Directory UMM :Data Elmu:jurnal:M:Mathematical Social Sciences:Vol37.Issue3.May1999:
Mathematical Social Sciences 37 (1999) 107–122
Parametric methods of apportionment, rounding and
production
´ b
Michel Balinski a , *, Victoriano Ramırez
a
´
´
´ , Ecole
C.N.R.S. and Laboratoire d’ Econometrie
Polytechnique, 1 rue Descartes, 75005 Paris, France
b
´
Departamento de Matematica
Aplicada, Universidad de Granada, Granada, Spain
Received 10 November 1997; received in revised form 30 April 1998; accepted 30 May 1998
Abstract
The class of ‘‘parametric’’ methods of apportionment, of rounding, or for minimizing the
variation of production rates in just-in-time production systems is characterized in several different
ways that depend on the underlying qualitative behavior of its solutions. 1999 Elsevier
Science B.V. All rights reserved.
Keywords: Apportionment; Rounding; Just-in-time production; Axiomatic characterization
JEL classification: C44; C69; C79; D71
1. Introduction
The motivation for studying parametric methods is threefold. To begin they play an
important role in electoral systems. In Japan the debate on what method should be
chosen to apportion the 512 seats of its Diet seems to have concentrated on parametric
methods (Balinski, 1993; Oyama, 1991). In Spain the House of Representatives is
elected by a system of proportional representation at the level of its provinces. Recent
experience has shown that this has strongly and increasingly favored solidly implanted
provincial political parties at the expense of the nation-wide parties of similar or smaller
size – indeed, so much so that one party, the Catalonian CiU (Convergencia i Union)
party, has held the balance of power since 1993, permitting the socialists (Partido
˜ to govern from 1993 to 1996, and then permitting the right
Socialista Obrero Espanol)
(Partido Popular) to govern since 1996. This has sparked a considerable debate on what
should be done that is regularly nourished by electoral data. The data naturally evoked a
*Corresponding author. Fax: 133 1 46343428; e-mail: balinski@poly.polytechnique.fr
0165-4896 / 99 / $ – see front matter 1999 Elsevier Science B.V. All rights reserved.
PII: S0165-4896( 98 )00027-4
108
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/ Mathematical Social Sciences 37 (1999) 107 – 122
M. Balinski, V. Ramırez
desirable property that a method of apportionment should have, which it turns out is met
only by the class of parametric methods. More generally, political scientists have for
years talked about the ‘‘thresholds’’ of votes that will assure a party of representation in
a proportional representation system (see, Lijphart, 1994; Rae, 1971): an answer to this
question leads to yet another characterization of parametric methods. This is the
‘‘apportionment’’ part of the title.
The term ‘‘rounding’’ refers to a second motivation. Given a list of real numbers the
sum of their roundings may well differ from the rounding of their sum. For example,
newspapers usually give the percentages of the votes won by each of a set of competing
candidates along with the actual vote totals, and they often fail to add to 100%. Another
current example comes with the introduction of the new common European Union
currency, the ‘‘euro’’, expected to be introduced on January 1, 1999. There will be a
period of transition in which financial transactions in banks will be given in the two
currencies of a participating country, the old and the new: e.g., in France, the franc and
´
` de l’Economie
the euro (Ministere
et des Finances, 1997). Sums of roundings will often
differ from the rounding of the sum, and though each single problem represents
practically nothing, the discrepancies summed over all problems could bring significant
windfall gains (or losses) due to the sheer number of transactions. Two questions present
themselves (Balinski, 1996). The first has been addressed elsewhere (Balinski and
Rachev, 1993): if one accepts the idea that sums of percentages may differ from 100%,
what ‘‘rule’’ of rounding should be used? The second is of concern here: if one insists
that the sums must always be exactly 100%, what ‘‘method’’ should be used (Balinski
and Rachev, 1997)? A particularly desirable property for methods of rounding
distinguishes the class of parametric methods (called ‘‘stationary’’ in Balinski and
Rachev, 1997) among the far wider class of ‘‘divisor’’ methods.
There is an extensive literature on minimizing the variation of production rates in
‘‘just-in-time’’ systems (e.g., Balinski and Shahidi, 1997; Bautista et al., 1996; Kubiak,
1993; Miltenburg, 1989; Steiner and Yeomans, 1993). In one guise – the ‘‘product rate
variation’’ problem – it is formally equivalent to the problem of apportionment, though
what constitutes a ‘‘good’’ solution must of course be evaluated in the context of the
actual problem. A particularly appealing property, often discussed in the literature (when
praising the qualities of a particular method or type of solution) – to wit, that an optimal
sequence in the production of different products should eventually repeat or cycle –
once again elects parametric methods as the only ones that meet the test. This is the third
motivation, and the reason for the existence of the word ‘‘production’’ in the title.
The answer to the question, ‘‘why a parametric method?’’ depends on the inherent
context of the problem to be treated, but several different fundamental properties,
persuasive in different contexts, yield the same methods: they are the subject of this
paper.
2. The problem
A problem is defined by any pair ( p, h), where p 5 ( pj ) . 0, j [ S, is a nonzero
vector of reals, S a finite set, and h . 0 a positive integer. The dimension of the problem
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M. Balinski, V. Ramırez
109
is uSu 5 s. A solution
for the problem ( p, h) is a vector of positive integers x 5 (x j ) $ 0,
def
j [ S satisfying x S 5o j [S x j 5h.
A method f is a point to set mapping that assigns at least one solution to each
problem.
In apportionment, p are the populations of s states or provinces (or the vote totals of
political parties) and x the total number of seats each is to be allocated, which must sum
to h. The goal is to find solutions that are as ‘‘fair’’ as possible, the unattainable ideal
being proportionality.
In rounding, p are data that are to be rounded and x the roundings, whose sum must
equal h (e.g. h5100%). The aim here is to find solutions that in some probabilistic sense
best ‘‘represent’’ the vectors p.
In the product rate variation or PRV problem, p are the relative demands for s
products to be produced (so pS 51) and x the cumulative production through period h.
Here it is assumed that producing one item of any product requires one period and that
Dp is integer valued for some integer D .0: what is wanted is the order of production in
succeeding periods, so a solution is necessary for every value of h, 0#h#D. For the
PRV problem an order of production is sought that best maintains cumulative
productions over the D periods that come ‘‘closest’’ to being proportional to the rates of
production.
3. Divisor and parametric methods
There is a particularly rich history of methods in the context of apportionment (see
Balinski and Young, 1982). Most, though not all, of the specific methods proposed or
used belong to a class called ‘‘divisor’’ methods.
A divisor function d is any monotone real valued function defined over the
nonnegative integers that satisfies d(k)[[k, k11] for all integers k, and for which there
exists no pair of integers a$0 and b$1 with d(a)5a11 and d(b)5b. A d-rounding of
a real number z$0 is
[z] d 5
H
if z 5 0;
if d(a 2 1) # z # d(a).
0,
a,
Thus [d(a)] d 5a or a11: at the threshold one can either round up or down.
The divisor method f d based on d is
f d ( p, h) 5 hx 5 (xj ), j [ S: xj 5 [ l pj ] d , l . 0 chosen so that x S 5 hj.
(1)
The seats of the U.S. House of Representatives have been apportioned by the divisor
]]]
method based on d(a) 5œa(a 1 1) since 1940; and in 1840 the method based on
1
d(a) 5 a(a 1 1) /(a 1 ]2 ) was proposed (among others).
An equivalent definition is the min–max condition
H
J
pj
pi
f d ( p, h) 5 x 5 (xj ), j [ S, x S 5 h: min ]]] $max ]] .
x i $1 d(x i 2 1)
x j $0 d(x j )
(2)
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M. Balinski, V. Ramırez
110
A parametric method is a divisor method f d baseddefon d(k)5k1d for all k, where
0#d #1. It is denoted by f d, and for convenience [z] d 5[z] d . Note that [k] 0 5k or k11,
and [k] 1 5k or k21. (In terms of the more usual notation, where [z] 2 and [z] 1 denote
the real number z rounded down and up, respectively, to the nearest integer: [z] 1 5[z] 2
and [z] 0 5[z] 1 .)
Various specific parametric methods have been proposed or used for apportionment:
J.Q. Adams suggested d 50, Condorcet d 50.4, Webster and Sainte-Lague¨ d 50.5, and
Jefferson and d’Hondt d 51.
The lemma that follows is trivial to verify.
Lemma 1. If x[ f a ( p, h) and x[ f b ( p, h) then for all d, a #d # b, x[ f d ( p, h).
Due to lemma 1, Table 1 gives all of the parametric method solutions for the problem
( p, h), where p is the vector of votes received by the respective parties in the 1993
Spanish elections and h5350 is the number of seats in its lower house. The quotas of a
problem ( p, h) are given by the vector q5hp /pS : it is the exact proportional vector that
sums to h. One representative value of d is given in every case; and each pair of
adjacent columns are alternate solutions for some one intermediate value of d. Thus, the
first solution in the table is the same for all methods f d when 0#d #(1461987 /
7994587)5d1 ¯0.182872 . . . . When d 5d1 there is a ‘‘tie’’, the first two solutions both
belong, and so on, pair by pair for the remaining solutions of the table. The successive
approximate values of d where there is a tie (given to the closest 10 23 ) are: 0.183,
0.195, 0.243, 0.457, 0.569, 0.765.
Notice also that as d increases from 0 to 1 solutions change, with lower x i ’s in the list
decreasing and higher x i ’s in the list increasing. This suggests the following concept: In
comparing two methods f and f *, f gives-up to f * if
Table 1
All parametric method solutions
Party
Votes
PSOE
PP
IU
CiU
CDS
PNV
CC
HB
ERC
PAR
EA
UV
9 150 083
8 201 463
2 253 722
1 165 783
414 740
291 448
207 077
206 876
189 632
144 544
129 293
112 341
Total
22 467 002
Quota
d5
0.09
0.19
0.22
0.35
0.51
0.66
0.88
142.55
127.77
35.11
18.16
6.46
4.54
3.32
3.23
2.95
2.25
2.01
1.75
141
126
35
18
7
5
4
4
3
3
2
2
141
127
35
18
7
5
4
3
3
3
2
2
142
127
35
18
7
5
3
3
3
3
2
2
142
128
35
18
7
5
3
3
3
2
2
2
143
128
35
18
6
5
3
3
3
2
2
2
143
129
35
18
6
4
3
3
3
2
2
2
144
129
35
18
6
4
3
3
3
2
2
1
350
350
350
350
350
350
350
350
Note: data from 1993 Spanish elections.
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M. Balinski, V. Ramırez
111
x [ f ( p, h), x* [ f *( p, h) and pi . pj implies x i # x i* or x j $ x j* .
