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# Electoral System Effects on Party Systems

## Abstract and Keywords

The question of how electoral institutions affect party systems has been central to the literature on elections. For a given electoral system configuration, how many parties earn votes and win seats? How large is the largest party’s share of all votes and seats? This varies from country to country, from election to election, and, inside the country, from district to district. Yet two institutional inputs—district magnitude and assembly size—determine the worldwide averages surprisingly well, and they do so for well-defined logical reasons. These worldwide averages supply benchmarks against which to compare individual countries, elections, and districts: given their two basic institutional inputs, do they have rather many or few parties, and by how much are they off, compared to logical average expectations? History, culture, and current politics account of course for which parties form and which of them is the largest, but institutions shape their number and sizes.

# Electoral System Effects on Party Systems

The question of how electoral systems shape party systems has been central to the literature on elections for about as long as there have been elections. At one time, elections were almost universally conducted by some form of majority voting system. Late in the nineteenth century, and especially throughout the twentieth century, more and more countries began to adopt other rules for distributing seats in national assemblies and other elected bodies. These systems include proportional representation (PR).

Some nineteenth-century writers promoted various PR systems on the grounds that they would produce assemblies more reflective of the range of opinion in the society (e.g., Droop 2012 [1869]; Hare 1859). As mass political parties developed, interest turned to whether the number of parties systematically varied across different electoral systems. The most famous statement of this connection between rules and parties is the set of propositions that have come to be known as Duverger’s law. The so-called law states that the use of plurality rule in districts electing one member tends to produce a two-party system. With various restatements and refinements, this basic proposition has been the dominant way of thinking about the nexus between electoral systems and party systems.

Notwithstanding its vaunted status as a “law,” this central proposition of the literature on electoral systems fails to meet the basic standards of a scientific law. In this chapter, rather than offer a review of the “law” itself, we will trace an alternative means of quantifying party systems and how they are connected to electoral systems and other institutions. We take the heretical stance that it is time to retire the notion of a “Duverger’s law” and to build and refine more quantitatively precise models instead. (p. 42)

# Counting and Measuring

In this section, we review two key quantitative indicators that are by now well established in the analysis of party systems: the “effective number of parties,” which is a size-weighted count of how many parties there are, and “deviation from proportionality,” which measures the degree to which votes and seats shares differ from one another in a given party system.

## Counting Parties: The Effective Number

Counting parties turns out to be not so simple as just counting parties. We may not find it very interesting that five parties won seats in Canada’s national election of 2015. After all, one of them had more than half the seats, and one of the remaining four had only 1 of the 338 total seats. The outcome of this election may or may not qualify as a “two-party system.” It depends on what one means by that concept. Two parties combined for 83 percent of the seats, but only 71.4 percent of the votes. However, the third party had 19.7 percent of votes and 44 seats. This is certainly not a “two-party system” in the same way that Jamaica’s is. In the latter country’s 2016 election, there were only two parties that won seats (one with 32, the other with 31), and these two together combined for 99.8 percent of the votes.

Perhaps we could say that in both Canada and Jamaica there are only two “significant” parties. Yet this too runs into trouble. Why is the third party in the Canadian example, with nearly a fifth of the votes and an eighth of the seats, “insignificant”?

The difficulty of making sense of real-world election systems by just a simple count of parties or arbitrarily deciding which subset of parties is “significant” leads to the desire to have some sort of weighted count. This is where the concept of an “effective number,” first introduced by Laakso and Taagepera (1979), comes in.

The effective number of parties is an index that aims to summarize the unequal-sized parties into a single number. It is a weighted count in that its calculation ensures that a large party contributes more to the index than any smaller one contributes. It accomplishes this aim by squaring each party’s share of either seats or votes. The formula for calculating the effective number of seat-winning parties (which we can designate NS) is

$Display mathematics$

In words, we square the seat shares for each of i parties—however many there are, starting with the seat share for the largest party, s1. Then we sum up all the squares. Once we have this sum, we take the reciprocal. In this way, the index weights each party by its own size. The squaring results in a large party contributing more to the final index value (p. 43) than does a small one. For instance, suppose the largest party has half the seats, s1 = 0.5; thus, we have 0.52 = 0.25. Now suppose among several remaining parties the smallest (the ith) one has only 5 percent of the seats. We take the share, 0.05, and square it, and get 0.0025. In this way, when we sum up the squared shares of all the parties, the smallest one has counted for much less than the largest. This is precisely what we want—a size-weighted count of how many parties there are.

Alternatively, we could calculate our index on vote shares, giving us the effective number of vote-earning parties (NV):

$Display mathematics$

Here vi stands for the fractional vote share of the ith party. Thus, for any given election result, we have two effective numbers: NS for the seats and NV for the votes. These numbers are sometimes referred to as ENPP (effective number of parliamentary parties)1 and ENEP (effective number of electoral parties), respectively. However, given our interest in systematically constructing logical models, we adopt the approach more typical of scientific notation: single symbols with subscripts.2

A useful feature of the index is that it will always yield the actual number of parties (or other components) if they are equal in size. If, as is usually the case, the parties vary in size, the effective number will be smaller than the actual number. For instance, take a case of four equal-sized parties, meaning each has 25 percent of the seats (or votes):

$Display mathematics$

Precisely as it should, the index yields N = 4 for this constellation of four same-sized parties. Now, suppose one of them splits in two, so that we have

$Display mathematics$

We now have five parties, but they are no longer equal in seats. The effective number logically should be some value greater than four but less than five. If we repeat the calculation procedure with this new constellation, we get NS = 4.57. Now suppose one of the small ones merges with one of the bigger parties. We are now back to four parties, but they are unequal:

$Display mathematics$

The calculation will result in NS = 3.55, suggesting, accurately, that this constellation has a degree of fragmentation somewhere roughly equidistant between one with three equal-sized parties and one with four equal-sized parties. (p. 44)

The effective number of seat-winning parties is directly related to the index of fractionalization (F) of party seat shares:

$Display mathematics$

The index, F, has a long pedigree in economics and was also the measure of fragmentation used by Rae (1967) in the first major cross-national quantitative study of the impact of electoral systems on party systems.3 It seems to have fallen into disuse. The effective N has become dominant.

Notwithstanding the dominance of N, it is not without its flaws, critics, and suggested alternatives (Molinar 1991; Dunleavy and Boucek 2003; Golosov 2009). No single measure can summarize everything we would want to know about a given constellation of party sizes. One drawback that is relevant to the electoral systems literature is the ambiguity of its measures of how closely a constellation conforms to a “two-party system” (Gaines and Taagepera 2013). Given how central this notion of two-party versus multiparty is to the literature on electoral systems, the fact that some distributions of party systems might have a considerable degree of “two-partyness” yet have N near 3 might lead to faulty interpretations. An example would be two parties each having 40 percent, and two others with 10 percent each. By any reasonable standard, this hypothetical constellation features competition between two equal parties, with the others clearly out of the running for the first position. Yet it yields N = 2.94.

