MMP as sub-category of two-tier PR–some basis for doubt

In yesterday’s review of the German election outcome, I used the extended Seat Product Model (SPM) formula for two-tier PR systems. I have done this many times, and Rein Taagepera and I (in our 2017 book, Votes from Seats) do explicitly include mixed-member proportional (MMP) in the category of two-tier PR systems.

However, there is one problem with that characterization. All other two-tier PR systems that I can think of entail a single vote, which is then used both for allocating seats in the basic tier and pooled across districts for national (or sometimes regional) compensation.

MMP, of course, usually entails two votes–a nominal (candidate) vote used only in the basic tier, and a second, party-list, vote used for determining overall proportionality. (In MMP, the basic tier is a “nominal tier” because the vote there is cast for a candidate, and the district winner earns the seat solely on votes cast for him or her by name.) This two-vote feature is a complex feature of MMP that is actually emphasized in my more recent coauthored book, Party Personnel Strategies, but which I may have tended to underplay in my comparative work on modeling the effects of electoral systems on party systems. Of course, by being two-tier, it is already a non-simple system, as Taagepera and I define that term. But we also say that two-tier PR, including MMP, is as simple as an electoral system can be and still be included in the complex category (see p. 263 and 299 of Votes from Seats).

Maybe that is not an accurate statement for two-vote MMP. Our definition of simple (pp. 31-36) concentrates on two features: (1) all seats allocated within districts, and (2) adherence to the rank-size principle, such that the largest party gets the first seat in a district, and remaining seats are allocated in a way that respects their relative sizes (i.e., by any of the common PR formulas). We further say that for simple PR, “the vote for candidate and for party is one act” (p. 35). This latter condition still holds for any two-tier list-PR system, because there is a list vote that applies both for allocating seats within a district, and also for the “complex” feature of the supra-district compensation mechanism. Obviously, however, MMP as used in Germany violates the principle that “the vote for candidate and for party is one act.” So maybe it is not “simple enough” to qualify as an almost-simple complex system. (Yes, that was a complex statement, but that’s kind of the point.)

If MMP were to tend to produce a party system more fragmented than expected from the extended SPM, it might be due to the “second” vote, i.e., the list vote. To test this, one could aggregate all the nominal votes and use them as the notional list votes in a simulated compensation. (This is how MMP in Germany worked in 1949, albeit with compensation only at state level. It is also how MMP now works in Lesotho.) The aggregation of basic-tier votes should work better from the standpoint of modeling the party system impact of the key features of a given MMP system–the size of the basic tier and the share of seats in the compensation tier.

The catch in all this is that, of course, till quite recently German MMP was under-fragmented, according to the SPM, despite using a separate list vote. Thus the issue did not arise. The New Zealand MMP system also has matched expectations well, after the first three post-reform elections were over-fragmented relative to model prediction. The graph below shows the relationship over time between the expectations of the SPM and the observed values of effective number of seat-winning parties (NS) in both Germany and New Zealand. For the latter country, it includes the pre-reform FPTP system. In the case of Germany, it plots NS alternately, with the CDU and CSU considered separately. As I noted in the previous discussion, I believe the “correct” procedure, for this purpose, is to count the “Union” as one party, but both are included here for the sake of transparency. In both panels, the dashed mostly horizontal line is the output of the extended SPM for the countries’ respective MMP systems1; it will change level only when the electoral system changes. (For New Zealand, the solid horizontal line is the expectation under the FPTP system in use before 1996.)

The German party system from 1953 through 2005 was clearly fitting quite poorly, due to how under-fragmented it was for the electoral system in use. The old CDU/CSU and SPD were just too strong and overwhelmed the considerable permissiveness of the electoral rules.2 So clearly the question I am raising here–whether the two-vote feature of MMP means it should not be modeled just like any (other) two-tier PR system–is moot for those years. However, perhaps it has become an issue in recent German elections, including 2021. The underlying feature of voter behavior pushing the actual NS to have risen to well above “expectation” would be the greater tendency of voters towards giving their two votes to different parties. At least that would be the cause in 2021, given that we saw in the previous post that the basic tier produced almost exactly the degree of fragmentation that the SPM says to expect. It is the compensation tier that pushed it above expectation, and the problem here (from a modeling perspective) is that the formula implicitly assumes the votes being used in the compensation mechanism are the same votes being cast and turned into seats in the basic (nominal) tier. But with two votes, they are not, and with more voters splitting tickets, the assumption becomes more and more untenable.

The previous planting on this matter emphasized that the SPM is actually performing well, even in this most recent, and quite fragmented, election. I am not trying to undermine that obviously crucial point! However, the marked rise in NS since 2009–excepting 2013 when the FDP failed to clear the threshold–may suggest that the model’s assumption that the two votes are pretty similar could be problematic.

Maybe two-vote MMP is more complex after all than its characterization as a two-tier PR system–the simplest form of complex electoral system–implies. In fact, maybe I should stop referring to MMP as a sub-category of two-tier PR. Yet for various reasons, it is a convenient way to conceptualize the system, and as yesterday’s discussion of the recent German election showed, it does work quite well nonetheless. It could be based on a flawed premise, however, and the more voters cast their nominal and list votes differently, the more that flaw becomes apparent.