This definition is well known in the literature (see Balinski and Young, 1982 page 118):
it formalizes the idea of one method favoring the bigger as versus the smaller parties.
Think of ‘‘gives-up’’ in terms of seats being ‘‘given-up’’ by the smaller parties, as the
method is changed from f to f *, in favor of the bigger parties . . . that are further ‘‘up’’
on the list of parties!
In the context of parametric (as versus the more general divisor) methods this concept
permits the characterization given in the following lemma.
Lemma 2. A parametric method f a gives-up to another parametric method f b if and
only if a , b.
Proof. Consider a problem ( p, h), with x[ f a ( p, h), x*[ f b ( p, h) and pi .pj . Then
pi /(x i 211 a )$pj /(x j 1 a ), so a , b implies pi /(x i 211 b ).pj /(x j 1 b ), meaning that
either x *
i $x i or x *
j #x j . However, if a . b it is easy to construct a problem ( p, h), with
a
x[ f ( p, h), x*[ f b ( p, h) and pi .pj , for which pi /(x i 211 a )$pj /(x j 1 a ) and
pi /(x i 211 b ),pj /(x j 1 b ), meaning that either x i* #x i 21 or x j* $x j 11. j
Thus, the parametric method f d that is most favorable to the smaller parties is that
with the smallest d, namely d 50; and that most favorable to the bigger parties is that
with the largest d, namely d 51.
The well known method of Hamilton (sometimes called that of Hare), used for
apportionment at various times in various countries including France, Israel, Mexico and
the USA, stands, in view of its properties, in opposition to the class of divisor methods.
It is easily described: (i) for each i set x i 5[qi ] 2 ; (ii) then increase by 1 each x i that
belongs to a set of cardinality h2o i q[ i ] 2 having the largest remainders qi 2[qi ] 2 .
There may, of course, be several solutions (when some of the remainders are the same).
4. Properties of methods
The three most fundamental properties that a method for any of the three problems
should enjoy are as follows. First, scale-invariancy: f ( p, h)5 f ( l p, h) for every l .0.
The problem is the same no matter what scale is used in presenting it. Second,
exactness: if p is integer valued and o i pi 5h then p is the unique solution, f ( p, h)5p.
If there is no ‘‘problem’’ then there is no problem! Third, anonymity: solutions depend
only on the values of the data, i.e., they are independent of the order in which the data is
presented. Every method must realize these three demands. In particular, divisor
methods as well as Hamilton’s method satisfy these properties.
Arguably the most important ‘‘nonobvious’’ property concerning methods is that they
be ‘‘consistent’’. In all of the applications of problems ( p, h) the ‘‘ideal’’ solution is the
proportional one. The fundamental underlying property of proportionality is that any part
of a proportional solution is itself proportional. To make this precise, if J is some subset
of S, J ,S, and J¯ is its complement, denote the corresponding subvectors of p by p(J)
112
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M. Balinski, V. Ramırez
and p(J¯ ), and similarly for x, and write x5(x(J), x(J¯ )) and p5( p(J), p(J¯ )) (modulo a
rearrangement of the order of the indices which, by anonymity, is of no consequence).
A method f is consistent if (x(J), x(J¯ ))[ f (( p(J), p(J¯ )), h) implies x(J)[ f ( p(J),
xJ ) for all J ,S; and if also y(J)[ f ( p(J), xJ ), then ( y(J), x(J¯ ))[ f (( p(J), p(J¯ )), h).
Any part of a best apportionment, rounding or production schedule should itself be a
best apportionment, rounding or production schedule. It is at once evident that every
divisor method, so every parametric method, is consistent. On the other hand, simple
examples show that Hamilton’s method is not consistent.
Indeed, consistency is a strong property. We begin by deducing several of its
consequences.
A method f is balanced if x[( p, h) and pi 5pj implies ux i 2x j u#1.
Lemma 3. A consistent, exact and anonymous method is balanced.
Proof. Suppose it were not so. There would then exist x[ f ( p, h) with (say) p1 5p2 5p
whose seats x 1 , x 2 differ by at least 2, (x 1 , x 2 )5(x1d, x), d $2.
Suppose d 52k. Then by consistency, (x12k, x)[ f (( p, p), 2x12k), contradicting
exactness. By the same reasoning, if x[ f (( p, . . . , p), h) and x i , x j are of the same
parity, then x i 5x j . It may therefore be assumed that
If f is not balanced, (x12k11, x)[ f (( p, p), 2x12k11) for some k$1. Consider
Then a y1(4k2 a )z52k(2x12k11), so that a ( y2z);2k (mod 4k). This equation
always has the solution a 52k. Moreover, if a is a solution then so is 4k2 a.
If a 52k, then y1z52x12k11. Applying consistency 2k times yields
Invoking consistency once more,
contradicting exactness.
If a ,2k solves the modular equation for some y, z, with y2z52l 11, l $1, then
a (z12l 11)1(2k2 a )z52kz1 a (2l 11), where a (2l 11) is divisible by 2k. By
consistency
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M. Balinski, V. Ramırez
113
which contradicts exactness. If, on the other hand, a .2k solves the modular equation, a
similar construction leads again to a contradiction. j
A method f is increasing if x[ f ( p, h) implies there exists x9[ f ( p, h11) with
x9$x. Simple examples show that Hamilton’s method is not increasing: this is the
‘‘Alabama paradox’’.
Lemma 4. A consistent, balanced and anonymous method is increasing.
Proof. To begin note that since f is consistent it is increasing if and only if (x 1 ,
x 2 )[ f (( p1 , p2 ), h) and (x 19 , x 29 )[ f (( p1 , p2 ), h11) implies x i9 $x i for i51, 2.
Suppose f is not increasing: say x 91 #x 1 21, so that x 92 $x 2 12. Consider ( y 1 , y 2 , y 91 ,
y 92 )[ f (( p1 , p2 , p1 , p2 ), 2h11). f balanced implies uy 1 2y 19 u#1 and uy 2 2y 29 u#1, so it
may be assumed that y 1 1y 2 5h and y 19 1y 29 5h11. Two appeals to consistency yields
(x 1 , x 2 , x 19 , x 92 )[ f (( p1 , p2 , p1 , p2 ), 2h11). But then by consistency once more (x 2 ,
x 92 )[ f (( p2 , p2 ), x 2 1x 29 ): this contradicts the fact that f is balanced because ux 29 2x 2 u$
2. j
Lemma 5. If a method f is consistent, balanced and anonymous then (x 1 , x 2 )[ f (( p1 ,
p2 ), x 1 1x 2 ) implies
, for any t$2.
Proof. Since f is balanced it may be assumed that
(3)
where 0#s,t.
Suppose x 91 1x 29 ±x 1 1x 2 , say x 19 1x 29 ,x 1 1x 2 . Then since f is increasing (x 19 , x 29 )#
(x 1 , x 2 ), so x 19 #x 1 and if x 29 ,x 2 we have x 92 11#x 2 , implying x 1 1tx 2 5x 19 1tx 92 1s#
x 1 1tx 2 2t1s, and thus s$t, a contradiction. One concludes that x 29 5x 2 , x 19 ,x 1 , and
since f is increasing, (x¯ 1 , x 92 ) [ fs( p1 , p2 ), x 1 1 x 29d for every x¯ 1 [[x 19 , x 1 ], in particular,
(x 91 11, x 92 )[ f (( p1 , p2 ), x 19 1x 29 11). But consistency and (3) imply that (x 19 , x 29 11)[
f (( p1 , p2 ), x 19 1x 29 11), so by consistency the first of these solutions may be substituted
in (3) to obtain
Repeating one obtains
with x 1* 1x 92 5x 1 1x 2 and x *1 1tx 92 5x 1 1tx 2 , so x *1 5x 1 and x 92 5x 2 . j
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M. Balinski, V. Ramırez
114
Corollary. If a method f is consistent, balanced and anonymous then (x 1 , x 2 )[ f (( p1 ,
p2 ), x 1 1x 2 ) implies
for any integers t 1 , t 2 .0.
Nevertheless, consistency (together with anonymity, exactness and invariancy) is not
sufficient to characterize the divisor methods, as shown by the following example. For
all k$0 and integer, define
izi 5
Hk or kk 1 1
if k , z , k 1 1
if z 5 k
and consider the method
c ( p, h) 5 hx 5 (xj ), j [ S: x j 5 i l pj i, l . 0 chosen so that xS 5 hj.
It is easy to verify that the method c meets the properties of scale-invariancy, exactness
and anonymity, and it is clearly consistent: but it is not a divisor method.
A method is responsive if
x [ f ( p, h) and pi . pj implies x i $ x j .
Theorem 1 (Balinski and Young, 1982, 1977). f is a divisor method if and only if it is
consistent, exact, anonymous, scale-invariant and responsive.
This theorem characterizes the class of divisor methods, and opens the door to a host
of characterizations of individual methods, but how and why the parametric methods
constitute a class of their own has remained an open question.
5. Cyclic characterization
In the contexts of production scheduling and of rounding a cyclic property seems to
be particularly compelling. Suppose, for example, that the relative demands for two
7
18
7
18
]
] ]
products are p5( ]
25 , 25 ) and that a method f yields f (( 25 , 25 ), 5)5(1, 4). Observe that
7
18
]
f (( ]
25 , 25 ), 25); f ((7, 18), 25)5(7, 18), the last by exactness, and it is evident that the
7
18
]
same order of production should repeat, so it should be true that f (( ]
25 , 25 ), 30)5(7,
18)1(1, 4)5(8, 22). And the same remark seems reasonable in the context of rounding.
For apportionment, however, the property seems uninteresting: by the very nature of the
problem, h is very small as compared with the large values of the vector p.
A method is cyclic if
x [ f ( p, h) and p integer implies x 1 p [ f ( p, h 1 pS ).
Hamilton’s method, for example, is clearly cyclic.
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M. Balinski, V. Ramırez
115
Theorem 2. A divisor method f is parametric if and only if it is cyclic.
Proof. To see that a parametric method is cyclic it suffices by (2) to note that
pj
pj
pi
pi
]]]
$ ]] if and only if ]]]]] $ ]]]]
xi 2 1 1 d xj 1 d
kpi 1 x i 2 1 1 d kpj 1 x j 1 d
for all integer k.
Assume then that f is a cyclic divisor method, say f d . Suppose it is not parametric,
so d(k)5k1d for all k satisfying 0#k,k* but d(k*)5k1d *, with d ±d *, say d .d *.
Take p1 , p2 to be any positive integers satisfying d *,p1 /p2 ,d.
