Is the effective number of parties rising over time?

I was recently having a conversation with another political scientist who showed me a graph that suggests the effective number of vote-earning parties in established democracies has been increasing over time. I was skeptical that it was, relative to baseline. Of course, if we do not have a baseline, we do not really know what is causing any such possible increase. The baseline should be the Seat Product Model, which tells us what we should expect the effective number of parties to be, given the electoral system. When we do the baseline, the increase over time remains, but is not significant.

Here is a graph with no baseline. It is just the the effective number of vote-earning parties (NV) in Western Europe (most countries–see below for notes on coverage). The scatterplot marks elections by a three-letter abbreviation for each country. The x-axis is years since 1945, the earliest election year in the dataset. The graph’s y-axis is unlogged, but the plotted regression curve and 95% confidence intervals are based on a logged NV.

(The regression is a GLS with random effects by country. It would not be much different if OLS were used.)

There does seem to be an increase over time. The regression estimates NV averaging around 3.42 in 1945 and around 4.60 in 2011. The 95% confidence intervals on those estimates are 2.96 – 3.96 and 3.98 – 5.32, respectively. So, yes, the vote is getting more fragmented over time in Western Europe!

But hold on a moment. We should look at the fragmentation relative to baseline. As shown in Votes from Seats, the seat product (mean district magnitude times assembly size; in a two-tier system, also taking into account the size of the compensatory tier) explains around 60% of the variance in key party-system outcomes, including the effective number of parties. It would be useful to know if the Seat Product Model (SPM) is on its way to being unable to account for party-system fragmentation if current trends continue. It would be useful to know if recent fragmentation is part of that other 40% (i.e., the amount of variance in NV that the SPM can’t account for). That is, are we witnessing some inexorable fragmentation of party systems that is resulting from the breakdown of existing party alignments in the electorate, and which electoral systems have begun to lose their ability to constrain? Should Western European countries go so far as to reduce their proportionality, in order to contain fragmenting trends?

So the next data visualization asks the question from a different perspective. Is the ratio of observed fragmentation to the SPM prediction increasing over time? We can take any given election’s actual NV, divided by the SPM-predicted NV to arrive at a ratio, which is equal to 1.00 for any election in which the result exactly matches the predicted value. (In other words, if R2=100%, all elections would have a ratio of 1.00.)

Here it is, for NV, again with the estimates from a GLS regression and the 95% intervals. In the regression, the ratio is entered as its decimal log, but the graph uses the underlying values for ease of interpretation.

What we see is indeed an increase (note the slope of the dashed line). However, the reference line at 1.00 (the log of which is, of course, zero) is easily within the 95% confidence interval of the regression throughout the six and a half decades of the data series. The regression estimates a ratio of actual to SPM of 0.911 in 1945 and 1.054 in 2011. The 95% confidence intervals are 0.782 – 1.062 and 0.905 – 1.228, respectively.

In other words, the increase is not statistically significant. There may in fact be an increase, which is to say that something in that other 40% is driving, over time, the SPM to be less successful at predicting the fragmentation of the vote. However, it could just be “noise”; we really can’t say, statistically, because of 1.00 remaining well within the confidence interval.

If it continues on current pace, then 1.00 (or rather its log) will be outside the confidence interval on NV as soon as the year 2065. I will put it on my calendar to check how we are doing at that time.

Independent of the statistical significance, there could be something of interest going on. Note that the regression trend does not cross the 1.00 line till about 42 years into the time series (i.e., 1987). This suggests that, prior to that time, the average election in Western Europe saw the vote be less fragmented than it “should have been”, according to its electoral system. That could suggest that major party organizations were partially overriding the electoral-system effect (producing party systems on average around 90% as fragmented as expected) in the early post-war years. In more recent times, the weakening of party alignments could be making the electoral system expectation finally be realized, with some tendency to exceed in recent times. But we really can’t say, given that the main conclusion is the SPM is all right, and should be for a little while yet, even if the current trend continues (which, of course, it might not).

I also wanted to checked the parliamentary party systems, that is, the effective number of seat-winning parties (NS).

Here it is even more clear that the SPM is doing all right! It is only about now that the regression estimate has finally reached 1.00, but the rate of increase is more minor than with NV, and clearly of minimal significance.

The regression estimates a ratio of actual NS to SPM prediction of around 0.909 in 1945 and 0.991 in 2011. Confidence intervals are 0.773 – 1.068 and 0.843 – 1.164, respectively.

It is somewhat interesting that the trend in the ratio for NV is rising above 1.00 before the ratio for NS. Perhaps there’s an explanation of interest in there. The electoral system more directly constrains NS, after all, and voters perhaps are more willing to “waste” votes as party alignments decrease. But it could just be noise.

(If I do a graph like the first one, with NS with the baseline, there is an increase, but less significant than for NV.)

