TOTAL FAVORING IN PROPORTIONAL APPORTIONMENTS

The notion of “total favoring” of large or of small beneficiaries in proportional apportionments of entities is defined as a particular case of favoring. It is proven that the number of known conditions of total favoring of beneficiaries in an apportionment (APP) can be considerably reduced. Thus, the volume of calculations to be performed for the respective computer simulation was reduced. In order to quantitatively estimate the total favoring of beneficiaries by APP methods, three indicators were used: the percentage of apportionments, in which large beneficiaries are totally favored; the percentage of apportionments, in which small beneficiaries are totally favored; the percentage of total favoring of large or of small beneficiaries, depending on the APP method applied. A total of five APP methods are being researched: Hamilton (Hare), Sainte-Laguë (Webster), d’Hondt (Jefferson), Huntington-Hill and Adapted Sainte-Laguë. Based on results of computer simulation, the total favoring of beneficiaries by these five APP methods was estimated, including comparatively. For example, it has been identified that the d’Hondt method does not always totally favors beneficiaries to a greater extent than the Huntington-Hill method. At the same time, the Adapted Sainte-Laguë method always totally favors small beneficiaries less compared to the Huntington-Hill method.


Introduction
It is often necessary to distribute a given number M of discrete entities of the same kind among n beneficiaries, in proportion to a numerical characteristic assigned to each of them V i, i = 1, n ����� . This is known as proportional apportionment (APP) problem [1 -3]. The integer character of this problem usually causes a certain disproportion of the apportionment xi, = 1, ����� [1,[4][5][6], some beneficiaries being favored at the expense of others. Favoring of beneficiaries leads to the increase of disproportionality and vice versa [6]. Therefore, reducing the favoring in question is one of the basic requirements when is choosing the APP method to be applied under concrete situations (free of bias condition [1,3]).
As it is well known, the d'Hondt method favors large beneficiaries (with larger Vi value) [1,4,6], and Huntington-Hill method favors the small ones (with smaller Vi value) [4,6]. But which of the two favors beneficiaries to a larger extent? Preferences, in this sense, between methods, can help. Par example, in [7], five APP methods are placed "in the order as they are known to favor larger parties over smaller parties". However, the best way is to estimate this property quantitatively. One approach in this aim is proposed in [8]. Another, a specific one, based on the definition of (total) favoring of large or of small beneficiaries by an apportionment method done in [1], is examined in this paper. Estimates of the frequency of total favoring in apportionments for the widely used Hamilton (Hare), Sainte-Laguë (Webster), d'Hondt (Jefferson), Huntington-Hill and Adapted Sainte-Laguë methods are obtained by computer simulation.