Then, (1, l p2 21)5 f d ((1, l p2 21), l p2 ) for every l .0 for which l p2 is integer,
since f d is exact. Also, (0, l p1 )5 f d ((1, l p2 21), l p1 ) for l large enough and l p1
integer. To see this it suffices to show 1 /d(0),( l p2 21) /d( l p1 21):
l p2 2 1
l p2 2 1 p2
1
1
1
]]]
$ ]]] 5 ] 2 ] . ] 5 ]],
l p1
p1 l p1 d d(0)
d( l p1 2 1)
which is true for l large enough.
f d cyclic implies that (k*, k*( l p2 21)1 l p1 )[ f d ((1, l p2 21), l p1 1k* l p2 ). But
this is impossible because for large enough l the min–max condition (2) is violated:
l p2 2 1
l p2 2 1
1
]]]]]]]
# ]]]]]]] 5 ]]]]
l p1 2 1
d(k*( l p2 2 1) 1 l p1 2 1) k*( l p2 2 1) 1 l p1 2 1
k* 1 ]]]
l p2 2 1
1
1
, ]]] 5 ]].
k* 1 d * d(k*)
A similar construction handles the case d *.d. j
In the PRV problem, the aim is that the cumulative productions at each period be
approximately proportional to the rates of demand. By and large the approach found in
the literature is to postulate a measure of disproportionality or of error and so turn the
problem into a sequence of minimization problems, one for each period. Most papers
impose a measure yielding solutions identical to those given by the method of Hamilton.
But this leads to an unfortunate difficulty since Hamilton’s method is not increasing: it is
possible for the cumulative production to decrease in going from one period to the next!
Thus the main effort has been to devise heuristics or other involved algorithms to
guarantee a feasible solution (which may or may not be cyclic). No attention has been
given to the idea that perhaps another objective function would do better: indeed, there
are infinite numbers of measures of disproportionality that one could use, but little to
nothing on why one is better than another. Cyclicity is a powerful reason for choosing to
minimize a measure of error whose solutions belong to parametric methods. The fact is
that x[ f d ( p, h) if and only if x solves
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M. Balinski, V. Ramırez
116
min
x
x 1d 2
O p S]]]
2 hD when O x 5 h, x $ 0.
p
1
]
2
i
2
i
i
i
i
i
i
This makes it tempting to chose the parametric method based on d 51 / 2. In fact there
is a persuasive reason for this choice: it is the unique parametric method for which the
average cumulative production of each product i over the full cycle of D periods is
exactly equal to one-half of its total demand Dpi (see Balinski and Shahidi, 1997). So
f 1 / 2 is the unique parametric method that achieves ‘‘proportionality’’ over the entire
production cycle.
6. Remainder characterizations
In Spain’s Parliament each province (or state) is allocated a certain number of
deputies, on the basis of its population, and within each province parties are allocated
seats according to their vote totals within the province by the method of Jefferson
(known in Europe as the method of d’Hondt), f 1 . For example, in the 1989 elections the
results within the province of Barcelona, which elects 32 deputies, were as in Table 2.
Lemma 2 shows that among the parametric methods – and this is true as well among
the class of all divisor methods – the method of Jefferson (or d’Hondt) is the most
favorable for the large parties, the least favorable for the small parties. In the interest of
reducing this advantage, and so obtaining a more ‘‘equitable’’ apportionment, the idea of
¨
using Webster’s method (or Sainte-Lague’s),
f 1 / 2 , was considered. The example of
Table 2 immediately fuelled the arguments against: why should the PP receive 4 seats
‘‘for’’ 3 and a remainder of 0.72 and the CDS 2 ‘‘for’’ 1 and a remainder of 0.60,
whereas the CiU only receives 10 ‘‘for’’ 10 and a remainder of 0.74? This suggests two
possible properties.
Letting q j 5hpj /pS 5n j 1r j , n j $0 and integer, a method f respects remainders at d if
for all x[ f ( p, h),
r k # d # rl , r k , rl and x k $ n k 1 1 implies xl $ nl 1 1.
The idea is that if party k receives more seats than its quota and its ‘‘claim’’ to an extra
seat on the basis of its remainder r k is at most d, whereas another party l’s ‘‘claim’’ on
Table 2
1989 Barcelona election results
Party
PSOE
CiU
PP
IC
CDS
Total
Votes
900 039
740 479
256 785
199 372
110 175
2 206 850
Quota
f 1/2
f1
13.05
10.74
3.72
2.89
1.60
13
10
4
3
2
13
11
4
3
1
32
32
32
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M. Balinski, V. Ramırez
117
the basis of rl is at least d, then surely party l should also receive more than its quota. It
is surprising that this property together with consistency, exactness and anonymity is
sufficient to characterize the parametric method f d, with no appeal to the more general
and more difficult characterization of divisor methods.
Theorem 3. The unique consistent, exact and anonymous method that respects
remainders at d is the parametric method f d.
Proof. The parametric method f d respects remainders at d because r k #d #rl and r k ,rl
imply
qk
ql
]]
# 1 # ]],
nk 1 d
nl 1 d
with at least one inequality strict. Therefore, pl /(nl 1d ).pk /(n k 1d ), and xl takes the
value nl 11 before x k takes the value n k 11. This proves that f d , f.
Since f respects remainders it must be balanced. To see this, suppose the contrary:
for some positive p and positive integer x, either (a) (x11, x21)[ f (( p, p), 2x), or (b)
(x12, x21)[ f (( p, p), 2x11). (a) Take q1 5x10, q2 5(x21)11, with 0#d #1. Since
x11$x11 holds (the first component), it must be that x21$x (the second component), a contradiction. (b) Take q1 5(x11)2 ]21 , q2 5(x21)1 ]23 , with 2 ]21 #d # ]23 .
Since x12$x12 holds (the first component), it must be that x21$x (the second
component), a contradiction.
Again it suffices to prove that f , f d over all 2-dimensional problems. So suppose
that for some 2-dimensional problem the statement is false, and let h* be the smallest h
where a difference is realized: (x 1 , x 2 )[ f (( p1 , p2 ), h*21)> f d (( p1 , p2 ), h*21) and
(x 1 11, x 2 )[ f (( p1 , p2 ), h*), but (x 1 11, x 2 )[
⁄ f d (( p1 , p2 ), h*). This means that
p1 /(x 1 1d ),p2 /(x 2 1d ), so p1 x 2 2p2 x 1 1 e 5( p2 2p1 )d with e .0.
Let t 1 , t 2 be positive integers chosen to satisfy the following inequalities
p2d
t1
p1d 1 e
]]]]
, ] , ]]].
p2 (1 2 d ) 1 e t 2 p1 (1 2 d )
By the corollary to lemma 5,
Let q1 , q2 be the respective quotas:
p1 (t 1 x 1 1 t 1 1 t 2 x 2 )
p1 t 1 1 ( p1 x 2 2 p2 x 1 )t 2 def
q1 5 ]]]]]] 5 x 1 1 ]]]]]]] 5 x 1 1 r 1 ,
t 1 p1 1 t 2 p2
t 1 p1 1 t 2 p2
and
p2 (t 1 x 1 1 t 1 1 t 2 x 2 )
p2 t 1 1 ( p2 x 1 2 p1 x 2 ) def
q2 5 ]]]]]] 5 x 2 1 ]]]]]] 5 x 2 1 r 2 .
t 1 p1 1 t 2 p2
t 1 p1 1 t 2 p2
118
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M. Balinski, V. Ramırez
A straightforward calculation shows that r 1 ,d ,r 2 , thus contradicting the fact that f
respects remainders at d. j
The example of Table 2 invokes a second possible property. A method f respects the
remainders of the large when for all x[ f ( p, h),
pi . pj , r i . r j , and x j . q j implies x i . qi .
In this case the concept of respecting remainders takes on a more ‘‘realistic’’ cast (if
realism coincides with the weight of the bigger parties that might induce the choice of a
method that satisfies the property): if some party receives more seats than its quota
‘‘for’’ a remainder of r, then only the bigger parties who have larger remainders are
guaranteed to receive more seats than their quotas (see the CiU in Table 2).
Theorem 4. f d respects the remainders of the large if and only if d $12(1 /s).
Proof. Suppose d $12(1 /s) and let i, j be a pair satisfying pi .pj , r i .r j . If r j ,d,
qj
[q j ] 2 1 r j [qi ] 2 1 r j [qi ] 2 1 r i
qi
]]] 5 ]]] # ]]] , ]]]
5 ]]],
[q j ] 2 1 d [q j ] 2 1 d
[qi ] 2 1 d
[qi ] 2 1 d [qi ] 2 1 d
showing that f d gives [qi ] 2 11 seats to i before it gives [q j ] 2 11 seats to j.
˜
˜
Otherwise, r i .r j .d. Let S5hk[S:
r k $d j, and suppose S5S.
This implies r S 5s21,
¯
so r k 5d 5121 /s for all k[S, a contradiction. Therefore, S˜ ,S and r S $s¯d .s21,
¯s5uSu, so since r S is integer valued, r S $s.
¯ But then o k [qk ] d #h, so any apportionment
d
˜ and the condition is verified.
of f must at least round-up every qk , k[S,
d
On the other hand, if d ,12(1 /s) then f does not necessarily respect the remainders
of the large. An example suffices to show this. Choose s$3 and e .0 to satisfy
e ,minh(121 /s2d ) /(112d ), 1 /sj, and consider the problem (q, h) where
is the vector of the quotas. r 1 5121 /s1 e .r 2 5121 /s, but (1, . . . , 1)[ f d (q, s) is the
unique apportionment since
1
1
1
1
1
22]1e 12]
12] 12]2e 22]1e
s
s
s
s
s
]]] . ]] 5 ? ? ? 5 ]] . ]]] . ]]]
.
d
d
d
d
11d
This means state 1 receives its quota rounded-down, state 2 its quota rounded-up. j
7. Threshold characterizations
A great deal of attention has been given to ‘‘thresholds of representation’’ in the
political science literature (Lijphart, 1994; Rae, 1971). This concerns problems of
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M. Balinski, V. Ramırez
119
apportionment ( p, h), where p represents the votes of parties (as in the example of the
previous section). The questions are these: when is a party assured of at least one
deputy? and when is a party denied any representation whatsoever? An answer leads to
yet other characterizations of parametric methods.
An inequality such as x#[z] d is unambiguous when [z] d has a unique value. But it
may have two values: in this case (and in similar cases) take the inequality to mean that
x is less than or equal to both of the values.
Lemma 6. If x$0 is integer and z.0 is real,
x # [z] d if and only if x , z 1 1 2 d
(and similarly x$[z] d if and only if x.z2d ).
Proof. Let z5k1r, where k is integer and 0#r,1.
Suppose, first, that d ±0, 1. If r,d, [z] d 5k and z112d 5k111r2d, with 21,r2
d ,0. If r.d, [z]d 5k11 and z112d 5k111r2d, with 0,r2d ,1. If r5d, [z] d 5k
or k11 and z112d 5k11. In each case the statement is verified.
