The bounds of votes of divisor electoral methods

Abstract

In proportional electoral systems, party vote counts must be converted to seat allocations within a parliament of fixed size. Divisor methods are the most common approach to this problem, but different divisor methods often give different seat allocations. To highlight these differences, the effects of various divisor methods on a party’s seat allocation are expressed as intervals of the party’s vote count within which the seat allocation is unchanged, assuming other parties’ votes are fixed. These bounds are applied to data from four recent European parliamentary elections, as well as one hypothetical dataset.

1 Introduction

Proportional electoral systems are used in many countries to convert the party vote count in an election to an allocation of parliamentary seats. The problem is that the numbers of seats assigned to each party must be integers, which means that the proportion of seats allocated to a party is almost always different from the party’s proportion of the votes. The same problem arises in federations when numbers of representatives of districts or states are apportioned according to their populations (Cortona et al. 1999; Still 1979).

Divisor methods are the most common approach to the allocation problem (Balinski and Young 2001; Carstairs 1980; Pukelsheim 2014; Mackie and Rose 1982). But different divisor methods can give different seat allocations from the same vector of party vote counts, so the choice of divisor method may determine whether a particular coalition of parties forms a parliamentary majority (Marošević and Soldo 2018).

We use the following notation:

• $n\ge 2$ - the total number of parties (or the total number of districts or member states in the country),

• $I=\{1,2,\ldots ,n\}$ - the set of parties (i.e., the set of districts or member states),

• S - the total number of seats in the parliament,

• ${{\textbf{v}}}=(v_1,v_2,\ldots ,v_n)$ - the vector of votes, where $v_k\in {{\mathbb {N}}}$ is the number of votes received by the party k (or the population of the district or the member state k), $k\in I$, and the total number of votes $\displaystyle P=\sum _{k=1}^{n} v_k$,

• ${{\textbf{s}}}=(s_1,s_2,\ldots ,s_n)$ - the vector of seat allocation, where $s_k\ge 0$ is the number of seats assigned to the party k, and $\displaystyle \sum _{k=1}^{n} s_k=S$.

The divisor methods are based on the increasing sequence of their divisors (i.e., the functions $d:{{\mathbb {N}}}\rightarrow [0,\infty \rangle$): $d(1)< d(2)< \ldots < d(S) \,.$ For each party and each divisor, the following ratios are observed (provided that $d(1)\ne 0$):

$$\begin{aligned} \frac{v_k}{d(1)}> \frac{v_k}{d(2)}> \ldots > \frac{v_k}{d(S)}, \qquad \forall k=1,\ldots ,n . \end{aligned}$$

Then the S seats are assigned to the parties that have got the S largest such ratios. In the case of ties, additional rules for seat allocation must be defined.

The most popular divisor methods are listed in Table 1 (Cortona et al. 1999; D’Hondt 1878; Hill 1911; Huntington 1921; Sainte-Laguë 1910).

Table 1 The divisor methods

Remark 1

The smallest divisors method (SD), the harmonic mean method (HM) and the equal proportions method (EP) all have $d(1) = 0$. The interpretation of a method with first divisor 0 is that every party automatically receives one seat, and all ratios $\frac{v_k}{d(1)}$ are dropped from the calculation. The remaining $S-n$ seats are allocated to the parties with the $S-n$ largest ratios. Note that these methods cannot be applied unless the condition $S\ge n$ is satisfied.

Another group of proportional electoral methods are the quota methods. By these methods seats are allocated on the basis of the integer part $\lfloor q_k \rfloor$ of the given quota $q_k$ and after that, the remaining seats are allocated on the basis of the largest remainders $r_k\in [0,1\rangle$, where $q_k=\lfloor q_k \rfloor + r_k$, $k=1,\ldots ,n$. The most popular quota methods are defined by the following quotas:

• (a) the largest remainders method (LAR), by the natural quota $\displaystyle {q_k=\frac{v_k}{P}\cdot S}$,

• (b) the Droop quota method (DRO), by the Droop quota $\displaystyle { \frac{v_k}{P}\cdot (S+1)}$,

• (c) the Imperiali quota method (IMP), by the Imperiali quota $\displaystyle {\frac{v_k}{P}\cdot (S+2)}$.