The conclusion is that there is indeed some truth to the notion that West European party systems are fragmenting. However, relative to the Seat Product Model, they are fragmenting at a slow and hardly significant pace. How can that be? Well, perhaps it is obvious, or perhaps it is not. But a country’s seat product tends to increase over time. Most countries included here have expanded their assemblies over time, and some have also increased district magnitudes and/or adopted upper (compensatory) tiers. So, the effective number of parties should increase to some degree over time, even if voters were just as moored to their party organizations and identities as they ever were!

_________

Appendix: some details.


On the last point above: Specifically, a GLS regression on expected NS says we should have seen on average NS=3.58 in 1945 but 3.67 in 2011. That is not much, but it means some increase is “baked in” even before we look at how actual voters behave. Some part of the increase is in the 60% rather than the 40%.


I dropped Belgium and Italy from the regressions, although they are included in the scatterplots for recent years. The reasons for dropping are that we could not obtain data for the share of seats allocated in upper (compensatory) tiers for the years when these countries used multi-tier PR systems; without that, we can’t calculate the extended version of the SPM (for 2-tier PR). In the later years in the Italy series, when we have such data, these are actually even more complex rules (involving a majoritarian component and alliance vote-pooling), and so the SPM really can’t predict them. In Belgium, the electoral system has been “simple” since 2003, but I think we can agree that there is no semblance of a national party system in that country.


France is also not included, partly due to the importance of the elected presidency (after 1965) and partly due to the two-round system for assembly (after 1958). We do show in Votes from Seats that the SPM works pretty well for France nonetheless. So I doubt its inclusion would have altered the results much. But I wanted to stick to the PR systems and FPTP, which the SPM is designed to handle.

Canadian provincial elections and the SPM: Those assemblies are too small

After posting my earlier overview of expectations from possible electoral reform in British Columbia, I was wondering how well the Seat Product Model has performed over time in Canadian provincial assembly elections. Spoiler alert: not nearly as well in the provinces as a whole as in BC, and better for votes than for seats. The latter is particularly puzzling; the model works by first estimating seats (which are more “mechanically” constrained by district magnitude and assembly size than are votes). That is why the book Rein Taagepera and I published in 2017 is called Votes from Seats. The key to the puzzle may be the serious under-sizing of Canadian provincial assemblies. As I will show in a table at the end of this entry, many provincial assemblies should be almost twice their current size, if we go by the cube root law.

I have a dataset originally constructed for my “to keep or change FPTP” project (published in Blais, ed., 2008). It has most provincial elections back to around 1960, although it stops around 2011. Maybe some day I will update it. For now, it will have to do.

The first graph shows the degree of correspondence between a given election’s observed effective number of vote-earning parties (NV) and the expectation from its seat product (i.e., for FPTP, the assembly size). The black diagonal is the equality line: perfect prediction would place an election on this line. The lighter diagonal is a regression line. Clearly, the mean Canadian provincial election exhibits NV higher than expected. On the other hand, the 95% confidence interval (dashed curves) includes the equality line other than very trivially near the middle of the x-axis range. Thus, in statistical terms, we are unable to reject the hypothesis that actual NV in Canadian provincial elections is, on average, as expected by the SPM. However, the regression-estimated line is systematically on the high side.

It might be noted that we never have NV expected to be 2.0, and in the largest provinces, the assemblies are large enough that we should expect NV>2.5. (“Large enough” here meaning independent of what they “should be”, by the cube root; this is referring only to actual assembly size.) So the classic “Duvergerian” outcome is really only expected in the one province with the smallest assembly. And such an outcome is more or less observed there, in PEI. Nonetheless, a bunch of elections are very much more fragmented than expected, with NV>3! And several are unexpectedly low; many of these are earlier elections in Quebec.

All individual elections are labelled; in a few cases the label generation did not work well (some elections in 2000s). The regression coefficient is significant, although the regression’s R2 is only 0.15. The basic conclusion is a marginally acceptable fit on average, but lots of scatter and some tendency for the average election to be more fragmented than expected.

Now, for the largest seat-winner in the assembly (s1). Here things get a little ugly.

This might be considered a rather poor fit. There is a systematic tendency for the largest party to be bigger than expected: note that the equality line is essentially never within the 95% confidence interval. When we expect, based on assembly size, the largest party to have 60% of the seats, it actually tends to have more like 68%. More importantly, the scatter is massive. In fact, the regression coefficient is insignificant here; please do not ask what the  R2 is!

The size of Canadian provincial parliaments is never so large that the leading party is expected to have only 50% of the seats (note the reference lines and where the equality line crosses the 50-50 point). Yet there is not a trivial number of elections with the largest party under 50%. More common, however, are the blowout wins, where the largest party has 80% or more of the seats. This has been a chronic feature of Canadian provincial politics, especially in a few provinces (notably Alberta, New Brunswick, and Prince Edward Island).