Similarly, values of N near 2 can result from constellations that are not in any meaningful sense “two party.” Consider a case of a party with 60 percent and two other parties with 20 percent each. This results in N = 2.27. Yet, given such a strongly dominant party, no one could call that a “two-party system.”

Despite these drawbacks and various efforts to propose alternatives, the effective number has become the dominant index, by far. It is easy to calculate, as sketched earlier, and has a solid logical foundation. Like any tool, it needs to be used for its proper purpose. It is a measure of fragmentation, not, for instance, a proxy for the number of “serious” parties.4 As long as we do not try to read more into it than it is capable of telling us (Taagepera 1999), it is a fine measure of how fragmented a party system is.5 For this reason, like most of the related literature, we will continue using it. We find that the effective number lends itself well to generalizations about the institutional effects on party systems, which is the main focus of this chapter. First, however, we briefly describe another important index that summarizes an aspect of how electoral systems affect political parties—measures of disproportionality.

## Measuring Deviation from Proportionality

As with quantitatively summarizing the variable sizes of parties in an assembly or in the electorate, there are also various measures in use for characterizing how “proportional” (p. 45) the seats–votes relationship is. We will refer to such indices as measures of “deviation from proportionality.” Two main measures dominate.6 Both start with the difference between seat and vote shares, for each party, but then they process these differences in different ways.

Loosemore and Hanby (1971) introduced into the literature on electoral systems an index of deviation that we’ll designate as D1, following the systematics of Taagepera (2007, 76–79). For deviation from proportionality, it is

$Display mathematics$

Here si is the ith party’s seat share, and vi is its vote share. The index can range in principle from 0 to 1 (or 100 percent). Note that |si-vi| = |vi-si| is never negative. D1 dominated until Gallagher (1991) introduced what we’ll designate as D2:

$Display mathematics$

It has often been designated as the “least square” index, but this is a misnomer. The index does involve squaring a difference but no minimization procedure so as to find some “least” squares. D2 can range from 0 to 1 (100 percent), but whenever more than two parties have nonzero deviations, the upper limit actually remains below 1. The value of D1 will be greater than or equal to that for D2.

Gallagher’s D2 rapidly displaced D1 during the 1990s as the more widely used index,7 despite grounds for doubting whether it is the best of the various measures (Taagepera and Grofman 2003; Taagepera 2007, 76–78).

# Institutional Effects

By now there exists a substantial body of work that seeks to explain how electoral systems and other institutional rules shape party systems. Most of this work has used the effective number of parties as its principal outcome variable, and we will focus our attention on that outcome as well. Many studies have recognized district magnitude as one of the key input variables, but various works differ in what other inputs are considered. We will review some of this prior work later, but the primary purpose of what follows is not to review the literature per se, but rather to demonstrate the substantial explanatory power of just two basic institutional parameters: district magnitude (the number of seats in an electoral district) and assembly size (the total number of seats in the main national representative body).

There really is no better way to demonstrate the power of a simple institutional effect than with a graph, and accordingly we will show several in this chapter. Strikingly, most works in the related literature have few or no graphs—or those they present include (p. 46) no actual data.8 Partly the reason for the absence of data graphs in many works is the preference of authors for multivariate regressions, which do not lend themselves well to graphing.9 More important, the regressions found in most of these works typically produce widely varying coefficient estimates on key variables like magnitude, depending on which other variables are included and on which specific sample selection criteria are used.10 The authors of these works typically are not troubled by such variance in their own estimates, because they are testing merely directional hypotheses—for instance: the effective number of parties increases as magnitude increases, conditional on (various factors). As political scientists interested in electoral systems effects, we can do better than this. We can offer a specific numerical estimate of the effect, grounded in logic.

To do so, we start with the fragmentation of seats and only then extend to predicting the fragmentation of votes. The reason for doing so is actually straightforward, but it is contrary to what most other authors have done. That is, it is typically the case that authors (e.g., Amorim Neto and Cox 1997; Cox 1997; Clark and Golder 2006) first estimate the contribution of various inputs, including the electoral system but also a measure of social diversity, to the effective number of vote-earning parties (NV) at the national level. This relationship is estimated through a regression. They then take the NV to be an input into a second regression, in which the electoral system is the primary independent variable and the outcome to be explained is the effective number of seat-winning parties (NS).

The method of estimating the votes first and then the seats is sensible. After all, parties earn votes before they earn seats. The votes are cast, and then the electoral authorities calculate how many seats each party wins by applying the electoral law to the known distribution of votes. However, as Taagepera (2007) notes, this method proved to be a dead end for theory building. For the logic of institutional effects on party systems, it is much more fruitful to start at the other end—the seats. The logic for doing so is elementary, in that it is found within the famous so-called Duverger’s law. Duverger (1951, 1954) claimed that the electoral system—specifically first past the post (FPTP)—first worked through a “mechanical” effect, whereby the available seats constrained which parties actually could win. Only then did a “psychological” effect kick in, encouraging voters and other actors to avoid “wasting” votes on parties that could not possibly win.

By the same logic, attempts to predict how electoral systems and party systems are related should start with the quantity that is more constrained—the seats. It is the number of available seats that directly limits the feasible number of (seat-winning) parties. The number of parties earning votes is only indirectly constrained. By this logic, we can develop a model of how seats—both in a district and in the national assembly as a whole—shape the party system through what we term the Seat Product Model. In the next section, we sketch the steps in this model, summarizing Shugart and Taagepera (2017). The inputs into the Seat Product Model are strictly institutional; later we will return to the question of whether inclusion of a factor like social diversity might improve the predictive power of the model. (p. 47)

# The Seat Product Model: How We Can Predict Party Systems from Seats

The concept of the seat product was introduced by Taagepera (2007) and refers to the mathematical product of a country’s mean district magnitude (M) and its assembly size (S). Through a series of logical steps and application of algebra, it allows us to derive formulas predicting what a given output quantity, such as the effective number of seat-winning parties (NS), can be expected to be, on average, for a given seat product. The formulas then can be tested, both via visual inspection through graphing and by statistical regression.

In Figure 3.1, we show a data plot in which the effective number of seat-winning parties (NS) is on the y-axis and the “seat product” (MS) is on the x-axis. The data points represent country-level means, for all democracies that have at least three post–World War II elections in our dataset,11 for which the electoral system meets Taagepera’s (2007) definition of “simple.” A simple electoral system is one in which all seats are allocated in districts, meaning there are no “upper tiers” (see Gallagher and Mitchell’s chapter).12 Moreover, the seat allocation formula must be a basic proportional one in a single round of voting. Importantly, FPTP is included in the definition of a simple system, because all proportional formulas that are used in party list systems13 reduce to FPTP (plurality) when M = 1.