Essence of favoring of beneficiaries in apportionments
The essence of favoring of beneficiaries in apportionments is described in different papers, including the [4,9,10] ones. In [6] they are distinguished three notions of favoring of beneficiaries by an APP method: a) favoring of a beneficiary in an apportionment; b) favoring of large or of small beneficiaries in an apportionment; c) favoring of large or of small beneficiaries overall by an apportionment method. It is considered that a beneficiary i is favored if a larger number xi of entities is distributed to him than would be due according to the Vi value, more precisely if xi > MVi /V, where M = x 1 + x2 + … + xn and V = V1 + V2 + … + Vn. Of course, the lack of favoring is possible only if the equalities MVi /V = MVi /V, = 1, ����� take place; here z means the integer part of the real number z. In practice, such equalities rarely occur and that is why some beneficiaries are favored and others, respectively, are disfavored.
In a formalized form, the first, probably, definition of favoring of large or of small beneficiaries in apportionments is done in [1].
Definition 2 (according to [1, p. 125]). An apportionment method favors large parties if and it favors small parties if where L and S are subsets of {1, 2, …, n} such that xi > xj whenever i ∈ L and j ∈ S [3].
If, when applying an APP method to any of possible initial data, requirement (1) or, respectively, requirement (2) always occurs, then it can be considered that this method "overall favors" large or, respectively, small beneficiaries (parties). But there are no known such methods that would be used in practice. In such a situation the Definition 2 can be used to identify the favoring of large or of small beneficiaries in particular apportionments.
At the same time, it is considered that d'Hondt method favors large beneficiaries, in sense that more frequently it favors large beneficiaries that it favors the small ones, and Huntington-Hill method favors small beneficiaries, in sense that more frequently it favors small beneficiaries that it favors the large ones in apportionments. Moreover, in one and the same apportionment may be favored some large beneficiaries and some small beneficiaries. The approach proposed in [8] can identify, if such an apportionment favors predominantly large or predominantly small beneficiaries. That's why in this paper the apportionments compliant with requirement (1) are considered "total favoring" large beneficiaries, and the ones compliant with requirement (2) are considered "totally favoring" small beneficiaries. These are particular cases of the "favoring" of beneficiaries -large (predominantly) or small (predominantly) in sense of [8].
Finally, to determine if an APP method totally favors (overall) large beneficiaries or it totally favors (overall) small beneficiaries, it is needed to have apportionments on infinity (sufficient large number) of cases of initial data. If the frequency of total favoring of large beneficiaries is larger than the frequency of total favoring of small beneficiaries, then it is considered that the APP method totally favors (overall) large beneficiaries and vice versa.

Number of restrictions to check the total favoring in apportionments
The frequency of total favoring of large (small) beneficiaries, on a sufficient large number of cases of initial data, can be determined by computer simulation. To do this, it is important to know how many of different inequalities (1) or, in case of favoring of small beneficiaries, of the (2) ones there are.
Without diminishing the universality of the approach, below it is considered that the n beneficiaries are ordered in non-ascending order of Vi, = 1, ����� , that is V1 > V2 > V3 > … > Vn. In proportional apportionments, if Vi > Vj then xi ≥ xj. Let's consider the apportionments for which However, if all cases, for which | L | + | S | ≤ n, L ≠ ∅, S ≠ ∅, to be taken into account, then the number K n of variants of different pairs of subsets L and S is considerably larger than n -1. Statement 1. In general case, the number Kn of variants of different pairs of subsets L and S of {1, 2, …, n}, such that x i > xj whenever i ∈ L and j ∈ S, is determined according to recurrent formula where K0 = K1 = 0. Similarly, when n = j beneficiaries: 2) in cases, in which subset L begins with beneficiaries from 2 to j -1, there are a summary number of possible variants of S equal to Kj -1; 3) for subsets {1,2}, {1, 2, 3}, …, {1, 2, …, j} of K j, the summary number of possible variants of subset S is equal to those for subsets Table 1 show that Kn value increases rapidly with the increase of n, becoming more than 2 mil at n = 20 beneficiaries. For approximate calculations, instead of recurrent formula (3) can be used the following one K n ≈ 2 × 10 3n/10 , n = 7÷68, where the absolute value of the relative error doesn't exceed 15%. The relative error is positive decreasing from 14.97% to 0.96% at n = 7÷10 and negative decreasing from -0.66% to -14.89% at n = 11÷68.

