LAWRENCE R. ERNST
Apportionment Methods for the House of Representatives

In addition to challenging the constitutionality of EP,
Montana had challenged the constitutionality of the
automatic apportionment law, claiming it deprived their
congressional delegation of their right to vote on an
apportionment. While not formally ruling on this claim,
the district court majority stated that it had merit. Justice
Stevens disagreed, stating that this claim had no merit.
He found that an automatic use of an otherwise con-
stitutional apportionment method is a sensible proce-
dure that removed apportionment from political con-
troversy. He found nothing in the Constitution that
prevented the adoption of an automatic procedure. Jus-
tice Stevens concluded his answer to this 200-year-old
constitutional question, stating:*

The decision to adopt the method of equal proportions was
made by Congress after decades of experience, experimentation,
and debate about the substance of the constitutional require-
ment. Independent scholars supported both the basic decision
to adopt a regular procedure to be followed after each census,
and the particular decision to use the method of equal pro-
portions. For a half century the results of that method have
been accepted by the States and the Nation. That history sup-
ports our conclusion that Congress had ample power to enact
the statutory procedure in 1941 and to apply the method of
equal proportions after the 1990 census.

’ Former address: Statistical Research Division, Bureau of the Census,
Washington, D.C. 20233. The views expressed in this paper are at-
tributable to the author and do not necessarily reflect those of the
Bureau of Labor Statistics or the Census Bureau. The author thanks
Bev Causey who did the programming that produced most of the
new empirical results in §4

Appendix

PROOF OF THEOREM 3.1. Let dj, i = 1, _ N, denote the HM
apportionment and dj, i= 1,...,N, any other apportionment sat-
isfying quota. Let D = {i: d, # d}}, and m denote the number of
elements in D. If i,j €D,i#j, then

min{d;, d)} <d-<_max{d;, d,}
since both apportionments satisfy quota. Consequently,
\d,-d\ + |d)—d| = \d, — 4)
Therefore, for i, | D,i#j,
\d;- dl + |dj— dl = |dj- djl < Idd} + |dj—-d),

from which it follows that

= ld-a

iep

=————~ © (id dl + ld dl) s & dia).
m(m—~ 1) ep ‘ep
isi

MANAGEMENT SCIENCE/Vol. 40, No. 10, October 1994

PROOF OF THEOREM 4.1, The ordered pair (p1, p2) is uniformly
distributed in a square with vertices (by\, bp), ((b, + 1)A, bad),
(br, (b2 + 1)A), ((br + 1)A, (ba + 1)A) and area d?, The region within
the square in which state 2 is favored over state 1, that is, for which
42/2 > 4/1, is the union of the following four regions:

(1) A triangle with vertices

(byA, bad), (by (bz + 0.5)X/b:, bed),
(bi(bz + 0.5)X/b2, (bz + 0.5)A),

and area 0.125b,\?/ bs, for which a = by, a2 = bp
(2) A rectangle with vertices

(bi(b2 + 0.5)X/b2, bod), ((bi + 0.5)d, bad),
(by (bz + 0.5)A/bz, (bz + 0.5)A), ((b; + 0.5)A, (bz + 0.5)A),

and area 0.25(1 — b; /bz)A?, for which a
(3) A square with vertices

by, a = by

(BA, (bz + 0.5)A), ((br + 0.5)A, (bz + 0.5)A),
(iA, (bz + 1)A), ((b; + 0.5)A, (b2 + 1)A),

and area 0.252, for which a;
(4) A triangle with vertices

by and a, = bz + 1.

((br + 1)(bz + 0.5)A/(b2 + 1), (bz + 0.5)A),
((b; + 1)A, (bz + 0.5)A), ((by + 1)A, (b2 + 1)A)

and area 0.125(b; + 1)A?/(bz + 1), for which a; = by + 1, a = by
+ 1. The total area of these four regions is

(0.5 + 0.125(b2 — by) /[b2(b2 + 1)])A?
and the probability that state 2 is favored is, therefore,
0.5 + 125(bz ~ by)/[ba(b2 + 1)] > 0.5

PROOF OF THEOREM 4.2. We drop all subscripts to obtain results
that apply to both states. Since for EP, d = p/b if p < Vo(b + 1)
and d = p/(b + 1) if p> Vo(b + 1)), it follows that

_1/ pe p pe p 7
a= 5(f, pet combi)"

for all b, where the 1/) term in (A.1) arises from the fact that
{b, (b + 1)A] is an interval of length d.
Similarly, for MF,

1 (eee ce »
E(d)=* 2a
(4) xf bat [a ®)

= (1+ 5055)

which is a decreasing function of b

(Al)

PRoor oF THEOREM 4.3, (a) The resullts for the ideal q, were
established in Balinski and Young (1982, pp. 132-133). The results
for s follow from these results and the fact that since |s; — s|
= |a; — 4i|/pi, bringing 5, closer to s and a; closer to q; are equivalent,

1225


LAWRENCE R. ERNST
Apportionment Methods for the House of Representatives

of each state to the nearest integer and excluded the 6
smallest states, which were those that originally had
modified exact quotas less than 1.5. New simulations
based on random variations from the rounded values
for the remaining 44 states then yielded results more in
accordance with their bias claims. It was also noted that
Balinski and Young’s exclusion of states with modified
exact quotas less than 1.5 was not consistent with their
usual exclusion criterion of 0.5.

The plaintiffs rebutted the defendants’ claim that the
exact quota cutoff for excluding states from bias com-
putations should be 1, by citing a hypothetical example
of a state with exact quota of 0.9 in one census that
grew in the next census to have an exact quota of 1.1
and received one seat each time. Such a state would be
fairly treated on average for the two censuses, they ar-
gued, but if it was excluded from the bias calculations
for the first census, then one would conclude incorrectly
that it had been unfairly treated because in the second
census, its allocation was less than its exact quota.

The defendants responded to this rebuttal by noting
that the plaintiffs’ position, unlike the defendants’, was
based on averaging of exact quotas for a state over more
than one census and assumed growth of a state to the
point where its exact quota was over 1. We noted that
the Constitution clearly requires that an apportionment
be based solely on the current census numbers, not on
averaging over past censuses or hypothesizing about
future censuses. We also noted that two of the three
states with exact quotas below 1 for the 1990 census,
Alaska and Wyoming, have never had exact quotas
above 1.

The defendants responded to the plaintiffs’ “near the
quota” claims in several ways. We first noted that while
it was true that MF was the only divisor method for
which a transfer of a seat between states can never bring
both states’ allocations closer to their exact quotas, as
measured by either |a; — q;| or |a; — q;| /q; for state i,
it is actually EP that is the only divisor method with
this property if relative difference, that is |a; — q;|/min
{4;, qi}, is used as the measure of discrepancy.

We also noted that the Balinski and Young’s concept
of “near the quota” can be generalized to “near the
ideal.” An apportionment method is said to be “near
the ideal” if a transfer of a seat between two states can
never bring both states’ allocations closer to the ideal.

MANAGEMENT SCIENCE/Vol. 40, No. 10, October 1994

We considered the three ideals q;, d, ands, and proved
the following results.

THEOREM 4.3. For the ideals q;, d, and s:

(a) MF is the only divisor method that is ‘‘near the
ideal” for q; and s as measured by |a; — q;| and |s; — s|,
or, equivalently, by |a; — qi|/qi and |s; —s|/s.

(b) HM is the only divisor method that is ‘near the
ideal” for d as measured by |d; — d|, or, equivalently, by
\d; —d|/d.

(c) EP is the divisor method that is ‘‘near the ideal’
for all three ideals, q;, d, and s as measured by relative
difference, that is |a; — qi|/min{a;, qi}, |d; — d|/
min{d;, d} and |s; — s|/min{s;,s}.

It was noted that the plaintiffs’ examples from the
1920 and 1970 censuses of ‘near the quota” violations
would not hold if relative difference is used as the mea-
sure of discrepancy. In addition, the 1870 census was
used to illustrate that MF was subject to “near the ideal”
violations. For that census, MF would have allocated
Illinois 20 seats and Florida 1. A transfer of a seat from
Illinois to Florida, as would have occurred under EP,
would have brought both states closer to the ideals q;,
d, and s as measured by relative difference and also
brought both states closer to d as measured by absolute
difference.

The defendants also remarked that the term “near
the quota’” was somewhat of a misnomer since these
words can be misinterpreted to imply more than the
actual property. It was noted that it is actually GR that
minimizes the overall measure of discrepancy of an ap-
portionment from exact quotas, (2.2), and that EP can
sometimes produce a smaller value for (2.2) than MF.

The plaintiffs responded to the defendants’ arguments
concerning “near the quota’ by claiming that relative
difference is not appropriate when comparing deviations
from an ideal, with examples to illustrate this point. A
key example concerned a city with an average annual
rainfall of 20 inches, which had a rainfall of 15 inches
one year. The plaintiffs noted that one might say that
the rainfall was 5 inches below normal (analogous to
the measure | 4; — q;|) or 25% below normal (analogous
to the measure |; — q;| /q;). However, they noted that
relative difference between 20 and 15 is 334%, which
says that normal rainfall was 33} % above the observed
value. They stated that if one is interested in deviations

1219


LAWRENCE R. ERNST
Apportionment Methods for the House of Representatives

Table 3. Allocations for Six Methods for Example with n = 100
a; for Method

State by 4 GR sD HM EP ME 6D
1 91,490 91.490 92 88 89 90 93 94
2 1,660 1.660 2 2 2 2 2 1
3 1,460 1.460 2 2 2 2 1 1
4 1,450 1.450 1 2 2 2 1 1
5 1,440 1.440 1 2 2 2 1 1
6 1,400 1.400 1 2 2 1 1 1
7 1,100 1.100 a 2 ml _ 4 1

Totals 400,000 100.000 100 100 100 100 100 100

Min 1,040 1,023 4,011 979 964

Max 1,051 1,033 1.018 989 973

and Arizona became states in 1912, fixed the House size
at 435. About the time of the 1920 census, Professor
Edward Huntington of Harvard refined and became the
principal champion of EP, which had first been devel-
oped by Joseph Hill of the Census Bureau in 1911.
Huntington (1921) is one of the earliest of his many
papers on this subject. The case for EP rested primarily
on the pairwise optimality tests. An apportionment is
said to be pairwise optimal with respect to a particular
measure of inequity if no transfer of representatives be-
tween any pair of states can decrease the amount of
inequity between these states. HM is pairwise optimal
with respect to absolute difference in average district
sizes, that is with respect to the measure, | p; /a; — p;/4)|
between states i and j. MF is pairwise optimal with
respect to the absolute difference in per capita shares
of a representative, that is |a;/p; — a;/p;|. However,
EP is pairwise optimal with respect to relative differences
in both district sizes and shares of a representative,
which became the key argument for EP. SD and GD
are pairwise optimal with respect to two other tests,
absolute representation surplus and absolute represen-
tation deficiency, respectively. (If a; /a; > p;/p;, then
absolute representation surplus for the pair i, j is a;
— (pi /p;) aj, that is the amount by which the allocation
for state i exceeds the number of seats it would have if
its allocation was directly proportional to the actual al-
location for state j. Similarly, absolute representation
deficiency is (p;/p;)a; — 4;.) Furthermore, every transfer
of seats between a pair of states from an apportionment

1210

obtained from the optimal method will actually strictly
increase, as opposed to merely not decrease, the cor-
responding measure of inequity for the pair of states
when the optimal method produces a unique appor-
tionment, that is when there are no ties for the last seat.