With regard to the principle of ’fair distribution’ of seats, one can look at the properties of underrepresentation and overrepresentation that a particular proportional method gives, and at the characterization of these properties by means of the Lorenz curve and the Gini index, which are considered in the field of welfare economics (Cortona et al. 1999).

One can also observe proportional methods as algorithms that give a solution to integer optimization problems of special objective functions that represent various measures of proportionality between the obtained votes and the assigned seats (Cortona et al. 1999).

In this paper, we consider the bounds of votes of divisor methods, i.e., we consider intervals of vote counts within which different divisor methods give identical seat allocations. The same question for quota methods as a class different from divisor methods is complex and not considered in this paper.

In Sect. 2, the conditions of specified seat allocation for different divisor methods are considered by means of the bounds of votes. In Sect. 3, a few cases of elections and examples are given that illustrate the bounds of votes of different divisor methods. Finally, a few concluding remarks are made at the end of the paper.

2 The bounds of votes of divisor methods

Given the input data on the obtained votes of parties (i.e., on the number of population of each member state), various proportional methods can allocate seats differently.

In this section, we look at the conditions under which a divisor method will give specified seat allocation by means of the bounds of votes, i.e., we consider the following question: Assume a vote count vector $v=(v_1,v_2,\ldots ,v_n)$ is given and the application of a divisor method based on $d(1),d(2),\ldots ,d(S)$ results in the seat distribution $s=(s_1,s_2,\ldots ,s_n)$. What are the maximum and minimum values of $v_k$ such that if $v_i$ does not change for all i ≠ k, then s (and, in particular, $s_k$) does not change after the same divisor method is applied?

The following theorem is the well-known min-max inequality, which describes the condition that a divisor method gives specified seat allocation (Balinski and Young 1975, 2001; Cortona et al. 1999).

Theorem 1

Let there be given the vector of obtained votes $(v_1,v_2,\ldots ,v_n)$, the vector of seat allocation $(s_1,s_2,\ldots ,s_n)$, where $\sum _{k=1}^{n} s_k=S$, and the M divisor method determined by its divisors d(m), $m=1,\ldots , S$.

Let us denote the set $I_0=\{j\in I:\, s_j=0\}$. Then the M divisor method gives seat allocation $(s_1,s_2,\ldots ,s_n)$ if and only if

$$\begin{aligned} \min _{k\in I\setminus I_0} \frac{v_k}{d(s_k)} > \max _{\ell \in I} \frac{v_\ell }{d(s_\ell +1)}. \end{aligned}$$

(1)

$\Box$

The next proposition follows from Theorem 1. It gives the bounds of votes, i.e., the vote intervals, within which one can change the number of votes $v_k$ of the party k such that the M divisor method gives the same seat allocation (provided that the other votes $v_i$, $i \ne k$, are unchanged). Note that when $v_k$ changes, but not $v_i$ for $i\ne k$, then the total number of votes in the election changes.

Proposition 1

Let there be given the vector of obtained votes $(v_1,\ldots ,v_n)$, the vector of seat allocation $(s_1,\ldots ,s_n)$, where $\sum _{k=1}^{n} s_k=S$, and the M divisor method determined by its divisors d(m), $m=1,\ldots , S$.

Let us denote the set $I_0=\{j\in I:\, s_j=0\}$. Then the M divisor method gives seat allocation $(s_1,\ldots ,s_n)$ if and only if

$$\begin{aligned} \max _{\ell \in I\setminus \{k\}} \frac{d(s_k)}{d(s_\ell +1)}v_\ell&< v_k <&\min _{\ell \in I\setminus (I_0\cup \{k\})} \frac{d(s_k+1)}{d(s_\ell )} v_\ell , \quad \forall k\in I\setminus I_0, \end{aligned}$$

(2)

$$\begin{aligned} \qquad&v_k <&\min _{\ell \in I\setminus I_0} \frac{d(1)}{d(s_\ell )} v_\ell , \qquad \forall k\in I_0. \end{aligned}$$

(3)

Proof

Proposition 1 has the same assumptions as Theorem 1. Inequality (1) holds if and only if the following inequalities hold:

$$\begin{aligned} d(s_k)\cdot \max _{\ell \in I}\frac{v_\ell }{d(s_\ell +1)}< v_k, \quad \text{ for } \text{ any } k\in I\setminus I_0, \end{aligned}$$