On logical grounds introduced in Taagepera (2007) and summarized later in this chapter, we expect

$Display mathematics$
(Equation 1);

in words, the effective number of seat-winning parties equals, on average, the seat product, raised to the power, one-sixth. Equation 1, it must be emphasized, is not a mere regression result. Rather, it is derived from logic, which we summarize later. When we do test it via regression, we almost perfectly confirm it;14 the solid line in Figure 3.1 represents Equation 1. Countries are labeled if their actual mean value is either greater than 1.33 times the value predicted by Equation 1 or less than three-quarters the predicted value.15

In Figure 3.1, we differentiate parliamentary and presidential systems, because much of the literature, starting from Amorim Neto and Cox (1997) and Cox (1997), argues that the assembly party system can be explained only via regression designs in which the observed effective number of presidential candidates is entered as an “independent” variable. Our data plot gives scant reason to claim the pattern is different for presidential systems. There is one significant outlier—Brazil, represented by the data point with the highest NS in Figure 3.1—but as a group, the presidential and parliamentary systems fit within a common trend represented by Equation 1.16 (p. 48)

Click to view larger

Figure 3.1. Relation of the nationwide effective number of seat-winning parties (NS) to the seat product (MS), both on logarithmic scales.

Note: Abbreviations for relatively deviant cases: BRA (Brazil), CAP (Cap Verde), CEY (Ceylon to 1970), HON (Honduras), FIN (Finland), LAT (Latvia), SKN (St. Kitts and Nevis), SPN (Spain), SWI (Switzerland), UK (United Kingdom), US (United States).

Source: Unless otherwise indicated, data in the tables and figures in this chapter are authors’ calculations on original data drawn from Li and Shugart (n.d.). Data points represent country averages over several elections.

Further, our symbol patterns in Figure 3.1 differentiate FPTP systems (where every district elects just one member, via plurality rule) from PR systems (in which mean M > 1). Once again, there is scant evidence for any need to treat FPTP and PR as if they were distinct categories. We see some intermixing of PR systems (open symbols) and FPTP systems (filled in) at moderate levels of MS. The range of MS from about 200 to 1,000 includes both FPTP and PR, as well as both parliamentary and presidential.

Thus, for example, were a given country to have MS = 625, we would predict from Equation 1 that it would tend to have NS = 2.92, regardless of its executive type and regardless of whether the system consisted of 625 single-seat districts (M = 1, S = 625),17 a PR system with a single 25-seat district (M = S = 25), or any intermediate combination (say, mean M = 5, S = 125). For any given country or election, we can expect some deviation from the predicted value, due to various factors other than the seat product.

We will refer to Equation 1 as the Seat Product Model (SPM). The section that follows describes its logical derivation. (p. 49)

## The Logic behind the Seat Product Model

The SPM for the effective number of seat-winning parties (NS) starts with a basic question: how many parties would we expect to win at least one seat in a district electing M seats? We can designate the number of parties, of any size, that win representation in a given district as N′S0, where the prime mark indicates that we are referring to a district-level quantity instead of nationwide and the zero indicates that this is the unweighted count (i.e., the actual number rather than the effective number, which is the size-weighted count).

Taagepera and Shugart (1993) first proposed that the relationship should be

$Display mathematics$
(Equation 2).

Figure 3.2 shows that it is a reasonable approximation. The figure plots our two quantities against each other on logarithmic scales; data points represent individual elections at the district level in a large number of democracies. The light gray diagonal line represents Equation 2. While data points are somewhat scattered, Equation 2 captures the average trend. Like Equation 1, it is important to emphasize, this is not a best-fit regression line. It is derived from logic, which we now explain.

The logic that leads to Equation 2 starts with the boundary conditions. What are the ranges in which the data could not possibly occur? Asking this basic question can be a good starting point for figuring out what the average relationship should be, on logical grounds. In our case, the minimum number of parties that could win a seat in any district is clearly one; the thick horizontal line in Figure 3.2 at N′S0 = 1 is thus a lower limit. For any given M the feasible maximum is N′S0 = M, in which case each party has exactly one of the district’s seats. This is the thick black diagonal line above which the graph space is shaded gray. Data points in the range N′S0 > M are thus impossible, hence the labeling of this region as a “forbidden area.”

Between these boundaries, any value of the number of parties between one and M is feasible. Yet we note that there is considerable white space in the feasible area, where no data points are found. We can express the average trend by taking the mean of our plausible extremes; in logarithmic space, this is the geometric average, and it leads us to Equation 2:

$Display mathematics$

If we then want to develop expressions for the nationwide outcomes, we can start with the simplest case, which is when there is a single nationwide district, so that we have M = S. A few countries have had such districts, including Israel and the Netherlands. By Equation 2 and the condition of M = S, we have18 (p. 50)

Click to view larger

Figure 3.2. The relationship between district magnitude (M) and the number of seat-winning parties (N’S0).

Data source: Belden and Shugart (n.d.).

$Display mathematics$
(Equation 3).

Equation 3 should be generalizable to any mean M and any S, including simple districted systems in which S seats in the assembly are divided across some plural number of districts with mean M.19 When tested graphically or by regression, it is confirmed.

Figure 3.3 shows the data plot. Clearly the fit is quite strong, despite a few prominent outliers.20 Similarly to Figure 3.1, countries are labeled if their actual mean value is either greater than 1.33 times the value predicted by Equation 3 or less than three-quarters the predicted value. All subsequent figures similarly have labels for countries outside this range of the predictions for the graphed quantity. Remarkably, the United Kingdom and Spain deviate from average expectation in both Figures 3.1 and 3.3, but in opposite directions. Unexpectedly, many parties win seats, yet the effective number of parties is low. This is the signature of a country with one or two unusually dominant parties and a profusion of tiny (often regional) ones. In contrast, the United States is low on both accounts. Inspecting our graphs helps in pinning down which countries share some oddities, and to what extent.

If we want to proceed from the number of parties (of any size) to the effective number, we can go by way of the share of the largest party, s1. This share is the one that most strongly affects NS, by virtue of the squaring (weighting by size). Taagepera (2007) first (p. 51) suggested that the same logic of considering boundary conditions, described earlier for deriving Equation 2, could work for estimating s1, as follows. The minimum s1 for any given NS0 is s1 = 1 / NS0. This minimum is reached when all parties are equal in size. The maximum is almost s1 = 1; more precisely, it is as close as possible to one party having all seats as can be, while leaving seats for each of the NS0 – 1 remaining parties. To keep it simple, we can take the approximation

Click to view larger

Figure 3.3. How the actual number of seat-winning parties (NS0) relates to the seat product (MS), national level.