If d 50 and r.0, [z] 0 5k11 and z112 5k1r11. If d 50 and r50, [z] 0 5k or k11
and z112d 5k11. Again, the statement is verified.
Finally, if d 51 and r.0, [z] 1 5k and z112d 5k1r. If d 51 and r50, [z] 1 5k or
k21 and z112 5k. So the statement is verified in all cases. j
Theorems 5 and 6 below characterize parametric methods f d in terms of upper and
lower bounds on the number of seats each single party i receives given as functions of
its percentage of the vote, the number of seats to be distributed, the number of parties in
competition and the value of d [[0, 1]. The upper bound answers the question: when is
the party denied any representation whatsoever (or is denied any given total number of
seats). The lower bound answers the question: when is the party sure to have at least one
seat (or is guaranteed any given total number of seats). And in the ‘‘gaps’’ in between
there is a doubt. An example below illustrates how these theorems may be used. Since
both characterize the parametric method f d the bounds are the best possible (for some
problem at least one inequality must be tight). On the other hand, there may be other
upper and lower bound functions that do the job differently.
Theorem 5. The unique consistent and anonymous method f that satisfies
pi
x [ f ( p, h) implies x i # (h 1 d s) ]
for all i [ S,
pS d
F
G
(4)
d
is the parametric method f .
Proof. Suppose f is an anonymous and consistent method that satisfies the upper bound
conditions (4), and let p¯ i 5pi /pS .
f d is clearly anonymous and consistent. Suppose it did not satisfy (4). Then by the
previous lemma for some i [S, x i $p¯ i (h1d s)112d, implying 1 /(h1d s)$p¯ i /(x i 211
d ). By (2)
120
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M. Balinski, V. Ramırez
p̄ j
p̄i
1
]] $ ]]]
$ ]] all j [ S, or x j 1 d $ p¯ j (h 1 d s),
h 1 d s xi 2 1 1 d xj 1 d
with at least one inequality strict (when j5i), so summing over j [S one deduces
h1d s,h1d s, a contradiction. This proves that f d , f.
Now note that f must be balanced. Otherwise, for some problem (x, x1 b, . . . )[
f (( p, p, . . . ), h) with 2 and integer. By consistency, (x, x1 b )[ f (( p, p), 2x1 b ). But
from the upper bound conditions (4) and lemma 5, x1 b , ]12 (2x1 b 12d )112d 5x1
11 b / 2, implying b ,2, a contradiction.
It remains to show that f , f d. Suppose the statement were true for all 2-dimensional
problems (( p1 , p2 ), h). Consider x[ f ( p, h) for an arbitrary problem. Since f is
consistent, (x i , x j ) (( pi , pj ), h) for every pair of indices i, j. But then (x i , x j )[ f d (( pi ,
pj ), h) for every pair of indices i, j, so the conditions (2) are satisfied and x[ f d ( p, h).
Therefore, to establish the theorem it is only necessary to show that f , f d overall
2-dimensional problems. Suppose the contrary and let h* be the smallest h where a
difference is realized, so that f (( p1 , p2 ), h), f d (( p1 , p2 ), h) for h,h*, but not for
h5h*. Since f is balanced and so increasing (lemma 4) it may be assumed that (x 1 ,
x 2 )[ f (( p1 , p2 ), h*21)> f d (( p1 , p2 ), h*21) and (x 1 , x 2 11)[ f (( p1 , p2 ), h*), but
(x 1 , x 2 11)[
⁄ f d (( p1 , p2 ), h*). This means that p1 /(x 1 1d ).p2 /(x 2 1d ). By lemma 5
But f satisfies the upper bound conditions (4), so
F
p2
x 2 1 1 # ]]] (tx 1 1 x 2 1 1 1 d s)
tp1 1 p2
G.
d
Using lemma 6 with s5t11,
p2
p2
x 2 1 1 , ]]] (t(x 1 1 d ) 1 x 2 1 1 1 d ) 1 1 2 d, or t , ]]]]]]],
tp1 1 p2
p1 (x 2 1 d ) 2 p2 (x 1 1 d )
a contradiction. j
Theorem 6. The unique consistent and anonymous method f that satisfies
F
pi
x [ f ( p, h) implies x i $ (h 2 (1 2 d )s) ]
pS
( for h$(12d )s) is the parametric method f
G
for all i [ S,
(5)
d
d
Proof. The argument parallels that for theorem 5. Let f be an anonymous, consistent
method that satisfies the lower bound conditions (5), and p¯ i 5pi /pS .
f d is clearly anonymous and consistent. Suppose it did not satisfy (5). Then for some
i [S, x i #p¯ i (h2s1d s)2d, implying 1 /(h2s1d s)#p¯ i /(x i 1d ). Thus
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M. Balinski, V. Ramırez
121
p̄ j
p̄i
1
]]] # ]]
# ]]] all j [ S, or x j 2 1 1 d # p¯ j (h 2 s 1 d s),
h 2 s 1 d s xi 1 d xj 2 1 1 d
with at least one inequality strict (when j5i), so summing over j [S one deduces
h2s1d s,h2s1d s, a contradiction. This proves that f d , f.
f must be balanced. Else, deduce (x, x2 b )[ f ( p, p), 2x2 b ) for an integer b $2.
But then from (5), x2 b . ]12 (2x2 b 2212d )2d, implying b ,2, a contradiction.
The argument that shows it suffices to prove that f , f d over 2-dimensional problems
is the same. So suppose that for some 2-dimensional problem the statement is false, and
let h* be the smallest h where a difference is realized: f (( p1 , p2 ), h), f d (( p1 , p2 ), h)
for h,h*, but not for h5h*. Since f is increasing (x 1 , x 2 )[ f (( p1 , p2 ), h*21)>
f d (( p1 , p2 ), h*21) and (x 1 , x 2 11)[ f (( p1 , p2 ), h*), but (x 1 , x 2 11)[
⁄ f d (( p1 , p2 ),
h*). This means that p1 /(x 1 1d ).p2 /(x 2 1d ). But
and since f satisfies the lower bound conditions (5),
F
p1
x 1 $ ]]] (x 1 1 tx 2 2 1 1 d t 1 d )
p1 1 tp2
G
d
p1
or t , ]]]]]]],
p1 (x 2 1 d ) 2 p2 (x 1 1 d )
a contradiction. j
These theorems generalize the following well known characterizations of the methods
of Jefferson and of Adams (Balinski and Young, 1982). The unique anonymous,
consistent method that satisfies: x[ f ( p, h) implies (i) x i $[hpi /pS ] 2 for all i [S is the
method of Jefferson, f 1 ; and (ii) x i #[hpi /pS ] 1 for all i [S is the method of Adams, f 0 .
In these particular cases the result is independent of the number of parties s.
For more of an idea as to the use of these results, consider the example given at the
beginning of Section 6: h532 seats are to be distributed among s55 political parties on
the basis of their respective percentages of the votes, pi /pS . With what percentage of the
vote is a party guaranteed to obtain at least one seat, no matter how the remaining votes
are distributed among the other parties, when the law of the land is the parametric
method f d ? Theorem 6 says when [(3225(12d ))p] d $1, that is, when the percentage
p.d /(3225(12d )). Thus, Adams (f 0 ) assures 1 seat for any positive percentage of the
vote; Condorcet (f 2 / 5 ) for p.2 / 145 or at least 1.38%; Webster or Sainte-Lague¨ (f 1 / 2 )
for p.1 / 59 or at least 1.69%; and Jefferson or d’Hondt (f 1 ) for p.1 / 32 or at least
3.125%.
Similarly: with what percentage of the vote is a party guaranteed to obtain no seats, no
matter how the remaining votes are distributed among the other parties, when the law of
the land is again the parametric method f d ? Theorem 5 answers when [(3215s)p] d #0,
that is, when p,d /(3215d ). Thus Adams denies representation only when a party does
not run at all; Condorcet when p,1 / 85 or a party has less than 1.18%; Webster when
122
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M. Balinski, V. Ramırez
p,1 / 69 or a party has less than 1.45%; and Jefferson when p,1 / 37 or a party has less
than 2.70%. (All percentages are reported to the nearest 0.01%.)
When a party has a percentage of the vote that falls in the gap between 1.45% and
1.69% of the vote and h532 seats are to be distributed among s55 parties by the
method of Webster the result is uncertain: it may receive either 0 or 1 seat (and similarly
for other parametric methods and other gaps).
The interest of these characterizations is that they are sparse and simple: only
anonymity and consistency together with the bounds are required, and the proofs are
relatively straightforward (and much simpler than those required to characterize the more
general class of divisor methods).
References
¨
Balinski, M.L., 1996. How should data be rounded? In: Ruschendorf,
L., Schweizer, B., Taylor, M.D. (Eds.),
Distributions with Fixed Marginals and Related Topics. Lecture Notes-Monograph Series, vol. 28. Institute
of Mathematical Statistics, Hayward, CA, pp. 33–44.
Balinski, M., 1993. The problem with apportionment. Journal of the Operations Research Society of Japan 36,
134–148.
Balinski, M.L., Rachev, S.T., 1997. Rounding proportions: methods of rounding. Mathematical Scientist 22,
1–26.
Balinski, M.L., Rachev, S.T., 1993. Rounding proportions: rules of rounding. Numerical Functional Analysis
and Optimization 14, 475–501.
Balinski, M., Shahidi, N., 1997. A simple approach to the product rate variation problem via axiomatics. To
appear in Operations Research Letters.
Balinski, M.L., Young, H.P., 1982. Fair Representation: Meeting the Ideal of One Man, One Vote. Yale
University Press, New Haven, CT.
Balinski, M.L., Young, H.P., 1977. On Huntington methods of apportionment. SIAM Journal on Applied
Mathematics (Part C) 33, 607–618.
Bautista, J., Companys, R., Corominas, A., 1996. A note on the relation between the product rate variation
(PRV) problem and the apportionment problem. Journal of the Operational Research Society 47, 1410–
1414.
Kubiak, W., 1993. Minimizing variation of production rates in just-in-time systems: a survey. European Journal
of Operational Research 66, 259–271.
Lijphart, A., 1994. Electoral Systems and Party Systems. Oxford University Press, Oxford.
Miltenburg, J., 1989. Level schedules for mixed-model assembly lines in just-in-time production systems.
Management Science 35, 192–207.
´
` de l’Economie
Ministere
et des Finances, mai 1997. Le passage a` l’euro: les arrondis.
Oyama, T., 1991. On a parametric divisor method for the apportionment problem. Journal of the Operations
Research Society of Japan 34, 187–221.
Rae, D.W., 1971. The Political Consequences of Electoral Laws, rev. ed. Yale University Press, New Haven.