Why is Nv so much better predicted (even if not exceptionally well) than s1? It is hard to say. It is an unusual situation to have both NV and s1 trend higher than expected. After all, normally the more “significant” parties there are the smaller the largest party should tend to be. In the set of predictive equations, s1 (and the effective number of seat-winning parties, NS) are prior to NV, because the seat-based measures are more directly constrained. This is why the book is called Votes from Seats. In our diagram (p. 149) deriving the various quantities from the seat product, we show NV and s1 coming off separate branches from “Ns0” (the actual number of parties winning seats), which is expected to be (MS)0.25, where M is the magnitude and S the assembly size. Thus for FPTP, it is S0.25. A parliament with 81 seats is expected to feature three parties; the other formulas would predict that NS=2.08, s1=0.577, and NV=2.52. If the parliament had 256 seats, we would expect four parties, NS=2.52, s1=0.50, and NV=2.92.

Unfortunately, the dataset I am using does not (yet) have how many parties won seats–actual or effective number. Thus I can’t determine whether this is the point at which the connections get fuzzy in the Canadian provincial arena. Nonetheless, there should be a relationship between NV and s1. It can be calculated from the formulas displayed in Table 9.2 (also on p. 149). It would be:

s1=(NV1.5 -1)-0.5.

In this last graph, I plot this expectation with the solid dark line, and a regression on s1 and NV from Canadian provincial elections as the lighter line (with its 95% confidence intervals in dashed curves).

The pattern is obvious: there are many elections in Canadian provinces in which the leading party gets a majority or even 60% or 70% or more of the seats despite a very fragmented electorate. We should not expect a leading party with more than 50% of the seats when NV>2.92. And yet 17 elections (around 15% of the total) defy this logically derived expectation. Six have a party with 2/3 or more of the seats despite NV>2.92 (in order of increasing NV: BC91, AB04, NB91, QC70, AB67, BC72; in the last one, NV=3.37!).

I think the most likely explanation is Cube Root Law violations! Canadian provincial assemblies are much too small for their populations. So, the cause of the above patterns may be that voters in Canadian provinces vote as if their assemblies were the “right” size, but these votes are turned into seats in seriously undersized assemblies, which inflates the size of the largest party. (Yes, votes come from seats in terms of predictive models, but obviously in any given election it is the reverse!)

There is some support for this. I can calculate what s1 and NV would be expected to be, if the assemblies were the “right” size, which is to say the cube root of the number of voters (which is obviously smaller than the number of citizens, but this is what I have to work with). I will call these s1cr and NVcr. Then I can take ratios of actual s1 and actual NV to these “expectations”. The mean ratios are: NV/NVcr=0.994; s1/s1cr=1.19. If the assemblies were larger, the votes–already with a degree of fragmentation about as expected from more properly sized assemblies–would probably have stayed about the same. However, with these hypothetically larger assemblies, the largest party in parliament would be less inflated by the mechanics of the electoral system.

Canadian provinces would have a greatly reduced tendency to have lopsided majorities if only they would expand their assemblies up to the cube root of their active voting population. Of course, this assumes they stick to FPTP. The other thing they could do is switch to (moderately) proportional representation systems, like BC is currently considering. That would be seem to be a good idea regardless of whether they also correct their undersized assemblies.

Below is a table of suggested sizes compared to actual, for several provinces. (“Current” here is for the latest election actually in the dataset; some of these have been increased–somewhat–subsequently.)

Prov. Current S Increased S
Alberta 83 121
British Columbia 79 152
Manitoba 57 94
New Brunswick 55 91
Newfoundland 48 81
Nova Scotia 52 95
Ontario 107 208
Prince Edward Island 27 55
Quebec 125 197
Saskatchewan 58 95

_________

(By the way, the R2 on that s1 graph that I asked you not to ask about? If you must know, it is 0.03.)

 

What could we expect from electoral reform in BC?

This week is the beginning of the mail voting period for the referendum on whether to reform the electoral system for provincial assembly elections in British Columbia. The ballot asks two questions: (1) Do you want to keep the current FPTP system or “a proportional representation voting system”: (2) If BC adopts PR, which of three types of PR do you prefer?

The second question offers three choices, which voter may rank: Mixed-Member Proportional (MMP); Dual-Member Proportional (DMP); Rural-Urban Proportional (RUP).

I have reviewed before what these options entails, and will not repeat in detail here. Besides, the official BC Elections site explains them better than I could. What I want to try to get it here is how we might expect BC’s provincial party system to change, were any of these options adopted. To answer that question, I turn, of course, to the Seat Product Model, including the extended form for two-tier systems developed in Votes from Seats.