Note: Abbreviations for relatively deviant cases: BRA (Brazil), CAP (Cap Verde), CZR (Czechia), DR (Dominican Republic), JAM (Jamaica), NEP (Nepal), NZ (New Zealand), POL (Poland), SKN (St. Kitts and Nevis), SPN (Spain), STP (Sao Tome and Principe), SVK (Slovakia), SWI (Switzerland), UK (United Kingdom), US (United States).

$Display mathematics$
(Equation 4).

In Figure 3.4, we see the scatterplot. Once again, the fit is overall quite strong, despite a few cases that are somewhat more distant from the line representing Equation 4 than the rest.21 The United Kingdom and Spain again stand out.

Having connected the largest seat share to the actual number of parties, which in turn connects to the seat product (Figure 3.3), we now have only one more step to get us back (p. 52) to Equation 1 (and Figure 3.1): how are the largest share and the effective number related? Taagepera (2007) offers a logic leading to the following average expectation:

Click to view larger

Figure 3.4. How the largest seat share (s1) relates to the number of seat-winning parties (NS0).

Note: Abbreviations for relatively deviant cases: LAT (Latvia), NEP (Nepal), SPN (Spain), UK (United Kingdom).

$Display mathematics$
(Equation 5).

Figure 3.5 shows this to be a strong approximation to our country means, with nearly all cases close to the line representing Equation 5.22

The series of steps and graphs shown so far complete the major steps in the logical chain extending from the number of parties to expect in a single district to the Seat Product Model of the nationwide effective number of seat-winning parties. What remains to be done is the connection to votes, which we take up in the next section.

So far we have summarized the derivation of Equation 1, the Seat Product Model for the effective number of seat-winning parties (NS). Here, we show that this can be extended to votes. When a given number of parties wins seats in a representative assembly, how many more are likely to try their luck? How are these two numbers connected logically? (p. 53) The answer that Shugart and Taagepera (2017) propose is that the number of “pertinent” vote-earning parties should tend to be the number of actual seat-winning parties, plus one close loser: strivers are winners plus one.

Click to view larger

Figure 3.5. Relationship of the effective number of seat-winning parties (NS) to the largest seat share (s1).

Note: Abbreviations for relatively deviant case: IND (India).

The now fairly well-established idea of an “M + 1 rule” for the number of “serious” or “viable” candidates offers a useful starting point. This rule was elaborated by Reed (1990, 2003) and Cox (1997). A recent review summarizes it as follows: “The M+1 rule, whereby the number of parties or candidates in a district is capped by the district magnitude (M) plus one” (Ferree, Powell, and Scheiner 2013, 812). This “parties or candidates” becomes problematic when M is large and the number of parties is thus sure to be smaller than M.

The original notion behind the M + 1 rule was that it generalized the so-called Duverger’s law. When M = 1, it hardly matters whether we think of the competing agents as parties or candidates—each party generally presents just one candidate, so the concepts merge. Duverger’s law predicts two parties (two candidates) when M = 1. Reed’s contribution was to say that under single nontransferable vote (SNTV) rules formerly (p. 54) used in Japan (see Nemoto’s chapter), the number of “serious” candidates also was near M + 1. Reed operationalized “serious” via the effective number,23 and he explicitly meant candidates, not parties. Under SNTV, larger parties typically nominate more than one candidate per district (but fewer than M—see Bergman et al. 2013). Because there are M winners, and they are those with the highest individual vote totals regardless of party, there are M + 1 viable candidates,24 according to Reed’s argument. In other words, the district has M winners and one close loser. The rest tend to fall farther behind, at least under certain conditions specified by Reed and extended by Cox.

Cox himself recognizes the limits of applicability of the M + 1 rule to larger magnitudes and to the (effective) number of parties, saying it specifies an “upper bound” rather than a prediction. It is obvious that it would be a poor prediction of the number of parties. For instance, in the single nationwide district of M = 150 seats in the Netherlands, 151 parties is overkill. Above some moderate level of MCox (1997, 100) suggests “about five” seats—the M + 1 rule is not the principal factor limiting proliferation of parties (Cox 1997, 122). Further, Cox (1997, 102n) states that the application of the M + 1 rule to lists in PR systems is “substantially less compelling.” Thus, he concludes, “something else” other than the strategic voting that leads to M + 1 viable candidates under FPTP and SNTV must be at work when we are concerned with the number of vote-earning parties in PR systems (Cox 1997, 110).25

For making sense of the number of viable vote-earning parties, Shugart and Taagepera agree that the “plus one” is an important logical building block. However, the “plus one” should be added to the number of seat-winning parties (NS0), not the total number of seats allocated in a district (M). The Reed and Cox notion of viability being conditioned by competition for seats is logically correct, but further progress comes when we put this “plus one” after the number of seat-winning parties, instead of after the magnitude. This leads to an expression,

$Display mathematics$
(Equation 6),

where NV0 is the “number of pertinent vote-earning parties” and NS0 is the actual (not effective) number of seat-winning parties. Then, given Equation 3, we can replace NS0 with the seat product:

$Display mathematics$
(Equation 7).

Only one step remains to get us to the outcome of interest, the effective number of vote-earning parties, NV0. We need to specify a relationship between NV and NV0. We can do so with the basic observation that the relationship should be, logically, the same as for seats, that is, between NS and NS0. As outlined in the preceding section, the equations26 already imply

$Display mathematics$
(p. 55)

and thus we should also have

$Display mathematics$

We can substitute in and derive the following prediction for NV:

$Display mathematics$
(Equation 8).

This is testable, and confirmed by both regression analysis and the data plot of Figure 3.6.

The data plot in Figure 3.6 shows that Equation 8 (the solid curve) is a reasonable fit, despite the scatter of the actual data. The dashed curve is the result of a regression on these data points. It is only slightly off, giving a slope of 0.75. The expected slope of two-thirds is grounded in logic, and as the data plot shows, the result is hardly different. Moreover, if we leave out the one extreme data point, Brazil, with the exceedingly high NV = 9.6, we obtain a nearly precise estimate of our coefficient, 0.682.27 Here the United Kingdom and Spain do not stand out as extreme outliers. This means their vote structure is closer to ordinary for the given seat product MS; it’s at the seat level that strong surprises emerge, compared to other FPTP and PR countries.