The opposition to the views of Huntington was led
by Professor Walter Wilcox of Cornell, who supported
MF. He was of the opinion that EP was biased in favor
of small states, while MF was mathematically neutral
between small and large states. Huntington disagreed,
contending that it is actually EP that is mathematically
neutral in this respect. Huntington’s argument was
based on the fact that among SD, HM, EP, MF, and
GD, all transfers of seats that result from the replacing
of one method with a method further to the right on
this list are to states that are larger than the states losing
seats. Thus, in a relative sense, EP favors smaller states
less than SD and HM, and larger states less than GD
and MF. This result alone does not establish anything
about bias beyond how the methods compare in relation
to each other.

Congress failed to reapportion the House at all after
the 1920 census, but in an attempt to resolve the tech-
nical dispute, the Speaker of the House requested that
the National Academy of Sciences (NAS) review the
mathematical aspects of the problem of reapportion-
ment. A NAS committee issued a report in 1929 (Bliss
et al.). The report considered the five divisor methods
discussed in this paper and focused on the pairwise
comparison tests described above. The committee

MANAGEMENT SCIENCE/Vol. 40, No. 10, October 1994


LAWRENCE R. ERNST
Apportionment Methods for the House of Representatives

(b) Clearly it suffices to establish this result for the measure
dj — dl.

Let b be a positive integer, 5(b) = 2b(b + 1)/(2b + 1), the rounding
function for HM, and p the population of a state, To prove that an
HM apportionment is always “near the ideal’ d for the measure
| d, — dl, first establish that

ifp/5(b)=d then |p/b—d|=|p/(b+1)—d|.  (A.2)
To do this, observe that
Py ip 2p
bt ost ab)
and hence
BL ke
pet (A3)

In addition, clearly p /b— d > p/(b+ 1)—d, and (A.2) then follows.

Similarly, it can be established that if both inequality signs in (A.2)
are replaced by either ’<” “>” or '<" the statement remains true.
Furthermore, (A.2) remains true if b is replaced by 0, with the con-
vention that p/0 = o, a convention that is used throughout this
proof.

Then consider an HM apportionment for which states i and j have
populations p;, p; and allocations a,, aj. Then

pi /8(a; — 1) = pj/8(a))

by the min-max inequality (Balinski and Young 1982, p. 100). Con-
sequently, if a seat is transferred from state i to state j then either p; /
5(a; — 1) = d, in which case the transfer does not bring the allocation
for state i closer to the ideal by (A.2) with b = a; — 1, or p/8(a)
sd, in which case the allocation for state j is not brought closer to
the ideal by A.2, with the inequality signs reversed and b = a).

To prove that any other divisor method, with rounding function
5*(b), is not always “near the ideal’ d for the measure | d, — d|, first
note that for some positive integers b, # b2,

5 (bo) | (bs)
5%(b2) ~ 5%) ©

(For otherwise, §*(b) = 6(b) for all b > 0. Then 6%(0) > 0, in which
case * can produce apportionments in which some states have 0
representatives and other states have more than 1 representative,
which would be a “near the ideal’ violation.)

Then consider a three-state problem, for which

Pr = 8(br) + & Pe = 5(b2)— 6 ps = by + bz + 2 ~ 8(b1) — (ba),

where ¢ > 0 satisfies the relations

Po PA

, AA

Hb 7 Fb) (a4)

mh

(be —1) > Hb)’ (a5)
Pa

wens! (A6)

1226

Note that since 6(1) — 1 = } and | < 6(b) — b < § for b > 1, it follows
that

l<p<i (A7)

Furthermore, by (A.5),

P2

imo?! (A8)

Let a;, i = 1, 2, 3, denote the apportionment corresponding to the
rounding function 4 for this set of populations and a house size of b;
+ b, + 2. Since by the definitions of p, and p2, and (A.7),

PL Po Ps
aby 717 molt ’ ti}

state 1 has a higher priority for b, + 1 seats than either state 2 has
for b; + 1 seats or state 3 has for 2 seats. Consequently a; = b, + 1.
Similarly, since by (A.5), (A.7) and (A.8),

Pa Pi Ps
—— > max ,
6(b2 — 1) (5 i]
it follows that a = bp. Finally, since a3 = 1, the HM apportionment
must be a; = by + 1, a = bz, a3 = 1. Furthermore, since

pi/8(a;—1)>1=d>pi/(a), 7=1,2,3,

it follows from the variations of (A.2) that | d; — d| is uniquely min-
imized for each i by this apportionment.

Now for the 6*(b) apportionment, by (A.4) state 2 has a higher
priority for b; + 1 seats than state 1 has for b, + 1 seats. Consequently,
the two apportionments are not identical, and hence the 5*(b) ap-
portionment must switch seats between at least two states from the
HM apportionment, increasing | d, — d| for these states.

(c) Since s; = a)/pi, 8 = qi/Pi, dy = pi/ai, d = pi/qu, all three
measures are identical. Therefore, it suffices to prove these results for
any one of the ideals, say d.

To prove that an EP apportionment is always “near the ideal’’ d
for the measure | d; ~ d|/min{ d,, d}, simply establish that, with 6(b)
= Vo(b+1),

Ip/b=d\_ __\p/(b +1) ~ di

ifp/d(b)=d then 7b, d) = min{d/(b + 1), dj

(A9)

and then proceed as in (b). To prove (A.9), note that

\p/b- dl
min{p/b, d}

ip/(b +1) = dl
min{p/(b +1), d}

= max|p/(bd), bd/p} — 12 p/(bd) — 1,

= max{p/[(b + 1)d],(b + 1)d/p}

Now, p/(ba) = (b + 1) /p can be obtained by algebraic manipulation
of p/5(b) = d, while clearly p /(bd) > p/{(b + 1)d]. (A.9) can then
be obtained by combining all the above relations

To prove the converse, that is that no other divisor method is always
“near the ideal’ d for the measure | d; — d|/min{d,, d}, proceed as
in (b)

MANAGEMENT SCIENCE/Vol. 40, No. 10, October 1994


LAWRENCE R. ERNST
Apportionment Methods for the House of Representatives

davits of Montana’s experts that either of them under-
stood that (5.1) is the proper measure of variance in
this context, since they had only associated the word
“variance” with (3.1). HM does indeed minimize (5.1)
among the six apportionment methods considered in
this paper, not only for 1990, but for all 21 censuses.
However, for the example in Table 6, the value of (5.1)
is 340 for HM and 320 for EP.

Montana responded to the Government's observation
on the fractional remainders of Montana and Washing-
ton with their central argument that quota, like share
of a representative and all factors other than absolute
difference in district sizes, is an inappropriate criterion
for measuring adherence to the ‘one person, one vote”
principle.

Massachusetts participated in the Supreme Court case
through a friend-of-the-court brief (Harshbarger et al.
1992), after an unsuccessful request that the Supreme
Court delay hearing the Montana case until a district
court decision had been issued in the Massachusetts
case, and then hear appeals of these two cases in tan-
dem. Massachusetts’ friend-of-the-court brief was filed
shortly before the district court ruling in their own case.
Anew affidavit from their expert, Peyton Young (1992),
accompanied it. They raised the same three issues as
had been raised in the district court. Their argument on
the bias and the ‘near the quota” issues were similar
to those in the district court. However, perhaps in rec-
ognition of the focus of Montana’s arguments, Massa-
chusetts presented new arguments to support their con-
tention that absolute deviations of shares of a repre-
sentative is the only proper measure of equity in the
interstate context.

Massachusetts presented an interesting verbal argu-
ment for the superiority of share of a representative
over district size as a test. They observed that: “To bring
all district sizes as near equity as possible is to treat all
representatives as equally as possible. The relevant
principle is to treat all citizens as equally as possible.”

Massachusetts also provided an example that they
stated showed a problem with average district size as a
measure of equity. They considered a state with a pop-
ulation of 750,000 in each of two censuses, with d
= 500,000 in both censuses. The state’s allocations were
1 seat in census 1, and 2 seats in census 2. They observed
that averaged over the two censuses, the state’s allo-

MANAGEMENT SCIENCE/Vol. 40, No. 10, October 1994

cations and its exact quotas are both 1.5. Also, the av-
erage share of a representative for the state averaged
over the two censuses and s = 1/500,000 are both the
same. However, the state’s average district size averaged
over the two censuses is greater than d = 500,000.

This example implied that exact quota and average
share of a representative are consistent measures and
that average district size is inconsistent with the other
two. The implication that average share of a represen-
tative is consistent with exact quota in this respect is
false. To see this, simply consider the same example
except suppose that the state’s population is 600,000 in
census 1 and 900,000 in census 2 (or any two numbers
P1, P2, respectively, with arithmetic mean 750,000, for
which 500,000 < p, < 750,000). Then the state’s allo-
cations and exact quotas averaged over the two censuses
remain the same, but the average share of a represen-
tative averaged over these two censuses is less than
1/500,000.

Massachusetts approached the absolute difference
versus relative difference issue somewhat differently
than previously. Citing case law, they claimed equity
should be measured by deviations from the ideal, not
pairwise comparisons. They repeated their claim that
relative difference is inappropriate in measuring devia-
tions from an ideal. They observed that MF minimizes
(2.4) with p = 2 and asserted that this measure is su-
perior to (2.3) with p = 2.

Massachusetts presented one new idea relating to the
“near the ideal” principle. EP for 1990 does not violate
this principle in any of the senses of Theorem 4.3.
However, for states i, j, if sj; = (a; + 4))/(p; + pj), that
is, the average share of a representative for the two
states combined, then the following result holds:

THEOREM 5.1. Fora unique MF apportionment, every
transfer of seats between states i, j increases both
|s; — | and |s; — sj| (and, equivalently, |s; — si;| /si
and |s; — si| / sj).

This is equivalent to saying that any apportionment not
agreeing with the MF apportionment for states i, j will
violate the ‘near the ideal” principle for the ideal 5,
with respect to the measures given in the theorem.
Massachusetts did not actually state this theorem, but
instead illustrated this result by demonstrating that for
the 1990 census, MF brings both Massachusetts and

1223


LAWRENCE R. ERNST
Apportionment Methods for the House of Representatives

Concerning share of a representative, they observed
(footnote 7):
The Constitution decreed that one house should be chosen on
the basis of population (persons per representative) and Con-
gress cannot ignore that mandate by choosing a method which
considers each person’s share of a representative.
This author does not fully comprehend either of these
quotes.

