For the second time in 2023, I have made a foray into podcast land. I was interviewed by Ben Raue for his podcast, The Tally Room: “From seats in to votes,” in which we discuss what the Effective Number of Parties is and how the Seat Product of an electoral system shapes party systems. We discuss Australia extensively, but also Canada and various other systems. Even St. Kitts & Nevis gets a reference.

# Tag Archives: Seat Product Model

# The effects (or their lack) of fused presidential–assembly ballots

A question that has arisen* is whether fused ballots–a single vote electing president and assembly, i.e., with no opportunity for ticket-splitting–suppress the number of parties, particularly when the president is elected by plurality and assembly by PR.

A challenge in addressing this question is that fused ballots are rather rare. Moreover, they may be adopted/abolished by ruling parties/coalitions based on expectations of advantage. In other words, the direction of causality between party-system outputs and rules is more ambiguous than usual. With such caveats reiterated, here is what I find.

This is for pure presidential systems, only because I am not aware of cases of semi-presidential systems that fuse presidential and assembly votes. (In parliamentary systems, the option does not arise, or in a sense the vote is always fused. I did not include the brief case in Israel of separate and direct election of an executive who was still responsible to the parliamentary majority.)

My outcome of interest is the ratio of expected effective number of seat-winning parties (*N _{S}*) or seat share of the largest party (

*s*

_{1}) to the expectation, given the seat product of the assembly (first chamber) electoral system.

For *N _{S}*, the ratio in non-fused cases is 1.13, for fused it is 0.927. This looks like good news for the hypothesis that fused ballots restrict party systems more than the separate vote does. However, the difference is not close to significant (p=0.12).

For *s*_{1}, the ratio in non-fused is 1.012, and in fused it is 1.047. Obviously that’s not significant. (Also, the seat product model is pretty good–even for presidential systems!)

Note that for *N _{S}*, the mean assembly party system in a presidential democracy tends to be

**more fragmented**than expected from its electoral system. Probably not what most people expect. Perhaps this is driven by the unusually fragmented case of Brazil. If I take it out, the ratios in non-fused are 1.083 for Ns and 1.031 for

*s*

_{1}. So not much impact.

Perhaps one should drop Uruguay from the set of fused cases. Not because ballots are not clearly fused, but because the electoral system is so different. Before 1999, parties could present multiple presidential candidates (and pool votes at party level for determining which party would win), and since then the fused ballot is only for the first round of a two-round presidential election. However, if we do this, we have only four cases left, so it is kind of meaningless. For the record, we would then have about a p=0.1 signifiant result in the expected direction. But I would put no stock in a result comparing four elections (in two countries) in one group to over 150 in the other group!

This is the list of cases with fused ballots that I am using. If I missed some, please let me know. (Angola, the case that prompted me to investigate this, is not in the dataset, nor are other countries that are not generally classified as democratic.)

```
country year
Dominican Rep 1978
Honduras 1993
Honduras 1997
Honduras 2001
Uruguay 1989
Uruguay 1994
Uruguay 1999
Uruguay 2004
Uruguay 2009
Uruguay 2014
Uruguay 2019
```

To this list could be added Bolivia. However, I did not include it because elections for president were not direct before 2005 (congress chose from top three if the popular vote did not yield a majority) and since 1997 the fusion has been only between the presidential vote and the party list vote of an MMP system.

(* A version of this text was originally posted as a comment in a thread on Angola, but it seemed to warrant a place in the center row of the virtual orchard.)

# No, the parties:largest-size relationship is not different for two-tier PR

Might as well graph it.

No reason here to doubt that the logical model, **N****_{S} = s_{1}^{–4/3}**, applies equally well to two-tier systems as it does to simple, single-tier systems. This was a question I raised in the earlier planting on the revision of the extended Seat Product Model (incorporating two-tier systems without an empirical constant).

Thus any deviations of regression output from the precise predictions of the models–as reported in that earlier post–are not caused by some systematic difference in this relationship for two-tier systems. Such deviations are just noise. For instance, the regression intercept on these 472 elections is significantly greater than zero. Yet a nonzero intercept is impossible. It can’t be that the effective number of parties is any different from 1.0 (the log of which is 0) if the largest party has 100% of the seats.* More to the point for the question I had, the regression shows no significant difference in slope (or intercept for that matter) between single-tier and two-tier systems. They behave the same in this sense, meaning that when the compensation tier increases the effective number of parties and reduces the seat share of the largest, it does so while preserving *N _{S}* =

*s*

_{1}

^{–4/3},

*on average*. And, by the way, for those who care about such things, the R

^{2}=0.899.

Bottom line: there is no statistically significant difference between single-tier and two-tier electoral systems in how the effective number of parties is related to the size of the largest.

* If I suppress the constant (while also eliminating the binary for “simple”) the coefficient is –1.341, or almost precisely the logically required –4/3. When run with the constant, it is –1.235, but the 95% confidence interval includes –1.333.

# The Austrian Question: Or how I corrected some data I’ve been using on two-tier systems

In the previous planting, I presented a revised version of the extended Seat Product Model. I noted that in the process of attempting to improve on the logical model, I discovered some inconsistencies in the treatment of remainder pooling systems in the dataset used in *Votes from Seats*. Here I describe the problem and how I corrected it. The changes here may still require further refinement, but at least they make the treatment of the cases internally consistent.

This first began to bother me even before *Votes from Seat*s had been published. Figure 17.2 in the book shows how well (or not) the extended seat product model accounts for the effective number of seat-winning parties (*N _{S}*) over time in several two-tier PR systems (plus Japan, included despite not fitting the category for reasons explained in the book). It plots every election in the dataset for this set of countries, with the observed value of

*N*shown with the solid grey line in each country plot. The expectation from the extended Seat Product Model (Equation 15.2) is marked by the dashed line. This equation is:

_{S}*N _{S}* = 2.5

*(*

^{t}*MS*

_{B})

^{1/6},

where *N _{S}* is the effective number of seat-winning parties (here, meaning the

*expected*

*N*),

_{S}*M*is the mean district magnitude of the basic tier,

*S*

_{B}is the total number of seats in the basic tier, and

*t*is the “tier ratio” defined as the share of the total number of assembly seats allocated in the compensatory tier.

For countries that changed from simple to complex, the plots also show the expectation in the era of the simple system with the solid dark line. The troublesome case here is Austria, but why? That is my “Austrian Question.” It led me down quite a rabbit hole, but I think I have it figured out, more or less.

It always seemed unlikely that the design of the Austrian electoral system was such that *expected* *N _{S}* could have reached well over 6 in the latter part of the time series! But that is what our data showed, supposedly. When you get an absurd result, generally you should impeach the data, not the model.

^{1}

The problem turned out to be that for several remainder-pooling systems, including Austria, some seats were effectively counted twice in the derivation of the extended seat product. We drew most of our data from Bormann and Golder’s *Democratic Electoral Systems Around the World*. However, the manner in which we did so handled remainder-pooling systems poorly. With one important exception that I will note below, the error was not in the original data, but in our application of it.

In a remainder-pooling system there is no fixed upper tier. Most two-tier compensatory systems have a fixed number of seats which are allocated nationally (or regionally) to “correct” for distortions in votes-to-seats allocation produced by the lower district magnitude of the basic tier. An example would be the system of Denmark, with 139 basic-tier seats and 40 compensatory seats. Others have a fixed minimum, such as the MMP systems of Germany and New Zealand (where the upper tier can expand if needed due to “overhang” seats, but it has a fixed starting size). In a remainder-pooling system, on the other hand, the “upper tier” can be as large *or as small* as needed to generate compensation. In theory, all seats could be allocated in the basic tier, and it would end up no different from a simple system. Typically these systems work by stipulating that parties earn seats based on full quotas (usually Hare quota, sometimes Hagenbach-Bischoff) in the districts. Any seats not filled are then “pooled” in a supra-district tier where they are allocated based on pooled votes, rather than being filled within districts. The upper tier is thus whatever number of remainder seats there are from all of the districts, which can vary from election to election depending on how votes are distributed among the parties and across the basic-tier districts.

A particular challenge in the analysis of these systems is that all seats may be attached to districts, and national reporting agencies vary in whether they indicate that a given seat is actually based on supra-district allocation. Thus a district might have, say, five seats, and in the determination of quotas, two parties may have obtained a total of three seats through quotas in the district. The other two seats go to the remainder pool. Maybe, once all the national seat allocation is complete, one of those two remainder seats goes to the largest party (bringing it up to three) and one goes to a third party that was short of a quota in the initial allocation. The complication is that while all five seats are assigned to candidates who were on party lists in the district, only three were assigned based solely on votes cast in the district. The other two were assigned according to the compensation mechanism, drawing upon the pooled votes from across multiple districts. Where is the upper tier? It is sort of a phantom, and if we count the two seats in our hypothetical example as part of the upper tier, and we also count them as part of the district (basic tier), we have double counted them!

Here is where the Austrian case comes in. If we look at the 1990 election, we see an assembly size of 183, with 9 basic-tier districts, averaging 20.3 seats apiece. Bormann and Golder report that 25 seats were allocated as “upperseats”. In our Equation 15.2, the input parameters were *MS*_{B}=20.3*(183–25)=3207 (rounding off) and *t*=(25/183)=0.137. That is, the basic-tier seat product is mean district magnitude multiplied by the size of the basic tier (which is total assembly size minus upper seats). But hold on! Those 25 upper-tier seats are taken out of the 20.3 seats per (average) district. Yet our original calculation takes them only out of the “*S*” part, but not out of the “*M*” part. They should not be counted in both tiers! Those 25 seats came from the 9 districts, so 25/9=2.8 remainder seats per district, on average. This gives us an adjusted basic-tier *M*=20.3–2.8=17.5. Now we have *MS*_{B}=17.5*(183–25)=17.5*158=2765, and *t*=0.137. This changes the “expected” *N _{S}* (based on Equation 15.2) from 4.35 to 4.25. Not a huge difference, but one that more accurately reflects how the system actually works.

Where things really went haywire was with the electoral reform that took place before the 1994 election. The Bormann and Golder dataset correctly notes that the number of basic-tier districts was increased to 43. With *S*=183 unchanged, this is a mean district magnitude of *M*=183/43=4.26, a figure which matches the description in Electoral System Change in Europe, maintained by Jean-Benoit Pilet and Alan Renwick. However, for some reason, the Bormann and Golder gives first-tier mean district magnitude for the post-1992 system as 17.2. The indicated values of “upper seats” range from 78 to 111 in the elections of 1994–2008. When we apply the same procedures of the preceding paragraph to elections in these years, we get a reduction in *MS*_{B} from the 2669 we used in the book to a more accurate 196.7. That is quite a change! It comes from the reduction in district magnitude to 4.26, which in turn greatly pushes up the number of seats allocated in upper tiers.^{2} When we stop double counting the remainder seats, we actually have an adjusted basic-tier magnitude of less than 2, and an upper tier ratio,* t*=0.5. This changes that rather absurd “expected *N _{S}*” depicted in Figure 17.2 as 6.3 for recent elections to a more reasonable 3.83. And, in fact actually observed

*N*in recent years has tended to be in the 3.4–4.2 range.

