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.

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