The majority never made clear what specific tests in-
volving absolute differences in district sizes should be
used. They quoted various tests from previous opinions,
including: “variances between the actual district and
the district size,” “range,” “average deviation from the
ideal district size,” and “‘maximum deviations above
and below the mean district size,” but never stated
which, if any, of these tests they were adopting.

However, the majority did state: “Courts traditionally
look to variances from the ideal district size to determine
whether a district is under or over represented’’ and,
additionally, ‘absolute difference from the ideal is the
proper criterion. . . ,” statements which would appear
to rule out range and pairwise difference tests. The ma-
jority did not directly address the defense point that it
is actually EP that minimizes variance among all dis-
tricts, although they may have been indirectly referring
to it when they stated: “The Hill method can never
meet the criteria proposed by Plaintiffs, because its ex-
press objective is to minimize the relative difference be-
tween the number of persons per representative and
the relative difference between each person’s share of
a representative.” It is not clear if the majority meant
by that statement that EP cannot possibly minimize
(2.3) with p = 2.

In any case, whatever specific tests the majority had
in mind, they considered that HM comes closer than
EP to satisfying the “one person, one vote’ principle,
and concluded that the use of EP is unconstitutional.
They did, however, reject SD from consideration based
on the quota violations.

Circuit Court Judge O’Scannlain of Oregon, while
agreeing with the majority on the justiciability of the
case, dissented on the merits, noting several points
(O’Scannlain 1991). He first found, as did the majority,
that SD is inconsistent with the constitutional require-
ment of allocating House seats by population, since it
results in quota violations for four states in 1990. He
cited the fact, from the defense declaration, that the

MANAGEMENT SCIENCE/Vol. 40, No. 10, October 1994

relative difference between Montana’s and Washing-
ton’s average district size is larger under HM than EP.
Judge O’Scanniain also stated that range of district sizes
is not the best test of disparity, employing the most
common argument against a range test, that it only con-
siders the largest and smallest of the 435 congressional
districts.

Judge O’Scannlain’s opinion focused on measures of
total error, which he referred to generically as variance.
He cited Gilford’s (1981) testimony in which it was
noted that different apportionment methods optimize
(2.2), (2.3), and (2.4) with p = 2 and p = 1, as evidence
that this is not a straightforward issue. Citing the defense
declaration, he observed that (2.3) with p = 2, not (3.1),
measures variance among all districts, and that EP, not
HM, minimizes the appropriate variance. Judge
O’Scannlain also quoted the majority statement that
“absolute difference from the ideal district is the proper
criterion. . . .” He interpreted this as requiring the test
(2.3) with p = 1. He calculated that EP for 1990 pro-
duces a lower value for this measure than HM. (As
noted in §2, it is actually GR that minimizes (2.3) with
p = 1, but Judge O’Scannlain prefaced his discussion
by stating that three methods were before the court,
EP, SD and HM.) The Judge concluded: “In sum, neither
of the formulae proposed by the State lead to less pop-
ulation variance than the Hill equal proportions formula
in use for the last fifty years. The State, in my view, has
failed to demonstrate that a better formula exists than
the one chosen by Congress.”

A number of questions naturally arise from the issues
raised in this case that were not answered in the court
documents. For example, the plaintiffs noted that for
1990, HM has a smaller value for (3.1) than EP, and
both HM and SD have a smaller range of district sizes
than EP. The example in Table 6 with N = 3,n =5
illustrates that none of these relationships always hold.

For this example, the EP allocation results in values
of 121,300 for (3.1) and 530 for the range, while the
HM and SD allocation results in values of 122,962.5 for
(3.1) and 650 for the range. The EP allocation is actually
optimal among all apportionments for these two mea-
sures for this example.

However, if the comparisons are limited to the 21
actual censuses, then among the nine censuses for which
HM and EP did not produce identical apportionments,
HM always yielded a lower value of (3.1) than EP, and

1215


LAWRENCE R. ERNST
Apportionment Methods for the House of Representatives

which in effect creates a constitutionally mandated bias
in favor of the small states. The bias ratio was obtained
by first computing for each census the number of pairs
of nonexcluded states i, j with p; < pj, where state i,
the smaller state, is favored in the sense that a; /p; > a;/
p;. The total of the number of pairs for which the smaller
state was favored, summed over the 19 censuses
through 1970, was then divided by the total number of
pairs of nonexcluded states in these 19 censuses to ob-
tain the bias ratio. Balinski and Young’s results for the
five historical divisor methods are presented in Table
4. The ideal ratio is, of course, 50%.

They computed percentage bias for each census by
first dividing the nonexcluded states into approximately
equal classes of large (L), middle, and small states (S),
with the middle class receiving the extra states when
the number of nonexcluded states was not divisible by
three. The percentage bias for each census is then

(20 /20)/(20/2n)-1

expressed as a percentage. Balinski and Young's (1982)
results, averaged over the 19 censuses through 1970,
are presented in Table 5. A positive percentage indicates
that small states are favored and a negative value in-
dicates that large states are favored.

Balinski and Young (1982) also presented a secondary
reason for their support of MF, the “near the quota”
property that they developed. They defined an appor-
tionment to be “near the quota” if no transfer of a seat
from one state to another can bring both states nearer
to their exact quotas. They proved that MF is the unique
divisor method that is “near the quota” for all appor-
tionments and noted that this result is trae whether
distance is measured in absolute or relative terms. (GR
apportionments also always satisfy this property.) By
absolute terms, they meant, of course, the measure
la; — q;|. By relative terms, they meant |a; — q;|/qi,

Table 4 Bias Ratio of Censuses Through 1970

SD HM EP MF GD
Bias ratio 77.2% 56.6% 54.6% 51.5% 25.0%
1212

Table 5 Percentage Bias Averaged over Censuses Through 1970

SD HM EP MF GD

Average bias 18.3% 5.2% 3.4% 0.3% 15.7%

not the relative difference between a; and q;, which is
la; — q;|/min {a;, q;}. Balinski and Young's result is
equivalent to saying that MF is the only divisor method
which can never produce an apportionment which
rounds up q; for a state i with q; — qi] < 0.5, while
rounding down q; fora state j with 4; — | q)]> 0.5. (Again,
as these authors note, because of the minimum require-
ment of one representative per state, this result is ac-
tually only true for an apportionment of the House of
Representatives if q; is replaced by the modified exact
quota, qj.)

A final set of properties of apportionment methods
are measures of total error of an apportionment. Let d;
= pi/aj,d =(= pj)/n, 8; = 1/d;, ands = 1/d. Three
classes of error measures are, for p = 1,

N
> lai - ail’, (2.2)
mt

N

Daild—d\’, and (2.3)

i
N
> pilsi — sl’. (2.4)
=

(2.2), (2.3) and (2.4) are, respectively, the sum of
the pth power of each state’s absolute deviation from
its exact quota, each district’s absolute deviation from
the national average district size, and each person's ab-
solute deviation from the national average share of a
representative. The assumption that the districts within
each state are of the same size is used in (2.3) and (2.4).

GR minimizes (2.2) for all p = 1 (Birkhoff 1976),
while for p = 2, EP minimizes (2.3) (Huntington 1928)
and MF minimizes (2.4) (Owen 1921). As observed by
Gilford (1981), for p = 1, (2.3) and (2.4) are minimized
by GR since they are constant multiples of (2.2) with
p=l.

Interestingly, the various measures of total error of
an apportionment have generally not been a major focal

MANAGEMENT SCIENCE/Vol. 40, No. 10, October 1994


LAWRENCE R. ERNST
Apportionment Methods for the House of Representatives

adopted Huntington’s reasoning that EP is preferred on
the basis of the pairwise tests for which it is optimal
and also concluded that EP ‘‘occupies mathematically
a neutral position with respect to emphasis on larger
and smaller states.”

The 1930 allocations for EP and MF were identical,
so Congress took no further action after that census.
Under the applicable law, the House was automatically
apportioned under the method last used, MF.

In 1940, however, EP and MF differed, with Arkansas
allocated 7 seats by EP and 6 by MF, while Michigan
was allocated 17 by EP and 18 by MF. In 1941, ona
mainly party line vote, legislation was enacted appor-
tioning the House by EP. This method has been used
ever since and, under the 1941 law, its continued use
is automatic until superseding legislation is enacted.

In 1948, a new NAS committee revisited the appor-
tionment issue and also endorsed EP (Morse et al.).
Their report included the new argument that among
the four pairwise comparison tests previously mentioned
for which EP, HM, or MF are optimal, EP is always
superior to each of the other four divisor methods for
at least three of them. For example, it can be shown
that EP is superior to MF with respect to absolute dif-
ference in district sizes, in the sense that no transfer of
seats resulting from the use of MF instead of EP can
ever lower this measure of inequity for any pair of states.
In this sense EP is, of course, also superior to MF and
all other methods, with respect to relative differences
in district sizes and shares of a representative, while
MF is superior to EP with respect to absolute difference
in shares of a representative. Analogously, EP is superior
to HM for all of these tests except absolute difference
in district sizes. The committee found the total score in
favor of EP using this approach “decisive.”

Much of the interest in the apportionment issue since
the mid 1970s is a result of the work of Michel Balinski
and H. Peyton Young. In their early writings on ap-
portionment (Balinski and Young 1975), they expressed
the view that an apportionment method should never
violate quota and should not be subject to the Alabama
paradox. None of the six methods considered in this
paper meet both of these conditions. However, Balinski
and Young (1975) developed a modification of GD that
they called the quota method, which does satisfy both
of these conditions. Still (1979), among others, gen-

MANAGEMENT SCIENCE/Vol. 40, No. 10, October 1994

eralized Balinski and Young's result and obtained mod-
ifications of all six methods considered in this paper
which satisfy these two conditions. Unfortunately, all
of these modified methods suffer from an unpleasant
property that none of these methods possesses in un-
modified form. They allow a form of ‘population par-
adox” in which one state can have a population increase,
while all other states and the total house size remain
fixed, and yet the growing state can lose seats. Balinski
and Young eventually abandoned their support of the
quota method and became proponents of MF. Their
main argument for MF was, like Wilcox’s decades ear-
lier, their belief that MF is the only divisor method that
is not biased in favor of either large or small states.
Their work, culminating in the book, Fair Representation
(Balinski and Young 1982), presented a number of new
theoretical and empirical results to support their view.