Steiner, G., Yeomans, S., 1993. Level schedules for mixed-model, just-in-time processes. Management Science
39, 728–735.
Parametric methods of apportionment, rounding and
production
´ b
Michel Balinski a , *, Victoriano Ramırez
a
´
´
´ , Ecole
C.N.R.S. and Laboratoire d’ Econometrie
Polytechnique, 1 rue Descartes, 75005 Paris, France
b
´
Departamento de Matematica
Aplicada, Universidad de Granada, Granada, Spain
Received 10 November 1997; received in revised form 30 April 1998; accepted 30 May 1998
Abstract
The class of ‘‘parametric’’ methods of apportionment, of rounding, or for minimizing the
variation of production rates in just-in-time production systems is characterized in several different
ways that depend on the underlying qualitative behavior of its solutions. 1999 Elsevier
Science B.V. All rights reserved.
Keywords: Apportionment; Rounding; Just-in-time production; Axiomatic characterization
JEL classification: C44; C69; C79; D71
1. Introduction
The motivation for studying parametric methods is threefold. To begin they play an
important role in electoral systems. In Japan the debate on what method should be
chosen to apportion the 512 seats of its Diet seems to have concentrated on parametric
methods (Balinski, 1993; Oyama, 1991). In Spain the House of Representatives is
elected by a system of proportional representation at the level of its provinces. Recent
experience has shown that this has strongly and increasingly favored solidly implanted
provincial political parties at the expense of the nation-wide parties of similar or smaller
size – indeed, so much so that one party, the Catalonian CiU (Convergencia i Union)
party, has held the balance of power since 1993, permitting the socialists (Partido
˜ to govern from 1993 to 1996, and then permitting the right
Socialista Obrero Espanol)
(Partido Popular) to govern since 1996. This has sparked a considerable debate on what
should be done that is regularly nourished by electoral data. The data naturally evoked a
*Corresponding author. Fax: 133 1 46343428; e-mail: balinski@poly.polytechnique.fr
0165-4896 / 99 / $ – see front matter 1999 Elsevier Science B.V. All rights reserved.
PII: S0165-4896( 98 )00027-4
108
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M. Balinski, V. Ramırez
desirable property that a method of apportionment should have, which it turns out is met
only by the class of parametric methods. More generally, political scientists have for
years talked about the ‘‘thresholds’’ of votes that will assure a party of representation in
a proportional representation system (see, Lijphart, 1994; Rae, 1971): an answer to this
question leads to yet another characterization of parametric methods. This is the
‘‘apportionment’’ part of the title.
The term ‘‘rounding’’ refers to a second motivation. Given a list of real numbers the
sum of their roundings may well differ from the rounding of their sum. For example,
newspapers usually give the percentages of the votes won by each of a set of competing
candidates along with the actual vote totals, and they often fail to add to 100%. Another
current example comes with the introduction of the new common European Union
currency, the ‘‘euro’’, expected to be introduced on January 1, 1999. There will be a
period of transition in which financial transactions in banks will be given in the two
currencies of a participating country, the old and the new: e.g., in France, the franc and
´
` de l’Economie
the euro (Ministere
et des Finances, 1997). Sums of roundings will often
differ from the rounding of the sum, and though each single problem represents
practically nothing, the discrepancies summed over all problems could bring significant
windfall gains (or losses) due to the sheer number of transactions. Two questions present
themselves (Balinski, 1996). The first has been addressed elsewhere (Balinski and
Rachev, 1993): if one accepts the idea that sums of percentages may differ from 100%,
what ‘‘rule’’ of rounding should be used? The second is of concern here: if one insists
that the sums must always be exactly 100%, what ‘‘method’’ should be used (Balinski
and Rachev, 1997)? A particularly desirable property for methods of rounding
distinguishes the class of parametric methods (called ‘‘stationary’’ in Balinski and
Rachev, 1997) among the far wider class of ‘‘divisor’’ methods.
There is an extensive literature on minimizing the variation of production rates in
‘‘just-in-time’’ systems (e.g., Balinski and Shahidi, 1997; Bautista et al., 1996; Kubiak,
1993; Miltenburg, 1989; Steiner and Yeomans, 1993). In one guise – the ‘‘product rate
variation’’ problem – it is formally equivalent to the problem of apportionment, though
what constitutes a ‘‘good’’ solution must of course be evaluated in the context of the
actual problem. A particularly appealing property, often discussed in the literature (when
praising the qualities of a particular method or type of solution) – to wit, that an optimal
sequence in the production of different products should eventually repeat or cycle –
once again elects parametric methods as the only ones that meet the test. This is the third
motivation, and the reason for the existence of the word ‘‘production’’ in the title.
The answer to the question, ‘‘why a parametric method?’’ depends on the inherent
context of the problem to be treated, but several different fundamental properties,
persuasive in different contexts, yield the same methods: they are the subject of this
paper.
2. The problem
A problem is defined by any pair ( p, h), where p 5 ( pj ) . 0, j [ S, is a nonzero
vector of reals, S a finite set, and h . 0 a positive integer. The dimension of the problem
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M. Balinski, V. Ramırez
109
is uSu 5 s. A solution
for the problem ( p, h) is a vector of positive integers x 5 (x j ) $ 0,
def
j [ S satisfying x S 5o j [S x j 5h.
A method f is a point to set mapping that assigns at least one solution to each
problem.
In apportionment, p are the populations of s states or provinces (or the vote totals of
political parties) and x the total number of seats each is to be allocated, which must sum
to h. The goal is to find solutions that are as ‘‘fair’’ as possible, the unattainable ideal
being proportionality.
In rounding, p are data that are to be rounded and x the roundings, whose sum must
equal h (e.g. h5100%). The aim here is to find solutions that in some probabilistic sense
best ‘‘represent’’ the vectors p.
In the product rate variation or PRV problem, p are the relative demands for s
products to be produced (so pS 51) and x the cumulative production through period h.
Here it is assumed that producing one item of any product requires one period and that
Dp is integer valued for some integer D .0: what is wanted is the order of production in
succeeding periods, so a solution is necessary for every value of h, 0#h#D. For the
PRV problem an order of production is sought that best maintains cumulative
productions over the D periods that come ‘‘closest’’ to being proportional to the rates of
production.
3. Divisor and parametric methods
There is a particularly rich history of methods in the context of apportionment (see
Balinski and Young, 1982). Most, though not all, of the specific methods proposed or
used belong to a class called ‘‘divisor’’ methods.
A divisor function d is any monotone real valued function defined over the
nonnegative integers that satisfies d(k)[[k, k11] for all integers k, and for which there
exists no pair of integers a$0 and b$1 with d(a)5a11 and d(b)5b. A d-rounding of
a real number z$0 is
[z] d 5
H
if z 5 0;
if d(a 2 1) # z # d(a).
0,
a,
Thus [d(a)] d 5a or a11: at the threshold one can either round up or down.
The divisor method f d based on d is
f d ( p, h) 5 hx 5 (xj ), j [ S: xj 5 [ l pj ] d , l . 0 chosen so that x S 5 hj.
(1)
The seats of the U.S. House of Representatives have been apportioned by the divisor
]]]
method based on d(a) 5œa(a 1 1) since 1940; and in 1840 the method based on
1
d(a) 5 a(a 1 1) /(a 1 ]2 ) was proposed (among others).
An equivalent definition is the min–max condition
H
J
pj
pi
f d ( p, h) 5 x 5 (xj ), j [ S, x S 5 h: min ]]] $max ]] .
x i $1 d(x i 2 1)
x j $0 d(x j )
(2)
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M. Balinski, V. Ramırez
110
A parametric method is a divisor method f d baseddefon d(k)5k1d for all k, where
0#d #1. It is denoted by f d, and for convenience [z] d 5[z] d . Note that [k] 0 5k or k11,
and [k] 1 5k or k21. (In terms of the more usual notation, where [z] 2 and [z] 1 denote
the real number z rounded down and up, respectively, to the nearest integer: [z] 1 5[z] 2
and [z] 0 5[z] 1 .)
Various specific parametric methods have been proposed or used for apportionment:
J.Q. Adams suggested d 50, Condorcet d 50.4, Webster and Sainte-Lague¨ d 50.5, and
Jefferson and d’Hondt d 51.
The lemma that follows is trivial to verify.
Lemma 1. If x[ f a ( p, h) and x[ f b ( p, h) then for all d, a #d # b, x[ f d ( p, h).
Due to lemma 1, Table 1 gives all of the parametric method solutions for the problem
( p, h), where p is the vector of votes received by the respective parties in the 1993
Spanish elections and h5350 is the number of seats in its lower house. The quotas of a
problem ( p, h) are given by the vector q5hp /pS : it is the exact proportional vector that
sums to h. One representative value of d is given in every case; and each pair of
adjacent columns are alternate solutions for some one intermediate value of d. Thus, the
first solution in the table is the same for all methods f d when 0#d #(1461987 /
7994587)5d1 ¯0.182872 . . . . When d 5d1 there is a ‘‘tie’’, the first two solutions both
belong, and so on, pair by pair for the remaining solutions of the table. The successive
approximate values of d where there is a tie (given to the closest 10 23 ) are: 0.183,
0.195, 0.243, 0.457, 0.569, 0.765.
Notice also that as d increases from 0 to 1 solutions change, with lower x i ’s in the list
decreasing and higher x i ’s in the list increasing. This suggests the following concept: In
comparing two methods f and f *, f gives-up to f * if
Table 1
All parametric method solutions
Party
Votes
PSOE
PP
IU
CiU
CDS
PNV
CC
HB
ERC
PAR
EA
UV
9 150 083
8 201 463
2 253 722
1 165 783
414 740
291 448
207 077
206 876
189 632
144 544
129 293
112 341
Total
22 467 002
Quota
d5
0.09
0.19
0.22
0.35
0.51
0.66
0.88
142.55
127.77
35.11
18.16
6.46
4.54
3.32
3.23
2.95
2.25
2.01
1.75
141
126
35
18
7
5
4
4
3
3
2
2
141
127
35
18
7
5
4
3
3
3
2
2
142
127
35
18
7
5
3
3
3
3
2
2
142
128
35
18
7
5
3
3
3
2
2
2
143
128
35
18
6
5
3
3
3
2
2
2
143
129
35
18
6
4
3
3
3
2
2
2
144
129
35
18
6
4
3
3
3
2
2
1
350
350
350
350
350
350
350
350
Note: data from 1993 Spanish elections.
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M. Balinski, V. Ramırez
111
x [ f ( p, h), x* [ f *( p, h) and pi . pj implies x i # x i* or x j $ x j* .
This definition is well known in the literature (see Balinski and Young, 1982 page 118):
it formalizes the idea of one method favoring the bigger as versus the smaller parties.