Compared to our criterion for major deviations (off by a factor of four-thirds), Figure 3.3 has the most deviations (fifteen). This implies that the actual number of seat-winning parties is most at the mercy of tiny parties winning a single seat or failing to do so. Effective numbers are more stable. In the logical chain of reasoning they are far removed from seat product MS, and deviations occur—ten cases in both Figures 3.1 and 3.6. Remarkably, proceeding from seats to votes does not increase scatter, as one might expect. As expected, scatter is much lower when factors graphed are just one step removed from one another in the logical chain—from NS0 to s1 in Figure 3.4 (four deviant cases), and from s1 to NS in Figure 3.5 (one deviant case).

When doing statistics of green peas the average size may be just an empirical given, and the extra large and small ones may be seen as mere anonymous statistical outliers. Here this is not so. First, the average output is not empirical but a logically grounded expectation, almost bafflingly confirmed by empirics. Second, countries have identities that peas don’t (at least for us). In our graphs the most frequent outliers, each in a somewhat different way, were Brazil, Cape Verde, Latvia, St. Kitts and Nevis, Spain, Switzerland, the United Kingdom, and the United States. While most of the forty-nine countries considered (because they have “simple” electoral systems) fall closer to expectations, why do those eight deviate in the various ways they do? Pinning down country-specific features is a major payoff of the study described here. We have not just an elegant completed edifice but also starting points for more country-specific studies, now that we know how these countries stand out against the benchmark. (p. 56)

Click to view larger

Figure 3.6. The effective number of vote-earning parties and the seat product.

Note: Abbreviations for relatively deviant cases: BRA (Brazil), CAP (Cape Verde), CEY (Ceylon to 1970), HON (Honduras), IND (India), LAT (Latvia), PER (Peru), SKN (St. Kitts and Nevis), SWI (Switzerland), US (United States).

# Complex Systems and Ethnic Factors

In this section, we consider two additional factors: (1) how institutions affect party fragmentation when the electoral system is not “simple” and (2) whether considering the ethnic fragmentation of a country can improve our ability to predict the party system beyond what we can get from the seat product model.

Considering these two additional factors is important for two reasons. First, a significant percentage of the world’s democratic electoral systems are not simple, but rather contain an “upper tier” of allocation (see Gallagher and Mitchell’s chapter for definition) or other complicating features. Second, most of the literature in the field of comparative electoral systems analysis over recent decades has included a variable for upper tiers and, in addition, argued that “permissive” electoral systems are associated with high degrees of party fragmentation only in the presence of high demand emanating from social diversity.28

A central argument in the literature since Ordeshook and Shvetsova (1994) is that electoral systems and social diversity jointly affect the effective number of parties. Amorim Neto and Cox (1997) and Cox (1997) were the first to articulate that the effect was interactive; see (p. 57) Moser et al. (this volume) for an extensive review and analysis. In the interactive formulation, “restrictive” electoral systems are associated with a low effective number of parties even when social diversity is high, whereas permissive systems are associated with a high effective number of parties only when social diversity is high. Absent demand from society, these authors claim, permissive systems do not result in increased number of parties. In these works, the degree of restrictiveness or permissiveness is measured by the mean district magnitude and the size of any upper tiers of allocation. Several subsequent works have rested their analysis on the same basic theoretical assumptions and variables but using different datasets and varying statistical estimating techniques or considering additional variables (Mozaffar et al. 2003; Clark and Golder 2006; Golder 2006; Hicken and Stoll 2011, 2012).

In the Seat Product Model the measure of permissiveness is the product of the mean29 district magnitude and the assembly size. However, as detailed in the preceding sections of this chapter, this model originally applied only to simple electoral systems—those in which all seats are allocated within districts with some proportional formula (or FPTP) in one round of voting. The model thus does not take account of upper tiers, which are found in several complex proportional representation systems, including two-tier PR (as in Denmark or South Africa30) and mixed-member proportional (MMP) systems (as in Germany and New Zealand31).

The possibility of incorporating upper tiers into the Seat Product Model was first proposed and tested by Li and Shugart (2016); it was tested further and confirmed by Shugart and Taagepera (2017). The theory behind the extended model is that the mean district magnitude and the total number of seats in the basic tier only—that is, excluding upper-tier seats—affect the party system much the same as it would were it the entire system (i.e., were the system simple). Then the upper tier inflates the basic-tier effective number of parties, according to how much of the entire assembly is contained in this tier.32 It is important to add that the theory assumes that the upper tier is compensatory; it is not expected to apply to noncompensatory allocation.33 The extended SPM is

$Display mathematics$
(Equation 9).

In this equation, MSB refers to the basic-tier seat product. The upper tier enters through the variable t, which is the “tier ratio,” calculated as the number of upper-tier seats divided by the total assembly size. The term J, the base of the tier ratio, is determined empirically. For the full dataset employed by Shugart and Taagepera, J = 2.5, and given that this chapter is built on the same comprehensive dataset, we will use that here, yielding

$Display mathematics$
(Equation 10).

A more complete test also requires inclusion of a measure of social diversity. As we have no good operational way to measure social diversity, the much narrower notion of ethnic diversity is usually substituted. Most works in this field have used the effective number of ethnic groups (NE), derived from Fearon (2003). Li and Shugart (2016) and (p. 58) Shugart and Taagepera (2017) carry out a regression test that parallels the statistical strategy used by Clark and Golder (2006). The regression includes a term for the log of NE, as well as a multiplicative interaction to test the combined effect of the electoral system and ethnic diversity.34 The result shows a small but significant impact of NE when MSB is high (especially if there is also a large upper tier), thus offering some support for the notion of an interactive effect of electoral system permissiveness and social demand for additional parties.

Nonetheless, when individual countries’ observed values are compared to both the regression estimates and the predictions of the institutions-only extended SPM, Shugart and Taagepera find that the inclusion of diversity improves the prediction for only a few cases. In Table 3.1 we summarize the results for each of the countries included in the statistical analysis of this chapter.

Table 3.1 breaks the cases into four categories: the first two include countries that have a below-median value of expected NS from the extended SPM (Equation 10), while the second two have above-median values of expected NS. Within the groups of low and high expected NS, a further division is between those with NE below or above the median. The table includes a column for the output of the regression that includes the interaction of electoral system permissiveness (i.e., the seat product and, where relevant, the tier ratio) with ethnic fragmentation. This is thus a “postdiction,” because by the nature of a regression equation, it is telling us what the pattern is in the actual data on which it was run. Then there is a column for the output of the Seat Product Model; this is a prediction in the sense that it relies on institutional inputs only and is not a regression equation. The last two columns show the ratio of the system’s observed mean NS to the regression postdiction and the SPM prediction.

If the interactive hypothesis is correct, we should see a tendency for the regression to perform substantially better when ethnic fragmentation is high, particularly when the seat product is also high. We can summarize the relative power of the two approaches by the medians indicated for the ratios in each group, as well as for the entire set at the bottom of the table. What we see is perhaps surprising: the institutions-only Seat Product Model performs better for the cases of high ethnic diversity despite the fact that ethnicity is not a parameter in the SPM! That is, the high seat product (and an upper tier, where present) appears to be sufficient to account for highly fragmented assembly party systems, on average.