(4)

$$\begin{aligned} \quad v_\ell < d(s_\ell +1)\cdot \min _{k\in I\setminus I_0} \frac{v_k}{d(s_k)}, \quad \text{ for } \text{ any } \quad \ell \in I. \end{aligned}$$

(5)

It can be easily seen that inequalities (4) and (5) are equivalent to inequalities (2) and (3) provided that $v_k$ does not have to be compared with itself. $\square$

Remark 2

If $I_0=\emptyset$, then expression (2) in Proposition 1 takes the form:

$$\begin{aligned} \max _{\ell \in I\setminus \{k\}} \frac{d(s_k)}{d(s_\ell +1)} v_\ell< v_k < \min _{\ell \in I\setminus \{k\}} \frac{d(s_k+1)}{d(s_\ell )} v_\ell , \qquad \forall k\in I. \end{aligned}$$

(6)

From Proposition 1 we can determine the corresponding bounds of votes related to particular divisor methods.

The bounds of votes for those divisor methods where there holds $I_0=\emptyset$ are listed in Table 2. (If $d(1)=0$, then $I_0=\emptyset$, i.e., then there holds $s_k\ge 1$, $\forall k=1,\ldots ,n$). In these methods, the bounds of votes are obtained by putting the divisor formula of a particular divisor method in expression (6).

Table 2 Bounds of votes $l_k<v_k<u_k$ for the divisor methods where $I_0=\emptyset$

The bounds of votes for those divisor methods where there holds $d(1)\not =0$ are listed in Table 3. In these methods, the bounds are obtained by putting the corresponding divisor formula for a particular divisor method in expressions (2) and (3).

Knowing the vote intervals within which different divisor methods give identical seat allocations could help in (the posteriori) analysis of elections, in future planning of an electoral strategy, when considering the properties of different divisor methods, and the like. For example, if an electoral system consists of several electoral districts, one can see in which districts some party would gain a seat if its vote count increased by a small percentage. So, the party could choose on which district it would focus its future election campaign more strongly, and vice versa.

Table 3 Bounds of votes $l_k<v_k<u_k$ for the divisor methods in which $d(1)\not =0$

Remark 3

Divisor methods can be compared according to the extent that they ‘favour large states’ (Balinski and Young 1975, 2001). In this approach, divisor method M”, with divisors $d_{M''}$, favours large states relative to divisor method M’, with divisors $d_{M'}$, if the following inequalities hold:

$$\begin{aligned} \frac{d_{M''}(k)}{d_{M''}(\ell )}< \frac{d_{M'}(k)}{d_{M'}(\ell )},\quad \forall k> \ell \ge 1. \end{aligned}$$

(7)

Note that if $d_{M'}(1)=0$, then one makes the convention that $\frac{d_{M'}(k)}{0}=+\infty$, $\forall k>1$, where the ’plus infinity’ symbol $+\infty$ is such that $\min \{+\infty ,a\}=a$ and $\max \{+\infty ,a\}=+\infty$ for every given real number a (Cortona et al. 1999).

For instance, with respect to the BE and SD methods, the following inequality holds:

$$\begin{aligned} \frac{d_{BE}(k)}{d_{BE}(\ell )}< \frac{d_{SD}(k)}{d_{SD}(\ell )}, \quad \forall k> \ell \ge 1 \,. \end{aligned}$$

So, compared to the SD method, the BE method tends to ’favour large states’. Note that inequality (7) does not hold for the DA and HM methods.

Divisor methods, ranging between the BE method and the SD method, are listed in Table 1 (Balinski and Young 1975, 2001; Marošević and Scitovski 2007). They can be compared by the following lemma.

Lemma 1

If the BE method and the SD method give equal seat allocation $s=(s_1,\ldots ,s_n)$, then each divisor method listed in Table 1, i.e., the DH, MS, SL, EP, HM and DA methods, gives the equal seat allocation. $\Box$

This lemma can also be proven by means of the bounds of votes for the corresponding divisor methods that are given in Tables 2 and  3. Lemma 1 is illustrated in Example 5.

3 Examples

In order to empirically study and illustrate the bounds of votes of different divisor methods, we look at a few cases of elections and an example with hypothetical data.

