N. R. Miller

3/9/03

rev. 10/17/04

VOTES AND SEATS IN DISTRICTED ELECTIONS

            Most elections have the direct effect (if voters are voting for parties) or the indirect effect (if voters are voting for candidates nominated by parties) of allocating seats in parliaments (or other legislative assemblies) to political parties. British general elections (indirectly) allocate seats in the House of Commons among parties (and thus determine which party leader becomes Prime Minister). U.S. Congressional elections (indirectly) allocate seats in the House of Representatives and the Senate to the Democratic and Republican (and possibly other) parties (and thus determine which party organizes Congress). Even U.S. Presidential elections can be interpreted as (indirectly) allocating electoral votes (“seats” in the “Electoral College”) to parties (and thus determine which party nominee becomes President).

            Most electoral systems are districted, with the result that this allocation takes place in two steps: first, some electoral formula translates votes into seats within each district and, second, seats allocated within each district are aggregated by simple addition across districts into an overall allocation of seats in the national parliament or other assembly. We focus here on such districted elections.

            The magnitude M of an election district is the number of seats it contains and therefore the number winning candidates in the district. English-speaking countries with “majoritarian” or “first-past-the-post” electoral systems commonly use single-member districts (SMDs) with M = 1. Proportional representation systems necessarily use multi-member districts (MMDs), and the greater the magnitude of the districts the more precisely proportional the seat allocation within a district can be.

            Let’s introduce some terminology classifying parties seeking seats within a district into three categories. Trailing parties have little electoral support and probably cannot win even one seat in the district. Leading parties have support such that they can expect to win one or more seats. Marginal parties stand between the two other types with respect to electoral strength and may or may not win a single seat. Given a more or less proportional electoral formula, a party with the support of distinctly less than about 1/(M+1) of the electorate are likely to be trailing, a party with the support of about 1/(M+1) or more of the electorate is likely to be leading, and a party with the support of about 1/(M+1) or somewhat less of the electorate is likely to be marginal. It is important to note that the actual level of voter support necessary to put a party into one or other category varies by district magnitude, as indicated by the 1/(M+1) expression.

1.         MMDs and Multi-Party Systems

            Within a large multi-member district (MMD), any list-PR electoral formula translates the vote for parties into seat allocations among the parties in a manner that can be (and usually is) highly proportional. For example, a party that receives 27.139% of the vote in an MMD with M = 38 seats would, under an ideally proportional system, be entitled to its quota of .27139 × 38 = 10.313 seats. Since seats must be awarded in whole numbers, it cannot receive precisely its quota of 10.313 seats but, regardless of whether it actually receives 10 seats (26.3% of 38 seats) or 11 seats (28.9%), the result is quite close to proportional. And if results within districts are close to proportional, the national results aggregated across districts must be close to proportional also. Indeed, a leading or marginal party that is slightly penalized by virtue of being “rounded down” in one district is likely to be slightly rewarded in a compensating fashion by virtue of being “rounded up” in another district, so national results with respect to such parties are likely to be more closely proportional than most district results. This tendency for the “rounding errors” that result from the fact that seats must be allocated in whole numbers to roughly balance out in the national seat allocation may be called the compensation effect. However, a party that is trailing in all districts cannot expect to be held even approximately harmless by the compensation effect; such a party is consistently “rounded down” to zero seats in all districts, even though national proportionality would entitle it to several seats in parliament. For example, a party that wins about 1% of the vote in each of 10 MMDs of magnitude 38 is ideally entitled to about 0.38 seats in each district but would probably be “rounded down” to zero seats in all (or almost all) districts and thus would be allocated zero seats nationally, even though national proportionality would entitle it to about 3 or 4 seats (i.e., 0.01 × 380 = 3.8 seats rounded down or up).

            Within a small (say magnitude 3-6) MMD, any list-PR or quasi-proportional formula can translate votes for parties into seats in a manner that is only roughly proportional. For example, a party that receives 27.139% of the vote in an MMD with 5 seats would, under an ideally proportional system, be entitled to .27139 × 5 = 1.357 seats. In practice, it must receive either one (20%) or two (40%) seats, and in either event the result not very proportional. Again leading and marginal parties are more or less equally likely to be “rounded up” or “rounded down,” so the compensation effect again implies that the national seat allocation among such parties will be quite proportional. But trailing parties winning no more than about 10-15% of the vote in any district will probably fail to win seats in any district, though such a party would win about its proportional share if large MMDs had been used. (The logic is identical to the large MMD case; the difference is that considerably larger parties may now have trailing status.)

2.         SMDs and Two-Party Systems

            Now let us consider a (“first-past-the-post”) electoral system made up exclusively of single-member districts (SMDs) for which M = 1. A common criticism of such electoral systems based on plurality voting is that they can generate highly unproportional national seat allocations. It is worth noting that all list-PR formulas (and some quasi-proportional formulas) when applied to SMDs are logically equivalent to the simple plurality formula, which is in fact as proportional as possible given that there is only one seat to be allocated. Given that the lack of proportionality of such systems derives from aggregation across districts, the criticism should focus on the districting system, not on the electoral formula applied within each district.