The punch line is that the various scenarios I ran on the options all suggest the effective number of parties in the legislative assembly would be, on average, somewhere in the 2.46 to 2.94 range, the effective number of vote-earning parties would tend to be in the 2.83 to 3.32 range, and the size of the largest party would be somewhere between 45% and 51% of the seats. Again, these are all on average. The ranges just provided do not mean elections would not produce a largest party smaller than 45% or larger than 51%. Actual elections will vary around whatever is the point prediction of the Seat Product Model for any given design that is adopted. And fine, yet important, details of whichever system is adopted (if FPTP is not retained) will remain to be fleshed out later.

The ranges I am giving are formula-predicted averages, given the inputs implied by the various scenarios. I explain more below how I arrived at these values. The key point is that all proposals on the ballot are quite moderate forms of PR, and thus the party system would not be expected to inflate dramatically. However, coalition governments, or minority governments with support from other parties, would become common; nonetheless, single-party majority governments would not likely disappear from the province’s future election outcomes. As we shall see, one of the proposals would make single-party governments reman as the default mean expectation.

Before going to the scenarios, it is important to see whether the real BC has been in “compliance” with the Seat Product Model (SPM). If it has tended to deviate from expectations under its actual FPTP system, we might expect it to continue to deviate under a new, proportional one.

Fortunately, deviations have been miniscule. For all elections since 1960, the actual effective number of vote-earning parties has averaged 1.117 times greater than predicted. That is really minor. More important is whether it captures the actual size of the largest party well. This, after all, is what determines whether a single-party majority government can form after any given election. For all elections since 1960, the average ratio of actual largest-party seat share to the SPM prediction is 1.068. So it is even closer. For an assembly the size of BC’s in recent years (mean 80.7 since 1991), the SPM predicts the largest party will have around 57.8% of seats. The mean in actual elections since 1991 has been 62.7%. That is a mean error on the order of 4 seats. So, the SPM captures something real about the current BC electoral system.

Going a little deeper, and looking only at the period starting in 1996, when something like the current party system became established (due to the emergence of the Liberals and the collapse of Social Credit), we find ratios of actual to predicted as follows: 1.07 for effective number of vote-earning parties; 1.07 for largest parliamentary party seat share; 0.905 for effective number of seat-winning parties. If we omit the highly unusual 2001 election, which had an effective number of parties in the assembly of only 1.05 and largest party with 96.2%, we get ratios of 0.98 for effective number of seat-winning party and 0.954 for largest party size. The 2017 election was the first one since some time before 1960 not to result in a majority party, and it is this balanced parliament that is responsible for the current electoral reform process.

As for the proposed new systems, all options call for the assembly to have between 87 seats (its current size) and 95 seats. So I used 91, the mean; such small changes will not matter much to the estimates.

The MMP proposal calls for 60% of seats to remain in single-seat districts (ridings) and the rest to be in the compensatory tier (which would be itself be regionally based; more on that later). So my scenarios involved a basic tier consisting of 55 seats and a resulting 36 seats for compensation. Those 36/91 seats mean a “tier ratio” of 0.395 (and I used the rounded 0.4). The formula for expected effective number of seat-winning parties (Ns) is:

Ns=2.5^t(MS)^0.167.

With t=.4, M=1 (in the basic tier) and S=91, this results in Ns=2.81. I will show the results for other outputs below.

For the DMP proposal, the calculations depend on how many districts we assume will continue to elect only one member of the legislative assembly (MLA). The proposal says “rural” districts will have just one, to avoid making them too large geographically, while all others will have two seats by combining existing adjacent districts (if the assembly size stays the same; as noted, the proposals all allow for a modest increase). In any case, the first seat in any district goes to the party with the plurality in the district, and the second is assigned based on province-wide proportionality. For my purposes, this is a two-tier PR system, in which the compensatory tier consists of a number of seats equivalent to the total number of districts that elect a second MLA to comprise this compensatory pool. Here is where the scenarios come in.

I did two scenarios, one with minimal districts classified as “rural” and one with more. The minimal scenario has 5 such districts–basically just the existing really large territorial ridings (see map). The other has 11 such districts, encompassing much more of the interior and north coast (including riding #72, which includes most of the northern part of Vancouver Island). I will demonstrate the effect with the minimal-rural scenario, because it turned out to the most substantial move to a more “permissive” (small-party-favoring) system of any that I looked at.

Of our 91 seats, we take out five for “rural” districts, leaving us with 86. These 86 seats are thus split into 43 “dual-member” districts. The same formula as above applies. (Votes from Seats develops it for two-tier PR, of which MMP is a subset.) The total number of basic-tier seats is 48 (the five rural seats plus the 43 DM seats). There are 43 compensation seats, which gives us a tier ratio, t=43/91=0.473. Ah ha! That is why this is the most permissive system of the group: more compensation seats! Anyway, the result is Ns=2.94.