Example 1

We use data related to elections for the European Parliament in Austria in 2019, where $n=5$ parties (or lists) passed the electoral threshold of $5\%$ in the whole country as a single electoral unit (euparl 2021; wikiped 2021). The obtained votes $v_k$, $k=1,\ldots , 5,$ are shown in descending order in Table 4. The allocation of seats $S=19$ is done by the d’Hondt method: ${{\textbf{s}}_{DH}}=(7,5,3,3,1)$.

For the purpose of comparison and empirical research, we have applied other different divisor methods, and the corresponding seat allocations are shown in Table 4.

Table 4 Elections in Austria for the EuP in 2019: allocation of seats to 5 parties

In this case, the SD, DA, HM and EP divisor methods give the same seat allocation $s=(6,5,3,3,2)$. According to expressions for these methods given in Tables 2 and 3, the corresponding lower bounds $l_k-v_k$ and upper bounds $u_k-v_k$ of votes are listed in Table 5.

Table 5 The lower bounds $l_k-v_k$ and upper bounds $u_k-v_k$ of votes for the corresponding divisor methods (Austria, EuP, 2019)

Intervals $\langle l_k-v_k, 0\rangle$ and $\langle 0,u_k-v_k\rangle$, $k=1,\ldots ,5$, green (the first bar) for the SD method and red (the second bar) for the EP method, are illustrated in Fig. 1. For the first party with the largest number of votes $v_1=1,305,956$, the SD method has got greater bounds than the EP method. A reverse relation holds for the fifth party with the smallest number of votes $v_5=319,024$. For instance, it can be seen that if the first party with the largest number of votes $v_1=1,305,956$ had slightly larger number of votes $v'_1=u_1=1,308,791$, then the EP method would give one seat more to the first party (i.e., $s'_1=7$), while the SD method would give the unchanged number of seats $s_1=6$ to the first party (provided that the other number of votes $v_i$, $i\not = 1$, are unchanged). This is in accordance with the property that the EP method ’tends to favour large states’ over the SD method.

Fig. 1

Intervals $\langle l_k-v_k, 0\rangle$, $\langle 0, u_k-v_k\rangle$, green (the first bar) for the SD method and red (the second bar) for the EP method, from Table 5 (Austria, EuP, 2019) (colour figure online)

Example 2

We use data related to elections for the European Parliament in Hungary in 2019, where $n=5$ parties (or lists) passed the electoral threshold of $5\%$ in the whole country as a single electoral unit (euparl (2021); wikiped (2021)). The obtained votes $v_k$, $k=1,\ldots , 5,$ are shown in descending order in Table 6.

The allocation of seats $S=21$ is done by the d’Hondt method: ${{\textbf{s}}_{DH}}=(13,4,2,1,1)$. According to expressions in Table 3 for the d’Hondt method, the corresponding lower bounds $l_k$ and upper bounds $u_k$ of votes are shown in Table 6. Furthermore, the corresponding intervals $\langle l_k-v_k, u_k-v_k\rangle$ are also shown in Table 6. For instance, one can see that if the second party obtains $v_2-l_2=35,876$ votes less than $v_2=557,081$ votes, then the second party will get one seat less (provided that the other votes $v_i$, $i\not = 2$, are unchanged). Note that the second party has the smallest lower bound ($v_2-l_2=35,876$). So, the party which is most sensitive to the lowering of its vote count is the second party, in this case for the d’Hondt method.

Table 6 The bounds of votes for the d’Hondt method ($s_{DH}=(13,4,2,1,1)$) (Hungary, EuP, 2019)

We have applied other different divisor methods, and the corresponding seat allocations are shown in Table 7.

Table 7 Elections in Hungary for the EuP in 2019:allocation of seats to 5 parties

In this case, the SD, HM and DA divisor methods give the same seat allocation $s=(11,4,2,2,2)$. Intervals $\langle l_k-v_k, 0\rangle$ and $\langle 0,u_k-v_k\rangle$, $k=1,\ldots ,5$, green (the first bar) for the SD method and yellow (the second bar) for the DA method, are illustrated in Fig. 2. One can see that for the first party with the largest number of votes $v_1=1,824,220$, the SD method has got greater bounds than the DA method. A reverse relation holds for the fifth party with the smallest number of votes $v_5=220,184$. This is in accordance with the property that the DA method tends to favour large states over the SD method.

