What does the Seat Product Model say about the Welsh electoral reform?

Wales will use a new electoral system at its next devolved assembly election (see the previous planting by JD). The new system will be a purely list-PR system using closed lists, replacing the mixed-member proportional (MMP) system in previous use. The PR system will consist of sixteen districts of six seats each, for a total of 96 seats. Although this is a more proportional system than the rather restrictive form of MMP that they currently have, it is still not very proportional by design, as we shall see.

Under the Seat Product Model for simple (single-tier) systems, we expect the seat share of the largest party (s₁) to be s₁=(MS)^–1/8, where M is the mean district magnitude (here, 6) and S is the assembly size (96). Thus the seat product is 6*96=570. This is a quite modest seat product by the standards of PR systems, owing to the rather small assembly size and moderate magnitude. Plug 570 into the preceding equation and you get expected s₁=0.45. As to the expected effective number of seat-winning parties (N_S), the formula is N_S=(MS)^1/6. This yields expected N_S=2.88.

How does this seat product, along with its resulting expected outputs, compare to the outgoing system in terms of its actual outputs and expectations? To answer this we have to turn to the extended Seat Product Model for two-tier PR. I have been working on an improvement over the version of this that appears in Votes from Seats. I debuted an earlier version of it here, and the current version that I am using is essentially the one previously discussed on this blog, but with a further refinement. For s₁ the expected value would be captured by the following formula:

s₁=(1–t)^c(MS_B)^–1/8,

where we are now using the basic-tier seat product (MS_B), which is to say the mean magnitude of that tier times the number of seats in that tier, with t being the tier ratio defined as the share of the total assembly that is elected in the compensation (“upper”) tier, and c is the compensation coefficient. This latter coefficient tends to be in the 0.2–0.33 range, but it depends on how big the basic-tier seat product is. If you have a large basic tier seat product, then compensation has little effect (c tending towards 0 so that compensation makes no difference¹) but if the basic tier is small, then the size of the compensation tier matters greatly (c tending towards 1), with ultimate compensation limited by the upper tier’s overall size relative to the total assembly. A good empirical fit for compensation factor is given by the following formula:

c = 1 – 0.38(MS_B)^1/10.

For the MMP system in Wales, the parameters are M=1, MS_B=40, and t=0.33 (based on there being 20 compensation seats in an assembly of 60). Thus we have c = 1 – 0.38(40)^1/10 = 1 – 0.38(1.45) = 0.45. Plug that into the expected-s₁ formula and you get s₁ = (1–t)^c(MS_B)^–1/8 = (1–0.33)^0.45(40)^–1/8 = 0.835*0.63 = 0.527. The expected size of the largest party under the MMP system Wales has had until now is 52.7% of the seats, which out of 60 means about 32 seats.

We can also calculate an “effective seat product” in order to compare the structure of this complex system to a simple one. If s₁=(MS)^–1/8, then it must be that, for a simple system, MS=s₁^–8. So if we expect a given complex system to produce s₁=0.527, then its effective seat product is 168. Thus Wales has rather substantially increased its (effective) seat product, from 168 to 576, and this increase is reflected in an expected decrease of the size of the largest party from about 53% of seats to about 45%.

Despite an increase in (effective) seat product from 168 to 576, the latter value still signals that the Welsh PR system will not be highly proportional. That is a seat product that is smaller than that of the UK House of Commons after all, even though the latter is FPTP (MS=650) rather than PR. The best example of an existing PR system with an effective seat products between the high 500s and mid 600s would be Ireland; in a similar ballpark are Cyprus and Peru. Still, this certainly puts them in more proportional range when we consider that their MMP system actually had an effective seat product of 168, a value that is more typical of FPTP countries like Zambia (159) and Malawi (193) or a small-assembly MMM like Moldova in 2019 (which also had an effective seat product of 168).

As for deviation from proportionality (D, using the Gallagher index), the formula for expected deviation is D=0.33(MS)_eff^–1/3, using our effective magnitude in case of a two-tier system. So for the former MMP system, given (MS)_eff=168, we would expect D=0.06. For the new PR system, with (MS)_eff=576, it should decrease to D=0.04.