If we do the 11-rural seat scenario, we are down to 80 seats in the DM portion of the system and thus 40+11 basic-tier seats. The tier ratio (40/91) drops to 0.44. The resulting prediction is Ns=2.61. This does not sound like much, and it really is not. But these results imply a difference for largest seat size between the first scenario (45%) and the second (49%) that makes a difference for how close the resulting system would be to making majority parties likely.

Finally, we have RUP. This one is a little complex to calculate because it is really two different systems for different parts of the province: MMP for “rural” areas and STV for the rest. I am going to go with my 11 seats from my second DMP scenario as my “rural” area. Moreover, I understand the spirit of this proposal to be one that avoids making the districts in rural areas larger than they currently are. Yet we need compensation seats for rural areas, and like the full MMP proposal, RUP says that that “No more than 40% of the total seats in an MMP region may be List PR seats”, so this region needs about 18 seats (the 11 districts, plus 7 list seats, allowing 11/18=0.61, thereby keeping the list seats just under 40%.) That leaves us with 91-18=73 seats for the STV districts. The proposal says these will have magnitudes in the 2-7 range. I will take the geometric mean and assume 3.7 seats per district, on average. This gives us a seat product for the STV area of 3.7*73=270.

In Votes from Seats, we show that at least for Ireland, STV has functioned just like any “simple” PR system, and thus the SPM works fine. We expect Ns=(MS)^.167=2.54. However, this is only part of the RUP system. We have to do the MMP part of the province separately. With just 11 basic-tier seats and a tier ratio of 0.39, this region is expected to have Ns=2.13. A weighted average (based on the STV region comprising 80.2% of all seats) yields Ns=2.46.

The key point from the above exercise is that RUP could result in single-party majority governments remaining the norm. Above I focused mainly on Ns expectations. However, all of the predictive formulas link together, such that if we know what we expect Ns to be, we can determine the likely seat-share of the largest party (s1) will be, as well as the effective number of vote-earning parties (Nv). While that means lots of assumptions built in, we already saw that the expectations work pretty well on the existing FPTP system.

Here are the results of the scenarios for all three output variables:

System Expected Ns Expected Nv Expected s1
MMP 2.81 3.19 0.46
DMP1 2.94 3.32 0.45
DMP2 2.61 3.01 0.49
RUP 2.46 2.87 0.51

“DMP1” refers to the minimal (5) seats considered “rural” and DMP2 to the one with 11 such seats. If we went with more such seats, a “DMP3” would have lower Ns and Nv and larger s1 than DMP2, and the same effect would be felt in RUP. I did a further scenario for RUP with the MMP region being 20 districts, and wound up with Ns=2.415, Nv=2.83, s1=.52; obviously these minor tweaks do not matter a lot, but it is clear which way the trend goes.And whether any given election is under or over s1=0.50 obviously makes a very large difference for how the province is governed for the following four years!

I would not really try to offer the above as a voter guide, because the differences across systems in the predicted outputs are not very large. However, if I wanted to maximize the chances that the leading party would need partners to govern the province, I’d probably be inclined to rank MMP first and RUP third. The latter proposal simply makes it harder to fit all the parameters together in a more than very marginally proportional system.

By the way, we might want to compare to the BC-STV proposal that was approved by 57% of voters in 2005 (but needed 60%, and came up for a second referendum in 2009, when no prevailed). That proposal could have been expected to yield averages of Ns=2.61, Nv=3.0, and s1=.49. By total coincidence, exactly the same as my DMP2 scenario.

A final note concerns the regional compensation in the MMP proposal vs. province-wide in DMP. In an on-line appendix to Votes from Seats, I explored whether regional compensation in the case of Scotland produces a less permissive system than if compensation were across all of Scotland. I concluded it made no difference to Ns or s1. (It did, however, result in lower proportionality.) Of course, if it made a difference, province-wide would have to be more favorable to small parties. Thus if this were a BC voter’s most important criterion, DMP might pull ahead of MMP. However, the benefit on this score of DMP is greater under a “low-rural” design. The benefit of DMP vanishes, relative to MMP, if the system adopted were to be one with a higher share of seats marked as rural. I certainly am unable to predict how the design details would play out, as this will be left up to Elections BC.

The bottom line is that all proposals are for very moderately proportional systems, with MMP likely the most permissive/proportional on offer.

Brazil’s open list is (a little bit of) a hybrid now

Brazil is a classic case of open-list proportional representation (OLPR): lists win seats in proportion to their collective votes in a district (state), but candidates within the list are ordered solely according to preference votes obtained as individuals. These rules can result in individual candidates elected with very small preference-vote totals.