Fig. 2

Intervals $\langle l_k-v_k, 0\rangle$, $\langle 0, u_k-v_k\rangle$, green (the first bar) for the SD method and yellow (the second bar) for the DA method (Hungary, EuP, 2019) (colour figure online)

Example 3

We use data related to elections for the European Parliament in Croatia in 2019, where $n=6$ parties (or lists) passed the electoral threshold of $5\%$ in the whole country as a single electoral unit (euparl 2021; wikiped 2021). The obtained votes $v_k$, $k=1,\ldots , 6,$ are shown in descending order in Table 8. The allocation of seats $S=12$ is done by the d’Hondt method: ${{\textbf{s}}_{DH}}=(4,4,1,1,1,1)$.

For the purpose of comparison, we have applied other different divisor methods, and the corresponding seat allocations are shown in Table 8.

Table 8 Elections in Croatia for the EuP in 2019: allocation of seats to 6 parties

In this case, the SL, MS, EP, HM and DA divisor methods give the same seat allocation $s=(4,3,2,1,1,1)$. According to expressions for these methods given in Tables 2 and 3, the corresponding lower bounds $l_k-v_k$ and upper bounds $u_k-v_k$ of votes are listed in Table 9.

Table 9 The lower and upper bounds of votes for the corresponding divisor methods (Croatia, EuP, 2019)

Intervals $\langle l_k-v_k, 0\rangle$ and $\langle 0,u_k-v_k\rangle$, $k=1,\ldots ,5$, yellow (the first bar) for the DA method and orange (the second bar) for the SL method are illustrated in Fig. 3. Let us note that for the first party with the largest number of votes $v_1=244,076$, the DA method has got greater bounds than the SL method. A reverse relation holds for the sixth party with the smallest number of votes $v_6=55,806$. This is in accordance with the property that the SL method ’tends to favour large states’ over the DA method.

Note that the fourth party has the smallest upper bound ($u_4-l_4=6,781$), in this case. So, the party which is most sensitive to the increment of its vote count is the fourth party (i.e., the fourth party would gain a seat if its vote count increased by the smallest amount, provided that the other votes $v_i$, $i\not = 4$, are unchanged).

Fig. 3

Intervals $\langle l_k-v_k, 0\rangle$, $\langle 0, u_k-v_k\rangle$, yellow (the first bar) for the DA method and orange (the second bar) for the SL method, from Table 9 (Croatia, EuP, 2019) (colour figure online)

Example 4

We use data related to elections for the European Parliament in Spain in 2019, that have got no electoral threshold. The allocation of seats $S=59$ was done by the d’Hondt method: ${{\textbf{s}}_{DH}}=(21,13,8,6,4,3,3,1)$ (euparl 2021; wikiped 2021). Therefore, we take into account $n=8$ parties (or lists) that have got at least one seat.

For the purpose of comparison, we have also applied other different divisor methods, and the corresponding seat allocations are shown in Table 10.

Table 10 Elections in Spain for the EuP in 2019: allocation of seats to 8 parties

In this case, the SL and EP divisor methods give the same seat allocation $s=(20,13,8,6,4,3,3,2)$. According to expressions for these methods given in Tables 2 and 3, the corresponding lower bounds $l_k-v_k$ and upper bounds $u_k-v_k$ of votes are listed in Table 11.

Table 11 The lower and upper bounds of votes for the corresponding divisor methods (Spain, EuP, 2019)

Intervals $\langle l_k-v_k, 0\rangle$ and $\langle 0,u_k-v_k\rangle$, $k=1,\ldots ,8$, red (the first bar) for the EP method and orange (the second bar) for the SL method are illustrated in Fig. 4. Let us note that for the first party with the largest number of votes $v_1=7,369,789$, the EP method has got greater bounds than the SL method. A reverse relation holds for the eighth party with the smallest number of votes $v_8=633,090$. This is in accordance with the property that the SL method ’tends to favour large states’ over the EP method.

Looking only at the EP method in this case, one can see that the second party has the smallest lower bound ($v_2-l_2=4,554$). So, the party which is most sensitive to the lowering of its vote count is the second party (i.e., the second party would lose a seat if its vote count decreased by the smallest amount).