_{S}Here is the corrected version of the figure. (I left Japan off this one.) In addition to using the corrected data, as just explained, it also uses the revision of the extended Seat Product Model:

*N _{S}* = (1–

*t*)

^{–2/3}(

*MS*

_{B})

^{1/6}.

Austria is no longer shown as system that should be “expected” to have an effective number of parties around six! It still has an observed *N _{S}* in most years that is smaller than expected, but that’s another story. We are not the first to observe that Austria used to have an unusually consolidated party system for its electoral system.

^{3}In fact, in recent years it seems that the revamped design of the system and the increasingly fragmented party system have finally come into closer agreement–provided we use the revised SPM (as explained in the previous planting) and the corrected electoral-system data, and not the inconsistent data we were using before.

And, here for the first time, is a graph of largest party seat share in these systems, compared to expectations. This seemed worth including because, as noted in the previous planting, the *s*_{1} model for two-tier works a little better than the one for *N _{S}*. Moreover, it was on

*s*

_{1}that the revised logic was based.

Note that the data plots show a light horizontal line at *s*_{1}=0.5, given the importance of that level of party seat share for so much of parliamentary politics.

Notes

1. Assuming the model is on solid grounds, which was very much not the case of the original version of application to two-tier PR. I hope it is now, with the revision!

2. Plural because the 9 provincial districts still exist but are now an intermediate compensation tier, and there is a single national final compensation tier. This additional complication should not affect our estimation of the system’s impact on party-system outputs. (It principally affects which candidates from which of a given party’s lists earn the various compensation seats.)

3. This is not unique to Austria. Several European party systems used to have effective number of parties smaller than expected for their electoral system. In recent decades, many have become more fragmented, although the fragmenting trend is not significant, when compared to the SPM baseline). The trend implies that, in many cases, their electoral systems are shaping their party systems more as expected now than in the early post-war decades. In the past, the full electoral system effect may have been tamped down by the stronger role of the major party organizations in society. This is a very big question that it far beyond the scope of my current tasks.

Further note

In order to attempt a further validation of the procedure, I calculated the number of quota seats expected in each district based on my district-level dataset, derived originally from CLEA. I can then sum this up across districts in a given election, and subtract the result from the total assembly size to arrive at an indicator of what the upper-tier size should have been in that election. When I do this, I usually come close to the value for “upperseats”in Bormann & Golder, although not always precisely. I do not know what explains the deviations, but in all but one election they are so small that I would not fret. For the two elections used as examples from Austria above, I get 24 remainder-pooled seats in 1990 (vs. B&G 25) and 111 in 2008 (identical to B&G). Ideally, we would be able to estimate what upper-tier seats should be, on average, for a given design of a remainder-pooling system. Then we could estimate the parameters needed for the extended SPM even if data sources do not separate out the seats allocated on district votes from those allocated via supra-district pooling. This would introduce some unknown error, given that the actual number of remainder-pooled seats can vary depending on election results, for constant institutions. For instance, for the current Austrian system, it has ranged from 81 to 111 between 1995 and 2008. Perhaps there is some mathematical relationship that connects this average (92) to fixed parameters of the electoral system, and that works across remainder-pooling electoral systems. If there is, it has not revealed itself to me yet.

# The Extended Seat Product Model: Getting rid of that annoying “2.5”

[**Update, late April, 2022:** I have continued to refine this method, and the specific values mentioned below no longer hold (due a revision of the estimation procedure outlined below), although the basic framework remains the same. In fact, the revision is based on what is described as “a further extension” towards the end of this post. This also means that *the datasets linked at the end of the linked post are not accurate.* I will upload corrected ones at some point.]

The extended version of the Seat Product Model (SPM), devised to be applicable to two-tier PR systems as well as simple electoral systems, states:

*N _{S}* = 2.5

*(*

^{t}*MS*

_{B})

^{1/6},

where *N _{S}* is the effective number of seat-winning parties (here, meaning the

*expected*

*N*),

_{S}*M*is the mean district magnitude of the basic tier,

*S*

_{B}is the total number of seats in the basic tier, and

*t*is the “tier ratio” defined as the share of the total number of assembly seats allocated in the compensatory tier. In the case of a simple (single-tier) system, this reduces to the basic SPM:

*N*=(

_{S}*MS*)

^{1/6}, given that for simple systems, by definition,

*t*=0 and

*S*

_{B}=

*S*, the total size of the elected assembly.

Ever since this formula first appeared in my 2016 *Electoral Studies* article with Huey Li (and later as Equation 15.2 in Shugart and Taagepera, 2017, *Votes from Seats*) I have been bothered by that “2.5.” The SPM for simple systems is a logical model, meaning its parameters are derived without recourse to the data. That is, the SPM is not an empirical regression fit, but a *deductive model* of how the effective number of seat-winning parties (and other electoral-system outputs) *should be* connected to two key inputs of the electoral system, if certain starting assumptions hold. When we turn to statistical analysis, if the logic is on the right track, we will be able to confirm both the final model’s prediction and the various steps that go into it. For simple systems, such confirmation was already done in Taagepera’s 2007 book, *Predicting Party Sizes*; Li and Shugart (2016) and Shugart and Taagepera (2017) tested the model and its logical antecedents on a much larger dataset and then engaged in the process of extending the model and its regression test in various ways, including to cover more complex systems. Yet the derivation of the “2.5” was not grounded in logic, but in an empirical average effect, as explained in a convoluted footnote on p. 263 of *Votes from Seats* (and in an online appendix to the Li-Shugart piece).

If one is committed to logical models, one should aim to rid oneself of empirically determined constants of this sort (although, to be fair, such constants do exist in some otherwise logical formulas in physics and other sciences). Well, a recent *Eureka! *moment led me to the discovery of a logical basis, which results in a somewhat revised formula. This revised version of the extended Seat Product Model is:

*N _{S}* = (1–

*t*)

^{–2/3}(

*MS*

_{B})

^{1/6}.

The variables included are the same, but the “2.5” is gone! This revision produces results that are almost identical to the original version, but stand on a firmer logical foundation, as I shall elaborate below.

Consider a few examples for hypothetical electoral systems.

MS_{B} | t | 1-t | (1–t)^{–}^{2/3} | 2.5^{t} | N (rev.) _{S} | N (Eq. 15.2)_{S} |

100 | .5 | .5 | 1.59 | 1.58 | 3.42 | 3.40 |

100 | .25 | .75 | 1.21 | 1.26 | 2.61 | 2.71 |

250 | .3 | .7 | 1.27 | 1.32 | 4.68 | 4.85 |

250 | .4 | .6 | 1.41 | 1.44 | 3.53 | 3.62 |

250 | .6 | .4 | 1.84 | 1.73 | 4.62 | 4.35 |

2500 | .3 | .7 | 1.27 | 1.32 | 4.68 | 4.85 |

2500 | .15 | .85 | 1.11 | 1.15 | 4.11 | 4.23 |

It may not work especially well with very high *MS _{B}*, or with

*t*>>.5. But neither does equation 15.2 (the original version); in fact, in the book we say it is valid only for

*t*≤0.5. While not ideal from a modelling perspective, it is not too important in the real world of electoral systems: cases we would recognize as two-tier PR rarely have an upper compensation tier consisting of much more than 60% of total

*S*; relatedly,

*S*much greater than around 300 is not likely to be very common. My examples of

_{B}*MS*=2,500 are motivated by the notion of

_{B}*S*=300 and a decently proportional basic-tier

_{B}*M*=8.3.

Testing on our dataset via OLS works out well, for both versions of the formula. Our largest-sample regression test of Equation 15.2, in Table 15.1 of *Votes from Seats*, regression 3, yields:

log *N _{S}* = –0.066 + 0.166log

*MS*+ 0.399

_{B}*t*.

Logically, we expect a constant of zero and a coefficient of 0.167 on the log of *MS _{B}*; the coefficient on

*t*is expected to be 0.398=log2.5 (but as noted, the latter is not logically based but rather expected only from knowledge of relationships in the data for two-tier systems). In other words, it works to almost point predictions for what we expected before running the regression! Now, let’s consider the revised formula. Using the same data as in the test of Equation 15.2 in the book, OLS yields:

log *N _{S}* = –0.059 + 0.165log

*MS*– 0.654 log(1–

_{B}*t*) .

Again we expect a constant at zero and 0.167 on log* MS _{B}* . Per the revised logic presented here, the coefficient on log(1–

*t*) should be –0.667. This result is not too bad!

^{1}

** OK, how did I get to this point?** Glad you asked. It was staring me in the face all along, but I could not see it.

I started the logical (re-)modeling with seat share of the largest party, **s**_{1}, as it was easier to conceptualize how it would work. First of all, we know that for simple systems we have *s*_{1}= (*MS*)^{–}^{1/8}; this is another of the logical models comprising the SPM and it is confirmed statistically. So this must also be the starting point for the extension to two-tier systems (although none of my published works to date reports any such extended model for *s*_{1}). Knowing nothing else about the components of a two-tier system, we have a range of possible impact of the upper-tier compensation on the basic-tier largest party size (*s*_{1B}). It can have no effect, in which case it is 1** s_{1B}*. In other words, in this minimal-effect scenario the party with the largest share of seats can emerge with the

*same*share of overall seats after compensation as it already had from basic-tier allocation. At the maximum impact, all compensation seats go to

*parties other than the largest*, in which case the effect is (1–

*t*)*

*s*_{1}

*. A fundamental law of compensation systems is that*

_{B}*s*

_{1}≤

*s*_{1}

*. (and*

_{B}*N*≥

_{S}*N*); by definition, they can’t enhance the position of the largest party relative to its basic-tier performance.

_{SB}^{2}

Let’s see from some hypothetical examples. Suppose there are 100 seats, 50 of which are in the basic tier. The largest party gets 20 of those 50 seats, for *s*_{1}*_{B}* = 0.4. If compensation also nets it 20 of the 50 compensation seats, it emerges with 40 of 100 seats, for

*s*

_{1}=0.4 = 1*

*s*_{1}

*. If, on the other hand, it gets none of the upper-tier seats, it ends up with 20 of 100 seats, for*

_{B}*s*

_{1}=0.2 = (1–

*t*)*

*s*_{1}

*. For a smaller*

_{B}*t*example… Suppose there are 100 seats, 80 of which are in the basic tier, and the largest gets 32 seats, so again

*s*_{1}

*= 0.4. If compensation nets it 8 of the 20 compensation seats (*

_{B}*t*=0.2), it emerges with 40 of 100 seats, for

*s*

_{1}=0.4 = 1*

*s*_{1}

*. If, on the other hand, it gets none of the upper-tier seats, it ends up with 32 of 100 seats, for*

_{B}*s*

_{1}=0.32 = (1–0.2)*

*s*_{1}

*= 0.8*0.4=0.32.*

_{B}In the absence of other information, we can assume the upper tier effect is the geometric average of these logical extremes (i.e, the square root of the product of 1 and 1–*t*), so:

*s*_{1}= (1–t)^{1/2}(*MS _{B}*)

^{–}^{1/8},

and then because of the established relationship of *N _{S}* =

*s*

_{1}

^{–4/3}, which was also posited and confirmed by Taagepera (2007) and further confirmed by Shugart and Taagepera (2017), we must also have:

*N _{S}* = (1–t)

^{–}^{2/3}(

*MS*)

_{B}^{1/6}.