For the most recent Brazilian election, a new provision has gone into force. There is now a threshold on preference votes that candidates must obtain to be elected. This means that, in a very technical sense, a hybrid element has been brought into the Brazilian system. However, the provision is not the usual hybrid seen in “preferential list” systems, whereby seats not filled on preference votes are filled instead according to a party’s (or coalition’s) pre-determined rank. That hybrid format is what is typically called a flexible list or a semi-open list. However, Brazilian lists remain unranked, except via the preference votes.

Rather, in Brazil, a list that has an insufficient number of candidates with above-threshold preference votes forfeits those seats to other lists in the district. The threshold is set at 10% of the electoral quota, which is a Hare quota (1/M, where M is district magnitude).

This provision changed the allocation of 8 seats. Given a Chamber of Deputies with over 500 seats, we should not exaggerate the significance of the change, although of course, some other parties might have adjusted either their nomination behavior or their “intra-party vote management” practices (defined below) to avoid being hit by the threshold.

The Chamber’s website has an article regarding the seat shifts, and a table with the details (in Portuguese). The PSL, which is the party of the likely next president, Jairo Bolsonaro, won 7 fewer seats than it would have without the threshold. All these seats were in São Paulo, which is the highest-magnitude district in Brazil (M=70). The threshold there is thus 0.143% of the votes cast in the state. The Novo list in Rio Grande do Sul (M=31) also lost 1 seat due to the intra-list threshold. (Novo, meaning “New”, is a small liberal party.)

In São Paulo, the seven PSL candidates who were not eligible to take seats the list otherwise would have won had vote totals ranging from 19,731 to 25,908. They were replaced by candidates on six different lists with preference votes ranging from 56,033 to 92,257, suggesting the replacements had, on average, about three times the votes of the forfeiting candidates. (The party that picked up two of these seats was the Democrats.) In Rio Grande do Sul, the seat Novo forfeited would have been won by a candidate with 11,003 votes, but was instead filled by a candidate the Brazilian Labor Party (PTB, not to be confused with the PT of Lula) who had 69,904 votes, a preference-vote total 6.35 times greater than that of the forfeiting candidate.

As is clear from the vote totals of those who lost under this provision and those who gained, if the intention was to prevent candidates with marginal personal followings from riding in on the “coattails” of strong list-pullers (whose popularity increases the votes of the collective list), then the reformers can declare “mission accomplished”.

I am personally quite excited by this provision, which I had missed when summarizing minor changes made to the electoral law in 2017, because I once wrote up a proposal for just such a hybrid. It is in some text that was going to be part of one of the chapters in Votes from Seats, but Rein Taagepera and I decided it was not directly germane to the book and left it out. The chapter it would have been part of compares OLPR to the single non-transferable vote (SNTV) with respect to vote shares of first and last winners, and regarding the extent to which parties do (or do not) manage their intra-party competition.

Managing intra-party competition refers to parties doing one or both of: (1) restricting the number of candidates nominated, or (2) intervening in the campaign in an effort to shift votes from non-viable candidates to viable ones.

Under SNTV, these intra-party competition-management practices are critical because the total number of seats a party (or set of cooperating parties) can elect is entirely dependent on how many individual candidates it has whose votes are in the top-M vote totals in the district. Under OLPR, parties have no incentive to do this, if their goal is simply to maximize list seats–a list under OLPR can never displace seats to another list due to having “too many” candidates or having the candidates’ vote totals be widely unequal. (Parties may have other reasons to care about which candidates win, and multiple parties running in alliance face an SNTV-like conundrum in that they are competing with one another inside the list to get their candidates into the top s, where s is the number of seats won by the list. But these are separate problems, and the latter is a problem covered in Votes from Seats).

The proposal I drafted was a hybrid of OLPR and SNTV (unlike flexible lists, which re a hybrid of OLPR and closed-list PR). A threshold would be set on preference votes, and if a list won more seats, via application of the inter-list allocation rule, than it had candidates over the threshold, it would forfeit these seats. Any such forfeited seats would go into an “SNTV pool” to be be won by the otherwise unelected candidates with the highest preference-vote totals, independent of which list they had run on. My intention in devising this proposal was to encourage parties to be more active in managing their intra-party competition–taking some aspects of SNTV as beneficial–in order to make victory by candidates with marginal personal popularity less likely. (I would have set the threshold a little higher than 10% of a Hare quota.)

The article on the Chamber website is not clear on the precise rule now used in Brazil for deciding on the replacement candidates. In any case, it certainly has a similar effect to my proposal. (From a comment by Manuel at the earlier thread, it seems the forfeited seats are assigned proportionally rather than SNTV-like.) I can’t claim credit, as there is no way any Brazilian official saw my unpublished proposal. But I am pleased that some such a provision has been adopted somewhere.

Thanks to Dr. Kristin Wylie (on Twitter) for calling my attention to this article.

