance to win a plurality, neither is going to stand down. (I suspect this was a factor in the US 1912 election also: Taft and Teddy Roosevelt both had plausible grounds — far more plausible than Ross Perot had in 1992, for example — to think they could win the Presidency, and to hell with the idea of standing aside for the other).
    The above comments are all aimed at the “first-past-the-post voting produces fewer viable parties” interpretation of Duverger. The variant — “single-member districts produce fewer viable parties” does have some force. “Your vote is wasted if you support the Greens/ UKIP instead of Labour/ the Conservatives” is a very powerful argument although not — as, I hope, I’ve shown above — an absolute deal-breaker. On the other hand, “If you support the Greens/ UKIP, you probably won’t elect one of them, but then you can transfer your vote to help elect a Labour/ Conservative candidate instead” is much less of a deterrent to sincere voting.

    • Regarding point #3. The effective number of parties can’t be approximated as M+1 (effective Nv=121 in Israel???). That’s why the Seat Product Model is more effective (so to speak).

      And please note: it is the number, not (necessarily) any given party, that is ‘effective’.

    • Also, on “regional parties”, it certainly is a concept that needs better articulation. In FPTP systems, in particular, parties may enter many places but win only in some–such as the UK Liberals for many years, as Tom notes. Is that a regional or national party?

      Duverger made the following observation in his classic piece: “For a new party to succeed in establishing itself firmly it must have at its disposal strong backing locally or great and powerful organization nationally.”* The “or” here is important. Obviously, the LibDems and predecessors have been a national party of longstanding whose strength outside of regional strongholds has ebbed and flowed. On the other hand, UKIP and Greens have pretense of being national parties, but are weakly organized except for a few constituencies, whereas SNP is also a new (relative to Liberals) party that has strong, and strengthening, local/regional backing and does not run outside of its region.

      Thus I am using “regional parties” here to refer to those that run only in some defined sub-part of the wider polity. The SNP clearly is a regional party, as is Plaid Cymru. But not UKIP or LibDems.

      There is not really a single accepted definition, and research in the area needs to begin with a clear articulation of what the concept is, and it could vary with a given research endeavor. But for purposes of the points I was making here, it is not too hard to define which are regional and which are not.

      * p. 108 of the reprint of Maurice Duverger, “The Number of Parties”, in Electoral Systems, ed. by David Farrell and Matthew S. Shugart.

      • MSS, good point. Intention/ aims vs outcomes/ success. Maybe some metric could be developed that combines…
        (a) in what proportion of districts overall does a “regional” party stand candidates (eg, Northern Ireland vs ROTUK being a paradigm clean read on this)
        (b) in what proportion of those districts does it win seats, and
        (c) how geographically close and contiguous are the above set and subset?
        The waters can however be muddied by:
        (d) socio-economic factors that recur in specific types of “regions” even though widely dispersed, eg the National Party is more likely to win seats in both far northern Queensland and far western Victoria than in the urban strips in between. Likewise Liberal [Democrat] MPs sitting for constituencies in the far south of Cornwall or Wales and the far north of Scotland are quite widely dispersed by British standards, even though they can be classed as “Celtic fringe”
        (c) how different a regional affiliate of a national party “really” is. Eg,
        * the Liberal National Party in Queensland is (for Commonwealth Electoral Act financial purposes) counted as the Qld division of the Liberal Party of Australia. Its federal MHRs and Senators are listed as “LNP” on the ballot paper but then decide individually whether they sit with the Liberal or the National caucus when they reach Canberra
        * the Bavarians seem quite insistent that the CSU is a whole separate party from the CDU, even though in federal elections their Zweitestimmer are pooled together as one nationwide total
        * But on the other hand, despite a rich history etc, I don’t know of many US political scientists who treat the Minnesota Democratic-Farmer-Labor Party these days as much more than the local branch of the national Democratic Party. Minnesota’s not treated as some kind of special case in federal elections on this account. Likewise I believe there have still in recent decades been some Scottish MPs elected as “Unionists” who nonetheless sit at Westminster as “Conservatives” for all intents and purposes.
        Relating this back to Duverger, what pol scientists care about is whether fluctuations in a party’s regional strength are such that it’s a viable contender under FPTP in some districts but a “joke”, “vote-wasting” minor party elsewhere.

      • Tom Round “the CSU is a whole separate party from the CDU, even though in federal elections their Zweitestimmer are pooled together as one nationwide total” No they’re not, their list votes are tabulated separately and their seats are allocated separately.

      • ‘Sectional party’ for parties that are local to a section rather than a region, perhaps? I’m thinking of the Nationals in Australia who basically run in rural and regional areas in NSW, Victoria and WA. There is a long standing campaign for a state of New England to be formed in northern New South Wales which would probably be a National lock, at least for a time.