Testing of the *s*_{1} formula on the original data used for testing Equation 15.2 is less impressive than what was reported above for *N _{S}*, but statistically still works. The coefficient on log(1–

*t*) is actually 0.344 instead of 0.5, but its 95% confidence interval is 0.098–0.591. It is possible that the better fit to the expectation of

*N*than that of

_{S}*s*

_{1}is telling us that these systems have a different relationship of

*N*to

_{S}*s*

_{1}, which I could imagine being so. This remains to be explored further. In the meantime, however, an issue with the data used in the original tests has come to light. This might seem like bad news, but in fact it is not.

The data we used in the article and book contain some inconsistencies for a few two-tier systems, specifically those that use “remainder pooling” for the compensation mechanism. The good news is that when these inconsistencies are corrected, the models remain robust! In fact, **with the corrections, the s_{1} model turns out much better than with the original data**. Given that

*s*

_{1}is the quantity on which the logic of the revised equation was based, it is good to know that when testing with the correct data, it is

*s*

_{1}that fits revised expectations best! On the other hand, the

*N*model ends up being a little more off.

_{S}^{3}Again, this must be due to the compensation mechanism of at least some of these systems affecting the relationship of

*s*

_{1}to

*N*in some way. This is not terribly surprising. The fact that–by definition–only under-represented parties can obtain compensation seats could alter this relationship by boosting some parties and not others. However, this remains to be explored.

_{S}A **further extension** of the extended SPM would be to allow the exponent on (1–*t*) to vary with the size of the basic tier. Logically, the first term of the right-hand side of the equation should be closer to (1–*t*)^{0}=1 if the basic tier already delivers a high degree of proportionality, and closer to (1–*t*)^{1}=1–*t* when the upper tier has to “work” harder to correct deviations arising from basic-tier allocation. In fact, this is clearly the case, as two real-world examples will show. In South Africa, where the basic tier consists of 200 seats and a mean district magnitude of 22.2, there can’t possibly be much disproportionality to correct. Indeed, the largest party–the hegemonic ANC– had 69% of the basic tier seats in 2009. Once the compensation tier (with *t*=0.5) went to work, the ANC emerged with 65.9%. This is much less change from basic tier to final overall *s*_{1} than expected from the equation. (Never mind that this observed *s*_{1} is “too high” for such a proportional system in the first place! I am simply focusing on what the compensation tier does with what it has to work with.) The ratio of overall *s*_{1} to the basic-tier *s*_{1}*_{B}* in this case is 0.956, which is approximately (1–

*t*)

^{0.066}, or very close to the minimum impact possible. On the other hand, there is Albania 2001. The largest party emerged from the basic tier (100 seats, all

*M*=1)

^{4}with 69% of the seats–just like in the South Africa example, but in this case that was significant overrepresentation. Once the upper tier (with

*t*=0.258) got to work, this was cut down to 52.1%. The ratio of overall

*s*

_{1}to the basic-tier

*s*_{1}

*here is 0.755, which is approximately (1–*

_{B}*t*)

^{0.95}, or very close to the maximum impact possible given the size of the upper tier relative to the total assembly.

These two examples show that the actual exponent on (1–*t*) really can vary over the theoretical range (0–1); the 0.5 proposed in the formula above is just an average (“in the absence of any other information”). Ideally, we would incorporate the expected *s*_{1} or *N _{S}* from the basic tier into the derivation of the exponent for the impact of the upper tier. Doing so would allow the formula to recognize that how much impact the upper tier has depends on two things: (1) how large it is, relative to the total assembly (as explained by 1–

*t*), and (2) how much distortion exists in the basic tier to be corrected (as represented by the basic-tier seat product,

*MS*).

_{B}However, incorporating this “other information” is not so straightforward. At least I have not found a way to do it. Nonetheless, the two examples provide further validation of the logic of the connection of the impact through 1–*t*. This, coupled with regression validation of the posited average effect in the dataset, as reported above, suggests that there really is a theoretical basis to the impact of upper-tier compensation on the basic-tier’s seat product, and that it rests on firmer logical grounds than the “2.5” in the originally proposed formula.

This a step forward for the scientific understanding of two-tier proportional representation!

In the next installment of the series, I will explain what went wrong with the original data on certain two-tier systems and how correcting it improves model fit (as it should!).

______

Notes.

1. The reported results here ignore the coefficients on the log of the effective number of ethnic groups and the latter’s interaction with the the log of the seat product. These are of no theoretical interest and are, in any case, statistically insignificant. (As explained at length in both Li & Shugart and Shugart & Taagepera, the interaction of district magnitude and ethnic fragmentation posited in widely cited earlier works almost completely vanishes once the electoral-system effect is specified properly–via the seat product and not simply magnitude.)

2. Perhaps in bizarre circumstances they can; but leave these aside.

3. This is what we get with the corrected data, First, for seat share of the largest party:

log * s_{1}* = 0.047 – 0.126log

*MS*+ 0.433 log(1–

_{B}*t*) .

(Recall from above that we expect a constant of zero, a coefficient of –0.125 on log* MS _{B}* and 0.5 on log(1–

*t*).)

For effective number of seat-winning parties:

log *N _{S}* = –0.111 + 0.186log

*MS*– 0.792 log(1–

_{B}*t*).

Both of those coefficients are somewhat removed from the logical expectations (0.167 and –0.667, respectively). However, the expectations are easily within the 95% confidence intervals. The constant term, expected to be zero, is part of the problem. While insignificant, its value of –0.111 could affect the others. Logically, it *must be* zero (if * MS_{B}*=1 and

*t*=0, there is an anchor point at which

*N*=1; anything else is absurd). If we suppress the constant, we get:

_{S} log *N _{S}* = 0.152log

*MS*– 0.713 log(1–

_{B}*t*).

These are acceptably close (and statistically indistinguishable from expected values, but then so were those in the version with constant). Nonetheless, as noted above, the deviation of this result from the near-precise fit of most tests of the SPM probably tells us something about the relationship between * s_{1}* and

*N*in these two-tier systems. Just what remains to be seen.

_{S}4. In other words, it was an MMP system, conceived as a subtype of two-tier PR.

# 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 (*N _{S}*) in both Germany and New Zealand. For the latter country, it includes the pre-reform FPTP system. In the case of Germany, it plots

*N*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 systems

_{S}^{1}; 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 *N _{S}* 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 *N _{S}* 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.

**Notes**

1. The “expected *N _{S}*” 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”

*N*of 0.36.

_{S} 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 *N _{S}*=3.08. This would be somewhat better for much of the earlier period in Germany. In fact, from 1953 through 2002, mean observed

*N*=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

_{S}*N*has been 3.14.

_{S}# The Germany 2021 result and the electoral system

The German general election of 2021 has resulted in a situation in which neither major party can form a government without either the other, or more likely, a coalition that takes in both the liberal FDP and the Greens. With the largest party, the social-democratic SPD, under 30% of seats, it is an unusually fragmented result compared to most German elections. Naturally, this being *Fruits & Votes*, attention turns to how much more fragmented this outcome is than expected, given the electoral system. The answer may be a bit of a surprise: **not all that much**. I expected this outcome to be a significant miss for the Seat Product Model (SPM). But it is really not that far off.

For a two-tier PR system, of which Germany’s MMP can be thought of as a subtype, we need to use the extended version of the SPM developed in *Votes from Seats*.

*N _{S}* = 2.5

*(*

^{t}*MS*

_{B})

^{1/6},

where *N _{S}* is the effective number of seat-winning parties (here, meaning the

*expected*

*N*),

_{S}*M*is the mean district magnitude of the basic tier,

*S*

_{B}is the total number of seats in the basic tier, and

*t*is the “tier ratio” defined as the share of the total number of assembly seats allocated in the compensatory tier. For Germany, basic-tier

*M*=1 and

*S*

_{B}=299. The tier ratio could be coded as 0.5, because the initial design of the system is that there are 299 list tier seats, allocated to bring the result in line with the overall party-list vote percentages of each party that clears the threshold. However, in Germany the electoral law provides that the list tier can be expanded further to the extent needed to reach overall proportionality. Thus

*t*is not fixed; we should probably use the ratio that the final results are based on, as

*N*would necessarily be lower if only 299 list seats had been available. In the final result, the Bundestag will have 735 seats, meaning 436 list seats, which gives us a tier ratio of

_{S}*t*=436/735=0.593. Plug all this into the formula, and you get:

*N _{S}* = 2.5

^{0.593}299

^{1/6}=1.72*2.59=

**4.45**.

Now, what was the actual *N _{S}* in the final result? We have to ask ourselves whether to count to two Christian “Union” parties, the CDU and the CSU, as one party or two. The answer really depends on the question being asked. They are separate parties, with distinct organization, and they bargain separately over portfolios and policy when they are negotiating a coalition with another party. However, for purposes of the SPM, I firmly believe that when two or more parties in a bloc do not compete against each other (or, alternatively, do so only within lists over which votes are pooled for seat-allocation

^{1}), they should be treated as one. The SPM does not “care” whether candidates of the bloc in question are branded as CSU (as they are in Bavaria) or as CDU (the rest of Germany). It simply estimates the effective number of “agents of the electorate” given the electoral rules. In terms of national politics, these are the same “agent”–they always enter government together or go into opposition together, and they jointly nominate a leader to be their Chancellor candidate.

Taking the CDU/CSU as a “party” for this purpose, we get actual *N _{S}* =

**4.84**in the 2021 election. So, given an expectation of 4.45, the actual outcome is just over 8.75% higher than expected. That is nothing too extraordinary. For comparison purposes, we can just take the ratio of actual

*N*to expected

_{S}*N*. Here are some elections in the dataset used for

_{S}*Votes from Seats*that are in the same range of over-fragmentation as Germany 2021:

```
country year simple Ns exp_Ns ratio
Barbados 1981 1 1.87 1.735597 1.077439
Norway 1965 1 3.51 3.255616 1.078137
Sri Lanka 1970 1 2.49 2.307612 1.079037
Dominican Rep 1990 1 3.05 2.810847 1.085082
Trinidad 2002 1 1.98 1.824064 1.085488
Iceland 1963 0 3.33 3.060313 1.088124
Israel 1961 1 5.37 4.932424 1.088714
Trinidad 2001 1 2 1.824064 1.096452
Trinidad 2000 1 2 1.824064 1.096452
Iceland 1999 0 3.45 3.146183 1.096567
Denmark 1950 0 3.98 3.624933 1.097951
```

(The table indicates as ‘simple’ those with a single tier; others are two-tier.)