Using ‘combomarginsplot’

In preparing graphs for a book I am working on, I have found a Stata package called ‘combomarginsplot’ very valuable. I wanted to share the experience in case it might help other reseachers. The package was written by Nicholas Winter, and I am indebted to him for this terrific tool.
The book is on how parties assign their MPs to legislative committees. The committees are divided into categories, including “high policy” (things like justice, foreign affairs, and defense), “public goods” (such as health and education), and “distributive” (principally agriculture, transport, and construction). The typology is based on one developed some years ago by Pekkanen, Nyblade, and Krauss (2006).
Covariates that we hypothesize are associated with assignment to a given committee category include gender, occupation, seniority, and various other personal and electoral variables.
One of the ways in which ‘combomarginsplot’ has come in handy is in setting covariates for purposes of simulating how various attributes of a politician are associated with increased or decreased probability that a member might be assigned to a given committee category.
Normally it would be fine to use standard ‘margins’ and ‘marginsplot’ for such purposes. For instance, if for a given party–let’s call it “Likud”–we want to know the odds that a candidate of a given gender is assigned to public policy (PG), we would run the logistic regression for this category, and then do the ‘margins’ command. It might look like this:
*
margins, at(female=(0 1) ) level(90)
  marginsplot, recast(scatter) plotopts(msiz(vsmall) mc(gs4)) ciopts(lw(medthick)) ///
xscale(range(0 1)) xsc(r(-.99 1.7)) ylabel(0(.2)1) ysc(r(0 1)) scheme(s1mono) ///
aspectratio(3) ytitle(“”) title(“Likud: PG”) name(Likud_pg_fem, replace)
*
A formatting note: Note the use of the “recast” option. The standard ‘marginsplot’ can produce some really dreadfully ugly graphs! The “recast(scatter)” option gives you these clean capped bars. (You can also use “recast(bar)”, but I find it less appealing.) The x-axis option, “xsc(r(-.99 1.7))”, is also useful because without doing something like this, the default has the bars right next to the box borders, and lots of white space in between. Obviously, Stata graph commands can be adjusted to user preference. This is mine; yours might vary.
The command above produces a graph like this:
Nice, right? No, not really. Please never accept a “result” like this! Look at the confidence interval for female MP. It goes above 1.00. But this is a logistic regression–the outcome can be, by definition, 1.00 or 0.00. It can’t be 1.2! Just because Stata says so, doesn’t make it so!
Sometimes setting ‘margins’ scenarios a certain way actually leads to the estimate being generated off a hypothetical case that is really unrealistic, given the data. That is, there may not be many real politicians who meeet the criteria. Then–especially if the overall sample is not very large–you can get utterly impossible “predictions”.
Go look at your data, and see what’s going on. In this case, it turns out that the problem is that a very small percentage of men in this party had what we term “high-policy occupations” (mostly lawyers), but a high percentage of the women did. When we run ‘margins’ without specifying values for any other covariates, we get an estimate at the sample means of the other variables on the right-hand side of the regression. So it is estimating men and women with the same likelihhod of also being of high-policy occupation, even though the male and female subsamples are rather different.
What we need is to estimate the men and women in separate ‘margins’ commands, each being more realistic on other covariates. However, we want the estimates for men and women to appear in the same box in our final graph. It won’t do to run separate ‘marginsplot’ commands and then do ‘graph combine’ because that will make two separate boxes in the space of one. So here is how you can make it look like the first graph, despite being based on two separate calls to ‘margins’:
*
margins, at(female=1 occu_hi=1) level(90) saving(“File1”, replace)
margins, at(female=0 occu_hi=0) level(90) saving(“File2”, replace)
  combomarginsplot “File2” “File1” , ///
    recast(scatter) plotopts(msiz(vsmall) mc(gs4)) ciopts(lw(medthick)) ///
ylabel(0(.2)1) ysc(r(0 1)) scheme(s1mono) ///
labels(0 1) xscale(r(.5 2.5)) xtitle(“Female MP”) ///
aspectratio(3) ytitle(“”) title(“Likud: PG”) name(Likud_pg_fem, replace)
*
When we do all the above, we get:
We see more plausible confidence intervals, because we are estimating on realistic politicians. There is essentially no difference in this party between the probabilities of men and women getting PG committees (or, more to the point, between women with high-policy occupations and men without them). We already knew from the first example plot above that there was not a significant difference. It was the confidence intervals that went haywire, due to the unrealistic scenario.
A challenging part of of this was getting the bars in the right place within the box. First, one needs to use the ‘labels’ option in order to have them marked “0” and “1” instead of the names of the saved file (e.g., “File1”, although you can name them just about anything you want). With a little–OK, a lot–of trial and error, it turned out that “xscale(r(.5 2.5))” worked about right.
I have found several other convenient uses for ‘combomarginsplot’ in this and other projects. A perhaps more common use than the one I demonstrated here would be when the plotted curves come from different regressions. You can save the results from each, then combine them into a single plot area. Another, which I have used, is combining multiple outcomes from one regression, such as a multinomial logit.
It is terrific that Stata has such a community of public goods providers to create tools like this!