Think of ‘‘gives-up’’ in terms of seats being ‘‘given-up’’ by the smaller parties, as the
method is changed from f to f *, in favor of the bigger parties . . . that are further ‘‘up’’
on the list of parties!
In the context of parametric (as versus the more general divisor) methods this concept
permits the characterization given in the following lemma.
Lemma 2. A parametric method f a gives-up to another parametric method f b if and
only if a , b.
Proof. Consider a problem ( p, h), with x[ f a ( p, h), x*[ f b ( p, h) and pi .pj . Then
pi /(x i 211 a )$pj /(x j 1 a ), so a , b implies pi /(x i 211 b ).pj /(x j 1 b ), meaning that
either x *
i $x i or x *
j #x j . However, if a . b it is easy to construct a problem ( p, h), with
a
x[ f ( p, h), x*[ f b ( p, h) and pi .pj , for which pi /(x i 211 a )$pj /(x j 1 a ) and
pi /(x i 211 b ),pj /(x j 1 b ), meaning that either x i* #x i 21 or x j* $x j 11. j
Thus, the parametric method f d that is most favorable to the smaller parties is that
with the smallest d, namely d 50; and that most favorable to the bigger parties is that
with the largest d, namely d 51.
The well known method of Hamilton (sometimes called that of Hare), used for
apportionment at various times in various countries including France, Israel, Mexico and
the USA, stands, in view of its properties, in opposition to the class of divisor methods.
It is easily described: (i) for each i set x i 5[qi ] 2 ; (ii) then increase by 1 each x i that
belongs to a set of cardinality h2o i q[ i ] 2 having the largest remainders qi 2[qi ] 2 .
There may, of course, be several solutions (when some of the remainders are the same).
4. Properties of methods
The three most fundamental properties that a method for any of the three problems
should enjoy are as follows. First, scale-invariancy: f ( p, h)5 f ( l p, h) for every l .0.
The problem is the same no matter what scale is used in presenting it. Second,
exactness: if p is integer valued and o i pi 5h then p is the unique solution, f ( p, h)5p.
If there is no ‘‘problem’’ then there is no problem! Third, anonymity: solutions depend
only on the values of the data, i.e., they are independent of the order in which the data is
presented. Every method must realize these three demands. In particular, divisor
methods as well as Hamilton’s method satisfy these properties.
Arguably the most important ‘‘nonobvious’’ property concerning methods is that they
be ‘‘consistent’’. In all of the applications of problems ( p, h) the ‘‘ideal’’ solution is the
proportional one. The fundamental underlying property of proportionality is that any part
of a proportional solution is itself proportional. To make this precise, if J is some subset
of S, J ,S, and J¯ is its complement, denote the corresponding subvectors of p by p(J)
112
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M. Balinski, V. Ramırez
and p(J¯ ), and similarly for x, and write x5(x(J), x(J¯ )) and p5( p(J), p(J¯ )) (modulo a
rearrangement of the order of the indices which, by anonymity, is of no consequence).
A method f is consistent if (x(J), x(J¯ ))[ f (( p(J), p(J¯ )), h) implies x(J)[ f ( p(J),
xJ ) for all J ,S; and if also y(J)[ f ( p(J), xJ ), then ( y(J), x(J¯ ))[ f (( p(J), p(J¯ )), h).
Any part of a best apportionment, rounding or production schedule should itself be a
best apportionment, rounding or production schedule. It is at once evident that every
divisor method, so every parametric method, is consistent. On the other hand, simple
examples show that Hamilton’s method is not consistent.
Indeed, consistency is a strong property. We begin by deducing several of its
consequences.
A method f is balanced if x[( p, h) and pi 5pj implies ux i 2x j u#1.
Lemma 3. A consistent, exact and anonymous method is balanced.
Proof. Suppose it were not so. There would then exist x[ f ( p, h) with (say) p1 5p2 5p
whose seats x 1 , x 2 differ by at least 2, (x 1 , x 2 )5(x1d, x), d $2.
Suppose d 52k. Then by consistency, (x12k, x)[ f (( p, p), 2x12k), contradicting
exactness. By the same reasoning, if x[ f (( p, . . . , p), h) and x i , x j are of the same
parity, then x i 5x j . It may therefore be assumed that
If f is not balanced, (x12k11, x)[ f (( p, p), 2x12k11) for some k$1. Consider
Then a y1(4k2 a )z52k(2x12k11), so that a ( y2z);2k (mod 4k). This equation
always has the solution a 52k. Moreover, if a is a solution then so is 4k2 a.
If a 52k, then y1z52x12k11. Applying consistency 2k times yields
Invoking consistency once more,
contradicting exactness.
If a ,2k solves the modular equation for some y, z, with y2z52l 11, l $1, then
a (z12l 11)1(2k2 a )z52kz1 a (2l 11), where a (2l 11) is divisible by 2k. By
consistency
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M. Balinski, V. Ramırez
113
which contradicts exactness. If, on the other hand, a .2k solves the modular equation, a
similar construction leads again to a contradiction. j
A method f is increasing if x[ f ( p, h) implies there exists x9[ f ( p, h11) with
x9$x. Simple examples show that Hamilton’s method is not increasing: this is the
‘‘Alabama paradox’’.
Lemma 4. A consistent, balanced and anonymous method is increasing.
Proof. To begin note that since f is consistent it is increasing if and only if (x 1 ,
x 2 )[ f (( p1 , p2 ), h) and (x 19 , x 29 )[ f (( p1 , p2 ), h11) implies x i9 $x i for i51, 2.
Suppose f is not increasing: say x 91 #x 1 21, so that x 92 $x 2 12. Consider ( y 1 , y 2 , y 91 ,
y 92 )[ f (( p1 , p2 , p1 , p2 ), 2h11). f balanced implies uy 1 2y 19 u#1 and uy 2 2y 29 u#1, so it
may be assumed that y 1 1y 2 5h and y 19 1y 29 5h11. Two appeals to consistency yields
(x 1 , x 2 , x 19 , x 92 )[ f (( p1 , p2 , p1 , p2 ), 2h11). But then by consistency once more (x 2 ,
x 92 )[ f (( p2 , p2 ), x 2 1x 29 ): this contradicts the fact that f is balanced because ux 29 2x 2 u$
2. j
Lemma 5. If a method f is consistent, balanced and anonymous then (x 1 , x 2 )[ f (( p1 ,
p2 ), x 1 1x 2 ) implies
, for any t$2.
Proof. Since f is balanced it may be assumed that
(3)
where 0#s,t.
Suppose x 91 1x 29 ±x 1 1x 2 , say x 19 1x 29 ,x 1 1x 2 . Then since f is increasing (x 19 , x 29 )#
(x 1 , x 2 ), so x 19 #x 1 and if x 29 ,x 2 we have x 92 11#x 2 , implying x 1 1tx 2 5x 19 1tx 92 1s#
x 1 1tx 2 2t1s, and thus s$t, a contradiction. One concludes that x 29 5x 2 , x 19 ,x 1 , and
since f is increasing, (x¯ 1 , x 92 ) [ fs( p1 , p2 ), x 1 1 x 29d for every x¯ 1 [[x 19 , x 1 ], in particular,
(x 91 11, x 92 )[ f (( p1 , p2 ), x 19 1x 29 11). But consistency and (3) imply that (x 19 , x 29 11)[
f (( p1 , p2 ), x 19 1x 29 11), so by consistency the first of these solutions may be substituted
in (3) to obtain
Repeating one obtains
with x 1* 1x 92 5x 1 1x 2 and x *1 1tx 92 5x 1 1tx 2 , so x *1 5x 1 and x 92 5x 2 . j
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M. Balinski, V. Ramırez
114
Corollary. If a method f is consistent, balanced and anonymous then (x 1 , x 2 )[ f (( p1 ,
p2 ), x 1 1x 2 ) implies
for any integers t 1 , t 2 .0.
Nevertheless, consistency (together with anonymity, exactness and invariancy) is not
sufficient to characterize the divisor methods, as shown by the following example. For
all k$0 and integer, define
izi 5
Hk or kk 1 1
if k , z , k 1 1
if z 5 k
and consider the method
c ( p, h) 5 hx 5 (xj ), j [ S: x j 5 i l pj i, l . 0 chosen so that xS 5 hj.
It is easy to verify that the method c meets the properties of scale-invariancy, exactness
and anonymity, and it is clearly consistent: but it is not a divisor method.
A method is responsive if
x [ f ( p, h) and pi . pj implies x i $ x j .
Theorem 1 (Balinski and Young, 1982, 1977). f is a divisor method if and only if it is
consistent, exact, anonymous, scale-invariant and responsive.
This theorem characterizes the class of divisor methods, and opens the door to a host
of characterizations of individual methods, but how and why the parametric methods
constitute a class of their own has remained an open question.
5. Cyclic characterization
In the contexts of production scheduling and of rounding a cyclic property seems to
be particularly compelling. Suppose, for example, that the relative demands for two
7
18
7
18
]
] ]
products are p5( ]
25 , 25 ) and that a method f yields f (( 25 , 25 ), 5)5(1, 4). Observe that
7
18
]
f (( ]
25 , 25 ), 25); f ((7, 18), 25)5(7, 18), the last by exactness, and it is evident that the
7
18
]
same order of production should repeat, so it should be true that f (( ]
25 , 25 ), 30)5(7,
18)1(1, 4)5(8, 22). And the same remark seems reasonable in the context of rounding.
For apportionment, however, the property seems uninteresting: by the very nature of the
problem, h is very small as compared with the large values of the vector p.
A method is cyclic if
x [ f ( p, h) and p integer implies x 1 p [ f ( p, h 1 pS ).
Hamilton’s method, for example, is clearly cyclic.
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M. Balinski, V. Ramırez
115
Theorem 2. A divisor method f is parametric if and only if it is cyclic.
Proof. To see that a parametric method is cyclic it suffices by (2) to note that
pj
pj
pi
pi
]]]
$ ]] if and only if ]]]]] $ ]]]]
xi 2 1 1 d xj 1 d
kpi 1 x i 2 1 1 d kpj 1 x j 1 d
for all integer k.
Assume then that f is a cyclic divisor method, say f d . Suppose it is not parametric,
so d(k)5k1d for all k satisfying 0#k,k* but d(k*)5k1d *, with d ±d *, say d .d *.
Take p1 , p2 to be any positive integers satisfying d *,p1 /p2 ,d.