The ratio variable has a mean of 1.021 in the full dataset and a standard deviation of 0.359. Its 75th percentile is 1.224 (and 25th is 0.745). So the German election of 2021 is actually very well explained by this method. The degree of fragmentation we saw in this election is not too surprising. It is about what should be expected with MMP consisting of 299 nominal-tier *M*=1 seats and a very generous and flexible compensation tier.

As an aside, if we used the initial tier size (299, so t=0.5) in the formula, we would get an “expected” *N _{S}*=4.09. This would mean a ratio of 1.183, still short of the 75th percentile of the 584 elections included in the book’s main statistical test. Here is the company it would be keeping in that neighborhood:

```
country year simple Ns exp_Ns ratio
Germany 2009 0 4.83 4.121066 1.172027
St. Kitts and Nevis 2000 1 1.75 1.491301 1.173472
Luxembourg 2009 1 3.63 3.077289 1.17961
Canada 2004 1 3.03 2.560218 1.183493
Denmark 1998 0 4.71 3.965222 1.187828
Venezuela 1963 0 4.32 3.63006 1.190063
Korea South 1988 0 3.55 2.981969 1.190488
Czech Republic 2010 1 4.51 3.767128 1.197199
Iceland 1991 0 3.77 3.146183 1.198277
```

This would put the German 2021 election about as “over-fragmented” as the Canadian election of 2004. In other words, still not a big deal. If we count the two “Union” parties separately, obviously the degree of over-fragmentation goes up considerably. As I have said already, I think for this purpose counting them as one is the correct decision.^{2}

As far as size of the largest seat-winning party is concerned, the SPD has 206 seats, for **28.03%**. The SPM would predict, given expected *N _{S}*=4.45, that the largest should have

**32.6%**(240 seats out of 735); that’s a ratio of 0.860 (which is a slightly bigger miss than the

*N*ratio of 1.088, the reciprocal of which would be 0.919). It is worth pausing on this for a bit. Polling before the election said the largest party might be only on a quarter of the votes. This was accurate, as the SPD won 25.7%. The advantage ratio (%seats/%votes) is 1.09, which is rather high for an electoral system that promises as near-perfect proportionality as Germany’s current system does, with its compensation for overhangs (cases in which a party has won more nominal-tier seats in a state than its list votes would have entitled it to). This bonus is a result of a rather high below-threshold vote. Not as high in 2013, of course, when two parties (FDP and AfD) narrowly missed the nationwide 5% threshold. But still considerably high, at 8.6% combined for all parties that failed to win a seat.

_{S}It is also worth asking whether the logic behind the extended SPM for two-tier systems holds for this German election. The formula says that the basic tier produces an initial allocation of seats consistent with the SPM for simple systems, and then inflates it based on the size of the compensation tier. So we can ask what the effective number of seat-winning parties is in the basic tier alone. It should be *N _{S}* =(

*MS*

_{B})

^{1/6}= 299

^{1/6}=

**2.59**. In fact, the basic-tier

*N*in this election was

_{S}**2.51**(as before, taking CDU/CSU as one party). The ratio of 0.969 is a pretty trivial miss. We should expect the largest party to have won 0.490 of these seats (about 146). Actually the Union parties, which together won the most single-seat districts, won 143 (0.478). Thus Germany’s MMP system, in the 2021 election, actually did produce a basic-tier (nominal-tier) party system pretty much just like it should, given 299 seats and

*M*=1 plurality, and then augmented this through a large compensatory national tier. The actual inflator is a factor of 1.93=4.84/2.51, rather than the expected 1.72=2.5

^{0.593}. Had it been 1.72 instead, the final effective number of seat-winning parties would have been 4.32, about “half a party” less than in reality, implying almost exactly one third of seats to the SPD instead of just 28%.

This surprised me (pleasantly, of course). When I saw that the Greens and AfD each had won 16 seats in the nominal tier, I thought that was too many! But in fact, it works out. Maybe sometimes even I think Duverger had a law, or something. But given 299 single-seat districts, this is pretty much in line with expectations.

The outcome is interesting in the many ways that it serves as a primer on details of the electoral system. Here I mean not only the substantial expansion of the Bundestag from 598 to 735 seats, due to the way the compensation mechanism works, but also the thresholds. One of the best known features of the German electoral system is the* 5% nationwide threshold*. But of course, the threshold is more complex than that. It is 5% of the national party-list vote *or* three single-seat wins, *except* if a party is an ethnic-minority party. All these provisions were on display. For instance, the Linke (Left) party fell below the 5.0% threshold, yet is represented at full proportionality. That is because it won three individual mandates, thus fulfilling the “or” clause of the threshold. There was a point on election night when it looked as if the Linke might hold only two single-seat districts. In that case, with less than 5% of the list votes nationwide, it would have held only those seats as its total. By winning three, it is entitled under the law to full proportional compensation, and as a result it was awarded 36 list seats. Then, for the first time in a very long time, an ethnic party has won a seat. The South Schleswig Voters’ Association (SSW), which had not contested federal elections in decades, ran in this one and was able to win a single (list) seat, because as a representative of the Danish and Frisian minorities, it is exempt from the usual threshold provisions, as long as its votes are sufficient to qualify it for a seat when the threshold is ignored. Its 0.1% of the national vote was good enough. The SSW has had some renewed success in state elections in Schleswig-Holstein recently, and now it has scored a seat in the federal parliament for the first time since 1949. In 1949, the MMP system was a bit different, in that the 5% threshold was determined state-by-state, rather than nationwide. If the threshold had been state-by-state in this election, one other party would have earned seats. The Free Voters won around 7.5% of party-list votes in Bavaria. However, they managed only 2.9% nationwide (and no district seat), so they are shut out.

Now attention turns to what the coalition will be. Two options are on the table: SPD+Greens+FDP (“traffic light”) or CDU/CSU+Greens+FDP (“Jamaica”). The possibility of a broad left coalition has been ruled out by the election results: SPD+Green+Linke is not a majority. It was never likely anyway; the SPD and Greens did not spend recent years convincing voters they were safe options near the center of German politics to team up with the far left. Nonetheless, had it been mathematically possible the SPD might have used it as leverage against the FDP. My guess is that the traffic light coalition will form. Despite some serious policy differences between the FDP and the other two, it would be a government made up of the winners of the election, as these three parties all gained votes compared to 2017. On the other hand, one led by the CDU/CSU would be led by a pretty big loser, even though it is mathematically possible and the Greens seem to have been positioning for it over the last several years.^{3} Following the election, the DW live blog has been reporting on comments by various prominent CDU and CSU politicians that could be interpreted as saying the bloc needs some time in opposition, after the disappointing result. I suspect this is the view that will prevail, and after a lot of intense and difficult bargaining, Germany will be led by a traffic light coalition for the first time.

____

Notes

1. Here I am thinking of cases like Chile, where alliance lists contain candidates of different parties, but for purposes of how the electoral system assigns seats between competing teams of candidates, we should count the alliances, not the component parties. The same condition applies in Brazil and Finland, only there it is essentially impossible to aggregate to a meaningful national alliance category because the combinations of parties are not always the same across districts. In Chile, and also in the FPTP case of India–as well as in the current case of Germany–there is no such problem, as the alliances are nationwide in scope and consistent across districts.

2. For the record, counting them separately yields *N _{S}*=5.51 in this election, which would put the ratio just barely above the 75th percentile.

3.To be clear, they are much happier working with the SPD, but what I mean is that their positioning for the possibility of a coalition with the CDU/CSU should make finding common ground with the FDP easier than it otherwise would have been.

# Canada 2021: Another good night for the Seat Product Model, and another case of anomalous FPTP

The 2021 Canadian federal election turned out almost the same as the 2019 election. Maybe voters just really do not want to entrust Justin Trudeau with another majority government, as he led from 2015 to 2019. The early election, called in an effort to turn the Liberal plurality into a Liberal majority, really changed almost nothing in the balance among parties.

The result in terms of the elected House of Commons is strikingly close to what we expect from the Seat Product Model (SPM). Just as it was in 2019. The predictive formulas of the SPM suggest that when your electoral system is FPTP and there are 338 total seats, the largest one should win 48.3% of the seats, or about 163. They further suggest that the effective number of seat-winning parties (*N _{S}*) should be around 2.64. In the actual result–with five districts still to be called–the largest party, Liberal, has won or is leading in 159, or 47.0%., and

*N*=2.78. These results are hardly different from expected. They also are hardly different from 2019, when the Liberals won 157 seats; in that election we had

_{S}*N*=2.79.

_{S}While the parliamentary balance will be almost what the SPM expects, the voters continue to vote as if there were a proportional system in place. The largest party again has only around a third of the votes, and the effective number of vote-earning parties (*N _{V}*) is around 3.8. For a FPTP system in a House the size of Canada’s, we should expect

*N*=3.04. Once again, the fragmentation of the vote continues to be considerably greater than expected.

_{V}Another bit of continuity from 2019 is the anomalous nature of FPTP in the current Canadian party votes distribution. For the second election in a row, the Conservative Party has won more votes than the Liberals, but will be second in seats. The votes margin between the two parties was about the same in the two elections, even though both parties declined a little bit in votes in 2021 compared to 2019. Moreover, as also has happened in 2019 (and several times before that), the third largest party in votes will have considerably fewer seats than the party with the fourth highest vote share nationwide. The NDP won 17.7% of the vote and 25 seats (7.4%), while the Bloc Quebecois, which runs only in Quebec, won 7.8% of vote and 33 seats (9.8%).

The Green Party and the People’s Party (PPC) more or less traded places in votes: Greens fell from 6.5% in 2019 to 2.3%, while the PPC increase from 1.6% to 5.0%. But the Greens’ seats fell only from 3 to 2, while the PPC remained at zero.

So, as in 2019, the 2021 election produced a good night for the Seat Product Model in terms of the all-important party balance in the elected House of Commons. However, once again, Canadians are not voting as if they still had FPTP. They are continuing to vote for smaller parties at a rate higher than expected–and not only in districts such parties might have a chance to win–and this is pushing down the vote share of the major parties and pushing up the overall fragmentation of the vote, relative to expectations for the very FPTP system the country actually uses.

It is worth adding that the virtual stasis at the national level masks some considerable swings at provincial level. Éric Grenier, at The Writ, has a table of swings in each province, and a discussion of what it might mean for the parties’ electoral coalitions. A particularly interesting point is that the Conservatives’ gains in Atlantic Canada and Quebec, balanced by vote loss in Alberta and other parts of the west, mirrors the old Progressive Conservative vs. Reform split. Current leader Erin O’Toole’s efforts to reposition the party towards the center may explain these regional swings.