For example, corresponding to a divisor ) and a div-
isor method based on 6, they considered intervals

[8(b —1)d, 6(b)A], B= 1,2,3+++, (2.1)

(where [a, 8] denotes {x: a < x < 6}), that is, popu-
lations for which b seats are assigned, and established
that MF is pairwise unbiased in the sense that if states
1 and 2 have independent populations p, and pp, re-
spectively, uniformly distributed in intervals

[8(br — 1)A, 5(br)AJ, [5(b2 — 1), 6(b2) A],

respectively, for positive integers b, > b,, then the
probability is 0.5 that state 2 is favored over state 1 in
the sense that b,/p2 > b,/p,. They also established
that MF is the only proportional divisor method with
this property, where proportional divisor methods are
a set of “reasonable” divisor methods, defined in Balin-
ski and Young (1982, p. 97), that include all five his-
torical divisor methods. They then generalized this result
from pairs of states to two groups of smaller and larger
states, obtaining the result that MF is the unique un-
biased proportional divisor method.

Their empirical results include comparisons of the
historical divisor methods for the “bias ratio’ and “‘per-
centage bias,” two measures of apportionment method
bias developed by these authors. For both measures they
excluded states with exact quotas below 0.5 as their
means of compensating for the constitutional require-
ment of at least one representative per state, a provision

1211


LAWRENCE R. ERNST
Apportionment Methods for the House of Representatives

Oklahoma closer to their combined average share of a
representative than EP does with respect to the measures
of Theorem 5.1. The difficulty with this result is that it
appears inconsistent to argue that deviations should be
measured from an ideal, not by pairwise comparisons,
and then use as an ideal s, which is a function of a
pair of states, instead of the traditional ideal s.

Interestingly, the result that would be analogous to
Theorem 5.1 for district size (that is with s;, s; and s;
replaced by dj, d; and dj; = 1/s,;) does not hold for
every HM apportionment. For example, for the 1990
HM apportionment, a transfer of a seat from Montana
to Washington would bring Washington’s district size
closer to the average district size for the two states com-
bined as measured by | d; — djj| or | d; — dij|/ dij.

The district court decision in the Massachusetts case
was released after the filing of Massachusetts’ brief, but
before the Government's reply brief was due. The Gov-
ernment chose not to respond to the Massachusetts
claims in detail, but simply to note that these claims
had been answered in the district court and had been
rejected by that court. The Government, in response to
the claims of both Montana and Massachusetts, did re-
view the expert support that EP has historically had, in
particular the reasoning in the 1948 National Academy
of Science report. The Government also cited the ex-
ample of Arkansas and Michigan for the 1940 census,
for which EP but not MF awarded the seat in dispute
to the state with the higher fractional remainder.

On March 31, 1992, only 27 days after oral argu-
ments, the Supreme Court unanimously upheld the
constitutionality of EP, in an opinion written by Justice
John Paul Stevens. Justice Stevens first considered and
rejected, as had both district courts, the Government's
argument that Congress’ selection of an apportionment
method is a “‘political question” not subject to judicial
review.

Justice Stevens then discussed the constitutional issues
in dispute. He observed that while the same principle
of equity that the Supreme Court requires in intrastate
districting might apply to interstate apportionment, he
did not find that the facts constituted a violation of the
Wesberry standard. He noted that there is no incom-
patibility within a state in minimizing both absolute and
relative differences, and that all districts within a state
can be brought closer to the ideal simultaneously.

1224

However, for 1990, HM, while bringing Montana’s av-
erage district size closer to the ideal district size as mea-
sured by absolute difference, brings Washington’s av-
erage district further away from the ideal district size
with respect to absolute difference, and moves both
states further from this ideal with respect to relative
difference. Justice Stevens also noted that it can be ar-
gued, as in Judge O’Scannlain’s dissent in the Montana
district court case, that a measure of deviation from the
ideal district size should take into account the number
of districts in each state.

Justice Stevens then made the critical observation that
“neither mathematical nor constitutional interpretation
provides a conclusive answer’ to the question of the
best measure of inequality among the four measures
obtained by pairing either absolute or relative difference
with either district size or share of a representative. As
had Judge Woodlock in the Massachusetts district court
case, he concluded: “The polestar of equal represen-
tation does not provide sufficient guidance to allow us
to discern a single constitutionally permissible course.”
These comments amounted to a complete rejection of
Montana’s entire argument and a rejection of one of
Massachusetts’ key issues.

The opinion further observed that the goal of math-
ematical equality, while appropriate in the intrastate
context, is illusory for interstate apportionment, since
each state must have at least one representative and
districts cannot cross state lines. In addition, since the
Constitution expressly authorizes Congress to enact
legislation to carry out its delegated responsibilities, its
choice of a method that apportions representatives “ac-
cording to their respective Numbers” commands far
more deference than a state redistricting decision that
can be required to meet a rigid mathematical standard.

The ‘near the quota” issue raised by Massachusetts
was not mentioned in the opinion. The bias issue was
discussed in a footnote, which first described Balinski
and Young’s (1982) views and then simply noted, citing
the opinion of the Massachusetts district court, that this
contention has been disputed. Later in the opinion, Jus-
tice Stevens returned to this issue, stating that a fair
apportionment required some compromise between the
interests of the smaller and larger states, and indicating
that Congress had been delegated the authority in the
Constitution to reach this compromise.

MANAGEMENT SCIENCE/Vol. 40, No. 10, October 1994


LAWRENCE R. ERNST
Apportionment Methods for the House of Representatives

After first reviewing the apportionment history, in-
cluding the 1929 NAS report, we responded to the gen-
eral argument that absolute differences in district sizes
is the only proper criterion for evaluating an appor-
tionment. We pointed out that it can be argued that a
test involving differences in shares of a representative
is a better test of “the one person, one vote” principle
for interstate apportionment than a test involving dif-
ferences in district sizes, since share of a representative
measures the portion of a vote to which a person is
entitled in the House. It was also observed that intrastate
redistricting and interstate apportionment are concep-
tually very different, since in the former case, the people
in each state are allocated to a fixed number of districts,
while in the latter case, districts are allocated to the
fixed number of people in the various states.

An artificial example was presented to illustrate the
distinction between absolute and relative difference,
which noted that if the national average district size is
600,000 then, as measured by relative differences, a
district of size 1,200,000, twice as large as the ideal, and
a district of size 300,000, twice as small as the ideal, are
equally inequitable. In addition, while the relative dif-
ference between a district of size less than 300,000 and
the ideal district of 600,000 is greater than the relative
difference between the 1,200,000 and 600,000, the op-
posite relationship holds for absolute difference, even
if the smaller district is of size 1.

It was noted that for 1990, as guaranteed by the op-
timality results for the pairwise difference tests, the rel-
ative difference between Washington’s and Montana’s
average district sizes and average shares of a represen-
tative under EP (48.0%) is smaller than under HM
(52.1%). It was also observed that the relative difference
between Montana’s average district size and the national
average district size is 40.4% under EP and 42.5% under
HM, while Washington’s is 5.4% under EP and 6.7%
under HM. EP always gives a higher priority to award-
ing a seat to a state that would be moved closer to the
ideal, as measured by relative difference, than to a state
for which the opposite is the case.

Although the plaintiffs declared the proper measure
of inequity in an apportionment is absolute population
variance among all districts and claimed that HM results
in the smallest such variance, the defendants observed
that it is actually EP that always minimizes this measure,
since it minimizes (2.3) with p = 2. The reason for the

1214

discrepancy in the claims is that the formula used by
the plaintiffs, (3.1), did not take into account the num-
ber of districts in each state. Their formula measures
variability among the mean district sizes of the 50 states,
not the variance of the sizes of the 435 districts. Now,
measuring variability among state average districts sizes
is not necessarily an inappropriate criterion for com-
paring apportionments; it is simply that (2.3) with p
= 2, not (3.1), reflects the criterion actually stated in
the plaintiffs’ briefs.

As for SD, it is indeed true that SD minimizes the
range of district sizes for the 1990 census among the
three methods considered by the plaintiffs and actually
minimizes this measure for 1990 among all possible ap-
portionments. The defense case against SD focused on
its tendency to violate quota. We noted that while, for
1990, California’s exact quota is 52.124 seats, SD only
allocates it 50 seats and also results in quota violations
for Illinois, New York, and Ohio. It was also noted that
if SD had been employed for all 21 censuses, quota
violations would have resulted for every census since
1820, with a total of 47 violations.

There is a requirement in federal law that redistricting
cases be heard before a three-judge panel, instead of a
single judge, with decisions of these panels generally
appealable directly to the Supreme Court. Although the
Government contended that this law does not apply to
interstate apportionment cases, such panels did hear
both the Montana and Massachusetts cases.

By a two-to-one majority, the court in the Montana
case upheld Montana’s position that equal proportions
was unconstitutional (Lovell 1991). Judges Lovell and
Battin, both from Montana, constituted the majority.
They agreed with the plaintiffs’ argument that the “one
person, one vote” principle applies to interstate appor-
tionment and, citing prior intrastate redistricting cases,
that absolute difference in district sizes is the only proper
standard for testing this principle. The judges provided
their rationale for rejecting tests involving representa-
tives per person or using relative differences in two
footnotes. First they dismissed relative difference (foot-
note 3) stating:

By arguing that proportions and percentages are the proper
criteria, rather than absolute numbers, Defendants ignore the
fact that each number represents a person whose voting rights
are potentially impacted by the population disparities.

MANAGEMENT SCIENCE/Vol. 40, No. 10, October 1994


Apportionment Methods for the House of Representatives and the Court
Challenges

Lawrence R. Ernst

Management Science, Vol. 40, No. 10 (Oct., 1994), 1207-1227.

Stable URL:
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Thu Mar 10 15:47:02 2005


LAWRENCE R. ERNST
Apportionment Methods for the House of Representatives

on the work of Balinski and Young (1982). Their major
claim was that EP is unconstitutionally biased on favor
of small states. They also found EP lacking because it,
unlike MF, can yield apportionments which violate the
“near the quota” principle. Finally, the plaintiffs claimed
that the “one person, one vote” principle in interstate
apportionment is best met by the pairwise test for which
MF is optimal, absolute difference in shares of a rep-
resentative.

Not surprisingly, the plaintiffs retained as their expert,
H. Peyton Young, who wrote three affidavits in support
of Massachusetts’ claims (Young 1991), which formed
the heart of their case. The plaintiffs and their expert
made the following key points on the bias issue. They
described the percentage bias test, mentioned in §2, and
presented the percentage bias figures averaged over all
21 censuses for EP and MF, which are virtually the same
as those in Table 5 for the first 19 censuses. They stated
that the percentage bias (in absolute value) for MF was
less than or equal to the percentage bias for EP for each
of the 21 censuses. They also noted that while the frac-
tional part of 6(b), which they called the “rounding
threshold,” increases for EP as b increases, the “round-
ing threshold” for MF is 0.5 regardless of the value of
b, and stated that this provides an intuitive reason why
EP is biased in favor of small states and MF is unbiased
in its treatment of small and large states. They also re-
ferred to Balinski and Young's theoretical result on the
unbiasedness of MF mentioned in §2.