Then, (1, l p2 21)5 f d ((1, l p2 21), l p2 ) for every l .0 for which l p2 is integer,
since f d is exact. Also, (0, l p1 )5 f d ((1, l p2 21), l p1 ) for l large enough and l p1
integer. To see this it suffices to show 1 /d(0),( l p2 21) /d( l p1 21):
l p2 2 1
l p2 2 1 p2
1
1
1
]]]
$ ]]] 5 ] 2 ] . ] 5 ]],
l p1
p1 l p1 d d(0)
d( l p1 2 1)
which is true for l large enough.
f d cyclic implies that (k*, k*( l p2 21)1 l p1 )[ f d ((1, l p2 21), l p1 1k* l p2 ). But
this is impossible because for large enough l the min–max condition (2) is violated:
l p2 2 1
l p2 2 1
1
]]]]]]]
# ]]]]]]] 5 ]]]]
l p1 2 1
d(k*( l p2 2 1) 1 l p1 2 1) k*( l p2 2 1) 1 l p1 2 1
k* 1 ]]]
l p2 2 1
1
1
, ]]] 5 ]].
k* 1 d * d(k*)
A similar construction handles the case d *.d. j
In the PRV problem, the aim is that the cumulative productions at each period be
approximately proportional to the rates of demand. By and large the approach found in
the literature is to postulate a measure of disproportionality or of error and so turn the
problem into a sequence of minimization problems, one for each period. Most papers
impose a measure yielding solutions identical to those given by the method of Hamilton.
But this leads to an unfortunate difficulty since Hamilton’s method is not increasing: it is
possible for the cumulative production to decrease in going from one period to the next!
Thus the main effort has been to devise heuristics or other involved algorithms to
guarantee a feasible solution (which may or may not be cyclic). No attention has been
given to the idea that perhaps another objective function would do better: indeed, there
are infinite numbers of measures of disproportionality that one could use, but little to
nothing on why one is better than another. Cyclicity is a powerful reason for choosing to
minimize a measure of error whose solutions belong to parametric methods. The fact is
that x[ f d ( p, h) if and only if x solves
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M. Balinski, V. Ramırez
116
min
x
x 1d 2
O p S]]]
2 hD when O x 5 h, x $ 0.
p
1
]
2
i
2
i
i
i
i
i
i
This makes it tempting to chose the parametric method based on d 51 / 2. In fact there
is a persuasive reason for this choice: it is the unique parametric method for which the
average cumulative production of each product i over the full cycle of D periods is
exactly equal to one-half of its total demand Dpi (see Balinski and Shahidi, 1997). So
f 1 / 2 is the unique parametric method that achieves ‘‘proportionality’’ over the entire
production cycle.
6. Remainder characterizations
In Spain’s Parliament each province (or state) is allocated a certain number of
deputies, on the basis of its population, and within each province parties are allocated
seats according to their vote totals within the province by the method of Jefferson
(known in Europe as the method of d’Hondt), f 1 . For example, in the 1989 elections the
results within the province of Barcelona, which elects 32 deputies, were as in Table 2.
Lemma 2 shows that among the parametric methods – and this is true as well among
the class of all divisor methods – the method of Jefferson (or d’Hondt) is the most
favorable for the large parties, the least favorable for the small parties. In the interest of
reducing this advantage, and so obtaining a more ‘‘equitable’’ apportionment, the idea of
¨
using Webster’s method (or Sainte-Lague’s),
f 1 / 2 , was considered. The example of
Table 2 immediately fuelled the arguments against: why should the PP receive 4 seats
‘‘for’’ 3 and a remainder of 0.72 and the CDS 2 ‘‘for’’ 1 and a remainder of 0.60,
whereas the CiU only receives 10 ‘‘for’’ 10 and a remainder of 0.74? This suggests two
possible properties.
Letting q j 5hpj /pS 5n j 1r j , n j $0 and integer, a method f respects remainders at d if
for all x[ f ( p, h),
r k # d # rl , r k , rl and x k $ n k 1 1 implies xl $ nl 1 1.
The idea is that if party k receives more seats than its quota and its ‘‘claim’’ to an extra
seat on the basis of its remainder r k is at most d, whereas another party l’s ‘‘claim’’ on
Table 2
1989 Barcelona election results
Party
PSOE
CiU
PP
IC
CDS
Total
Votes
900 039
740 479
256 785
199 372
110 175
2 206 850
Quota
f 1/2
f1
13.05
10.74
3.72
2.89
1.60
13
10
4
3
2
13
11
4
3
1
32
32
32
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M. Balinski, V. Ramırez
117
the basis of rl is at least d, then surely party l should also receive more than its quota. It
is surprising that this property together with consistency, exactness and anonymity is
sufficient to characterize the parametric method f d, with no appeal to the more general
and more difficult characterization of divisor methods.
Theorem 3. The unique consistent, exact and anonymous method that respects
remainders at d is the parametric method f d.
Proof. The parametric method f d respects remainders at d because r k #d #rl and r k ,rl
imply
qk
ql
]]
# 1 # ]],
nk 1 d
nl 1 d
with at least one inequality strict. Therefore, pl /(nl 1d ).pk /(n k 1d ), and xl takes the
value nl 11 before x k takes the value n k 11. This proves that f d , f.
Since f respects remainders it must be balanced. To see this, suppose the contrary:
for some positive p and positive integer x, either (a) (x11, x21)[ f (( p, p), 2x), or (b)
(x12, x21)[ f (( p, p), 2x11). (a) Take q1 5x10, q2 5(x21)11, with 0#d #1. Since
x11$x11 holds (the first component), it must be that x21$x (the second component), a contradiction. (b) Take q1 5(x11)2 ]21 , q2 5(x21)1 ]23 , with 2 ]21 #d # ]23 .
Since x12$x12 holds (the first component), it must be that x21$x (the second
component), a contradiction.
Again it suffices to prove that f , f d over all 2-dimensional problems. So suppose
that for some 2-dimensional problem the statement is false, and let h* be the smallest h
where a difference is realized: (x 1 , x 2 )[ f (( p1 , p2 ), h*21)> f d (( p1 , p2 ), h*21) and
(x 1 11, x 2 )[ f (( p1 , p2 ), h*), but (x 1 11, x 2 )[
⁄ f d (( p1 , p2 ), h*). This means that
p1 /(x 1 1d ),p2 /(x 2 1d ), so p1 x 2 2p2 x 1 1 e 5( p2 2p1 )d with e .0.
Let t 1 , t 2 be positive integers chosen to satisfy the following inequalities
p2d
t1
p1d 1 e
]]]]
, ] , ]]].
p2 (1 2 d ) 1 e t 2 p1 (1 2 d )
By the corollary to lemma 5,
Let q1 , q2 be the respective quotas:
p1 (t 1 x 1 1 t 1 1 t 2 x 2 )
p1 t 1 1 ( p1 x 2 2 p2 x 1 )t 2 def
q1 5 ]]]]]] 5 x 1 1 ]]]]]]] 5 x 1 1 r 1 ,
t 1 p1 1 t 2 p2
t 1 p1 1 t 2 p2
and
p2 (t 1 x 1 1 t 1 1 t 2 x 2 )
p2 t 1 1 ( p2 x 1 2 p1 x 2 ) def
q2 5 ]]]]]] 5 x 2 1 ]]]]]] 5 x 2 1 r 2 .
t 1 p1 1 t 2 p2
t 1 p1 1 t 2 p2
118
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M. Balinski, V. Ramırez
A straightforward calculation shows that r 1 ,d ,r 2 , thus contradicting the fact that f
respects remainders at d. j
The example of Table 2 invokes a second possible property. A method f respects the
remainders of the large when for all x[ f ( p, h),
pi . pj , r i . r j , and x j . q j implies x i . qi .
In this case the concept of respecting remainders takes on a more ‘‘realistic’’ cast (if
realism coincides with the weight of the bigger parties that might induce the choice of a
method that satisfies the property): if some party receives more seats than its quota
‘‘for’’ a remainder of r, then only the bigger parties who have larger remainders are
guaranteed to receive more seats than their quotas (see the CiU in Table 2).
Theorem 4. f d respects the remainders of the large if and only if d $12(1 /s).
Proof. Suppose d $12(1 /s) and let i, j be a pair satisfying pi .pj , r i .r j . If r j ,d,
qj
[q j ] 2 1 r j [qi ] 2 1 r j [qi ] 2 1 r i
qi
]]] 5 ]]] # ]]] , ]]]
5 ]]],
[q j ] 2 1 d [q j ] 2 1 d
[qi ] 2 1 d
[qi ] 2 1 d [qi ] 2 1 d
showing that f d gives [qi ] 2 11 seats to i before it gives [q j ] 2 11 seats to j.
˜
˜
Otherwise, r i .r j .d. Let S5hk[S:
r k $d j, and suppose S5S.
This implies r S 5s21,
¯
so r k 5d 5121 /s for all k[S, a contradiction. Therefore, S˜ ,S and r S $s¯d .s21,
¯s5uSu, so since r S is integer valued, r S $s.
¯ But then o k [qk ] d #h, so any apportionment
d
˜ and the condition is verified.
of f must at least round-up every qk , k[S,
d
On the other hand, if d ,12(1 /s) then f does not necessarily respect the remainders
of the large. An example suffices to show this. Choose s$3 and e .0 to satisfy
e ,minh(121 /s2d ) /(112d ), 1 /sj, and consider the problem (q, h) where
is the vector of the quotas. r 1 5121 /s1 e .r 2 5121 /s, but (1, . . . , 1)[ f d (q, s) is the
unique apportionment since
1
1
1
1
1
22]1e 12]
12] 12]2e 22]1e
s
s
s
s
s
]]] . ]] 5 ? ? ? 5 ]] . ]]] . ]]]
.
d
d
d
d
11d
This means state 1 receives its quota rounded-down, state 2 its quota rounded-up. j
7. Threshold characterizations
A great deal of attention has been given to ‘‘thresholds of representation’’ in the
political science literature (Lijphart, 1994; Rae, 1971). This concerns problems of
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M. Balinski, V. Ramırez
119
apportionment ( p, h), where p represents the votes of parties (as in the example of the
previous section). The questions are these: when is a party assured of at least one
deputy? and when is a party denied any representation whatsoever? An answer leads to
yet other characterizations of parametric methods.
An inequality such as x#[z] d is unambiguous when [z] d has a unique value. But it
may have two values: in this case (and in similar cases) take the inequality to mean that
x is less than or equal to both of the values.
Lemma 6. If x$0 is integer and z.0 is real,
x # [z] d if and only if x , z 1 1 2 d
(and similarly x$[z] d if and only if x.z2d ).
Proof. Let z5k1r, where k is integer and 0#r,1.
Suppose, first, that d ±0, 1. If r,d, [z] d 5k and z112d 5k111r2d, with 21,r2
d ,0. If r.d, [z]d 5k11 and z112d 5k111r2d, with 0,r2d ,1. If r5d, [z] d 5k
or k11 and z112d 5k11. In each case the statement is verified.