Academic writing styles

I am working on two books this summer/fall. I hope both will be done by the end of December, although that may be over-optimistic. As a result of being engaged in these writing processes, questions of academic writing style have been on my mind.

I owe many debts of gratitude to my mentor and frequent coauthor, Rein Taagepera. But the most recent one was his suggestion that every empirical chapter in our new book (Votes from Seats, 2017) start with a presentation of the key result. Don’t drag the reader through prior literature and a bunch of “hypotheses” (a practice he hates, and I tend to agree) before getting to the point. Start with the point, and then explain how you got there, and only then why others did not get there. But the thing is, this almost never works with a journal article (and maybe doesn’t work with books for most scholars not named Shugart or Taagepera), because reviewers impose a standard format that just makes for plodding reading. And writing.

For probably the best demonstrations of our preferred presentation, if you have access to the book, see Chapter 7, which has an overview of “Duverger’s law” near its end, but starts with the Seat Product formula for effective number of seat-winning parties and a graph showing the payoff. Also Chapter 12, in which the previously proposed concept of “proximity” is discussed at the end of a chapter that opens with some data plots showing our preferred “elapsed time“. Other empirical chapters in the book mostly follow this format as well.

Mexico, 2018

Mexico has its elections for President, Chamber of Deputies, and Senate on 1 July. It has been clear for a while that, barring a big surprise, Andrés Manuel López Obrador (popularly known as AMLO) will win.

AMLO’s support has risen steadily out of what looked like a tight three-way contest some months ago into a strong lead. When voters responding “no preference” are removed, it even looks likely that AMLO could win a clear majority of votes. Mexico elects its presidency via nationwide plurality, and no Mexican president has earned half the votes since 1994 (at a time when most experts still considered the regime authoritarian, albeit increasingly competitive).

Assuming AMLO wins, it will highlight the competitive three-party nature of the system. When the center-right National Action Party (PAN) won the presidency in 2000, it broke decades of continuous control by the Institutional Revolutionary Party (PRI). The PAN won again in 2006, on less than 37% of the votes in a very tight race, with AMLO close behind (and refusing to acknowledge defeat). The PRI returned to the presidency in 2012, and now AMLO will give the left its chance. (AMLO was with the Party of the Democratic Revolution, PRD, but in recent years has set up a new party, MORENA, while the remnant PRD is backing the PAN candidate this time.)

I would be very interested in seeing an analysis of AMLO’s own manifesto (and his party’s, if separate). There is much hand-wringing over his leftist “populism”. However, when he ran in 2006, he staked out a centrist economic platform well to the right of his own party–a clear case of what “presidentialization” does to parties. (See the discussion of the general point, and also the 2006 Mexican campaign, in my book with David Samuels, Presidents, Parties, and Prime Ministers). Is he doing so this time? I can’t claim to have followed closely enough to know.

As for the Chamber of Deputies, if the pattern of recent Mexican elections holds, the party winning the presidency will win fewer votes for its congressional candidates. That could mean MORENA (and pre-election allies) will not have a majority of seats. On the other hand, as noted above, these previous presidents have not themselves won majorities. Moreover, the electoral system is mixed-member (with the voter having a single vote). It is sometimes erroneously categorized as mixed-member proportional (MMP), but it is actually leans much more to the majoritarian category (MMM). Seats won based on nationwide votes for party are added to single-seat districts won (by plurality).

The allocation is not compensatory, but it is also not strictly parallel. There are caps on allowable over-representation (unlike in a “pure” MMM system). The most important cap is that no party can have a final seat percentage that is more than eight percentage points above its vote percentage. Thus if a party wins under 42% of the votes, it is unable to have a majority of seats. If it gets over 42% it is not guaranteed a majority, but a majority becomes likely, due to the non-compensatory nature of the allocation. This cap kept the PRI from retaining its majority in the midterm election of 1997, and I believe it has been hit in several subsequent elections, as well. This is what I will be watching most closely: Will MORENA (and allies) get a Deputies majority?

The Senate is also elected in a mix of regional and nationwide seats. Each state has three senators, elected by closed list, limited-nominations plurality. The largest list gets two seats and the runner up gets one. Then there are 32 seats elected by nationwide proportional representation (allocated in parallel, not compensatory manner).

These provisions, combined with the regionalization of party support in Mexico, make it difficult for a party (or alliance) to win a majority of the Senate’s 128 seats. AMLO is unlikely to have majorities in both houses, but it is worth noting that the federal budget must clear only the Chamber. There is no Senate veto on the spending side of the budget, although both houses must pass all other types of bills. Thus the left will be in a strong, but not unchecked, position to implement its program for the first time in Mexican democratic history.