Fig. 4

Intervals $\langle l_k-v_k, 0\rangle$, $\langle 0, u_k-v_k\rangle$, red (the first bar) for the EP method and orange (the second bar) for the SL method, from Table 11 (Spain, EuP, 2019)

Example 5

We use data from Marošević et al. (2013), where $n=5$ hypothetical electoral units in Croatia are suggested on the basis of cluster analysis. The corresponding populations $v_k$, $k=1,\ldots , 5$ of these 5 hypothetical electoral units are given in descending order in Table 12. (The Croatian Parliament has a total of 151 seats: $S=140$ seats for 10 actual electoral units, 8 seats for national minorities and 3 seats for Croatian citizens living abroad.) In order to apportion $S=140$ seats to the 5 hypothetical electoral units proportionally, based on their populations, we applied different divisor methods. In this case, the condition of Lemma 1 is satisfied. Therefore, the BE and SD methods give equal seat allocation ${{\textbf{s}}}=(35,29,27,26,23)$, and all other mentioned divisor methods give the same seat apportionment. According to expressions in Tables 2 and 3, the corresponding lower and upper bounds of populations are also shown in Table 12.

Table 12 The lower $l_k$ and upper bounds $u_k$ of populations for divisor methods

In Table 12, one can see that intervals of populations by particular divisor methods are in accordance with the property of ’favouring large states’. For instance, for the first electoral unit with the largest population $v_1=1,013,125$, the BE method (that has the tendency of ’favouring large states’ over other divisor methods) gets smaller bounds of population, while the SD method gets slightly greater bounds of population. On the other hand, a reverse relation holds for the fifth electoral district with the smallest population $v_5=677,313$.

Intervals $\langle l_k-v_k, 0\rangle$ and $\langle 0,u_k-v_k\rangle$, $k=1,\ldots ,5$, green (the first bar) for the SD method and brown (the second bar) for the BE method are illustrated in Fig. 5. For instance, it can be seen that if the first electoral unit with the largest population $v_1=1,013,125$ had slightly smaller population $v'_1=1,001,245$, then the SD method would give one seat less to the first electoral unit, while the BE method would give the unchanged number of seats $s_1=35$ to the first unit (provided that the other populations $v_i$, $i\not = 1$, are unchanged). This is in accordance with the property that the BE method ’tends to favour large states’ over the SD method.

Fig. 5

Intervals $\langle l_k-v_k, 0\rangle$, $\langle 0, u_k-v_k\rangle$, green (the first bar) for the SD method and brown (the second bar) for the BE method, from Table 12

Looking only at the SD method and five hypothetical electoral units in this case, one can see that the most sensitive upper bound of the population is in the fifth electoral unit (i.e., the smallest upper bound is $u_5-v_5=8,037$). So, the fifth electoral unit would gain a seat if its population increased by a small percentage, provided that the other populations $v_i$, $i\not = 5$, are unchanged. Note that the least sensitive upper bound of the population is in the first electoral unit (i.e., the largest upper bound is $u_1-v_1=48,431$).

4 Concluding remarks

Seat allocation is based on votes obtained by parties (i.e., seat apportionment is based on populations of electoral units). Various proportional methods can give different seat allocations. We consider the following proportional divisor electoral methods: the method of smallest divisors (SD), the Danish method (DA), the method of harmonic mean (HM), the method of equal proportions (EP), the Sainte-Laguë method (SL), the modified Sainte-Laguë method (MS), the d’Hondt method (DH), and the Belgian method (BE).

We look at divisor methods on the basis of reframing the well-known theorem on the min-max inequality. By using bounds of votes, we give conditions under which different divisor methods give specified seat allocation. These conditions are expressed by means of the lower and upper bounds of votes, i.e., by means of the intervals of vote counts, within which one can change the number of votes $v_k$ of the party k such that the specified divisor method gives the same seat allocation (provided that the other votes $v_i$, $i\ne k$, are unchanged). Knowing these intervals of vote counts could help in the analysis of elections, planning of an electoral strategy and understanding of different divisor methods.

For the purpose of empirical research and illustration, we consider the bounds of votes of a specified seat assignment for divisor methods in a few cases of elections (in Austria, Hungary, Croatia and Spain). Comparing the cases when particular divisor methods give equal seat allocation, we can conclude that the bounds of votes for these divisor methods are in line with their tendency of ’favouring large states’.

By connecting the obtained proposition and assertions, one can analogously state similar assertions about bounds of votes for specified seat allocation of the divisor methods that would be taken into account. The related problem for quota methods may be an area for future research.