A surprising finding is that the categories where the regression including NE performs its best are those with low ethnic fragmentation—especially the first category in the table, where we have both electoral system restrictiveness and low ethnic fragmentation. Such a finding is quite contrary to the standard interactive hypothesis, as the latter claims high NS is a joint effect of electoral system permissiveness and high NE. Perhaps a more accurate claim would be that when the seat product is low, NS tends to be low, as expected from the SPM; the presence of low ethnic fragmentation tends to result in a still lower value of NS.

It is worth considering a few specific countries that are not surprises from the standpoint of the SPM. Canada, which is often alleged to be exceptional from the perspective of the so-called Duverger’s law, actually has a long-term mean NS that is ever so slightly below what we should expect, given its large assembly size (actual NS = 2.50 vs. expected (p. 59) (p. 61) (p. 60) (p. 62) NS = 2.65). The very high fragmentation in Israel, with its nationwide PR, has tended to average around the SPM expectation.35 Strikingly, the only country that has an above-median NE and is explained substantially better by the regression that includes NE (and its interaction with the seat product) is India.36

Table 3.1 Actual, Postdicted, and Predicted Effective Number of Seat-Winning Parties

Country

System Type

Seat Product

Tier Ratio

Eff. No. of Ethnic Groups

Eff. No. of Seat-Winning Parties

Regression Postdiction

Seat Product Model (SPM) Prediction

Ratio of Actual to Regression Postdiction

Ratio of Actual to SPM Prediction

Below median on both seat product and ethnic diversity

Jamaica

Simple

57.8

0

1.20

1.67

1.77

1.97

0.945

0.850

New Zealand (1945–1993)

Simple (FPTP)

86.5

0

1.57

1.97

1.94

2.10

1.012

0.935

Costa Rica

Simple

440.7

0

1.31

2.66

2.53

2.76

1.053

0.965

Cyprus

Two tier

522.5

0

1.56

3.63

2.72

2.84

1.334

1.277

United Kingdom

Simple

639.4

0

1.48

2.16

2.78

2.94

0.777

0.737

New Zealand (1996–2011)

Two tier

68.7

0.43

1.57

3.29

3.18

3.01

1.033

1.091

Honduras

Simple

932.9

0

1.23

2.26

2.82

3.13

0.800

0.722

Denmark

Two tier

1,027.0

0.23

1.15

4.62

3.91

3.93

1.181

1.174

Germany

Two tier

270.3

0.49

1.10

3.38

3.69

4.00

0.917

0.846

Median

1.022

0.950

Below median on seat product, above median ethnic diversity

Simple (FPTP)

36.8

0

2.83

1.79

1.69

1.82

1.063

0.984

Ceylon (1960–1970)

Simple (FPTP)

151

0

1.75

3.46

2.19

2.31

1.577

1.497

Nepal (1991–1999)

Simple (FPTP)

205

0

3.10

2.55

2.56

2.43

0.997

1.052

Chile

Simple

240

0

1.99

2.05

2.46

2.49

0.833

0.821

Simple (FPTP)

281.6

0

2.48

2.50

2.65

2.56

0.944

0.976

United States

Simple (FPTP)

435.1

0

1.96

1.94

2.77

2.75

0.701

0.705

India

Simple (FPTP)

536

0

5.29

3.77

3.69

2.85

1.022

1.322

Dominican Rep

Simple

641.0

0

1.63

2.42

2.86

2.94

0.848

0.825

Peru

Simple

884

0

2.76

3.38

3.53

3.10

0.957

1.090

Venezuela

Two tier

1,003.2

0.21

1.93

3.37

3.63

3.83

0.930

0.880

Median

0.950

0.980

Above median on seat product, below median ethnic diversity

Norway (1949–1985)

Simple

1,149.2

0

1.11

3.21

2.83

3.24

1.131

0.991

Norway (1989–2009)

Two tier

1,259.0

0.07

1.11

4.44

3.13

3.50

1.415

1.266

Bulgaria

Two tier

1,857.6

0

1.43

3.12

3.34

3.51

0.934

0.890

Sweden (1948–1968)

Simple

1,912.6

0

1.23

3.11

3.19

3.52

0.973

0.882

Portugal

Simple

2,628.3

0

1.04

2.85

3.16

3.71

0.902

0.768

Finland

Simple

2,660

0

1.15

5.11

3.29

372

1.554

1.373

Czechia

Simple

2,858

0

1.47

3.76

3.65

3.77

1.029

0.998

Poland (2005-2011)

Simple

5,161.2

0

1.05

3.36

3.52

4.16

0.955

0.808

Poland (2001)

Two tier

5,878.8

0

1.05

3.60

3.59

4.25

1.002

0.847

Sweden (1970-2010)

Two tier

3,395.5

0.11

1.23

3.74

4.04

4.30

0.927

0.871

Austria

Two tier

2,260.1

0.23

1.14

2.71

3.66

4.47

0.741

0.607

Netherlands

Simple

20,625

0

1.08

4.88

4.44

5.24

1.098

0.931

Slovak Republic

Simple

22,500

0

1.50

4.93

5.37

5.31

0.918

0.927

Median

0.973

0.890

Above median on both seat product and ethnic diversity

Switzerland

Simple

1,551.5

0

2.35

5.22

3.82

3.40

1.369

1.535

Latvia

Simple

2,000

0

2.41

5.37

4.07

3.55

1.321

1.514

Croatia

Simple

2,124.5

0

1.60

2.99

3.56

3.59

0.840

0.835

Spain

Simple

2,355.5

0

2.01

2.67

3.95

3.65

0.676

0.732

Macedonia

Simple

2,415

0

2.15

3.14

4.07

3.66

0.770

0.856

Brazil

Simple

9,677.9

0

2.22

8.70

5.55

4.62

1.568

1.884

Israel

Simple

14,400

0

2.11

5.15

5.89

4.93

0.875

1.044

Median

0.875

1.044

Median (overall)

1.023

1.009

Notes: Systems are sorted within categories by increasing SPM prediction.

Dates are indicated where the system is not a current one, or where there has been a change from simple to complex (or reverse).

Seat product refers to basic tier only, in case of a two-tier system.

If a system is indicated as two tier but no tier ratio is given, it is a remainder-pooling system (no fixed upper tier).

When we compare the entire set of countries in Table 3.1, we see that the median ratio of actual NS to the SPM’s prediction is 1.01. It is thus as good as the regression (1.02), despite the fact that the regression is almost guaranteed to be accurate, on average, because its parameters were estimated directly from these data. That is why we refer to the output of the regression as a postdiction. The SPM, by contrast, is built on deductive logic. It is derived before going to the data, and is therefore genuinely a prediction, based only on institutional parameters.

Extensions to votes and to disproportionality have been done (Shugart and Taagepera, 2017), but we leave the discussion here on seat fragmentation. It is, after all, through the seats in the assembly that legislation is passed and, if the executive is parliamentary, cabinets made and unmade. We have shown that two parameters of simple electoral systems, district magnitude and assembly size, can predict the average trend in the effective number of parties. The addition of the parameter of the share of seats in a compensatory upper tier can encompass a wider set of nonsimple systems. Moreover, Shugart and Taagepera (2017) show that even the systems in France (two-round majority plurality37) and Ireland (single transferable vote38) are broadly predictable, despite having electoral system features that are more complex.

# Conclusion

When S is the number of seats in the assembly and M is the number of seats in the average electoral district, four basic laws of party seats and votes emerge from our model and graphs (directly or indirectly):

1. 1. Law of largest party seats:

The most likely seat share of the largest party in an elected assembly is s1= (MS)−1/8.

2. 2. Law of number of assembly parties:

The most likely effective number of parties in an elected assembly is NS= (MS)1/6.

3. 3. Law of largest party votes:

The most likely vote share of the largest party in assembly elections is v1= [(MS)1/4+ 1]−1/2.

4. 4. Law of number of electoral parties:

The most likely effective number of parties in assembly elections is NV= [(MS)1/4+ 1]2/3.

In principle, they apply only to “simple” electoral systems and (with the adjustment term in Equation 10) to two-tier systems. Actually, even complex electoral systems (two rounds, legal thresholds, and single transferable vote) fit to a surprising degree. These laws also enable us to take the seat product of a given electoral system and from that (p. 63) estimate the deviation from PR and the effective number of presidential candidates—see Shugart and Taagepera (2017). Through the inverse square law of cabinet duration—C = 42 yrs / NS2 (Taagepera 2007, 165–175)—which we have not discussed here, the impact of electoral institutions reaches the workings of governments in parliamentary democracies.

But a big surprise awaits us when we try applying these laws in individual districts: They underestimate the effective number of parties. This is so because national politics boosts the number of local parties. In this sense, “all politics is national”! The nationwide impact on the district level is quite systematic, but we will bypass it here.39

The present overview lacks space to cover all this. Rather than skimming over too much new ground that goes beyond the four basic laws, we have focused on explaining the logic behind the four laws and presenting empirical evidence, in the form of graphs and statistical testing. Building on the so-called Duverger’s law, fuzzy and quasi-quantitative as it is, the four laws raise the Duvergerian quest to a truly quantitative level, making specific prediction possible.

The permissiveness of electoral systems toward small parties has traditionally been measured mainly by district magnitude alone; the product of district magnitude and assembly size now must replace this. Indeed, 625 single-seat districts lead to as many parties as 25 districts of 25 seats each. This means there is little point in treating single-seat and multiseat systems as different species, for many purposes.

The reach of the seat product is powerful, doing away with the need for various other inputs. Presidential regimes fit in; indeed, Shugart and Taagepera (2017) show that the effective number of presidential candidates, rather than being a separate noninstitutional input factor, can be estimated from the seat product of the assembly. Social diversity, to the extent it can be reduced to mere ethnic diversity, finds its outlet in seat product, so that including the effective number of ethnic groups improves our predictive ability only when MS is very low.

A hallmark of developed sciences is that they go beyond establishing quantitative connections among various factors: they furthermore establish connections among such connections. This is the level electoral studies have now reached, interconnecting institutional inputs such as district magnitude and assembly size, and political outputs such as number of seat-winning parties, effective number of parties, the largest seat and vote shares, and even cabinet duration through a network of equations, of which the equations in this chapter are samples.

When an individual election or country deviates from the expected value, this is an opportunity for further research into what it is about the politics that causes the deviation. Is it something in the history and culture of the country, a specific issue or party leader dominating the campaign of a specific election, or something else? If we lack a deductive baseline for what the outcome should be, given the institutional inputs, we are unable to ask these questions in a scientifically meaningful way. However, we now have a baseline, grounded in logic, and tested on a worldwide sample of elections. Having a baseline is not the endpoint. It is where the hard work begins—when seeking to understand the way context sometimes results in deviations from expectations, even the (p. 64) average of numerous elections across most countries conforms.40 Few subfields of social sciences have reached such a degree of interconnected knowledge.

# Author Note

All figures are drawn by the authors specially for this chapter.

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## Notes:

(1.) Or sometimes effective number of legislative parties (ENLP).

(2.) Moreover, ENP risks to be mistaken for multiplication of the quantities E, N, and P.

(3.) As with the effective number, so could F be calculated on votes rather than seats.

(4.) Nor should it be called the “number of effective parties,” as one sometimes sees. It is the number that is “effective”; judgment on whether the parties per se are effective might be beyond the scope of a mathematical index.

(5.) Caution needs to be taken if there is a substantial percentage of either votes or seats listed as “others”; these should not be treated as if they were one party, nor ignored when calculating effective N. See Taagepera (1997) for guidance; see also Gallagher and Mitchell (2005, 598–602).

(6.) For further detail on various indices of deviation from proportionality, see Taagepera (2007, 65–82) and Taagepera and Grofman (2003).

(7.) It even entered popular media discourse in Canada in late 2016, when a parliamentary committee charged with considering alternative electoral systems released its report. See “The Problem with Maryam Monsef’s Contempt for Metrics,” McCleans’s, December 3, 2016, http://www.macleans.ca/politics/ottawa/the-problem-with-maryam-monsefs-contempt-for-metrics/ (accessed December 15, 2016).

(8.) For example, they might show graphs of “marginal effects” of the interaction of one variable with another, but not show an underlying bivariate relationship of theoretical interest through the plotting of actual election outcomes.

(9.) Even with many input (or “independent”) variables, one could graph the regression-estimated outcome against the actual outcomes from elections (or country averages). Yet this is only infrequently done in political science.

(10.) See, for example, Table 3.2 in Clark and Golder (2006), in which the coefficient estimate on the log of magnitude ranges from –0.08 to 0.33, depending on what other variables are included and the precise sample. Even their interaction with ethnic fragmentation—testing their preferred hypothesis (discussed later in this chapter)—has a similarly wide range of coefficient values and is statistically significant in some specifications and insignificant in others.

(11.) In our larger work (Shugart and Taagepera, 2017), we plot and test regressions with data from individual elections. Here we report only the country averages.

(12.) Later we will discuss the impact of more complex systems.

(13.) For instance, D’Hondt, Ste.-Laguë, Hare, or Droop quota and largest remainders. See the introduction to this volume for definitions of common formulas.