            Given simple plurality voting within each district, the single leading party is “rounded up” to one seat while all other (marginal or trailing) parties are “rounded down” to zero seats. But with SMDs, the scope of the compensation effect is quite restricted. If the same party is leading in all (or almost all) districts, it is always (or almost always) “rounded up” while all other parties are consistently “rounded down,” so the leading party wins all (or almost all) seats nationally.

            However, according the “Duverger’s Law” (to be discussed in the next handout), a SMD system produces a two-party system, so that the same two parties are likely to be more or less tied for leading status (i.e., are both marginal) both nationally and in almost all districts. If so, the compensation effect will operate (at least approximately) between the two leading parties but not with respect to trailing parties, which are (as always) consistently “rounded down.”

            Let us now consider the SMD case in more detail, assuming that Duverger’s Law is operating so perfectly that only two parties (call them D and R) are contesting elections. We shall see that, in so far as “rounding errors” do not tend to balance out between the two parties, the imbalance is likely to favor of whichever party leads with respect to national vote totals and to do so in proportion to the magnitude of that lead.

            Consider the district-by-district vote shares for D and R in a particular election and the frequency distribution of districts with respect to the percent D% of votes won by the D party (where of course R% = 100% -D%). The districts range potentially from 0% for the D party (100% for the R party) to 100% for the D party (0% for the R party). The height of each frequency curve in Figures 1-7 indicates relatively how many districts give each level of support to the D party. Since in each of Figures 1-3 the curve is highest at the 50% mark, the greatest concentration of districts is in the vicinity of 50%, i.e., districts which the D party wins or loses by a small margin. According to the curve shown Figure 1, there are almost no districts in which the D party gets less than 25% or more than 75% of the vote, whereas in Figure 2 there are a few districts that safe for each party and in Figure 3 there are still more.

            Any such frequency distribution may be characterized with respect to its center, its spread, and its shape.

            The center (or mean) of the distribution reflects how well the D party does in the election with respect to national vote totals. Indeed, if the same number of votes are cast in each district, the center is identical to the percent of the national vote received by the D party. Thus Figures 1-3 all depict elections in which the D party receives 50% of the national vote. In each of Figures 4-6, the curve has shifted 5% upwards compared with Figures 1-3, so Figures 4-6 all depict elections in which the D party receives 55% of the national vote.

            The magnitude of the spread (or standard deviation) of the distribution increases from Figures 1 to 3 and also from Figures 4 to 6 (while the center remains the same in each column). The spread reflects how much D (and R) party strength varies from district to district. If the spread is very small, the districts are pretty much little replicas of each other (and of the nation as a whole) with respect to partisan inclinations (and probably with respect to demographic and sociological characteristics that shape such inclinations). If the spread is large, the districts are diverse with respect to partisan inclinations (perhaps because some districts are overwhelmingly working-class while others are overwhelmingly middle class). Figures 1 and 4 show distributions with a small spread (standard deviation of 5 percentage points). Figures 2 and 4 show distributions with moderate spread (SD of 12 percentage points) while Figures 3 and 6 show distributions with a large spread (SD of 20 percentage points). If the spread is large and the center is close to 50%, there are a lot of districts that the D party wins or loses by large margins and there are relatively few districts that the D party wins or loses by close margins). If the spread is small and the center is close to 50%, few districts are safe for either party and many districts are very close.

            With respect to the shape of the distribution, we initially assume that it is symmetric and bell-shaped (e.g., a normal curve), as shown in each of Figures 1-6. Whether the D party wins a majority of seats depends on whether the D party carries the median district, i.e., the district such that the D party does better in half of the remaining districts and worse in the other half. If the district distribution is symmetric, as in Figures 1-6, the location of the mean and median coincide, so whichever party wins the most votes also wins the most seats.

            If, as in each of Figures 4-6, the D party wins 55% of the national vote, a proportional electoral formula applied nationwide or to sufficiently large MMDs would allocate the same number of seats to the D party regardless of how its strength varies from one district to another and the D party would win about 55% of the seat in any event. However, in a SMD system, the percent of seats won by the each party depends not only on the location of the center (i.e., the national vote shares) of the district distribution but also on its spread (i.e., how vote shares vary by districts). The proportion of seats won by the D party by definition equals the proportion of the districts in which the D party won more than 50% of the vote, which in turn is given by the proportion of the area under the frequency curve that lies above the 50% mark (the crosshatched area). Clearly this area and the resulting proportions differ considerably across Figures 4-6, in a manner that reflects the differing spread of the three distributions. In Figure 4, the D party wins about 84% of the seats, in Figure 5 about 66%, and Figure 6 about 60%. (These calculations assume that the distributions are normal.)

            So long as the shape of the district distribution remains bell-shaped (with a single peak), the national seat allocation will be in at least some degree supraportional, giving the winning party a share of seats that exceeds its share of votes. (Proportionality occurs if and only if the parties split the vote 50%-50%, as in Figures 1-3.) If the district distribution spreads out so as to become essentially uniform (i.e., so that the “curve” is essentially a horizontal line), national seat shares will be approximately proportional. If the district distribution becomes polarized with peaks at either extreme, the national seat distribution becomes subproportional, giving the winning party a share of seats that is smaller proportion than its vote shares (but still a majority) proportion.