Redefining the notion of total favoring in apportionments
The Kn value determined according to (3) can be considerable, especially at large values of n. Thus, for computer simulation, it is important to reduce the number of requirements (1) and (2). A solution is done by Statement 2.
Statement 2. In case of x 1 > x2 > x3 > … > xn, the necessary and sufficient conditions for compliance with all Kn inequalities (1) are the n -1 ones and with all the Kn restrictions (2) are the n -1 ones Indeed, the necessity of conditions (4) is evident. They belong to the Kn ones and cover all n(n -1)/2 variants of pairs {L, S} for |L| = |S| = 1. At the same time, they establish only n -1 relations for the total of n beneficiaries -the minimal possible number. A similar situation is with the necessity of conditions (5). ▼ Regarding the sufficiency of inequalities (4), let's begin with proving the following inequalities , for it to take place (6), it is sufficient to prove that Let's consider the equality , then the inequality (9) occurs. But the inequality x1/V1 > x2/V2 takes place, then (9) occurs and therefore (6) occurs, too. Evidently, based on same considerations, take place Also, by induction it is easy to show that occur where L is any subset of {1, 2, …, j}, = 1, − 1 ���������� and = 2, ����� . Indeed, noting x1,2 = x1 + x2 and V 1,2 = V1 + V2 (one new conventional beneficiary in place of two former ones) and based on (6) one has x1,2/V1,2 > x3/V3 and, following same steps when proving (6) , for it to take place (7), it is sufficient to show that Let's consider the equality From (14) one has z = (x2V2 , then the inequality (13) occurs. But the inequality x3/V3 < x2/V2 takes place, then (13) occurs and therefore relation (7) occurs, too.

Journal of Engineering Science
March, 2021, Vol. XXVIII (1) Similarly as proving the sufficiency of conditions (4) compliance with all Kn inequalities (17), that is with the (1) ones, can be proved the sufficiency of conditions (5) compliance with all K n inequalities (18) where L is any subset of {1, 2, …, r} and S is any subset of {r + 1, 2, …, n}, that is with the (2) ones for the case of x 1 > x2 > x3 > … > xn.■ Based on Statement 2, can be simpler redefined the Definition 2 regarding the total favoring of large/small beneficiaries in an apportionment. Based on Definition 3, can be defined the total favoring of large or of small beneficiaries by an apportionment method overall, on an infinity of apportionments. Evidently, the probability p L of total favoring of large beneficiaries in an apportionment is determined as where N is the total number of apportionments, and NL is the number of apportionments compliant with requirements (19). Similarly, the probability p S of total favoring of small beneficiaries in an apportionment is determined as where NS is the number of apportionments compliant with requirements (20). At the same time, indicator p L alone does not determine the total favoring of large beneficiaries by apportionment methods, just as indicator pS alone does not determine the total favoring of small beneficiaries by apportionment methods. It is well known that, in a particular apportionment, used methods can favor both some large and some small beneficiaries. This is why, when talking about the total favoring of beneficiaries by apportionment methods, it is needed to take into account both indicators: p L and pS.

Total favoring the beneficiaries by apportionment methods
Evidently, the compliance with requirements (19), or the (20) ones, for all n beneficiaries of an apportionment, especially when n is large, is rare. For example, it is sufficient only in one of the n -1 cases to take place V j < Vixj/xi and requirements (19) are not compliant. To determine, by computer simulation, the apportionment methods total favoring of large or of small beneficiaries, in sense of Definitions 4 and 5, the SIMAP application has been elaborated and respective calculations have been made. .2%] at n = 5 and is very close to 0% at n ≥ 7. So, along with n = 2, many cases of apportionments with totally favored large (small) beneficiaries are only at n = 3 (17.5÷18.3%) and no so many at n = 4 (3.7÷5.1%). .5%] at n = 3, to the range [5.0%; 6.3%] at n = 4, to the range [1.1%; 1.8%] at n = 5 and is very close to 0% at n ≥ 7.    Similarly, for 11 ≤ M ≤ 501, the FL(d'H) value ( Figure 5 and the results of calculations) belong to the range [38.6%; 47.3%] at n = 2, to the range [32.1%; 34.4%] at n = 3, to the range [14.6%; 19.0%] at n = 4, to the range [5.7%; 8.5%] at n = 5, to the range [0.6%; 1.2%] at n = 7 and is very close to 0% at n ≥ 10. Thus, Figure 5 clearly show that on average the d'Hondt method totally favors large beneficiaries, the percentage of total favoring being considerable at small values of n, especially at n ≤ 5 beneficiaries.