In a follow up, I will explore what this tendency towards vote fragmentation implies for the sort of electoral system that would suit how Canadians actually are voting.

Below are the CBC screen shots of election results for 2021 and 2019. As of Thursday afternoon, there remain a few ridings uncalled.

# Why 1.59√Ns?

In the previous planting, I showed that there is a systematic relationship under FPTP parliamentary systems of the mean district-level effective number of vote-earning parties (*N*‘* _{V}*) to the nationwide effective number of seat-winning parties (

*N*). Specifically,

_{S}*N*‘* _{V}* =1.59√

*N*.

_{S}But why? I noticed this about a year after the publication of *Votes from Seats* (2017) while working on a paper for a conference in October, 2018, honoring the career of Richard Johnston, to which I was most honored to have been invited. The paper will be a chapter in the conference volume (currently in revision), coauthored with Cory Struthers.

In *VfrS* Rein Taagepera and I derived *N*‘* _{V}* =1.59

*S*

^{1/12}. And as explained in yesterday’s planting, it is simply a matter of algebraic transformation to get from expressing of

*N*‘

*in terms of assembly size (*

_{V}*S*) to its expression in terms of

*N*. But perhaps the discovery of this connection points the way towards a logic underlying how the nationwide party system gets reflected in the average district under FPTP. In the paper draft, we have an explanation that I will quote below. It is on to something, I am sure, but it remains imperfect; perhaps readers of this post can help improve it. But first a little set-up is needed.

_{S}To state clearly the question posed in the title above, **why would the average district-level effective number of vote-winning parties in a FPTP system tend be equal to the square root of the nationwide effective number of seat-winning parties, multiplied by 1.59?**

We can deal with the 1.59 first. It is simply 2^{2/3}, which should be the effective number of vote-earning party in an “isolated” district; that is, one that is not “embedded” in a national electoral system consisting of other seats elected in other districts (this idea of embedded districts is the key theme of Chapter 10 of *VfrS*). The underlying equation for *N*‘* _{V}*, applicable to any simple districted electoral system, starts with the premise that there is a number of “pertinent” parties that can be expressed as the (observed or expected) actual (i.e., not ‘effective’) number of seat-winning parties, plus one. That is, the number of parties winning at least one seat in the district, augmented by one close loser. For

*M*=1 (as under FPTP), we obviously have one seat winning party, and then one additional close loser, for two “pertinent” parties. Thus with

*M*=1 it is the same as the “

*M*+1 rule” previously noted by Reed and Cox, but Taagepera and I (in Ch. 7 of our 2017 book) replace it with an “

*N*+1″ rule, and find it works to help understand the effective number of vote-earning parties both nationwide and at district level. Raising this number of pertinent vote-earning parties to an exponent (explained in the book) gets one to

*N*(nationwide) or

_{V}*N*‘

*(district-level). When*

_{V}*M*=1, the number of pertinent parties is by definition two, and for reasons shown by Taagepera in his 2007 book, the effective number of seat-winning parties tends to be the actual number of seat-winning parties, raised to the exponent, 2/3. The same relationship between actual and effective should work for votes, where we need the “pertinent” number only because “actual number of parties winning at least one vote” is a useless concept. Hence the first component of the equation, 2

^{2/3}=1.5874.

As for the second component of the equation, *S*^{1/12}, it is also an algebraic transformation of the formula for the exponent on the quantity defined as the number of seat-winning parties, plus one. At the district level, if *M*>1, the exponent is itself mathematically complex, but the principle is it takes into account the impact of extra-district politics on any given district, via the assembly size. The total size of the assembly has a bigger impact the smaller the district is, relative to the entire assembly. Of course, if *M*=1, that maximizes the impact of national politics for any given *S* –meaning the impact of politics playing out in other districts on the district of interest. And the larger *S* is, given all districts of *M*=1, the more such extra-district impact our district of interest experiences. With all districts being *M*=1, the exponent reduces to the simple 1/12 on assembly size (shown in Shugart and Taagepera, 2017: 170). Then, as explained yesterday we can express *N*‘* _{V}* in terms of

*N*via the Seat Product Model. It should be possible to verify

_{S}*N*‘

*=1.59√*

_{V}*N*empirically; indeed, we find it works empirically. I showed a plot as the second figure in yesterday’s post, but here is another view that does not add in the Indian national alliances as I did in yesterday’s. This one shows only Canada, Britain, and several smaller FPTP parliamentary systems. The Canadian election mean values are shown as open squares, and several of them are labelled. (As with the previous post’s graphs, the individual districts are also shown as the small light gray dots).

_{S}It is striking how well the Canadian elections, especially those with the highest nationwide effective number of seat-winning parties (e.g., 1962, 2006, and 2008) conform to the model, indicated with the diagonal line. But can we derive an explanation for *why* it works? Following is an extended quotation from the draft paper (complete with footnotes from the original) that attempts to answer that question:

Equation 4 [in the paper, i.e. *N*‘* _{V}* =1.59√

*N*] captures the relationship between the two levels as follows: If an additional party wins representation in the national parliament, thus increasing nationwide

_{S}*N*to some degree, then this new party has some probabilistic chance of inflating the district-level voting outcome as well. It may not inflate district-level voting fragmentation everywhere (so the exponent on

_{S}*N*is not 1), but it will not inflate it only in the few districts it wins (which would make the exponent near 0 for the average district in the whole country). A party with no seats obviously contributes nothing to

_{S}*N*, but as a party wins more seats, it contributes more.

_{S}^{[1]}According to Equation 4, as a party emerges as capable of winning more seats, it tends also to obtain more votes in the average district.

As Johnston and Cutler (2009: 94) put it, voters’ “judgements of a party’s viability may hinge on its ability to win seats.” Our logical model quantitively captures precisely this notion of “viability” of parties as players on the national scene through its square root of *N _{S}* component. Most of the time, viability requires winning seats. For a new party, this might mean the expectation that it will win seats in the current election. Thus our idea is that the more voters see a given party as viable (likely to win representation somewhere), the more they are likely to vote for it.

^{[2]}This increased tendency to vote for viable national parties is predicated on voters being more tuned in to the national contest than they are concerned over the outcome in their own district, which might even be a “sideshow” (Johnston and Cutler 2009: 94). Thus the approach starts with the national party system, and projects downward, rather than the conventional approach of starting with district-level coordination and projecting upward.

[Paragraph on the origin of 2^{2/3} =1.5874 skipped, given I already explained it above as stemming from the number of pertinent parties when *M*=1.^{3}]

Thus the two terms of the right-hand side of Equation 4 express a district component (two locally pertinent parties) and a nationwide one (how many seat-winning parties are there *effectively* in the parliament being elected?) Note, again, that only the latter component can vary (with the size of the assembly, per Equation 2, or with a given election’s national politics), while the district component is always the same because there is always just one seat to be fought over. Consider some hypothetical cases as illustration. Suppose there are exactly two evenly balanced parties in parliament (*N _{S}* =2.00), these contribute 1.41=√2 to a district’s

*N’*, while the district’s essential tendency towards two pertinent parties contributes 1.59=2

_{V}^{2/3}. Multiply the two together and get 1.59*1.41=2.25. That extra “0.25” thus implies some voting for either local politicians (perhaps independents) not affiliated with the two national seat-winning parties or for national parties that are expected to win few or no seats.

^{[4]}On the other hand, suppose the nationwide

*N*is close to three, such as the 3.03 observed in Canada in 2004. The formula suggests the national seat-winning outcome contributes √3.03=1.74 at the district level; multiply this by our usual 1.59, for a predicted value of

_{S}*N’*=2.77. […] this is almost precisely what the actual

_{V}*average*value of

*N’*was in 2004.

_{V}^{[5]}

^{[1]} The formula for the index, the effective number, squares each party’s seat share. Thus larger parties contribute more to the final calculation.

^{[2]} Likely the key effect is earlier in the sequence of events in which voters decide the party is viable. For instance, parties themselves decide they want to be “national” and so they recruit candidates, raise funds, have leaders visit, etc., even for districts where they may not win. Breaking out these steps is beyond the scope of this paper, but would be essential for a more detailed understanding of the process captured by our logic.

^{[3]} Because the actual number of vote-earning parties (or independent candidates) is a useless quantity, inasmuch as it may include tiny vanity parties that are of no political consequence.

^{[4]} A party having one or two seats in a large parliament makes little difference to *N _{S}*. However, having just one seat may make some voters perceive the party a somehow “viable” in the national policy debate—for instance the Green parties of Canada and the UK.

^{[5]} The actual average was 2.71.

# Small national parties in Canada in the 2021 election and the connection of district voting to national outcomes

One of the notable trends in polling leading up to the Canadian election of 20 September is the increasing vote share of the Peoples Party of Canada (PPC). At the same time, polls have captured a steady decline of the Green Party as the campaign reaches its end. These two small parties’ trends in national support appear to be happening in all regions of the country, albeit to different degrees (see the graphs at the previous link). That is, while these parties have different *levels* of support regionally, their *trends* are not principally regional. Rather, all regions seem to be moving together. This will be a key theme of this post–that politics is fundamentally national, notwithstanding real difference in regional strengths^{1} and the use of an electoral system in which all seat winning is very local (in each of 338 single-seat districts or “ridings”).

The PPC is a “populist” party of the right. It seems that the Conservatives’ attempt to position themselves closer to the median voter during this campaign has provoked some backlash on the party’s right flank, with increasing numbers of these voters telling pollsters they will vote PPC.

At *The Writ*, Éric Grenier offers a look into what the polls say about the type of voter turning to the PPC, and whether they might cost the Conservatives seats. The PPC vote share ranges widely across pollsters but in the CBC Poll Tracker (also maintained by Grenier) it currently averages 6.7%. This would be quite a strikingly high figure for a party that is not favored to win even one seat and probably very unlikely to win more than one.^{2} The Poll Tracker shows a stronger surge in the Prairies region than elsewhere (3.6% on 14 Aug. just before the election was called to 10.9% when I checked on 19 Sept.) and Alberta (4.6% to 9.0% now), but it is being picked up in polling in all regions (for example, from 2.2% to 4.4% in Quebec and 2.9% to 6.1% in Atlantic Canada).

What I wish I knew: *Is a voter more likely to vote PPC if he or she perceives that the party is likely to win at least one seat?* This question is central to the “all politics is national” model developed in Shugart & Taagepera (2017) *Votes from Seats*, in chapter 10. We do not mean “all” to be taken literally. Of course, regional and local political factors matter. We mean that *one can model the average district’s effective number of parties based on the national electoral system.* More to the point, we argue that the way to think of how party systems form under FPTP (or any simple districted system) is not to think in terms of local “coordination” that then somehow gets projected up to a national party system, but rather that the national electoral system shapes the national party system, which then sets the baseline competition in the district contests.

If the PPC or Greens are perceived as likely to have a voice in parliament–and perhaps especially if the parliament is unlikely to have a majority party– voters who like what a small party proposes may be more inclined to support it, even though few voters live in a district where it has any chance of winning locally. Below I will show two graphs, each based on a mathematical model, showing a *relationship of local votes to national seats*. The first is based on the total available seats–the assembly size–while the second will be based on the seat outcome, specifically the nationwide effective number of seat-winning parties. The formula derived in the book for the connection to assembly size states the following for FPTP systems (every district with magnitude, *M*=1, and plurality rule):

** N‘_{V}=1.59S^{1/12}**,

where *N*‘* _{V}* is the mean district-level effective number of vote-earning parties and

*S*is the assembly size. Please see the book for derivation and justification. It may seem utterly nuts, but yes, the mean district’s votes distribution in FPTP systems can be predicted when we know only how many districts there are (i.e., the total number of seats). In the book (Fig. 10.2 on p. 156) we show that this sparse model accurately tracks the trend in the data across a wide range of FPTP countries, particularly if they are parliamentary. Here is what that figure looks like:

Of course, individual election averages (shown by diamonds) vary around the trend (the line, representing the above equation), and individual districts (the smear of heavily “jittered” gray dots) have a wide variation within any given election. But there is indeed a pattern whereby larger assemblies tend to be associated more fragmented district voting than is the case when assembly size is smaller. At *S*=338, Canada has a relatively large assembly (which happens to be almost precisely the size it “should be,” per the cube root law of assembly size).

The model for *N*‘* _{V}* under FPTP is premised on the notion that voters are less attuned to the likely outcome in their own district than they are to the national scene. There is thus a systematic relationship between the national electoral system and the average district’s votes distribution.

Moreover, by combining the known relationship between the national electoral system and the national party system, we can see there should be a direct connection of the district vote distribution to the national distribution of seats. The Seat Product Model (SPM) states that:

*N _{S}*=(

*MS*)

^{1/6},

where *N _{S}* is the nationwide effective number of seat-winning parties. For FPTP, this reduces to

*N*=

_{S}*S*

^{1/6}, because

*M*=1. In terms of a FPTP system, this basically just means that because there are more districts overall, there is room for more parties, because local variation in strengths is, all else equal, likelier to allow a small party to have a local plurality in one of 400 seats than in one of 100. So, more seats available in the assembly (and thus more districts), more parties winning seats. The model, shown above, connecting district-level votes (

*N*‘

*) to the assembly size (*

_{V}*S*) then suggests that the more such seat-winning opportunities the assembly affords for small parties, the more local voters are likely to give their vote for such parties, pushing

*N*‘

*up. The process probably works something like this: Voters are aware that some small parties might win one or more seats somewhere, providing these parties a voice in parliament, and hence are likelier to support small parties to some degree regardless of their local viability. It is*

_{V}*national viability*that counts. “All politics is national.” The posited connection would be more convincing if it could be made with election-specific seat outcomes rather than with the total number of available seats. We can do that! Given the SPM for the national seat distribution (summarized in

*N*) based on assembly size, and the model for district-level votes distribution (

_{S}*N*‘

*), also based on assembly size, we can connect*

_{V}*N*‘

*to*

_{V}*N*algebraically:

_{S} ** N‘_{V}=1.59N_{S}^{1/2}**.

(Note that this comes about because if *N _{S}*=

*S*

^{1/6}, then

*S*=

*N*

_{S}^{6}, giving us

*N*‘

*=1.59(*

_{V}*N*

_{S}^{6})

^{1/12}, in which we multiply the exponents in the final term of the equation to get the exponent, 1/2, which is also the square root. A full discussion and test of this formula may be found in my forthcoming chapter with Cory Struthers in an volume in honor of Richard Johnston being edited by Amanda Bittner, Scott Matthews, and Stuart Soroka. Johnston’s tour de force,

*The Canadian Party System*likewise emphasizes that voters think more in terms of national politic than their local contest.)

Here is how this graph looks:

*N*, which is useful because it makes clear just how well India, in the era of competing alliances, follows the

_{S}*S*model–the one in the first graph. It obviously would not fit the

*N*model if we did not use the alliances, but again, it is the alliances that it should track with if the model is correct in its grounding district-level vote outcomes in the

_{S}*national balance of seats among the national political forces*–parties elsewhere, including Canada, but alliances in India.)

^{4}

By implication, this connection of district-level *N*‘* _{V}* to national

*N*may arise because voters have some estimate of how the national parliament is going to look when they decide whether or not to support a party other than one of the two leading national parties. For instance, a voter wavering between the NDP and the Liberals might be more likely to support the NDP if she estimates that there will be no majority, thereby allowing a smaller party like the NDP to be more influential than if one of the big parties has a majority on its own.

_{S}A vote for a much smaller party, like the PPC, might be simply expressive–“sending a message” to the Conservatives that they are not sufficiently right wing or populist. For a purely expressive voter, the national seat outcome may be irrelevant. Such a voter simply wants to register a protest. There still might be a connection to expected national *votes*: If such a voter thinks the PPC can get 7% he might be likelier to vote for it than if it’s only 3%.^{3} If, however, the connection runs through thinking about the national parliament, and whether one’s party will have voice there, it should help the party win votes around the country if its potential voters perceive that it will win one or more seats–in other words, that it is viable somewhere. I hope there is some polling data that I can find some day that allows us to get at this question, as it would connect the aggregate outcome demonstrated here with individual-level voter behavior. As the Canadian 2021 campaign has developed, it would be an especially good test of the model’s underlying individual-voter premise, given the surge of a small national party that is probably not likely to have a voice in the House of Commons. (But maybe its voters believe it will! They might even turn out to be correct.)

I do not, however, currently know if any polling or voter surveys exist to get at these questions. Such a survey ideally would ask the respondent how many seats they believe the various parties will get in the election. This would allow a rough construction of voter-expected effective number of seat-winning parties even though no voter actually has to know what that concept means or how to calculate it for the premise of the model to work. Minimally, as noted, it would at least be useful to know if voters choosing a small party think that party will indeed get one or more seats.

I have so far focused on the PPC in the Canadian 2021 election, as a possible example of a wider phenomena connecting local voting to the (expected) national seat outcome. A similar logic on the left side of politics should apply for the Green Party. Does its perceived viability for seats in parliament affect the tendency of voters to vote for it outside the specific districts where it is locally viable? The very big wrinkle this time around for the Greens, however, is that the party is struggling mightily, with an ongoing conflict between its leader and much of the rest of the party. It is currently projected to win no more than two seats, and perhaps none. It might be expected to retain the former leader’s seat in British Columbia, but even that may be in jeopardy with the national party in such disarray.

It is even questionable whether the Green Party still meets the criteria of a “national” party this time around; I do not (yet) have a really precise working definition of how many districts the party must be present in to qualify as “national.” The Green Party has not fielded a candidate in about a quarter of the ridings nationwide. Grenier has reviewed the 86 Green-less constituencies and whether their absence could affect outcomes among the contesting parties. Obviously the connection between expected seat winning nationally and obtaining votes in contests around the country is broken in any district in which there is no candidate running for the party. No candidate, no possibility of the local voters augmenting the party’s aggregate vote total. In any case, the party has dropped in national polls from 5.4% on 14 August to 3.2% now.

Further emphasizing now the Greens may not be a “national” party in this election is the campaign behavior of the leader. The CBC recently noted that the leader, Annamie Paul, is not exactly campaigning like the leader of a national party:

Asked why she hasn’t campaigned in more ridings, Paul acknowledged Friday that some candidates may want her to steer clear. She has campaigned outside of her home riding of Toronto Centre twice so far — once in a neighbouring riding and then Monday, in P.E.I.

Candidates distancing themselves from the leader is not normally a good sign for a party, particularly in a parliamentary system. “All politics is national,” after all. As explained in *Votes from Seats* (ch. 10), the impact of national politics on local voting is likely enhanced by parties bringing resources into districts to “show the flag” even where they are not likely to win a seat. (The PPC leader is certainly doing this.) If your leader remains mostly ensconced in her own district, the party is not deploying what is normally one of its best resources–the leader making the case for her party.

Nonetheless, it still might matter for the party’s ability to get votes, even in ridings it surely will not win, whether its potential voters believe it is viable for seat-winning somewhere. The good news for the party–and there is little of that–is that the province where it currently holds two seats, BC, is one of those where its polling has declined least: 7.0% on 14 August to 6.3% now. So, politics is still at least a bit more regional for the Greens than for other “national” parties, perhaps.

In conclusion, the district-level extension of the Seat Product Model states that in FPTP systems, district-level effective number of vote-earning parties can be predicted from the national electoral system–specifically, the assembly size. By further extension (in the aforementioned chapter I am working on with Struthers for the volume honoring Johnston), it should also be tied to the nationwide effective number of seat-winning parties, and to voter perceptions in the campaign as to how parties are doing at the national level. The result would be that voters are more likely to vote for even a small party under FPTP to the extent that they expect it to have a voice in parliament, and to the extent that the parliament may not have a majority party. The Canadian 2021 election, with a surging small party (the PPC) and another one declining (the Greens) offers an excellent case study of the phenomenon that is behind these models.

___________

Notes:

1. Obviously, things are different for an explicitly regional party (one that does not present candidates outside its region) like the Bloc Quebecois, which I will leave aside for this current discussion.

2. Perhaps it has some chance of winning the leader’s riding of Beauce (in Quebec), but as Grenier notes in a post the day before the election:

There’s nothing about Bernier’s Beauce riding that makes it particularly open to a party that has been courting the anti-vaxxer, anti-vaccine mandates and anti-lockdowns crowd. It’s hard to know where in the country that crowd would be big enough to elect a PPC MP.

He does also note that one poll, by EKOS, has put the party second in Alberta, albeit with only 20% of the vote. Maybe they could get a local surge somewhere and pick up a seat there.

3. Indeed, it might seem that we could make a similar algebraic connection. The Seat Product Model expects national effective number of vote-earning parties to be *N _{V}*=[(

*MS*)

^{1/4}+1]

^{2/3}. This is confirmed in

*Votes from Seats*. However, this can’t easily be expressed in terms of just

*S*(even for FPTP, where the term for

*M*drops out) and therefore is complicated to connect to the

*N*‘

*formula. In any case, the theoretical argument works better from seats–that voters key on the expected outcome of the election, which is a distribution of seats in parliament and whether one or another party has a majority or not. These outcomes are summarized in the effective number of*

_{V}*seat-winning*parties.

4. This graph is a version of the one that will be shown in the previouysly mentioned Shugart & Struthers chapter.

# “Effective Seat Product” for two-tier PR (including MMP) and MMM

[**Update, late April, 2022:** I have continued to refine this method, and the specific values mentioned below no longer hold (due a revision of the estimation of the exponent in the model for two-tier systems), although the basic framework remains the same. User beware! This also means that *the datasets linked at the end of this post are not accurate.* I will upload corrected ones at some point.]

The seat product for a simple electoral system is its assembly size (*S*) times its mean district magnitude (*M*) (Taagepera 2007). From this product, *MS*, the various formulas of the Seat Product Model (SPM) allow us to estimate the effective number of parties, size of the largest, disproportionality, and other election indicators. For each output tested in Shugart and Taagepera (2017), *Votes from Seats*, we find that the SPM explains about 60% of the variance. This means that these two institutional inputs (*M* and *S*) alone account for three fifths of the cross-national differences in party system indicators, while leaving plenty for country-specific or election-specific factors to explain as well (i.e., the other 40% of the variance).

The SPM, based on the simple seat product, is fine if you have a single-tier electoral system. (In the book, we show it works reasonably well, at least on seat outputs, in “complex” but still single-tier systems like AV in Australia, majority-plurality in France, and STV in Ireland.) But what about systems with complex districting, such as two-tier PR? For these systems, Shugart and Taagepera (2017) propose an “extended seat product model”. This takes into account the basic-tier size and average district magnitude as well as the percentage of the entire assembly that is allocated in an upper tier, assumed to be compensatory. For estimating the expected effective number of seat-winning parties (*N _{S}*), the extended SPM formula (Shugart and Taagepera, 2017: 263) is:

N=2.5_{S}^{t}(MB)^{1/6},

where *MB* is the basic-tier seat product, defined as the number of seats allocated in the basic tier (i.e., assembly size, minus seats in the upper tier), and *t* is the tier ratio, i.e., the share of all assembly seats allocated in the upper tier. If the electoral system is simple (single tier), the equation reduces to the “regular” seat product model, in which *MS*=*MB* and *t*=0.

(Added note: in the book we use *MS _{B}* to refer to what I am calling here

*MB*. No good reason for the change, other than blogger laziness.)

We show in the book that the extended seat product is reasonably accurate for two-tier PR, including mixed-member proportional (MMP). We also show that the logic on which it is based checks out, in that the basic tier *N _{S}* (i.e., before taking account of the upper tier) is well explained by (

*MB*)

^{1/6}, while the multiplier term, 2.5

^{t}, captures on average how much the compensation mechanism increases

*N*. Perhaps most importantly of all, the extended seat product model’s prediction is closer to actually observed nationwide

_{S}*N*, on average, than would be an estimate of

_{S}*N*derived from the simple seat product. In other words,

_{S}*for a two-tier system, do not just take the basic-tier mean M and multiply by S and expect it to work!*

While the extended seat product works quite well for two-tier PR (including MMP), it is not convenient if one wants to scale such systems along with simple systems. For instance, as I did in my recent planting on polling errors. For this we need an “effective seat product” that exists on the same scale as the simple seat product, but is consistent with the effect of the two-tier system on the effective number of parties (or other outputs).

We did not attempt to develop such an effective seat product in Shugart and Taagepera (2017), but it is pretty straightforward how to do it. And if we can do this, we can also derive an “effective magnitude” of such systems. In this way, we can have a ready indicator of what simple (hypothetical) design comes closest to expressing the impact of the (actual) complex design on the party system.

The derivation of effective seat product is pretty simple, actually. Just take, for the system parameters, the predicted effective number of seat-winning parties, *N _{S}*, and raise it to the power, 6. That is, if

*N*=(

_{S}*MS*)

^{1/6}, it must be that

*MS*=

*N*

_{S}^{6}. (Taagepera 2007 proposes something similar, but based on

*actual*output, rather than expected, as there was not to be a form of the seat product model for two-tier systems for almost another decade, till an initial proposal by Li and Shugart (2016).)

Once we do this, we can arrive at effective seat products for all these systems. Examples of resulting values are approximately 5,000 for Germany (MMP) in 2009 and 6,600 for Denmark (two-tier PR) in 2007. How do these compare to simple systems? There are actual few simple systems with these seat products in this range. This might be a feature of two-tier PR (of which MMP could be considered a subtype), as it allows a system to have a low or moderate basic-tier district magnitude combined with a high degree of overall proportionality (and small-party permissiveness). The only simple, single-tier, systems with similar seat products are Poland (5,161), with the next highest being Brazil (9,747) and Netherlands before 1956 (10,000). The implication here is that Germany and Denmark have systems roughly equivalent in their impact on the party system–i.e., on the 60% of variance mentioned above, not the country-specific 40%–as the simple districted PR system of Poland (*S*=460, *M*=11) but not as permissive as Brazil (*S*=513, *M*=19) or pre-1956 Netherlands (*M*=*S*=100). Note that each of these systems has a much higher magnitude than the basic-tier *M* of Germany (1) or larger assembly than Denmark (*S*=179; *M*=13.5). Yet their impact on the nationwide party system should be fairly similar.

Now, suppose you are more interested in “effective district magnitude” than in the seat product. I mean, you should be interested in the seat product, because *it tells you more about a system’s impact on the party system than does magnitude alone*! But there may be value in knowing the input parameters separately. You can find *S* easily enough, even for a complex system. But what about (effective) *M*? This is easy, too! Just take the effective seat product and divide it by the assembly size.

Thus we have an effective *M* for Germany in 2009 of 7.9 and for Denmark in 2007 of 36.9. These values give us an idea of how, *for their given assembly sizes*, their compensatory PR systems make district magnitude “effectively”–i.e., in terms of impact on the inter-party dimension–much larger than the basic-tier districts actually are. If we think low *M* is desirable for generating local representation–a key aspect of the intra-party dimension–we might conclude that Germany gets the advantages *M*=1 in local representation while also getting the advantages of the proportionality of 8-seat districts. (Best of both worlds?) By comparison, simple districted PR systems with average *M* around 8 seats include Switzerland and Costa Rica. (The Swiss system is complex in various ways, but not in its districting.) Eight is also the *minimum* magnitude in Brazil. Denmark gets whatever local representation advantages might come from an actual mean *M* of 13.5, yet the proportionality, for its assembly size, as if those districts elected, on average, 37 members. Actual districts of about this magnitude occur only in a relatively few districts within simple systems. For instance, the district for Madrid in Spain has *M* in the mid-30s, but that system’s overall average is only 6.7 (i.e., somewhat smaller than Germany’s *effective* *M*).

Now, what about **mixed-member majoritarian** (MMM) systems. Unlike MMP, these are not designed with a compensatory upper tier. In *Votes from Seats*, Taagepera and I basically conclude that we are unable to generalize about them. Each MMM system is *sui generis*. Maybe we gave up too soon! I will describe a procedure for estimating an effective seat product and effective magnitude for MMM systems, in which the basic tier normally has *M*=1, and there is a list-PR component that is allocated in “parallel” rather than to compensate for deviations from proportionality arising out of the basic tier.

The most straightforward means of estimating the effective seat product is to treat the system as a halfway house between MMP and FPTP. That is, they have some commonality with MMP, in having both *M*=1 and a list-PR component (not actually a “tier” as Gallagher and Mitchell (2005) explain). But they also have commonality with FPTP, where all seats are *M*=1 plurality, in that they reward a party that is able to win many of the basic seats in a way that MMP does not. If we take the geometric average of the effective seat product derived as if it were MMP and the effective seat product as if it were FPTP, we might have a reasonable estimate for MMM.

In doing this, I played with both an “effective FPTP seat product” from the basic tier alone and an effective FPTP seat product based on assuming the actual assembly size. The latter works better (in the sense of “predicting*,” on average* for a set of MMM systems, what their actual *N _{S}* is), and I think it makes more logical sense. After all, the system should be more permissive than if were a FPTP system in which all those list-PR component seats did not exist. So we are taking the geometric average of (1) a hypothetical system in which the entire assembly is divided into a number of single-seat electoral districts (

*E*

_{eff}) that is

*E*

_{eff}=

*E*

_{B}+

*tS*, where

*E*

_{B}is the actual number of single-seat districts in the basic tier and

*S*and

*t*are as defined before, and (2) a hypothetical system that is MMP instead of MMM but otherwise identical.

When we do this, we get the following based on a couple sample MMM systems. In Japan, the effective seat product becomes approximately 1,070, roughly equivalent to moderate-*M* simple districted PR systems in the Dominican Republic or pre-1965 Norway. For South Korea, we would have an effective seat product of 458, or very roughly the same as the US House, and also close to the districted PR system of Costa Rica.

Here is how those are derived, using the example of Japan. We have *S*=480, with 300 single-seat districts and 180 list-PR seats. Thus *t*=0.375. If it were two-tier PR (specifically, MMP), the extended seat product would expect *N _{S}*=3.65, from which we would derive an effective seat product, (

*MS*)

*=3.66*

_{eff}^{6}=2,400. But it is MMM. So let’s calculate an effective FPTP seat product.

*E*

_{eff}=

*E*

_{B}+

*tS*=300+180=480 (from which we would expect

*N*=2.80). We just take the geometric mean of these two seat-product estimates: (2400*480)

_{S}^{1/2}=1,070. This leads to an expected

*N*=3.19, letting us see just how much the non-compensatory feature reduces expected party-system fragmentation relative to MMP as well as how much more permissive it is than if it were FPTP.

_{S}How does this work out in practice? Well, for Japan it is accurate for the 2000 election (*N _{S}*=3.17), but several other elections have had

*N*much lower. That is perhaps due to election-specific factors (producing huge swings in 2005 and 2009, for example). As I alluded to above already, over the wider set of MMM systems, this method is pretty good on average. For 40 elections in 17 countries, a ratio of actual

_{S}*N*to that predicted from this method is 1.0075 (median 0.925). The worst-predicted is Italy (1994-2001), but that is mainly because the blocs that formed to cope with MMM contained many parties (plus Italy’s system had a partial-compensation feature). If I drop Italy, I get a mean of 1.0024 (but a median of only 0.894) on 37 elections.

_{S}If we want an effective magnitude for MMM, we can again use the simple formula, *M _{eff}*=(

*MS*)

*/*

_{eff}*S*. For Japan, this would give us

*M*=2.25; for Korea

_{eff}*M*=1.5. Intuitively, these make sense. In terms of districting, these systems are more similar to FPTP than they are to MMP, or even to districted PR. That is, they put a strong premium on the plurality party, while also giving the runner-up party a considerable incentive to attend to district interests in the hopes of swinging the actual district seat their way next time (because the system puts a high premium on M=1 wins, unlike MMP). This is, by the way, a theme of the forthcoming

_{eff}*Party Personnel*book of which I am a coauthor.

(A quirk here is that Thailand’s system of 2001 and 2005 gets an effective magnitude of 0.92! This is strange, given that magnitude–the real kind–obviously has a lower limit of 1.0, but it is perhaps tolerable inasmuch as it signals that Thailand’s MMM was really strongly majoritarian, given only 100 list seats out of 500, which means most list seats would also be won by any party that performed very well in the *M*=1 seats, which is indeed very much what happened in 2005. The concept of an “effective” magnitude less than 1.0 implies a degree of majoritarianism that one might get from multi-seat plurality of the MNTV or list-plurality kind.)

In this planting, I have shown that it is possible to develop an “effective seat product” for two-tier PR systems that allows such systems to be scaled along with simple, single-tier systems. The exercise allows us to say what sort of simple system an actual two-tier system most resembles in its institutional impact on inter-party variables, like the effective number of seat-winning parties, size of the largest party, and disproportionality (using formulas of the Seat Product Model). From the effective seat product, we can also determine an “effective magnitude” by simply dividing the calculated effective seat product by actual assembly size. This derivation lets us understand how the upper tier makes the individual district *effectively* more proportional while retaining an actual (basic-tier) magnitude that facilitates a more localized representation. Further, I have shown that MMM systems can be treated as intermediary between a hypothetical MMP (with the same basic-tier and upper-tier structure) and a hypothetical FPTP in which the entire assembly consists of single-seat districts. Again, this procedure can be extended to derive an effective magnitude. For actual MMP systems in Germany and also New Zealand, we end up with an effective magnitude in the 6–8 range. For actual MMM systems, we typically get an effective magnitude in the 1.5–3 range.

I will post files that have these summary statistics for a wide range of systems in case they may be of use to researchers or other interested readers. These are separate files for MMM, MMP, and two-tier PR (i.e, those that also use PR in their basic tiers), along with a codebook. (Links go to Dropbox (account not required); the first three files are .CSV and the codebook is .RTF.) [As noted at the top of this article, *these files should no longer be used.* At some point I will upload corrections. Sorry for the inconvenience.]

Added note: In the spreadsheets, the values of basic-tier seat product (*MB*) and tier ratio (*t*) are not election-specific, but are system averages. We used a definition of “system” that is based on how Lijphart (1994) defines criteria for a “change” in system. This is important only because it means the values may not exactly match what you would calculate from the raw values at a given election, if there have been small tweaks to magnitude or other variables during an otherwise steady-state “system”. These should make for only very minor differences and only for some countries.

# Chile 2017: Meet your new seat product

As discussed previously, Chile has changed its electoral system for assembly elections (and for senate). The *seat product* (mean district magnitude times assembly size) was increased substantially. Now that the 2017 Chilean election results are in, did the result come close to the Seat Product Model (SPM) predictions?

The old seat product was 240 (2 x 120). The new seat product is 852.5 (5.5 x 155). This should yield a substantially more fragmented assembly, according to the SPM (see *Votes from Seats* for details).

I will use the effective number of parties (seats and votes) based on alliances. The reason for this choice is that it is a list PR system, and the electoral system works on the lists, taking their votes in each district and determining each list’s seats. Lists are open, and typically presented by pre-election alliances, and the candidates on a list typically come from different parties. But the question of which parties win the seats is entirely a matter of the* intra-list* distribution of preference votes (the lists are open), and not an effect of the electoral system’s operation on the entities that it actually processes through seat-allocations formula–the lists. However, I will include the calculation by sub-alliance parties, too, for comparison purposes. [*Click here for an important correction on the intra-list allocation*. Nonetheless, the error in the above does not affect any of the calculations in this post.]

The predicted values with the new system, for effective number of seat-winning lists (*N _{S}*) and effective number of vote-earning lists (

*N*), given a seat product of 825.5, are:

_{V}N

_{S}=3.08 (SPM, new system)N

_{V}=3.45 (SPM, new system).

The actual result, by alliance lists, was:

N

_{S}=3.09N

_{V}=4.02.

**So the Chamber of Deputies is almost exactly as fragmented as the SPM predicts!** In the very first election under the new system! The voting result is somewhat more fragmented than expected, but not wide of the mark (about 14%). It is not too surprising that the votes are more off the prediction than the seats; voters have no experience with the new system to draw on. However, the electoral system resulted in an assembly party system (or more accurately, alliance system) fully consisted with its expected “mechanical” effect. The SPM for *N _{S}* is derived from the constraints of the number of seats in the average district and the total number of seats, whereas the SPM for

*N*makes a potentially hazardous assumption about how many “pertinent” losers will win substantial votes. We can hardly ask for better adjustment to new rules than what we get in the

_{V}*N*result! (And really, that

_{S}*N*result is not too shabby, either.)

_{V }Now, if we go by sub-alliance parties, the system seems utterly fragmented. We get *N _{S}*=7.59 and

*N*=10.60. These results really are meaningless, however, from the standpoint of assessing how the electoral system constrains outcomes. These numbers should be used only if we are specifically interested in the behavior of parties within alliances, but not for more typical inter-party (inter-list) electoral-system analysis. It is a list system, so in systems where lists and “parties” are not the same thing, it is important to use the former.

_{V}To put this in context, we should compare the results under the former system. First of all, what was expected from the former system?

N

_{S}=2.49 (SPM, old system)N

_{V}=2.90 (SPM, old system).

Here is the table of results, for which I include Np, the effective number of presidential candidates, as well as *N _{V}* and Ns on both alliance lists and sub-alliance parties.

By alliance | By sub-list party | ||||

year | N_{S} |
N_{V} |
N_{P} |
N_{S} (sub) |
N_{V} (sub) |

1993 | 1.95 | 2.24 | 2.47 | 4.86 | 6.55 |

1997 | 2.06 | 2.54 | 2.47 | 5.02 | 6.95 |

2001 | 2.03 | 2.33 | 2.19 | 5.94 | 6.57 |

2005 | 2.02 | 2.36 | 3.01 | 5.59 | 6.58 |

2009 | 2.17 | 2.56 | 3.07 | 5.65 | 7.32 |

mean | 2.05 | 2.41 | 2.64 | 5.41 | 6.79 |

We see that the old party (alliance) system was really much more de-fragmented than it should have been, given the electoral system. The party and alliance leaders, and the voters, seem to have enjoyed their newfound relative lack of mechanical constraints in 2017!

Can the SPM also predict *N _{P}*? In

*Votes from Seats*, we claim that it can. We offer a model that extends form

*N*

_{V }to

*N*; given that we also claim to be able to predict

_{P}*N*

_{V}from the seat product (and show that this is possible on a wide range of elections), then we can also connect

*N*to the seat product. We offer this prediction of

_{P}*N*from the seat product as a counterweight to standard “coattails” arguments that assume presidential candidacies shape assembly fragmentation. Our argument is the reverse: assembly voting, and the electoral system that indirectly constraints it, shapes presidential fragmentation.

_{P}There are two caveats, however. The first is that *N _{P}* is far removed from, and least constrained by, assembly electoral systems, so the fit is not expected to be great (and is not). Second, we saw above that

*N*in this first Chilean election under the new rules was itself more distant from the prediction than

_{V}*N*was.

_{S}Under the old system, we would have predicted Np=2.40, so the actual mean for 1993-2009 was not far off (2.64). Under the new system, the SPM predicts 2.62. In the first round election just held, *N _{P}*=4.17. That is a good deal more fragmented than expected, and we might not expect future elections to feature such a weak first candidate (37% of the vote). It is unusual to have

*N*>

_{P}*N*, although in the book we show that Chile is one of the countries where it has happened a few times before. Even the less constraining electoral system did not end this unusual pattern, at least in 2017.

_{V}In fact, that the assembly electoral system resulted in the expected value of *N _{S}*, even though

*N*was so high, is pretty good evidence that it was

_{P}**not**coattails driving the assembly election. Otherwise, Ns should have overshot the prediction to some degree. Yet it did not.

# Netherlands, compared to the Seat Product expectation

The recent election in the Netherlands was noteworthy for its high fragmentation. But was it higher than we should expect, given an extremely proportional electoral system? If so, how much higher?

Fortunately, from Taagepera’s Seat Product Model, we have a baseline against which to compare any given election. For “simple” electoral systems–those with a single tier of allocation and a basic PR formula (or FPTP)–we expect:

N

_{S}=(MS)^{1/6}

and

s_{1}=(MS)^{-1/8}.

N_{S} is the effective number of seat-winning parties, whereas s_{1} is the seat share of the largest party. MS is the seat product, defined as the mean district magnitude, times the assembly size. The derivation of these models for expectations may be found in Taagepera (2007), and is also summarized in Li and Shugart (2016) and my forthcoming book with Taagepera, *Votes from Seats*.

Two important points about these models: (1) They are not mere regression estimates, bur rather are derived deductively; (2) On average, they are remarkably accurate. For long-term European democracies, the mean ratio of actual N_{S} to the model expectation is 1.007; for s_{1} it is 1.074. (They are not substantially less accurate for other regions or younger democracies, but given the topic of this post, the longer-run European democracies are the most relevant comparison set. The ratios reported are based on 219 individual elections.)

For the Netherlands, with a single nationwide district, MS=150*150=22,500. This means we should expect, *on average*, in an electoral system like that of the Netherlands, that N_{S}=5.31 and the largest party has 28.6% of the seats (s_{1}=0.286). In other words, *we should expect the Dutch party system to be quite fragmented*.

In the graphs below, we compare the actual values in each election since 1945 to the Seat Product expectation.

Strikingly, both values are well off the expectation now and have been in some other recent elections–but not so much as recently as 2012 or 2006. The 2017 election appears to be a continuation or acceleration a trend, but that trend has been somewhat irregular. Note, however, a bit farther back in the past there were elections in which N_{S} was much *lower* than expected, and s_{1} much *higher*–in other words, when fragmentation was less than expected. (Note to readers: On 31 March I revised this paragraph to better reflect the recent trends shown in the graph.)

Over the entire period, the mean effective number of seat-winning parties has been 5.08, and the mean largest seat share has been 0.29. In other words, the Netherlands has not been exceptional in its long-term averages, given its extremely high seat product.

A key question is whether fragmentation will again come back closer to expectation. This is not a question the Seat Product Model can answer. But note that if we had been running this test in about 1986, I might have said, “will the Dutch electoral system ever again fragment, like we’d expect?” Sometimes things even out, sometimes they don’t.

Obviously, the fragmentation inn 2017 is far higher, relative to the baseline than it was during the previous (and brief) fragmenting around 1970. Perhaps that means the Dutch party system has entered a new phase from which there is no turning back. The very high proportionality of the system means it can sustain this level of fragmentation without anyone being seriously under-represented. On the other hand, one might want to be careful about assuming recent trends can’t reverse themselves. Parties could merge, or voters could tire of voting for small parties that are only bit players in policy-making.

The value of the Seat Product Model is it lets us go beyond simply saying “the Netherlands uses PR, so fragmentation is no surprise” or, alternatively, “fragmentation is out of control in the Netherlands”. It lets us say just how much the fragmentation in the Netherlands is out of whack with expectation. In 2017, the precise answer is that N_{S} is 1.62 times the expectation, while s_{1} is 77% of expectation. That degree of divergence from the expectation is almost at the 99th percentile for N_{S} among European countries; the divergence for s_{1} is at about the 18th percentile.

Will actual and expected values again converge in the Netherlands? Stick around for a few more elections and see.