If d 50 and r.0, [z] 0 5k11 and z112 5k1r11. If d 50 and r50, [z] 0 5k or k11
and z112d 5k11. Again, the statement is verified.
Finally, if d 51 and r.0, [z] 1 5k and z112d 5k1r. If d 51 and r50, [z] 1 5k or
k21 and z112 5k. So the statement is verified in all cases. j
Theorems 5 and 6 below characterize parametric methods f d in terms of upper and
lower bounds on the number of seats each single party i receives given as functions of
its percentage of the vote, the number of seats to be distributed, the number of parties in
competition and the value of d [[0, 1]. The upper bound answers the question: when is
the party denied any representation whatsoever (or is denied any given total number of
seats). The lower bound answers the question: when is the party sure to have at least one
seat (or is guaranteed any given total number of seats). And in the ‘‘gaps’’ in between
there is a doubt. An example below illustrates how these theorems may be used. Since
both characterize the parametric method f d the bounds are the best possible (for some
problem at least one inequality must be tight). On the other hand, there may be other
upper and lower bound functions that do the job differently.
Theorem 5. The unique consistent and anonymous method f that satisfies
pi
x [ f ( p, h) implies x i # (h 1 d s) ]
for all i [ S,
pS d
F
G
(4)
d
is the parametric method f .
Proof. Suppose f is an anonymous and consistent method that satisfies the upper bound
conditions (4), and let p¯ i 5pi /pS .
f d is clearly anonymous and consistent. Suppose it did not satisfy (4). Then by the
previous lemma for some i [S, x i $p¯ i (h1d s)112d, implying 1 /(h1d s)$p¯ i /(x i 211
d ). By (2)
120
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M. Balinski, V. Ramırez
p̄ j
p̄i
1
]] $ ]]]
$ ]] all j [ S, or x j 1 d $ p¯ j (h 1 d s),
h 1 d s xi 2 1 1 d xj 1 d
with at least one inequality strict (when j5i), so summing over j [S one deduces
h1d s,h1d s, a contradiction. This proves that f d , f.
Now note that f must be balanced. Otherwise, for some problem (x, x1 b, . . . )[
f (( p, p, . . . ), h) with 2 and integer. By consistency, (x, x1 b )[ f (( p, p), 2x1 b ). But
from the upper bound conditions (4) and lemma 5, x1 b , ]12 (2x1 b 12d )112d 5x1
11 b / 2, implying b ,2, a contradiction.
It remains to show that f , f d. Suppose the statement were true for all 2-dimensional
problems (( p1 , p2 ), h). Consider x[ f ( p, h) for an arbitrary problem. Since f is
consistent, (x i , x j ) (( pi , pj ), h) for every pair of indices i, j. But then (x i , x j )[ f d (( pi ,
pj ), h) for every pair of indices i, j, so the conditions (2) are satisfied and x[ f d ( p, h).
Therefore, to establish the theorem it is only necessary to show that f , f d overall
2-dimensional problems. Suppose the contrary and let h* be the smallest h where a
difference is realized, so that f (( p1 , p2 ), h), f d (( p1 , p2 ), h) for h,h*, but not for
h5h*. Since f is balanced and so increasing (lemma 4) it may be assumed that (x 1 ,
x 2 )[ f (( p1 , p2 ), h*21)> f d (( p1 , p2 ), h*21) and (x 1 , x 2 11)[ f (( p1 , p2 ), h*), but
(x 1 , x 2 11)[
⁄ f d (( p1 , p2 ), h*). This means that p1 /(x 1 1d ).p2 /(x 2 1d ). By lemma 5
But f satisfies the upper bound conditions (4), so
F
p2
x 2 1 1 # ]]] (tx 1 1 x 2 1 1 1 d s)
tp1 1 p2
G.
d
Using lemma 6 with s5t11,
p2
p2
x 2 1 1 , ]]] (t(x 1 1 d ) 1 x 2 1 1 1 d ) 1 1 2 d, or t , ]]]]]]],
tp1 1 p2
p1 (x 2 1 d ) 2 p2 (x 1 1 d )
a contradiction. j
Theorem 6. The unique consistent and anonymous method f that satisfies
F
pi
x [ f ( p, h) implies x i $ (h 2 (1 2 d )s) ]
pS
( for h$(12d )s) is the parametric method f
G
for all i [ S,
(5)
d
d
Proof. The argument parallels that for theorem 5. Let f be an anonymous, consistent
method that satisfies the lower bound conditions (5), and p¯ i 5pi /pS .
f d is clearly anonymous and consistent. Suppose it did not satisfy (5). Then for some
i [S, x i #p¯ i (h2s1d s)2d, implying 1 /(h2s1d s)#p¯ i /(x i 1d ). Thus
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M. Balinski, V. Ramırez
121
p̄ j
p̄i
1
]]] # ]]
# ]]] all j [ S, or x j 2 1 1 d # p¯ j (h 2 s 1 d s),
h 2 s 1 d s xi 1 d xj 2 1 1 d
with at least one inequality strict (when j5i), so summing over j [S one deduces
h2s1d s,h2s1d s, a contradiction. This proves that f d , f.
f must be balanced. Else, deduce (x, x2 b )[ f ( p, p), 2x2 b ) for an integer b $2.
But then from (5), x2 b . ]12 (2x2 b 2212d )2d, implying b ,2, a contradiction.
The argument that shows it suffices to prove that f , f d over 2-dimensional problems
is the same. So suppose that for some 2-dimensional problem the statement is false, and
let h* be the smallest h where a difference is realized: f (( p1 , p2 ), h), f d (( p1 , p2 ), h)
for h,h*, but not for h5h*. Since f is increasing (x 1 , x 2 )[ f (( p1 , p2 ), h*21)>
f d (( p1 , p2 ), h*21) and (x 1 , x 2 11)[ f (( p1 , p2 ), h*), but (x 1 , x 2 11)[
⁄ f d (( p1 , p2 ),
h*). This means that p1 /(x 1 1d ).p2 /(x 2 1d ). But
and since f satisfies the lower bound conditions (5),
F
p1
x 1 $ ]]] (x 1 1 tx 2 2 1 1 d t 1 d )
p1 1 tp2
G
d
p1
or t , ]]]]]]],
p1 (x 2 1 d ) 2 p2 (x 1 1 d )
a contradiction. j
These theorems generalize the following well known characterizations of the methods
of Jefferson and of Adams (Balinski and Young, 1982). The unique anonymous,
consistent method that satisfies: x[ f ( p, h) implies (i) x i $[hpi /pS ] 2 for all i [S is the
method of Jefferson, f 1 ; and (ii) x i #[hpi /pS ] 1 for all i [S is the method of Adams, f 0 .
In these particular cases the result is independent of the number of parties s.
For more of an idea as to the use of these results, consider the example given at the
beginning of Section 6: h532 seats are to be distributed among s55 political parties on
the basis of their respective percentages of the votes, pi /pS . With what percentage of the
vote is a party guaranteed to obtain at least one seat, no matter how the remaining votes
are distributed among the other parties, when the law of the land is the parametric
method f d ? Theorem 6 says when [(3225(12d ))p] d $1, that is, when the percentage
p.d /(3225(12d )). Thus, Adams (f 0 ) assures 1 seat for any positive percentage of the
vote; Condorcet (f 2 / 5 ) for p.2 / 145 or at least 1.38%; Webster or Sainte-Lague¨ (f 1 / 2 )
for p.1 / 59 or at least 1.69%; and Jefferson or d’Hondt (f 1 ) for p.1 / 32 or at least
3.125%.
Similarly: with what percentage of the vote is a party guaranteed to obtain no seats, no
matter how the remaining votes are distributed among the other parties, when the law of
the land is again the parametric method f d ? Theorem 5 answers when [(3215s)p] d #0,
that is, when p,d /(3215d ). Thus Adams denies representation only when a party does
not run at all; Condorcet when p,1 / 85 or a party has less than 1.18%; Webster when
122
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M. Balinski, V. Ramırez
p,1 / 69 or a party has less than 1.45%; and Jefferson when p,1 / 37 or a party has less
than 2.70%. (All percentages are reported to the nearest 0.01%.)
When a party has a percentage of the vote that falls in the gap between 1.45% and
1.69% of the vote and h532 seats are to be distributed among s55 parties by the
method of Webster the result is uncertain: it may receive either 0 or 1 seat (and similarly
for other parametric methods and other gaps).
The interest of these characterizations is that they are sparse and simple: only
anonymity and consistency together with the bounds are required, and the proofs are
relatively straightforward (and much simpler than those required to characterize the more
general class of divisor methods).
References
¨
Balinski, M.L., 1996. How should data be rounded? In: Ruschendorf,
L., Schweizer, B., Taylor, M.D. (Eds.),
Distributions with Fixed Marginals and Related Topics. Lecture Notes-Monograph Series, vol. 28. Institute
of Mathematical Statistics, Hayward, CA, pp. 33–44.
Balinski, M., 1993. The problem with apportionment. Journal of the Operations Research Society of Japan 36,
134–148.
Balinski, M.L., Rachev, S.T., 1997. Rounding proportions: methods of rounding. Mathematical Scientist 22,
1–26.
Balinski, M.L., Rachev, S.T., 1993. Rounding proportions: rules of rounding. Numerical Functional Analysis
and Optimization 14, 475–501.
Balinski, M., Shahidi, N., 1997. A simple approach to the product rate variation problem via axiomatics. To
appear in Operations Research Letters.
Balinski, M.L., Young, H.P., 1982. Fair Representation: Meeting the Ideal of One Man, One Vote. Yale
University Press, New Haven, CT.
Balinski, M.L., Young, H.P., 1977. On Huntington methods of apportionment. SIAM Journal on Applied
Mathematics (Part C) 33, 607–618.
Bautista, J., Companys, R., Corominas, A., 1996. A note on the relation between the product rate variation
(PRV) problem and the apportionment problem. Journal of the Operational Research Society 47, 1410–
1414.
Kubiak, W., 1993. Minimizing variation of production rates in just-in-time systems: a survey. European Journal
of Operational Research 66, 259–271.
Lijphart, A., 1994. Electoral Systems and Party Systems. Oxford University Press, Oxford.
Miltenburg, J., 1989. Level schedules for mixed-model assembly lines in just-in-time production systems.
Management Science 35, 192–207.
´
` de l’Economie
Ministere
et des Finances, mai 1997. Le passage a` l’euro: les arrondis.
Oyama, T., 1991. On a parametric divisor method for the apportionment problem. Journal of the Operations
Research Society of Japan 34, 187–221.
Rae, D.W., 1971. The Political Consequences of Electoral Laws, rev. ed. Yale University Press, New Haven.
Steiner, G., Yeomans, S., 1993. Level schedules for mixed-model, just-in-time processes. Management Science
39, 728–735.