Total favoring of beneficiaries by Huntington-Hill method
The graphs of PS(HH), PL(HH) and FS(HH) indicators dependence to M and n, when using Huntington-Hill method, are shown in Figures 6, 7 and 8, respectively.
According to Figures 6 and 7, the on M dependence of the P L(HH) indicator is decreasing and of the PS(HH) indicator is increasing, but the on n dependence are both strongly decreasing. So, for 11 ≤ M ≤ 501, the PS(HH) value ( Figure 6    An another situation is regarding the graphs of FS(HH) indicator dependence to M and n, when using the Huntington-Hill method.
According to Figure 8, the on M dependence of the F S(HH) is decreasing, but the on n dependence of it is increasing in the range from n = 2 to n= 3 and is decreasing for n ≥ 3.
So, for 11 ≤ M ≤ 501, the FL(HH) value ( Figure 8  Thus, Figure 8 clearly show that on average the Huntington-Hill method totally favors small beneficiaries, the percentage of total favoring being considerable at small values of n, especially at 3 ≤ n ≤ 5.

Total favoring of beneficiaries by Adapted Sainte-Laguë method
The graphs of FS(ASL) indicator dependence to M and n, when using Adapted Sainte-Laguë (ASL) method, are shown in Figure 9. If to not take into account the case of M = 6, the on M dependence of F S(HH) is decreasing, but the on n dependence of it is increasing in the range from n = 2 to n = 3 and is decreasing for n ≥ 3.

Comparative analyses of apportionment methods
As expected, for all examined APP methods, the on n dependence of P L(•) and PS(•) indicators are strongly decreasing (see Figures 1 -4, 6 and 7), while those of FL(d'H) and F S(HH) and FS(ASL) (see Figures 5, 8 and 9) are different -they are increasing for some segments and decreasing for the others. Comparing Figures 1 and 2   Although it is considered that d'Hondt method favors large beneficiaries strongly, and Huntington-Hill method favors small beneficiaries slightly, with refer to total favoring of beneficiaries, in many cases relation F L(d'H) < FS(HH) occur (Figure 10 no alternatives for the difference FS(HH) -FS(ASL) -it is always positive, that is FS(HH) > F S(ASL) (see Figure 12). Thus, Adapted Sainte-Laguë method rarer, than the Huntington-Hill one, implies the total favoring of beneficiaries -of the small ones.

Conclusions
The conditions of favoring large or small beneficiaries (parties) by an apportionment method defined in [1] (Definition 2) are very strong. There are no known such methods that would be used in practice. But these conditions can be used to identify the favoring of large or of small beneficiaries in particular apportionments. At the same time, in one and the same apportionment may be favored some large beneficiaries and some small ones and, however, predominantly to be favored large or, on the contrary, small beneficiaries. Therefore it is proposed to use two different notions: "favoring" of large or of small beneficiaries and "total favoring" of large or of small beneficiaries, the second one being a particular case of the first. The compliance of an apportionment with conditions (1) or with the (2) ones is referred to "total favoring" of large or, respectively, of small beneficiaries. The larger notion of favoring of large or of small beneficiaries is used when in an apportionment are predominantly favored large or, on the contrary, small beneficiaries in sense of [8].
There has been obtained the formula for determining the number K n of conditions (1) or (2) for computer simulation. But this number is growing very fast with the growth of the number n of beneficiaries, exceeding 2 mil at n = 20. Fortunately, it was possible to overcome this situation. Thus, the volume of needed calculus for computer simulation was considerably reduced.
In order to estimate quantitatively the total favoring of beneficiaries, three indicators were used: ; sample size N = 10 6 . As expected, for all five methods the on n dependence of indicators PL(•) and PS(•) is strongly decreasing, becoming approx. 0 at n ≥ 7÷10. With refer to the on n dependence of indicators F L(d'H), FS(HH) and FS(ASL) it is increasing for some of n = 2÷3 segments and is decreasing for the others.
Also, take place the relations: