Apportionment
2. Fairness and
Apportionment
The Constitution does not specify what fairness criteria should be used in comparing
two different proposed ways to solve an apportionment problem. From the beginning
there were both political and equity considerations in choosing an apportionment
system but there are some principles that one can call attention to in evaluating
apportionment systems. One fairness idea is that each state should get either
its lower quota or upper quota. A second approach to fairness is to look at
pairwise equity between states. The empirically discovered Alabama paradox (one
can get fewer seats in a larger house, with fixed population) called to the
attention of politicians and others interested in apportionment that one had
to worry about the fairness properties of the methods that one might use to
solve the apportionment problem. However, little seems to have been done in
a systematic way to see what the consequences for fairness were of adopting
different apportionments. In the United States, what became a matter of concern
was the perception that different methods of apportionment might display bias.
The idea of bias is that a method might systematically reward states with specific
characteristics or groups of such states. For example, were some methods more
likely to give smaller states more than their fair share of seats, say, as measured
by exact quota? One way to try to answer this question was to turn to statistics.
It is not hard to see that the different methods of apportionment treat small
and large states in different ways. This can be seen in contrived artificial
examples as well as using data that has arisen from the censuses. However, there
are many different ways of thinking and models for deciding what constitutes
bias of an apportionment method. As just one example, the Constitution itself
has a bias in favor of small states because it will give every state one seat
regardless of how undeserving the state might be. In computing bias of a method,
how should this "minimum of one seat" condition be taken into account?
Huntington was a pioneer in using mathematical ideas to compare the different
apportionment from a fairness point of view. His writings have an attractive
conversational style, though like many good debaters he does not shy from leaving
out things that do not help his case. Huntington realized that there were two
major approaches to evaluating fairness:
1. Global optimization
This approach sets up a measure of fairness and for a given apportionment method
sums up this measure for all the states. The goal is to select that method which
minimizes the sum for the given measure over all the states. This approach involves
various discrete optimization techniques and has not been pursued vigorously
until relatively recently. Some of the optimization problems that one is led
to may be computationally very difficult. Huntington rejects such methods but
they are mathematically interesting nonetheless.
2. Pairwise comparison for states.
This approach involves making sure that switching a seat from one state to another
does not diminish the fairness of the apportionment as given by some measure
of fairness. Huntington championed the study of this way of approaching the
apportionment problem. He showed, rather surprisingly, that each of the historic
methods was best under at least one fairness measure. This created the troublesome
situation that one had to make a value judgment as to which of these fairness
criteria was the best in order to justify the choice of a particular method,
and it is not clear on mathematical grounds how to do this.
First, Huntington observed that for each measure of equity between states one
could look at this measure in absolute terms or in relative terms. Here is an
example illustrating this, taken from one of Huntington's papers using data
from the 1940 census: Using all of the census data, it turns out that Webster
assigned Michigan (population 5,256,106) 18 seats and Huntington-Hill assigned
it 17 seats, while Webster assigned Arkansas (population 1,949,387) 6 seats
and Huntington-Hill assigned Arkansas 7 seats. The two methods agree except
for the number of seats that they give to these two states. With these numbers,
the size of the congressional district for Michigan under Webster was 292,006,
while the size under Huntington-Hill was 309,183. The equivalent figures for
Arkansas were 324,898 (Webster) and 278,484 (Huntington-Hill). Thus, the absolute
difference between the two states for Webster was 32,892 (324,898-292,006).
The absolute difference between the two states for Huntington-Hill was 30,699.
By this measure Huntington-Hill did a better job. However, there is also the
perspective of the size of the difference relative to the size of the populations
of the states involved. Michigan is a much bigger state than Arkansas. In relative
terms the Webster method resulted in a relative difference of 11.26 percent,
while Huntington-Hill resulted in a relative difference of 11.02 percent. The
relative difference for u and v is given by |u-v|/(min (u, v)). This example
should not mislead one into thinking that Huntington-Hill always does a better
job. Measuring the absolute difference in representatives per person would be
the basis for defending why the Webster apportionment is better. I use this
example only to illustrate the distinction between relative and absolute difference
ideas.
Here is a summary of what Huntington discovered:
a. For relative differences, Huntington-Hill is the optimal method for
all
the fairness measures about to be listed. Recall that a
i and P
i
are the number of seats for state i and its population, respectively. For a
fixed divisor method, the formula can be used to compare whether or not giving
the next seat to state i or state j is more justified, as measured by the the
given fairness formula.
b. For the measure
Adams method is optimal.
For the measure
Dean's method is optimal.
For the measure
Huntington-Hill is optimal.
For the measure
Webster is optimal.
For the measure
Jefferson's method is optimal.
Again, it might
not be obvious that these seemingly similar measures of absolute pairwise fairness
would give rise to such different methods to achieve optimality. Yet Huntington
showed that if one is concerned with relative differences, all the criteria
formulas are optimal only for Huntington-Hill.
- Apportionment Systems
- Fairness and Apportionment
- Balinski and Young's Contribution
- Where to Next?
- References