The table below shows the actual indicators for MMP elections in Wales. We see that the reality of Welsh politics has been a good deal more fragmented than expected. In other words, Welsh voters have been voting as if their electoral system were more proportional than it really is. In this sense, the shift to a simple proportional system (even if the seat product is only moderate rather than really large) may be just what Wales needs. My general take is if voters in a majoritarian or majoritarian-leaning system vote as if they have PR, then by all means given them PR! Yes, MMP is a “PR” system, but when the compensation tier is only one third, and the basic tier has a seat product of only 40, it can’t be very proportional. We see that from the expectations of a largest party with 52.7% of seats, even if the actual mean has been only 48%. Note, crucially, that this actual mean of just under half the seats comes despite a mean largest party vote share of only 34%! (Using the list votes, but even if we use the district-level nominal votes, that 48% is coming on just 38% of the vote.) This big gap between party votes and seats is, of course, why the deviation from proportionality is rather high, at D=0.11.² With voters producing an effective number of vote-earning parties that has average 4.42, more than one and a half times the expectation for their MMP system’s effective seat product,³ clearly they have implied a desire for (and perhaps a belief that they actually have) a more strongly proportional system.

election	Seats of 1st party	s1	v1-list	v1_nom	Ns	Nv	D
1999	28	0.47	0.354	0.376	3.03	3.82	0.0861
2003	30	0.50	0.366	0.4	3.00	4.38	0.1039
2007	26	0.43	0.296	0.322	3.33	5.08	0.1136
2011	30	0.50	0.369	0.423	2.90	4.36	0.1047
2016	29	0.48	0.315	0.347	3.11	4.95	0.1302
2021	30	0.50	0.362	0.399	2.71	3.91	0.1096
mean	28.83	0.48	0.34	0.38	3.01	4.42	0.11
expected	32	0.527	0.47	?	2.36	2.79	0.06
ratio of actual to expected		0.91	0.73		1.28	1.58	1.80
expected from new system	43 (of 96)	0.45	0.41	—	2.88	3.26	0.04

It bears noting, however, that the new electoral system that will be used starting in 2026 should be expected to yield an effective number of parties that is still lower than the actual average over the six elections since 1999 (2.88 by seats, which would imply 3.26 by votes). The expected values from applying the Seat Product Model to the newly adopted system are shown in the last row of the table. Thus even this new system may not be permissive enough for representing the party system Welsh voters have been actually voting for. But it is certainly a significant step in the right direction.

Finally I want to entertain an idea that was not formally considered, which would have kept MMP but with a much bigger compensation tier, and hence greater (effective) seat product. Given the demonstrated willingness to have a significant increase in assembly size, what if they had kept the new number of Westminster districts (32) as their basic tier of single-seat Welsh Assembly districts, with the rest for countrywide compensation? By using the above formulas on a 32-seat basic tier of M=1 and a 64-seat compensation tier⁴ (giving us t=0.33 and c=0.46), we get an expected s₁=0.39, an effective seat product of 1,824, and an implied largest vote-earning party of around 36%, as well as effective numbers of N_S=3.5 and N_V=3.9. Notice how close to the actual averages since 1999 those expected outputs would be! Expected deviation from proportionality for this hypothetical reformed MMP system would be around 0.027. So there was a way to give Welsh voters what they seem to want, while still having single-seat constituency elections, albeit for only a third of the total seats (based on assumption of not wanting to redistrict in order to have more than 32 constituencies). Had they asked me, I would have suggested they at least put some such model on the table. Oh well. I suspect Welsh voters might be a little disappointed with the PR model they will be getting because of the fairly restrictive seat product (by PR standards)–and I have not even entertained the issue of closed lists versus open lists or STV (which generated controversy). While their seat product will remain pretty small for PR, this reform is still a large step towards giving Wales greater proportionality.

1. Because c=0 turns the first term of the right-hand side of the formula to 1. ↩︎
2. Such a deviation from proportionality is quite high for “PR” systems, whether mixed-member or “pure” list. For all PR systems in my dataset, the mean is 0.038. For MMP systems the mean is about the same, at 0.034 (0.037 if the very large effective seat product of Germany is removed from the calculation). In other words, 0.11 is high! In fact, it is higher than in any of 41 MMP elections in the dataset and beyond the 90th percentile for all PR systems (677 elections). ↩︎
3. For expected effective number of vote-earning parties (N_V), I used the formula developed in Votes from Seats: N_V =(N_S^1.5 +1)^2/3. ↩︎
4. That would be a quite large tier with (presumably) close lists, but it could have been broken down into regional lists while still using countrywide compensation. ↩︎