Finally, the plaintiffs described computer simulations
of Balinski and Young (1984) in which the allocations
for each state were averaged over 1,000 randomly gen-
erated populations and compared to the state’s modified
exact quota. They stated that the results showed that
EP tended to allocate more seats on average than the
modified exact quotas for smaller states and to produce
the opposite result for larger states, while MF produced
no pattern of favoritism towards the smaller or larger
states.

On the “near the quota” property, the plaintiffs, in
addition to explaining this property, provided illustrative
examples from the 1970 and 1920 censuses. They noted
that in 1970, EP rounded up South Dakota's exact quota
of 1.435 while rounding down Connecticut's exact quota
of 6.503. MF produced the opposite results for these
states, and thus produced an allocation which brought

MANAGEMENT SCIENCE/Vol. 40, No. 10, October 1994

both states closer to their exact quotas in absolute terms
and relative terms (that is, with respect to the measures
la; — q;| and |a; — q;|/q;, as noted in §2) (Actually,
this is not a good example since Connecticut's modified
exact quota was 6.493). In 1920, they observed, three
states with exact quotas with fractional part less than
0.5 would have been rounded up by EP and down by
MF, while three states with exact quotas with fractional
part greater than 0.5 would have been rounded down
by EP and up by MF. (This also would have been true
for modified exact quotas.)

To support their claim that absolute difference in
shares of a representative is the best pairwise test, the
plaintiffs essentially used the same reasoning that the
defendants had used in the Montana district court to
argue the superiority of share of a representative over
district size as a test of the “one person, one vote” prin-
ciple. However, in their initial brief they had no real
argument to support the claim that absolute difference
is a better measure of inequity than relative difference.
They were handicapped on this point because their ex-
pert was unable to provide support, since his view had
always been that the choice between absolute and rel-
ative differences in pairwise comparisons is a “question
of preference” (Balinski and Young 1982, p. 102). The
plaintiffs initially were only able to argue that their pre-
ferred pairwise test is best since MF is optimal for it and
MF is, in their opinion, unbiased. The defendants crit-
icized the logic of this argument. In their final brief, the
plaintiffs developed a new argument that will be de-
scribed later in this section.

The technical arguments used by the defense (Gerson,
Budd et al. 1991) were based primarily on three dec-
larations written by this author (Ernst 1991b). Several
points were raised in response to the plaintiffs’ claims
on the bias issue. The key point of contention was the
plaintiffs’ assumption that only states with exact quotas
less than 0.5 should be excluded in bias measures, an
assumption on which much of their empirical and theo-
retical results were based. The defense claimed that it
would be more appropriate to exclude all states with
exact quotas less than 1, since even though all such
states are overrepresented, this is an overrepresentation
mandated by the Constitution.

The following are some of the changes in the empir-
ical results that we noted occurred with this change in

1217


LAWRENCE R. ERNST
Apportionment Methods for the House of Representatives

Table 6 Example for Which EP Minimizes Both Range of District
Sizes and Variance Formula (3.1)
HM and SD EP
State Bi G 4 q, 4a d,
1 2,370 2.37 2 1,185 3 790
2 1,320 1.32 2 660 1 1,320
3 1,310 1.31 1 1,310 1 1,310

a range of district sizes less than or equal to EP. EP did
yield a smaller range of district sizes than SD for the
1810 and 1840 censuses. Furthermore, it can be argued
that states with exact quotas less than 1 should be ex-
cluded from range computations, since they must re-
ceive one seat due to the constitutional provision of at
least one representative per state. With such states ex-
cluded, EP produced a smaller range of district sizes
than SD as recently as the 1940, 1950, and 1960 cen-
suses.

Judge O’Scannlain computed that (2.3) with p = 1 is
smaller for EP than HM for 1990. This is also true for
the other eight censuses for which these two methods
did not yield identical apportionments. However, for
the example in Table 7 with N = 3, n = 5, the EP al-
location has a value of 1344 for (2.3) with p = 1, while
the corresponding value for the HM and SD allocation
is 1320. Furthermore, since HM, SD and GR have the
same allocation for this example, HM and SD are op-
timal for this measure.

Finally, a measure of total error that the plaintiffs
never mentioned is

Because of the pairwise difference test that HM opti-
mizes, it might appear that HM is optimal for this mea-
sure. This is true with the qualifications given in the
following theorem. (The proofs of all theorems are given
in the Appendix.)

THEOREM 3.1. If HM produces an apportionment for
which there are no quota violations, then HM minimizes
(3.2) among all apportionments which do not violate quota.

In particular, since HM has not violated quota for any
of the 21 censuses, this result is applicable to all the
actual censuses.

However, the example in Table 8, with N = 9, n = 20,
illustrates that without the quota restriction, HM does
not always optimize (3.2). The value of (3.2) is 1210
for HM and 1095 for SD. However, SD violates quota
for state 1.

4. Massachusetts District Court
Case

The Massachusetts case was much more complex than
the Montana case in terms of the technical issues in-
volved. The EP 1990 allocation of 10 seats to Massa-
chusetts is one less than its 1980 allocation. Massachu-
setts would have received 11 seats for 1990 if either
MF, GD or GR had been used. The plaintiffs chose only
to claim that MF is constitutionally superior to EP
(Harshbarger et al. 1991). MF, in addition to increasing
Massachusetts’ EP allocation, would reduce Oklahoma’s
EP allocation of six seats to five seats, but would
produce the same apportionment for the remaining 48
states as EP.

The plaintiffs claimed that EP is unconstitutional for

N
> |d:- dl. (3.2) three separate reasons, the first two of which were based
isl

Table 8 Example for Which (3.2) is Smaller for SD than HM
Table 7 Example for Which HM and SD Minimize (2.3) with p = 1
HM SD
HM and SD EP OT OT
—_— State Dy % a, di a gi

State D Gi a dg; a 1;

1 4,320 4.32 4 1,080 3 1,440

1 2,328 2.328 2 1,164 3 776 2 2,970 2.97 2 1,485, 3 990
2 1,340 1.340 2 670 1 1,340 3-6 1,820 1.82 2 910 2 910
3 1,332 1.332 1 1,332 1 1,332 7-9 1,810 1.81 2 905 2 905
1216 MANAGEMENT SCIENCE/Vol. 40, No. 10, October 1994


LAWRENCE R. ERNST
Apportionment Methods for the House of Representatives

point in the selection of a method for apportioning the
House, perhaps because no apportionment method
minimizes all these measures. As will be seen in the
next two sections, these measures did become an issue
in both apportionment cases.

3. The Montana District Court Case
The Montana lawsuit was primarily based on the fol-
lowing legal reasoning (Racicot et al. 1991). In Wesberry
v. Sanders (1964), the U.S. Supreme Court had declared
that the intrastate redistricting of congressional districts
must be accomplished to provide “equal representation
for equal numbers of people,” that is the “one person,
one vote” principle. That case did not set any test for
meeting this principle, but in subsequent decisions, such
as the Karcher v. Dagget (1983), the Supreme Court
ruled that this principle required that “districts be ap-
portioned to achieve population equality as nearly as
practicable.” The plaintiffs concluded from the intrastate
redistricting cases they cited, that the courts required
this principle be met in the intrastate context by mini-
mizing ‘‘absolute population variances between dis-
tricts’’ and that this requirement also applied to inter-
state apportionment. Of course, no court had previously
ruled that the “one person, one vote” principle applied
to interstate apportionment, much less that a certain
test was superior to another for interstate apportion-
ment. In fact, even for intrastate redistricting, no court
had specifically ruled that a test based on district sizes
is a better test than one based on shares of a represen-
tative, or that absolute difference is a better measure
than relative difference. Furthermore, this issue of the
best test would not even be relevant for intrastate re-
districting since differences, at least in theory, can be
made as close to zero as desired for any of these methods
of measurement. Finally, the plaintiffs never offered any
specific reasons why absolute difference in district sizes
is the only appropriate test beyond citing these redis-
tricting cases.

After using these prior cases as their rationale for their
view that absolute difference between district sizes is
the only appropriate test, the plaintiffs noted in their
briefs and the affidavits of their experts that the pairwise
test for which HM is optimal and Dean’s original mo-
tivation for HM are both criteria that they considered

MANAGEMENT SCIENCE/Vol. 40, No. 10, October 1994

consistent with the cited cases. In addition, in the affi-
davits of the plaintiffs’ experts, Hill (1991) and Tiahrt
(1991), it was observed that for the 1990 census, among
EP, HM, and SD, HM produces the smallest variance
while SD produces the smallest range, and the plaintiffs
declared that either of these is an appropriate test of
inequity among district sizes. Furthermore, the Hill af-
fidavit included the formula used in computing the
variances, namely, using the notation of §2,

50
D (di ~ d)?/49,

ist

(3.1)

and also an alternative formula with 27°; [p; /(50a;)]
replacing d in (3.1).

The defendants’ reply to the plaintiffs’ assertions
(Gerson, Poppler et al. 1991) contained a number of
legal arguments, including the argument that appor-
tionment of the House is a political question to be de-
cided by Congress, and that it should not be considered
by the courts. It was also argued that in carrying out its
constitutionally mandated duty to apportion the House,
Congress should be allowed broad discretion by the
courts even if the issue is considered justiciable. In ad-
dition, it was observed that interstate apportionment is
very different from intrastate redistricting, since large
differences in district sizes between states are inevitable
because districts cannot cross state lines and each state
must have at least one representative. Consequently,
the defendants claimed that the redistricting cases cited
by the plaintiffs are not applicable to interstate appor-
tionments.

The (U.S.) Government used these arguments before
each of the courts that considered the two apportion-
ment cases. As will be seen, the success with these ar-
guments varied except for the political question argu-
ment, which was unsuccessful. In addition to the above
arguments, substantive arguments were presented to
demonstrate the advantages of EP, based primarily on
the declaration of this author (Ernst 1991a), which will
be the focus in this paper. There was no attempt to
demonstrate that EP is clearly superior to all other ap-
portionment methods, or the only constitutional
method, but instead that neither of these claims is true
for any other apportionment method or set of appor-
tionment methods which exclude EP.

1213


LAWRENCE R. ERNST
Apportionment Methods for the House of Representatives

PROOF OF THEOREM 5.1, Let a;, a; be the MF allocation for states i
and j, and b;, b; be an allocation obtained by transferring seats between
these two states. Then b; + b; = a; + 4), and hence

bt [bi pj — bjpil
Pi Pity) pilPs + Pi)
_ Pipjlbs/pi = B)/ Pil Pipilai/Pi = %/ Pil
pi(pi + Pi) Pi(pi + Pi)

References

Balinski, M. L. and H. P. Young, “The Quota Method of Apportion-
ment,” American Mathematical Monthly, 82 (1975), 701-730.

——, “Apportionment Schemes and the Quota Method,’’ American

Mathematical Monthly, 84 (1977), 450-455,

-, Fair Representation, Yale University Press, New Haven, CT, 1982.