    • The Afrikaner proposal is in fact not proportional at all. ‘supra-proportonal’ is as misleading as calling Greece ‘reinforced proportional’.

      • Consociational wouldn’t be a bad term for it. Certainly more fitting than when Lijphart called the Netherlands consociational. Also, isn’t it very similar to what was used in Bolivia for their constitutional convention, and to the Phillippines’ ‘party list’ tier?

      • Equal representation of all co-operating parties, is actually not uncommon as a once only way to constitute a transitional assembly. The difference in the NP proposal in South Africa was that it was to be permanent.

        The NP was extraordinarily productive at setting up proposals to hold universal suffrage elections that would have zero effect on the formation, legislation or policy of the government. The tricameral parliament was another example.

      • reinforced proportional representation seems as it means as if the proportionality is to be as if it were perfect as possible to proportional representation. Greece should call it plurality winner bonus at large proportional representation; but that seems awkward.

      • I don’t think the two are really very comparable, Alan. Under the tricameral parliament, however, the vast majority of people were still completely unrepresented, as blacks did not have their own house, and even if they did the president’s council would have been able to exclude them on all major decisions. Under the NP proposal, the lower house would still have been elected by PR, as it is today. if properly enforced*, it would simply have forced the ANC to work with other parties to pass laws in the upper house – so not unlike in Australia**. If represented by enough parties (easy enough to pull off, I think), blacks would undoubtedly have been able to gain a majority in both houses.

        *So assuming parties weren’t able to set up multiple lists, thereby undermining the system.
        **Except, of course, that Australia’s parliament is not House-proportional and Senate-consociational but H-majoritarian and S-proportional. But I expect it would look similar in terms of effective number of parties per house.

      • JD the NP position was (1) any party with more than 5% would qualify to appoint a deputy president who would exercise an unreviewable veto and (2) the presidency would rotate at regular intervals among the deputy presidents. Under that regime the only significant number would be the number of parties who qualified for a deputy presidency. Actual seats in the national assembly would be pretty much irrelevant because any qualifying party could veto any law.

        The NP justified the absence of a black house in the tricameral parliament by arguing that blacks were represented in the Bantustan legislatures. As you note, the tricameral legislative process could be effectively suspended whenever the presidential council so decided.

      • The problem with the NP proposal would be that the ANC could just run multiple lists/dummy parties, and win a majority. The NP clearly were struggling with the problem of how to ensure power for minority groups in a country where the largest party would win under any electoral system, and this rather poorly thought through system was the outcome.

      • Well, I suppose there must have been different proposals on the table at different points in time during the negotiations. My understanding, like Tom, was that there was a proposal concerning a consociationally-elected senate, and that’s what I was commenting on.

      • It is probably a thought for another thread, and as fas as I know none of the negotiators in ZA proposed a directly elected president, but I wonder if the parliamentary presidency was a really good idea when the ANC was going to dominate the parliament into the foreseeable future.

      • JD

        It’s possible to imagine a consociational senate for South Africa, The electorates could be the groups that use the 11 official languages with provision for the 2% of the population who do not use an official language as their home language. It would be Penrose allocated. You might require a concurrent majority for important questions but I cannot really see a case for an individual group veto.

  5. “some anecdotal evidence from British observers of 1983 and 1987 suggested that, while Alliance voters detested Thatcher, they also disliked Michael Foot-era Labour nearly as intensely, and may well have exhausted their ballots ¬- or even preferenced the wetter Tory candidates ¬- had Britain used a preferential system of voting.” As I pointed out in another recent thread, projections of 1983 and 1987 had they been held under AV, using some polling from the time, indicate that the preconception that a vast majority of Alliance voters hated Thatcher more than Labour might have been very wrong: In both elections, it seems more Alliance voters would have ranked Conservatives second than Labour, and Thatcher would still have won a majority in those elections.

      • Thanks, JD. Maybe “disliked Thatcher but really detested Foot Labour” would be more accurate, then. I read somewhere (Anthony Sampson, perhaps) that the Alliance hoped that if they could deny the Tories an absolute majority, the latter would drop Thatcher and replace her as leader with a wet of the Sir Iain Gilmour stripe, whom the Liberals at least would have been more than happy to work with.
        Also, a lot of the Alliance rhetoric in the 1980s focused on “the British political system is broken” and since at that time it was the Tories who were the immediate beneficiaries of that system, this set them at odds with the Tories on that point. Had it been forty years earlier, it would have been Attlee Labour whom they would have been at odds with over this point.
        The New Statesman was strongly behind the Charter 88 campaign at this time which drew together a lot of Labour supporters (not necessarily more left-wing or right-wing over policy, but more willing to consider constitutional reform) with Alliance figures.

      • Ie, it’s not inconsistent to say “I prefer Margaret Thatcher over Michael Foot as prime minister but I want to see her with a lower house majority based on a 52% majority instead of a 43% plurality, and checked by a PR-elected upper house and a supreme court enforcing a bill of rights.”