(14.) On thirty-eight country means from simple systems, we obtain log(NS) = 0.029 + 0.156log(MS); the intercept is insignificant and the expected 0.167 lies firmly within the 95 percent confidence interval of the estimated coefficient on log(MS). R2 = 0.665. Given that it must be that NS = 1 (and thus log NS = 0) if we had any actual cases of assembly elections with MS = 1, we can re-estimate the regression with the constant term suppressed. If we do so, our coefficient on logMS is 0.1664. For general principles of avoiding regression results that yield absurdities, see Shugart and Taagepera (in press) and Taagepera (2008).

(15.) This means being high or low by a factor of four-thirds. In other words, log(3/4) = –0.125 is the negative of log1.33 = log(4/3) = +0.125.

(16.) While several presidential systems are on the low side of the line representing Equation 1, they are not to any degree outside the general data range for nonpresidential systems. See Shugart and Taagepera (in press) for a detailed discussion of why presidential systems need not be treated separately. See also Li and Shugart (2016).

(17.) Our three actual cases of M = S (nationwide PR) are all, on average, very close to the line representing Equation 1. Note the three points closest to the right margin of the graph; these are Israel, the Netherlands, and Slovakia.

(18.) In this equation, we have dropped the prime mark from NS0 because we are again referring to nationwide results.

(19.) For M < S, we have M1/2 as the expectation for the mean district and S1/2 as the mean for the assembly. The geometric average of these conditions leads directly to Equation 3: NS0 = (M1/2 × S1/2)1/2 = (MS)1/4.

(20.) The ordinary least squares (OLS) regression result is logNS0 = 0.0702 + 0.237log(MS). The constant is insignificant, and R2 = 0.670. It is not surprising that the data plot shows some more outliers than does Figure 3.1, showing NS. A few very small parties unexpectedly getting a seat or two—or, conversely, an expected party getting no seats—make a full integer jump in NS0 for any given election. Yet such small deviations from the expected number have less consequence for NS, due to its being a size-weighted count. The surprise is that the fit is so good, despite the bluntness of the actual number of parties (NS0) as a measure.

(21.) The OLS regression result is logs1 = –0.000518 – 0.4562(NS0). The constant is effectively zero, and R2 = 0.705.

(22.) The OLS regression is logNS = –0.0493 – 1.238logs1. The constant is actually significantly different from zero, at 95 percent confidence. However, it is logically impossible to have a constant in this equation that is different from zero, as it would imply a value of NS > 1 when s1 = 1, which is an absurdity. Given that the expected –1.333 is within the 95 percent confidence interval of the estimated coefficient, we can consider Equation 5 (which is based on deductive logic) to be confirmed.

(23.) As we noted earlier in this chapter, using the effective number in this manner can be somewhat misleading.

(24.) Cox (1997, 99) defines viable as “proof against strategic voting.”

(25.) Cox goes on to suggest the answer is “economies of scale” that lead actors to “coordinate” around a smaller number of party lists.

(26.) We found that NS0 = (MS)1/4 and NS = (MS)1/6; algebraically, then, we have to have NS = NS02/3.

(27.) Leaving out the Brazilian case would be justified, as the effective number of vote-earning parties at the national level is an overcount, due to the existence of multiparty alliances within the country’s competing electoral lists. See Shugart and Taagepera (2017) for details. The same measurement problem affects seats, although less so. The regression without the Brazilian case is logNV = 0.0141 + 0.682log[(MS)1/4 +1]; the constant is insignificant and R2 = 0.648.

(28.) In addition, most standard regression treatments include variables found only in presidential systems: the effective number of presidential candidates (NP) and how “proximate” the assembly election is to a presidential election. A key problem with these treatments is that it requires an unrealistic value of NP = 0 for parliamentary systems for these cases to be in the same regression (with nonmissing values on NP) with presidential systems. Yet the feasible minimum value of NP is 1.00. Moreover, it is untenable to assume, as these works implicitly do, that a parliamentary system is no different from a presidential system that has only midterm elections (like the Dominican Republic from 1998 to 2010); both situations incur a “proximity” value of zero in these models. For extended critique of such approaches, see Li and Shugart (2016) or Shugart and Taagepera (2017); see also Elgie et al. (2014).

(29.) Results are not different when median is used instead of mean. Moreover, Shugart and Taagepera (2017) do not find evidence that wide variation in magnitude affects the accuracy of the predictions from the SPM. This does not preclude the possibility that such variance may have other effects, for instance, favoring certain parties over others (Monroe and Rose 2002; Kedar et al. 2016).

(30.) On South Africa, see the chapter by Ferree in this volume. On Denmark and two-tier systems more generally, see Elklit and Roberts (1996).

(31.) See, respectively, the chapters by Zittel and Vowles in this volume.

(32.) Through disaggregation of two-tier systems into their district level and separately accounting for the impact of the upper tier, Shugart and Taagepera (2017) are able to confirm the logic proposed by Li and Shugart (2016).

(33.) Noncompensatory upper tiers are found, for example, in the mixed-member majoritarian systems (as in Japan—see Nemoto’s chapter in the volume). The extended SPM also would not apply if the system used in the basic tier did not meet the criteria of “simple” (e.g., the two-round systems found in the basic tiers of Hungary and Lithuania).

(34.) The regression equation is logNS = α + β1log(MSB) + β2t + β3log(NE) + β4[log(MSB) * log(NE)]. For details, see Li and Shugart (2016) or Shugart and Taagepera (2017).

(35.) In recent years it has been well above, balancing out past periods when it was below. It is worth noting that the country’s recent surge in party fragmentation is also well above what is predicted when including its ethnic fragmentation in the regression. See Hazan et al. (this volume) for a detailed treatment. See also Stoll (2013).

(36.) India’s recent period of very high party system fragmentation has seen most parties agglomerate into a smaller number of alliances that structure national-level competition and provide government and opposition organization of the assembly (see Ziegfeld, this volume). Shugart and Taagepera (2017) show that the effective number of alliances in India since 1999 closely matches the value predicted from the SPM. (The model itself is “agnostic” about whether the entities in question are called by different party names in different districts or are a nationwide entity; in most countries the entities at the district level and nationwide levels are the same, but not in India, in the era of alliances.)

(37.) See the chapter in this volume by Hoyo.

(38.) See the chapter in this volume by Marsh.

(39.) What is the adjusted formula for such “embedded” districts? For effective number of seat-winning parties in districts it is N’S = M2k/3, where (hold your breath!) k = 0.5 + 0.2076log(S/M) / M.25—most of it logically grounded. See proof and evidence in Shugart and Taagepera (2017).

(40.) Several chapters in this volume explore how specific electoral systems operate in their country contexts.