, "Evaluation of Apportionment Methods,” Report prepared under

Contract No, CRS 84-15 for the Congressional Research Service

of the Library of Congress, Washington, D.C., 1984.

Birkhoff, G., “House Monotone Apportionment Schemes,” Proceedings
of the National Academy of Sciences, U.S.A. 73 (1976), 684-686.

Bliss, G. A., E. W. Brown, L. P. Eisenhart, and R. Pearl, Report to the
President of the National Academy of Sciences, February 9, 1929.
In Congressional Record, 70th Congress, 2nd Session, 70, 4966-
4967, March 2, 1929.

Ernst, L. R., Declaration in Montana v. United States Department of
Commerce, CV 91-22-H-CCL, Helena, MT, 199 1a.

——, Declarations in Massachusetts v. Mosbacher, CA 91-11234-WD,
MA, 1991b.

Gerson, S. M., D. M. Poppler, D. G. Linder, K. A. McLean, S. M.
Schraibman, S. L. Korytkowski, and Mark H. Murphy, Defen-
dants’ briefs in Montana v. United States Department of Commerce,
CV 91-22-H-CCL, Helena, MT, 1991.

——,W. A. Budd, J.S. Yogman, S. M. Schraibman, and M. H. Murphy,
Defendants’ briefs in Massachusetts v. Mosbacher, CA 91-11234-
WD, MA, 1991.

Gilford, L., Testimony at hearing before the Subcommittee on Census
and Population, 97th Congress, Ist Session, on H.R. 1990, 97-
18, 58-70, 1981,

Harshbarger, S., D. Golann, S. Berenson, J. P. Driscoll, Jr., E. P. Lei-
bensperger, and N. P, Motenko, Plaintiffs’ briefs in Massachusetts
v. Mosbacher, CA 91-11234-WD, MA, 1991.

——,, Massachusetts’ amicus curiae brief in United States Department
of Commerce v. Montana, U.S. Supreme Court, 91-860, 1992.

Hill, D. M., Affidavit in Montana v. United States Department of Com-
merce, CV 91-22-H-CCL, Helena, MT, 1991

Huntington, E. V., “A New Method of Apportionment of Represen-
tatives,” Quart. Publication of the American Statistical Association,
17 (1921), 859-870.

—, "The Apportionment of Representatives in Congress,” Trans-
actions of the American Mathematical Society, 30 (1928), 85-110.

Lovell, C. C. (Judge), Majority opinion in Montana v, United States
Departinent of Commerce, 775 F. Supp. 1358 (U.S.D.C. Mt), 1991.

Morse, M., J. von Neumann, and L. P. Eisenhart, Report to the Pres-
ident of the National Academy of Sciences, May 28, 1948.

O’Scanniain, D. F. (Judge), Minority opinion in Montana v. United
States Department of Commerce, 775 F. Supp. 1358 (U.S.D.C. Mt),
1991.

Owens, F. W., “On the Apportionment of Representatives,” Quart.
Publication of the American Statistical Association, 17 (1921), 958~
968.

Racicot, M., C. R. Smith, and E. S. Baker, Plaintiffs’ briefs in Montana
v. United States Department of Commerce, CV 91-22-H-CCL, Hel-
ena, MT, 1991.

—, Appellees’ briefs in United States Department of Commerce v.
Montana, U.S. Supreme Court, 91-860, 1992.

Still, J. W., “A Class of New Methods for Congressional Apportion-
ment,” SIAM J. Appl. Math, 37, 2 (1979), 401-418.

Starr, K. W., S. M. Gerson, J. G. Roberts, Jr., E. S. Kneedler, M. J.
Singer, M. B. Stern, and M. S. Raab, Appellants’ briefs in United
States Department of Commerce v. Montana, U.S. Supreme Court,
91-860, 1992.

Stevens, J. P. (Justice), Opinion in United States Department of Com-
merce v. Montana, 112 S. Ct. 1415, 1992.

Tiahrt, K. J., Affidavit in Montana v. United States Department of Com-
merce, CV 91-22-H-CCL, Helena, MT, 1991.

Woodlock, D. P. (Judge), Opinion in Massachusetts v. Mosbacher, 785
F, Supp. 230 (U.S.D.C. Ma.), 1992.

Young, H. P., Affidavits in Massachusetts v. Mosbacher, CA 91-11234-
WD, MA, 1991

—, Affidavit in United States Department of Commerce v. Montana,
U.S. Supreme Court, 91-860, 1992

Cases Cited

Franklin v. Massachusetts (1992), 112 S. Ct. 2767,

Karcher v. Dagget (1983), 426 U.S. 725.

Massachusetts v. Mosbacher (1992), 785 F. Supp. 230 (U.S.D.C. Ma.).

Montana v. United States Department of Commerce (1991), 775 F. Supp.
1358 (U.S.D.C. Mt).

United States Department of Commerce v. Montana (1992), 112 S. Ct.
1415,

Wesberry v. Sanders (1964), 376 U.S. 1

Accepted by Bruce L. Golden; received March 1, 1993. This paper has been with the author 2 months for 1 revision.

MANAGEMENT SCIENCE/Vol. 40, No. 10, October 1994

1227


LAWRENCE R. ERNST
Apportionment Methods for the House of Representatives

with d = 1,000, which corresponds to n = 100, and with
the divisor \ = d used for each of the divisor methods,
the allocations for each of the historical divisor methods
are given in Table 2. Note that with \ = 1,000, none of
these divisor methods gives a total allocation of 100
seats. To obtain an allocation of 100 seats, it is necessary
to adjust the divisor \ upward for SD, HM and EP, and
downward for MF and GD. The allocations for each of
the divisor methods for n = 100 and the minimum and
maximum integer values of \ which yield these allo-
cations are presented in Table 3, along with the GR
allocation.

An alternative to adjusting \ to obtain an apportion-
ment for the House of Representatives with a fixed
number of seats n, for a divisor method based on the
function 6, is to use the following recursive algorithm.
Let ax,i=1,...,N,k=N,N+1,...,n denote the
allocation to state i with k seats. Then let ajy = 1, i = 1,
...,N. Fork > N choose i, satisfying

Pi, / 5 (igeay) = max {pi / big: i=1,...,N},
and then let

Bie = Aigaryt 1, Mix = digay for i # iy.

4i,, i= 1,...,n is then a 6 apportionment for n seats.
Note that in the rare case when i, is not unique then
there is a “tie” for the nth seat and the apportionment
is not unique.

State i is said to satisfy quota if |q)] = a; <|qiJ+ 1.
Note that a quota violation occurs for i = 1 for each of
the five divisor methods for the allocations in Table 3,

Table 2 Divisor Methods Allocations for Example with \ = 1,000
a, for Method

State Di sD HM EP MF GD
1 91,490 92 91 co cl 91
2 1,660 2 2 2 2 1
3 1,460 2 2 2 1 1
4 1,450 2 2 2 1 1
5 1,440 2 2 2 1 1
6 1,400 2 2 1 1 1
7 1,100 2 1 t 1 4
Totals 100,000 104 102 104 98 7

MANAGEMENT SCIENCE/Vol. 40, No. 10, October 1994

since a, < 91 or a, > 92. GR, however, can never violate
quota. (Actually, because of the minimum requirement
of one representative per state, this last statement is
theoretically only true for an apportionment of the
House of Representatives if the exact quota q; is replaced
by the modified exact quota G; = max {1, tq; }, where ¢
satisfies }, max {1, tq; } = N, as discussed in Balinski
and Young (1982).) Furthermore, although all five his-
torical divisor methods can violate quota in theory, EP,
HM and MF would never have violated quota for any
of the 21 censuses through 1990, while SD and GD
would have violated quota for at least one state for each
census since 1820. For example, for California for 1990,
a; = 50 for SD and a; = 54 for GD, while q; = 52.124.

GD was used to apportion the House for the first five
censuses through 1830. Eventually, Congress became
dissatisfied with this method because it appeared to fa-
vor large states, allocating 40 seats to New York in 1830,
for example, despite an exact quota of 38.593. SD, MF,
and HM were developed as alternatives by John Quincy
Adams, Daniel Webster, and James Dean (a professor
at the University of Vermont), respectively. MF was
used in 1840. GR was the specified method from 1850-
1900, although as Balinski and Young (1982) note, for
some of these censuses the GR allocation was altered
so that no method was really used. However, Congress
became disenchanted with GR because under this
method, unlike any divisor method, it is possible, with
a fixed set of state populations, for a state to lose seats
if the House size is increased. This anomaly is known
as the ‘Alabama paradox” because it was observed that
for the 1880 census, Alabama would have received 8
seats with a House size of 299 and 7 seats with a House
of 300. This occurred because Alabama, Illinois and
Texas had exacts quotas of 7.646, 18.640, and 9.640,
and allocations of 8, 18, and 9 seats, respectively, for a
House size of 299, but these states had exact quotas of
7.671, 18.702, and 9.672, and allocations of 7, 19, and
10 seats, respectively, for a House size of 300. This was
a particularly unpleasant property since the House was
not automatically fixed by law during the period of use
of GR, but was decided upon by Congress following
each census, after reviewing the allocations with various
House sizes.

Congress returned to MF for the 1910 census. Con-
gress also passed legislation which, after New Mexico

1209


LAWRENCE R. ERNST
Apportionment Methods for the House of Representatives

from the mean, then the 33} % relative difference is not
relevant. The plaintiffs also observed, continuing this
example, that if the following year the rainfall was 25
inches, then the rainfall that year was 5 inches or 25%
above normal, and that the differences above and below
the mean balance out. However, the relative difference
between 25 and 20 is 25%, and hence the relative dif-
ferences do not balance out.

This example and similar examples accompanied the
last round of briefs and consequently, the defendants
did not have an opportunity to fully respond to them.
However, given the opportunity, the defendants could
have observed, concerning the rainfall example, that in
addition to talking about absolute amounts or percent-
ages above or below normal, it is common to say that
if the annual rainfall was 40 inches, it was twice the
normal amount and if it was 10 inches, it was one-half
the normal amount. Such phraseology corresponds to
thinking in terms of ratios, which is the whole point of
relative differences. That is, to measure discrepancy by
dividing the smaller number into the larger number, as
relative difference does, is as valid as measuring dis-
crepancy by subtracting the smaller number from the
larger, as absolute difference does.

The defendants also could have noted that the fact
that the absolute differences over two years balance out
while the relative differences do not, just establishes
that absolute differences are different from relative dif-
ferences, not that they are better. To illustrate, two
numbers above and below 20 have the same absolute
difference from 20 if and only if the arithmetic mean
of the numbers is 20. Similarly, the relative distance
between 20 and each of two numbers above and below
20 is the same if and only if the geometric mean of the
two numbers is 20. (For example, the relative difference
between 40 and 20 is 100% as is the relative difference
between 10 and 20, and the geometric mean of 40 and
10 is 20.) Thus, the plaintiffs’ claim that relative differ-
ence is not appropriate when comparing deviations from
an ideal is essentially equivalent to the claim that the
geometric mean is an inappropriate statistic for this
purpose. In addition, the defendants could have noted
that Balinski and Young (1982) had observed, without
criticism, that EP minimizes the relative difference
between each state’s average district size and the
divisor i.