A work in progress… in other words (fair warning), more such nerdy posts on this topic are likely coming.


1. The “expected NS” line for Germany takes the tier ratio to be 0.5, even though as I argued in the previous entry, we really should use the actual share of compensation seats in the final allocation. This would have only minimal impact in the elections before 2013; in 2021, it makes a difference in “expected” NS of 0.36.

2. Partly this is due to the 5% list-vote threshold, which is not a factor in the version of the SPM I am using. In Votes from Seats, we develop an alternate model based only on a legal threshold. For a 5% threshold, regardless of other features, it predicts NS=3.08. This would be somewhat better for much of the earlier period in Germany. In fact, from 1953 through 2002, mean observed NS=2.57. In the book we show that the SPM based only on mean district magnitude and assembly size–plus for two-tier PR, tier ratio–generally performs better than the threshold model even though the former ignores the impact of any legal threshold. This is not the place to get into why that might be, or why the threshold might have “worked” strongly to limit the party system in Germany for most of the postwar period, but the permissiveness of a large assembly and large compensation tier is having more impact in recent times. It is an interesting question, however! For New Zealand, either model actually works well for the simple reason that they just happen to arrive at almost identical predictions (3.08 vs. 3.00), and that for the entire MMP era so far, mean NS has been 3.14.

12 thoughts on “MMP as sub-category of two-tier PR–some basis for doubt

  1. Simple PR —



  2. If Germany wasn’t so generous in allocating overhang seats, would it’s MMP system mutate to a MMM system or MMM with partial proportionality?

      • Well, Germany isn’t ‘generous’ with overhang seats so much as it is ‘generous’ with levelling seats that compensate for the overhangs and thus ensure complete proportionality and proper apportionment between the states. If Germany had retained the rule it had until a decade ago, which did not have levelling seats, the overhangs could have continued to increase disproportionality, especially with the increasing fragmentation in the party vote. Under the current system, full proportionality is assured, so parties have no incentive to pursue dummy list strategies. If they did, it could lead to a massive ballooning of seat numbers far beyond the current situation, because parties with dummy lists would be likely to have very large overhangs (overhang = #district seats won – #seats allocated according to party vote).

  3. Is there an MMP system that has yet to be invented? What can be done to improve MMP? It is obvious that the German and NZ systems are the gold standard for MMP use in national elections.

    I have been reading about the Dual Member Mixed Proportional, is this an improvement on MMP? We may have talked about the Dual Member Mixed Proportional system, this was one of the choices for the PEI electoral reform referendum, this is a MMP related system that has been invented, but yet to be used. This system is confusing to me seems a bit similar to the Limited Vote system.

    • As noted in the article cited by Rob, Dual Member PR is similar to the 1-vote, best-losers version of MMP in Baden-Wurttemberg. Modify B-W to have: equal numbers of SMD and list members; list populated by losing candidates in order of their percentage SMD vote; no more than 1 list member from each SMD.
      This version of MMP would give the same party standings and mostly the same personnel as Dual Member PR. Smooth out some sharp edges with secondary candidates and Reserve fractions and you pretty much have the Full Monty DMPR.

      Dual Member PR would have the same overhang potential as MMP. Like 1-vote MMP, there is no vote-splitting and, in consequence, reduced incentive to run dummy candidates to harvest party votes.

      • I think I am understanding it. The problem with Dual Member PR is that the legislative body has to be of an even number size in order for it to work, but then I would imagine that 10% of the legislative body has to be reserve for at large seats, let says we have a 90 member parliament, 10% of that is 9 more seats that are at large, so it is 99 seats.

        Could this system be adapted to a preferential vote system, but doing that reminds me of a two member STV system? I can see why electoral reform is hard for many countries as the new system takes some time to learn.

      • I see Dave’s point, but for me, it still stretches the definition for inclusion of “DMP” as a subtype of MMP. I don’t say that to disparage it. The system might work OK. It just seems distinct from MMP to me.

      • Seems like a subtype of MMP to me. Isn’t it just using regional lists for the compensation with one seat each, but with party seat counts being determined nationwide?

      • To be honest, I am never entirely clear on what exactly it is. I understand it as two-tier PR, and this whole thread started out with my having some doubts about whether MMP is really a subtype of two-tier PR. So now I am trying to figure out whether a special (hypothetical) type of two-tier PR is a type of MMP. And… I don’t know!

  4. Rob,

    There are no “at large” seats in Dual Member PR. Every seat is tied to a single district, either the FPTP winner or the “PR” second member.

    The Dual Member PR hold-back of “Reserve” seats is intended to give small parties a better chance of seating their better-performing candidates. There is no pre-election designation of particular seats as “Reserved”. Which (if any) districts/candidates are affected by the Reserve fraction will depend entirely on how votes get distributed among the parties and their candidates.

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