Lijphart & Grofman - Electoral Laws and their political consequences(2003) CHAPTER 10 Degrees of Proportionality of Proportional Representation Formulass Avend tt iiss aa wweellll--estatablblisishedd propositition in the liliteteraratuturere on elelectotoraral syss-- Itetems thatt proportitional representatatition (PR) iiss generarallllyy quititee cessful in achieving its principal goal --aa rreeaassoonnaabbllyy prooporrtitioonall translation ooff votes into seats--especiallyy inin comparison with plurality and majorityy formulas. One ooff RRaaee’’s ((11997711, 96)) ”differentialal proposi-- tions” isis that “proportional representation formulae tend to allocate seats more proportionally than do majority and plurality formulae.”.”ItIt isis alsoo known that different PPRR formulas are not equally proportional, but students ooff PPRR disagree aboutt which ooff ththee formulas are more and d which are less proportional. TThhee ppuurrppoossee ooff tthhiiss aannaallyyssiiss iiss ttoo eessttaabblliisshh a rraannk oorrddeer ooff tthhee principal PR formululas aaccccoorrdidinngg ttoo tthheeiirr degree oof f proportionality. II shalalll usee BBlloonnddeell’’ss ((11996699,, pp. 118866--119911)) rarankiningg ooff the ablele votete,, Saintee--LLaagguuee,, aanndd llaarrggeesstt rreemmaaiinnddeer ssyysstteemmss--ththee mosstt importrtant attttempt ttoo rarank order PPRR formululas undertakenn so far --aass mmyy pprreelliimmiinnaarryy hypotthessiiss.. IInn aaddiittiion ttoo ththee four formuulalass aannaallyyzzeedd bbyy II sshhaalll aallsso ttrry ttoo iinncclluuddee tthhee ttwwo IImmppeerriiaallii formululas iinn mmyy rarankining.g. Thee degree ooff proportitionalalitityy mmaayy be defifined in tetermss ooff two elelee-- Proportionality of PR Formulas 272 degree to which large and small parties are treated equally. It is the second element that provides a clear criterion for judging the propor - tionality of PR formulas, because deviations from proportionality are not random: They tend to systematically favor the larger and to dis- criminate against the smaller parties. Blondel's Ranking Blonde1 (1969, p. 191) ranks four PR formulas in the following de- creasing order of proportionality: 1. Single transferable vote (STV) 2. Sainte-Lague 3. 4. Largest remainders STV is therefore the most, and the largest remainder method the least, proportional formula, according to Blondel. Other, more limi- ted, attempts to determine the proportionality of PR systems tend to (1)ignore the STV method, (2) agree with Blondel's judgment that the formula is less proportional than the Sainte-Lague formula, and (3) disagree with Blondel's placement of the largest remainder formula at the bottom of the list. Loosemore and Hanby (1971)consider three of Blondel's four formu- las, and they arrive at the following rank order: 1. Largest remainders 2. Sainte-Lague 3. Their relative placement of Sainte-Lagueand is in agreement with Blondel's, but they conclude that the largest remainder formula is the most proportional of the three. Rae (1971, p. 105)also finds that the largest remainder method is more proportional than the other two lumped together. Balinski and Young (1980) confirm the Loosemore- Hanby finding that the Sainte-Lague formula, equivalent to the ster method of apportionment of the U.S. House of Representatives, yields more proportional results than the formula, which is the equivalent of the Jeffersonmethod of apportionment. Most of the literature confines itself to a comparison of the 172 Arend Lijphart est remainder formula is more proportional and more favorable to the smaller parties (Van den Bergh, 1955, pp. 24-26; Mackenzie, 1958, 78-80; 1974, 93-97; Berrington, 1975, 366-368; Nohlen, 1978, pp. 77-80; Bon, 1978, This consensus deviates from Blondel’s ranking. In the next section, I shall show that the consensus is right and that Blonde1is wrong. Comparing d‘Hondt and Largest Remainders A comparison of the d’Hondt and largest remainder for - mulas is a good starting point for our exercise because it also supplies us with the key we need for the ranking of the other PR methods. How can we explain the different results of these two basic PR formu- las? The initial difficulty is that the d’Hondt and largest remainder for - mulas appear to use completely different methods for allocating seats on the basis of the parties’ votes. Table 10.1gives a concrete illustra- tion. The largest remainder formula first calculates the electoral quota or quotient (often called the Hare quota): the total number of valid votes cast divided by the total number of seats in the district (s). The parties’ votes are divided by this quota, and each party receives a seat for every whole number in the result. The remaining seats are then awarded to the largest of the unused ”remainders” or remaining votes. The d’Hondt formula does not require the calculation of an electoral quota. As Table 10.1shows, each party’s votes are divided by the series of divisors 1, 2, 3, and so forth, and the seats are given successively to the highest of the resulting values, usually referred to as “averages.” It is possible, however, to interpret the highest average formula in such a way that it becomes comparable to the largest re- mainder formula. The purpose of the d’Hondt formula may be said to be the improvement of the largest remainder formula by finding an electoral quota, lower than the Hare quota, which allows us to allocate all of the seats exactly according to the largest remainder rule but without having to take any remaining votes into account (Van den Bergh, 1955, 68-72). This lower d’Hondt quota is equal to the last of the ”averages” to which a seat is awarded. In the example of Table 10.1, the d’Hondt quota is 14 votes. When the parties’ votes are di- vided by this quota, party A is entitled to 2 seats, to 2, C to 1,and and E to none; all of the seats have been allocated and the remaining Proportionality of PR Formulas TABLE 10.1. Hypothetical Example of the Operation of the Largest Remain- der, and Pure Sainte-Lague Formulas a District with 100 Votes, 5 Seats, and 5 Parties Largest Initial Allocation Remaining Allocation of Final Seat Parties Votes of Seats Votes Remaining Seats Allocation A 36 1 16 1 2 B 30 1 10 0 1 C 14 0 14 1 1 D 12 0 12 1 1 E 8 0 8 0 0 Parties u12 u13 Total A 36 (1) 18 (3) 12 2 B 30 (2) 15 (4) 10 2 C 14 (5) 7 1 D 12 0 E 8 0 Pure Parties u13 u15 Total A 36 (1) 12(4 ) 7.2 2 B 30 (2) 10 1 C 14 (3) 4.67 1 D 12 (5) 4 1 E 8 0 electoral quota is 10015 = 20. numbers in parentheses indicate the sequential order of the allocation of seats. The reason for the disproportionality of now becomes clear. The remaining votes that it disregards are a relatively small portion of the votes of the larger parties but a very large portion of the small parties’ votes -and, of course, the entire vote total of a party that does not receive any seats. As a result, the seat shares of the larger parties will tend to be systematically higher than their vote shares, and the smaller parties will tend to receive seat shares that are system- atically below their vote shares. In contrast, the largest remainder method treats large and small parties equally: The initial allocation of at i tl ti al and all and la rtie et fo Proportionality of PR Formulas 179 largest remainders. However, both methods suffer from a more serious weakness: They may encourage party splits. In the example of largest remainders in Table 10.1, party B could win an additional seat, at the expense of party D, if it would present two separate lists, each of which would get 15votes. Pure Sainte-Lague would have the same effect. This tendency declines as PR formulas become less proportional, and it disappears entirely under the rule-certainly a very powerful argu- ment in favor of this not maximally proportional formula (cf. Balinski and Young, 1983)..