1220

The criticism of relative difference, which was in
Young's last affidavit, only applied to deviations from
an ideal, not pairwise comparison tests. The distinction
was not made in the plaintiffs’ accompanying brief,
however. They used their expert's claims concerning
deviations from an ideal to argue that absolute differ-
ence between average shares of a representative is the
best pairwise test. The defendants pointed out in oral
arguments that the plaintiffs had failed to note this dis-
tinction.

The defendants responded to the plaintiffs’ claim that
absolute difference in average shares of a representative
is the best test of the ‘‘one person, one vote” principle
in several ways. We noted that for 1990, as guaranteed
by the theory, the relative difference between Okla-
homa’s and Massachusetts’ average district sizes and
average shares of a representative, and the absolute dif-
ference between the two states’ average district sizes
are smaller under EP than MF. (It is 14.6% under EP
and 15.2% under MF for the two relative difference
tests, and 76,638 under EP and 83,425 under MF for
the absolute difference in average district sizes.)

The defendants noted the symmetry in the fact that
among the four pairwise tests for which either EP, MF,
or HM are optimal, the plaintiffs in this case consider
the one test for which MF is superior to EP to be the
only appropriate test, just as the plaintiffs in the Mon-
tana case consider absolute difference in average district
sizes to be the only appropriate test since it the only
one of these four tests for which HM is superior to EP.
The defendants expressed concurrence with Balinski
and Young's (1975, p. 709) rhetorical question: “Why
choose. . . one divisor criterion [rather] than another?”

The three-judge panel in this case, in a unanimous
decision, written by Judge Woodlock (1992) of Mas-
sachusetts, upheld the constitutionality of EP. Although
the judges agreed with the plaintiffs that the “one per-
son, one vote” principle applies to interstate apportion-
ment, they rejected each of the three major substantive
issues raised by the plaintiffs.

On the bias issue, the court observed that with states
with exact quotas below 1 excluded “the historical bias
showing made by plaintiffs all but evaporates,” and
that for the 1990 census, EP yields an apportionment
with a percentage bias closer to 0 than MF would. The
court, noting the exclusion cutoff of 1.5 in Balinski and

MANAGEMENT SCIENCE/Vol. 40, No. 10, October 1994


LAWRENCE R. ERNST
Apportionment Methods for the House of Representatives

However, this example was not mentioned by the de-
fendants, because (2.2) for all p > 1 would have been
lower for MF than for EP for each of the other 11 cen-
suses for which these two methods produced different
apportionments.

In the plaintiffs’ briefs, the assertion that MF is op-
timal with respect to absolute difference in shares of a
representative, was generally not qualified to be limited
to the pairwise comparison test. While MF is optimal
with respect to the absolute difference in shares of a
representative when applied to the pairwise comparison
test, it is not necessarily optimal when absolute differ-
ence is employed with other tests. Other possible tests
include: range of s; (2.4) with p = 1,

N
D |si—s| and (4.4)
ia

max|s; — s|. (4.5)
MF is not always optimal for any of these measures. EP
would have produced a smaller range for s; than MF
for the first census in 1790. (Delaware had the maximum
value for s; under either method. However, the mini-
mum value for s; was 0.00002772 for Pennsylvania un-
der EP and 0.00002338 for Vermont under MF.) As
noted in §2, GR minimizes (2.4) with p = 1. The EP
and GR apportionments for 1940 coincided, and hence
both yielded a lower value for this measure than MF.
The optimality of MF for (4.4) is analogous to the op-
timality of HM for (3.2), as stated in the following
theorem, which is applicable to all 21 censuses.

THEOREM 4.4, If MF produces an apportionment for
which there are no quota violations, then MF minimizes
(4.4) among all apportionments which do not violate quota.

The proof of this theorem is essentially identical to
the proof of Theorem 3.1.

However, without the quota restrictions, MF does not
always minimize (4.4). The value of (4.4) for the ex-
ample in Table 8 is 0.00111 for MF and 0.00103 for SD.
The MF apportionment is identical to the HM appor-
tionment for this example.

Although MF minimizes (4.5) among all six appor-
tionment methods for all 21 censuses, for the example
in Table 9 with N = 3, n = 7, the value of (4.5) is
0.00026 for EP, the optimal value, and 0.00037 for MF.

1222

Table 9 Example for Which EP Minimizes (4.5)
MF EP
State Di a Si a Ss
1 2,710 3 0.00111 3 0.00111
2 2,690 3 0.00112 2 0.00074
3 1,600 1 0.00063 2 0.00125

5. The Supreme Court Case

The (U.S.) Government appealed the decision of the
three-judge district court in the Montana case to the
Supreme Court, which granted an expedited review.
Generally, new factual information is not introduced
on appeal, and for the most part, both sides did adhere
to this rule. The Government did note (Starr et al. 1992)
that for 1990, HM rounded up Montana’s exact quota
of 1.404 while rounding down Washington's exact quota
of 8.538, even though Washington had the higher frac-
tional remainder. This issue had not been brought up
previously because in the Massachusetts case for the
two states for which EP and MF disagree, Oklahoma,
with an exact quota of 5.516, and Massachusetts, with
an exact quota of 10.532, it is MF that rounds up the
state with the higher fractional remainder.

The Government did reiterate most of the issues
raised with the district court. In addition, the Govern-
ment vigorously argued that EP unquestionably ap-
portions representatives among the states “according to
their respective Numbers,” which is all that the Con-
stitution requires.

Montana, while raising no real new points, did clarify
their position on the issue of variance among district
sizes (Racicot et al. 1992). They stated that by ‘‘vari-
ance” in redistricting cases, the courts have not meant
mathematical variance at all, in the sense of either (2.3)
with p = 2, or (3.1), but instead have meant

max|d; — d|. (5.1)
This had indeed been mentioned as one criterion in
Montana’s district court briefs, but this measure has not
been clearly referred to as a measure of variance pre-
viously. Furthermore, it does not appear from the affi-

MANAGEMENT SCIENCE/Vol. 40, No. 10, October 1994


LAWRENCE R. ERNST
Apportionment Methods for the House of Representatives

case was appealed to the Supreme Court (United States
Department of Commerce v. Montana 1992), with Mas-
sachusetts filing a friend-of-the-court brief before the
Supreme Court in order to present their position in favor
of MF. On March 4, 1992, the Supreme Court heard
the case and 27 days later unanimously overruled the
decision of the three-judge panel in the Montana case.

This paper discusses the mathematical and statistical
issues in these cases. This author wrote the declarations
that served as a basis for many of the technical argu-
ments used by the defense in these cases, and this paper
is in part an outgrowth of that work. Section 2 of the
paper provides an historical background on the appor-
tionment issue and a discussion of the properties of the
major apportionment methods. Balinski and Young
(1982), the major source of the material in that section,
provides a more detailed treatment of these matters.
Sections 3 and 4 discuss the issues debated before the
three-judge panels in the Montana and Massachusetts
cases, respectively. Finally, the Supreme Court appeal
is discussed in §5.

2. Historical Background and
Properties of Methods

Six apportionment methods are considered here. They
are the four methods mentioned in the Introduction,
the method of greatest divisors (GD), also known as
the Jefferson method, and the method of greatest re-
mainders (GR), also known as the Hamilton or Vinton
method.

All of these methods except GR are members of a
class of apportionment methods known as divisor
methods. Although there are an infinite number of pos-
sible divisor methods, only the five considered here have
had any significant role in apportionment history. They
will be referred to as the historical divisor methods. With
a divisor method, the number of seats assigned to a
state is a function of its population, p, and a divisor, \,
which can be thought of as a target district size. The
same value of \ must be used for each state. If |p /J
= b (where |x| denotes the integer portion of x), then
the state receives either b or b + 1 seats. It receives b
+1 seats if p/ > 5(b), and b seats if p/A < 5(b),
where 6, the function that determines the rounding, de-
pends on the particular method. If p/d = 6(b), the

1208

rounding is not unambiguously defined. 6 is a strictly
increasing function of b satisfying b < 6(b) < b + 1 for
all nonnegative integers b. In Table 1, 6(b) is presented
for each of the five historical divisor methods.

Thus, SD rounds up and GD rounds down all frac-
tional remainders, while MF rounds up fractional re-
mainders greater than 0.5. HM rounds up quotients that
exceed the harmonic mean of b and b + 1. This can be
shown to be equivalent to rounding up if the absolute
difference between ) and the state’s average district size
is minimized with b + 1 seats, that is if

|p/(b+1)—Al <|p/b-XI.

This was Dean’s original motivation for HM. Similarly,
EP rounds up quotients that exceed the geometric mean
of b and b + 1, which is equivalent to minimizing the
relative distance between ) and the average district size
for the state. (The relative difference between two pos-
itive numbers x, y is | x — y| /min{ x, y} or, equivalently,
(max {x,y} /min{x, y}) — 1.)

Note also that for MF and GD, the modification 5(0)
= 0 is required to insure that all states, no matter how
small, receive at least one representative.

A GR apportionment is obtained slightly differently.
Begin with a fixed house size n, and a set of N states
with populations p;,i=1,...,N.Letd = Mi p/n,
the national average district size; q; = p; /d, the exact
quota for state i; and a; denote the number of seats
allocated to state i under any method. Then for GR,
either a; = 4; ] or a; = q;|+ 1, with a; = (9;]+ 1 for
the n — ©, [gi J states with largest fractional remain-
ders, q; — | 4; |. To illustrate how these six methods pro-
duce apportionments, consider the example in Tables
2 and 3 for which N = 7 and the populations are as
given in the p; column of each of these two tables. Then

Table 1 Rounding Criteria for Historical
Divisor Methods
Method 4(b)

SD 6

HM 2b(b + 1)/(2b + 1)

EP vob + 1)

MF b+ 5

GD b+1

MANAGEMENT SCIENCE/Vol. 40, No. 10, October 1994


Apportionment Methods for the House of
Representatives and the Court Challenges

Lawrence R. Ernst
Bureau of Labor Statistics, Office of Compensation and Working Conditions, Research Group,
2 Mass. Ave., N.E., Room 3160, Washington, D.C. 20212

our different methods have been used to apportion the seats in the United States House of

Representatives among the states following the decennial census. The current method, the
method of equal proportions, has been used for each census since 1940. In 1991, for the first
time in U.S. history, the constitutionality of an apportionment method was challenged in court,
by Montana and Massachusetts in separate cases. Montana proposed two methods as alternatives
to equal proportions, the methods of harmonic means and smallest divisors, while Massachusetts
proposed the method of major fractions. On March 31, 1992, in a unanimous decision, the U.S.
Supreme Court upheld the constitutionality of equal proportions. This author wrote the dec-
larations on the mathematical and statistical issues used by the defense in these cases. The
declarations in the Massachusetts case contain several new theoretical and empirical results.
This paper discusses the technical issues in these cases together with a brief history of the

apportionment problem.