  6. JD ¬- thanks for the CDU/ CSU correction. Is that a relatively recent change? I thought I read elsewhere in the 1980s, although it was one or another news report (TIME, perhaps), and we all know how precise those can be when it comes to PR systems.
    Apart from counting votes, comments on German elections treat the CDU and CSU as a unit in terms of seats. I can’t remember any speculation since the 1960s that the Bavarians might take their toys and leave the alliance, or vice versa by the ROG even when Strauss was the CDU/CSU candidate for Federal Chancellor.
    Alan ¬- “supra-proportional” meaning “more than proportional,” “overly generous to the smaller players than a proportional system would be,” but like “extra[-]ordinary,” it can imply the opposite. Also the numbers of seats need not be exactly equal and, indeed, in the Afrikaners’ proposal, couldn’t always have been equal. If a Province had, say, 10 Senate seats and 4 parties met the 5% threshold. I assume the top two would in that case have been allocated 3 seats each and the other two 2 seats each.
    The short-lived Senate of the Republic of South Vietnam worked on a similar basis.
    The experience of SNTV suggests that all that would have happened under the Afrikaner proposal is that the ANC would have run (say) five separate lists, divided its supporters among them so that each polled around 11-13%, and won a super-majority of seats that way anyway.
    Australia usually uses the equal-numbers principle between government and opposition MPs when federal and state parliaments are appointing delegations to a constitutional convention (1973 and 1998).

    • Most commentary on German elections and party behavior implies that the CDU and CSU are essentially one party, but not all.

      Strangely, however, even I had been under the impression that the CDU and CSU were permitted to pool their list votes at the national level for purposes of calculating overall proportionality. Re-reading pp. 213-14 of Thomas Saalfeld’s chapter in The Politics of Electoral Systems (ed. Michael Gallagher and Paul Mitchell, Oxford University Press, 2005), I see that it says quite clearly that the CSU is treated as a fully separate party in seat allocation.

      • I suppose with an assembly of 500-600 seats, using (most of the time) largest remainder allocation, there is no great incentive for the CSU to pool its votes nationwide with the CDU. Bavaria has about 15% of the German population and the CSU regularly polls absolute majorities of votes, so surmounting the nationwide 5% threshold is not a concern and, in any event, the CSU would certainly win at least three district seats to qualify for list seats in its own right.
        I think most of the early (1980s) books I read about Germany and/or electoral systems gave a single total of votes for the CDU/ CSU, but that would be just for convenience.

  7. “The New Republic takes the prize for perhaps the single worst argument for two-party politics in history: ”

    Considering that the argument includes discussing a secessionist party, the argument has merit. FPTP is vulnerable to secessionists, and having to deal with them is unpleasant. When such parties dominate their region, they hijack the politics of the entire country and consume all the political capital.

  8. MSS said: “Note how, in the third graph, Canada’s long-term average is almost precisely what the Seat Product Model expects. This is striking, given how often in the political-science literature it is alleged that Canada is an exception to Duverger’s Law.”

    It isn’t that striking since the 3rd graph is Taagepera, not Duverger.

    The graph includes quite a few legislatures that are so small that comparison is strained. Exclude MS<50 and what we see is everyone is below the line and right around 2, except Canada and the south asians. One with a civil war and another the most complicated democracy in the world. Canada then looks like an outlier, especially among the English-speaking countries (except of course, it's not just an English-speaking country, and that contributes to the effective # of parties). There's an interesting thought – Canada in some respects is more like South Asia than it is like the US or UK.

    It would be interesting to swap out the M*S dependent variable in favour of comparing district size, leg size/cube root of population, or effective mother tongue count. Given the US's undersized legislature, large district size and low language count, it may be less of an outlier for these variables. In other words, I wonder if these variables might help partially account for the US's two party dominance.

    Aside: Contrary to what many may think, the US has a relatively low mother tongue diversity compared to most countries.

    • Ross: Some points.

      Yes, the third graph is Taagepera, not Duverger, but Taagepera calls his Seat Product model the extension of the “Duvergerian agenda” in the literature on electoral-system effects. I agree.

      Small parliaments are not a problem for generating predictions of “Duverger’s Law”. In fact, a core principle of Taagepera is that the constraints of the small are a logical-modeling opportunity not to be thrown away.

      Swapping out S for the cube root is an interesting idea. We know from other work that most legislatures are close the cube root, so it would not change the overall picture. However, we also know that the UK has one of the most over-sized legislatures, relative to the cube root of population, whereas India has a quite under-sized one (as does the US).

      “Effective mother tongue count”. That is very intriguing!

  9. Pingback: UK 2015 and Duverger’s Law | Fruits and Votes

  10. Pingback: The UK 2017 result–Comparative data forays | Fruits and Votes

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