            Let as now return to the more plausible bell-shaped distribution but allow its shape to become skewed, as in Figure 7. In a skewed distribution, the median and mean no longer not coincide — the mean is pulled in the direction of the longer “tail” (to the right in Figure 7). In fact, the magnitude of the interval between them is commonly used as a measure of skewness. If this interval happens to span the 50% mark, as in Figure 7, the districted election produces a reversal of winners (or a “wrong winner” or “misfire”), in which the party that wins the most votes fails to win the most seats. In Figure 7, the D party again wins 55% of the vote but only 45% of the seats; this comes about because it wins a minority of districts by (on average) large margins while it loses a majority of districts by (on average) small margins.

            We can generalize all these considerations by allowing the D party to win any percentage of the national vote, rather than fixing this at 50% (as in Figures 1-3) or 55% (as in Figures 4-6). We do this by interpreting the district distribution as representing not the actual district vote results in a particular election but the “normal vote” for the D party in a given electoral alignment. “Short-term forces,” reflecting circumstances (events, issues, identity of party nominee/leaders, etc.) that pertain to particular elections, produce a “swing of the pendulum” that causes the support of the D party in each district to move up or down by a (more or less) uniform amount from one election to the next. (In effect, the entire district distribution slides up and down the horizontal scale, up in good D years and down in bad.) We can then trace out the expected proportion of seats won by the D party for any D party national vote percentage. Statistically, this votes-into-seats function is simply the cumulative district distribution. Figure 8 shows several such votes-into-seats functions, including those resulting from the symmetric distributions depicted in Figures 1-6. Figure 8 also shows the votes-into-seats function resulting from a skewed distribution as depicted in Figure 7.

            The slope of the votes-into-seats function in the vicinity of 50% (which is where the slope is both steepest and close to being a straight line if the district distribution, and therefore the votes-into-seats function, is symmetric) is commonly called the swing ratio. A swing ratio of 3, for example, means that, if from one election to next the D party’s national vote percentage increases (or decreases) by 1 percentage point, its seat percentage will increase (or decrease) by about 3 percentage points.

            In national legislative elections based on SMDs, the swing ratio is typically about 2 to 3. But we might ask what determines the magnitude of the swing ratio? The immediate answer, of course, is the spread of the district normal votes (the greater the spread, the smaller the swing ratio). But what determines the latter? Three substantive political factors may be identified: (i) the size and regional diversity of the society that constitutes the national electorate, (ii) the practices employed in demarcating district boundaries, and (iii) the size and number of districts and thus the size of the parliament or other national assembly.

            Point (i) has already been presented at some length. Note that what matters is diversity among districts, not diversity within districts or in the electorate as a whole. For example, any electorate is likely to be equally diverse with respect to gender (50% male, 50% female) and geography (50% easterners, 50% westerners). If almost all females vote D and almost all males vote R, the swing ratio will be extremely high, because all districts have virtually the same 50-50% ratio between males and females, and the district normal vote distribution has almost no spread. But if almost all easterners vote D and almost all westerners vote R, the swing ratio will be almost zero, because almost all geographically defined districts are either all eastern or all western and the district normal vote distribution has almost maximum spread.

            It is well known that district boundaries are often drawn with electoral considerations in mind. (In U.S. politics, this is referred to as gerrymandering.). If both parties have veto power over the districting process, the result is likely that they draw the boundaries to create a lot of safe D districts and a lot of safe R districts, with very few close districts. The result is a high spread and a low swing ratio. Non-partisan districting might create many more close districts, i.e., low spread and a high swing ratio. If the D party unilaterally controls the districting process, it will probably “pack” as many R voters as possible into a relatively few districts that are very (and “wastefully”) safe for the R party, while creating a large majority of districts that are fairly close but lean in the D direction. This will probably produce moderate swing ratio but it also will produce a skewed distribution that may allow the D party to win a majority of seats with a minority of votes — that is, to benefit from a reversal of winners. More generally, it means a given vote share won by the D party will give it more seats than the R party would get with the same vote share.

            Generally speaking small districts are likely to be less diverse internally than large districts (a basic argument in James Madison’s Federalist 10). But this also means there will more diversity across small districts than large ones, i.e., the smaller the districts, the greater their spread, and the lower the swing ratio. This implies, for example, that the winning party will usually control the U.S. Senate (larger districts, smaller spread, higher swing ratio) by larger margins than the U.S. House of Representatives (smaller districts, larger spread, lower swing ratio). (This comparison in practice is made less clear-cut by staggered Senate elections and the phenomenon of “incumbency advantage” that is especially conspicuous in the House.) It also implies that if Presidential electors were selected from SMDs, rather than statewide on a general ticket, the electoral vote in Presidential elections would be much less lopsidedly in favor of the winning candidate.