(Bias; Divisor Methods; Equal Proportions; Optimality; Pairwise Tests; U.S. Constitution)

1. Introduction

Article I, Section 2 of the U.S. Constitution requires that
the House of Representatives “shall be apportioned
among the several States—according to their respective
Numbers,” and that “each State shall have at least one
Representative.” That section also includes the require-
ment that an enumeration of the population for the
purpose of apportioning the House be conducted every
10 years. The quoted words obviously do not explicitly
state what method should be used for apportionment,
and for over 200 years the issues of which is the “best”
method and which methods are constitutional have
been debated. In fact, apportionment of the House was
the subject of George Washington’s first veto.

The “best” method issue is, in this author's opinion,
unresolvable, since it depends on the criteria employed.
However, the constitutional question was at least par-
tially resolved on March 31, 1992, when Justice Stevens
delivered an opinion for a unanimous Supreme Court
upholding the constitutionality of the currently used
apportionment method, equal proportions (EP), also
known as the Hill or Huntington method.

0025-1909/94/4010/1207$01.25
Copyright © 1994, The Institute of Management Sciences

The path to this resolution began in 1991 when the
states of Montana and Massachusetts initiated separate
lawsuits in federal court (Montana v. United States De-
partment of Commerce 1991; Massachusetts v. Mosbacher
1992) challenging, for the first time in U.S. history, the
constitutionality of the current method. Montana pro-
posed two methods as alternatives to EP. Their preferred
methods are the method of harmonic means (HM), also
known as the Dean method, and the method of smallest
divisors (SD), also known as the Adams method, both
of which would have given Montana two seats instead
of the single seat allocated by EP, but would not have
increased Massachusetts’ EP allocation of ten seats.
Massachusetts proposed, using different arguments, the
use of the method of major fractions (MF), also known
as the Webster method, which would have allocated
eleven seats to Massachusetts, and one to Montana.

The two cases were considered by separate three-
judge panels. The panel in the Montana case, by a two-
to-one vote, declared EP unconstitutional, while the
judges in the Massachusetts case unanimously upheld
the constitutionality of EP. The ruling in the Montana

MANAGEMENT SCIENCE/Vol. 40, No. 10, October 1994 1207


LAWRENCE R. ERNST
Apportionment Methods for the House of Representatives

the set of excluded states. While the bias ratio in Table
4 is 54.6% for EP and 51.5% for MF, this ratio for the
same 19 censuses with all states with exact quotas below
1, instead of only those below 0.5, excluded is 50.8%
for EP and 47.4% for MF. Similarly, the average per-
centage bias in Table 5 is 3.4% for EP and 0.3% for MF,
while with all states with exact quotas below 1 excluded,
it is 1.8% for EP and —0.9% for MF. Furthermore, with
all states with exact quotas less than 1 excluded, the
plaintiffs’ assertion that the percentage bias for MF never
exceeded the percentage bias for EP for each of the 21
censuses does not hold. In fact, with these states ex-
cluded, the percentage bias for the 1990 census is —.6%
for EP and —1.0% for MF. That is, by this measure, the
1990 EP apportionment favors the large states and sub-
stitution of MF would simply increase the magnitude
of the favoritism. Finally, we noted that, with states
with exact quotas below 1 excluded, EP favored small
states 13 times among the 21 censuses through 1990
using the percentage bias test, but MF favored large
states 15 times.

Two new theoretical results were obtained by the de-
fendants. First it was observed that Balinski and Young's
(1982) result that MF is pairwise unbiased is dependent
on use of the partition (2.1) (ignoring the overlap of
endpoints), which for MF reduces to

[0.5A, 1.5A], [1.5A, 2.5A], [2.5A, 3.5A],.... (4.1)
However, we argued that the alternate partition
[A, 2A], [2A, 3A], [3A, 4A], 0+ (4.2)

would be more consistent with the exclusion of all states
with exact quotas less than 1. Partition (4.2) leads to
the following, very different result than (4.1) on the
pairwise bias of MF.

THEOREM 4.1. For a divisor X, if states 1 and 2 have
independent populations p,, p2 uniformly distributed in
intervals [b,X, (b; + 1)A] and [b2A, (b, + 1)AJ, respec-
tively, for positive integers b, > b,, and the states have
allocations a, and az, respectively, then for an MF appor-
tionment the probability is greater than 0.5 that a2 /p2
>a /pr.

Thus, in the sense of Theorem 4.1, MF is pairwise
biased in favor of large states.

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The defendants’ second theoretical result on bias is:

THEOREM 4.2. With the assumptions and notation of
Theorem 4.1, E(d) = E(d;) for EP and E(d,) < E(d;) for
MF.

Thus, in the sense of Theorem 4.2, EP is unbiased
and MF is biased in favor of large states.

We also developed an alternate approach to measur-
ing percentage bias consistent with Theorem 4.2, namely

(P4/24)-»

where S and L are as in §2, but with all states with exact
quotas below 1 excluded. Positive values for (4.3) in-
dicate that large states are favored and negative values,
that small states are favored. Averaged over all 21 cen-
suses through 1990, (4.3) is —1.0% for EP and 2.7%
for MF. For 1990 alone, (4.3) is 3.9% for EP and 4.5%
for MF. Among all 21 censuses, EP favored small states
by this measure 13 times, while MF favored large states
17 times.

The defendants responded to the plaintiffs’ argument
that the “rounding thresholds” provided intuitive evi-
dence to support their bias claims by simply pointing
out that “rounding thresholds” are not always indicative
of how the exact quota of a state would be rounded. It
was noted that for 1990, New York, New Jersey, Mas-
sachusetts, and Oklahoma have exact quotas of 31.521,
13.536, 10.532, and 5.516, respectively. The fractional
portion of the exact quota for each of these states is
above the “rounding thresholds” for both MF and EP.
Yet MF and EP round down the exact quota for each
of these states, with the exception of Massachusetts for
MF and Oklahoma for EP.

The defendants’ response to the claims concerning
the computer simulations of Balinski and Young (1984)
was to note that the results of the original simulations
done by these authors, which were based on random
variations from the modified exact quotas computed
from Census Bureau projections of the 1990 data,
showed that for MF the average allocation exceeded the
modified exact quota for each of the largest 22 states,
while the opposite was true for each of the 6 smallest
states. These results did not support the claim that MF
is unbiased. They then rounded the modified exact quota

(4.3)

MANAGEMENT SCIENCE/Vol. 40, No. 10, October 1994


LAWRENCE R. ERNST
Apportionment Methods for the House of Representatives

Young (1984), stated that any exclusion, at least up to
1.5, is a reasonable means of accounting for the special
constitutional treatment of very small states.

Judge Woodlock observed with regards to the plain-
tiffs’ “near the quota” claims, that the showing was “at
best mixed.” He noted that it is GR that satisfies (2.2),
and while MF satisfies the ‘‘near the quota” property
with respect to the measures |a; — q;| and |a; — q;|/
qi, it is EP that possesses this property with respect to
relative difference. The opinion explicitly addressed not
only Massachusetts’ claim that the pairwise test that
best meets the “one person, one vote” principle in in-
terstate apportionment is absolute difference in average
shares of a representative, but also the claim in the
Montana case that absolute difference in average district
sizes is the only constitutional test. Judge Woodlock
stated simply, “we can find nothing in the Constitution
mandating a particular mathematical formula be em-
ployed to the exclusion of others.” He expressed agree-
ment with Balinski and Young's (1975) view that there
is no reason to choose one divisor criterion over another.

The decision also noted that courts in the intrastate
context have consistently measured equity by relative
departures from the ideal district. (It is not clear whether
that meant | d; — d|/d or | d; — d| /min{d;, d}. In the
intrastate context, the ratio of these two measures would
generally be near 1 anyway, since d; and d should al-
ways be close.) The judges found relative measurement
to be a mathematically acceptable means of making eq-
uity comparisons and that nothing in case law or the
Constitution prohibited its use.

The court, summarizing their views, stated: “The
Constitution does not prescribe a particular formula, a
specific methodology or a set standard to embody the
‘one person, one vote’ principle in this complex setting.”
The judges concluded that EP does satisfy this principle,
and hence the courts have no authority to interpose a
different method than the one adopted by Congress. It
is clear from the opinion that their ruling would have
been the same if they had the Montana case before
them.

Massachusetts did not appeal this decision to the Su-
preme Court. The Supreme Court's decision in the
Montana case had made an appeal on this issue futile.
However, as detailed in the next section, Massachusetts
did present their views on apportionment methods to

MANAGEMENT SCIENCE/Vol. 40, No. 10, October 1994

the Supreme Court through a friend-of-the-court brief
in the Montana case.

Massachusetts also argued a second issue before the
district court. They contended, using several arguments,
that U.S. government employees working overseas,
military and civilian, and their dependents, should not
have been included in the apportionment counts. Mas-
sachusetts won on this issue before the district court on
one of these points, namely that the decision to allocate
overseas military personnel to the state designated as
their “home of record’ was arbitrary and capricious
under the standards of the Administrative Procedures
Act (APA). This decision, if upheld in its entirety, would
have increased Massachusetts’ allocation to 11 seats and
reduced Washington’s to 8 seats. However, the Gov-
ernment appealed this decision to the Supreme Court,
which unanimously reversed the district court’s decision
to exclude overseas federal employees from the appor-
tionment counts (Franklin v. Massachusetts 1992).

In overturning the district court ruling on the overseas
employees issue, five of the justices, in an opinion by
Justice O’Conner, held that the APA was not applicable,
since the apportionment law required an action by the
President, namely the transmittal of the apportionment
to Congress. These justices found that the APA did not
apply to the President and hence never reached the
merits of this issue. The other four justices, in an opinion
by Justice Stevens, did find the APA to be applicable,
reasoning that it is the Secretary of the Commerce, who
is covered by the APA, that is authorized to conduct
the decennial census, and that the President's role is
purely ministerial. However, these justices did not find
the “home of record” method of allocating the overseas
military personnel to be arbitrary and capricious.

As in the Montana case, some issues arose in the
Massachusetts case that were not answered in the court
documents. For example, although the defendants noted
that EP can sometimes produce a lower value for (2.2)
than MF, no illustrative example was provided. We were
well aware of a very important example. For the 1940
census, the one which resulted in the adoption of EP,
Arkansas’ exact quota was 6.473 and Michigan’s was
17.453. The EP allocation of 7 seats and 17 seats, re-
spectively, to these two states yielded a lower value for
(2.2) than the MF allocation of 6 seats and 18 seats,
since the two apportionments were otherwise identical.

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