# [EM] Apportionment

Joseph Malkevitch malkevitch at york.cuny.edu
Fri Jan 19 07:03:17 PST 2007

```Dear Elections List,

1. The major contribution of E.V. Huntington  to the study of
apportionment methods was to call attention to fairness questions
with regard to moving one seat assigned to some state to another
state. This led him to study "divisor methods" and their properties.
Among several contributions of Balinski and Young was to call
attention to problems created by ties. Ties (using a particular
method) can occur not only when two states have the same population
but also when they have different populations but the house size is a
particular value.

Consider the example of where Webster is applied to a house size h =
4 for two states with populations of 300 and 500, and each state
should have, as usual, at least one seat. Since the total population
is 800 and 4 seats are to be apportioned, the ideal district size
should be 200 people. Thus, the state with population 300 is entitled
to 300/200 = 1.5 seats which using the Webster rounding rule means 2
seats while the state with 500 people is entitled to 500/200 = 2.5
seats which using the Webster rounding rule means 3 seats. Thus, we
have not assigned the proper number of seats. However, if one changes
the divisor 200 up or down a little bit Webster will not assign
exactly 4 seats. Thus, for a house size of 4 with these populations
the "Webster method" must have a prevision for how to choose to give
the extra seat to either the 300 state or the 500 state. It is not
difficult to find examples and house sizes where the Webster method
may have many simultaneous ties similar to the example just provided.
For house sizes of 3 of 5 the situation is straightforward.

From a mathematical point of view, therefore, one can not describe a
method by saying "by trial and error" find a "value" of some
parameter that distributes exactly h seats; no such number may exist.
(Find a fraction a/b (b not zero, a and b integers) whose square is
2. No such fraction exists.) One must show that a solution exists or
describe the situations where one gets into trouble and resolve what
to do in those cases. Huntington did this, and Balinski and Young did
it even more carefully.

2. Huntington, Balinski, and Young have a very precise meaning for
what they call a "divisor method." There are definitely infinitely
many divisor methods in the technical sense and many methods that are
described in some very different way from, say, the rounding rule
approach to divisor methods, turn out to be a divisor method. On the
other hand, just because one does some "divisions" that does not mean
that one has produced a divisor method in the sense that Huntington,
Balinski, and Young use this term. Furthermore, suppose that one
describes some apportionment and it really is a divisor method in the
sense of Huntington, Balinski, and Young, then it is subject to
conclusions that were discovered by Huntington. Balinski and Young
discuss the idea of "stability" with regards to transfer of a seat
between two states using one of the sixteen different ways of
expressing the ways that state i can be better off than than state j
with respect to inequalities involving the population of the states
and the number of seats they are assigned. (See page 101 and 102 of
Fair Representation by Balinski and Young, where they explain
Huntington's ideas in a fairly nontechnical way.) Quoting Balinski
and Young:

"Not every measure of inequality gives stable apportionments: for
some measures there exist problems for which every apportionment can
be improved upon by some transfer. Huntington showed that, except for
four such "unworkable" measures, all others resulted in the methods
of either Adams, Dean, Hill, Webster or Jefferson."

What this means in practice is that if one develops a method which is
actually a divisor method, however, it is described, if it is not one
of the 5 "classical divisor methods" it has the unattractive property
of not behaving nicely for having transfers between pairs of states
which is "stable." Now if one wants to one can say that issues of
transfer equity between pairs of states does not matter, just as one
can say that violation of quota does not matter or that violation of
population monotonicity does not matter. There are different views on
what make methods reasonable and fair.

3. Whereas in America the practical debates about how to apportion
the House of Representatives has turned on discussions of measuring
inequities between pairs of states, in Europe's attempts to cope with
apportionment problems this approach did not surface. Rather, there
was a tradition of using methods of finding a "global" discrete
optimal answer. For a detailed look at this point of view you should
consult:

Evaluation and Optimization of Electoral Systems, by Pietro Grilli di
Cortona, Cecilia Manzi, Aline Pennisi, Federica Ricca, and Bruno
Simeone, SIAM Monographs on Discrete Mathematics and Its
Applications, Philadelphia, 1999. (All of the authors of this book
teach at universities or work in Rome, Italy.)

Perhaps not surprisingly, the 5 classical methods also show up in
this very different approach involving "global optimization."

4. The word "biased" means many things in common parlance and it also
can be given many technical definitions. To be "proportional' a
method should give states with increasingly large populations more
seats than what it gives to smaller states. A method which does not
do so has a form of "bias." There are different ways to try to
measure "bias" in failing to be "proportional." On the other hand if
a method tends to give groups of larger states more than their fair
share as compared with groups of smaller states (using some
definition of small and larger) then this is also a form of "bias."
Again there are different approaches to measuring such bias. Bias of
methods from different perspectives can be looked at in an abstract
theoretical way or it can be looked at as an empirical question for
data which has been used in the historical context.

5. For many years I have been teaching a fairness and equity course
both as part of my college's program in Humanities, in our Liberal
Studies Program, Honors Program, and mathematics courses.

http://www.york.cuny.edu/~malk/courses/fairness-outline.html
http://www.york.cuny.edu/~malk/courses.html

Often, at the start of the course, students ask me what insights
mathematics can have into these questions. By the end of the course
they typically have some answers. For many fairness situations there
are so many properties of fairness that one would like a "fair
method" to have in solving the problem that mathematicians have been
able to prove that there is no method that obeys all of the desired
fairness conditions. One reason there is so much acrimony about
fairness questions is that there are groups of people who say
conditions X, Y, and Z are the essential ones and others who say
conditions A, B, and C are the essential ones. Since there are no
methods that obey X, Y, Z, A, B, and C these groups endless argue
even though often there are methods that they might reach a consensus
on as being better than the methods currently in use.

Another thing mathematics can do is for each "reasonable" method Z
find a list of axioms that are satisfied by method Z but no other
method. When one has such axioms for each method that one might think
is appealing one can see more clearly what one gains and loses by
adopting one method rather than another.

Cheers,

Joe

------------------------------------------------
Joseph Malkevitch
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451

Phone: 718-262-2551 (Voicemail available)

My new email is:

malkevitch at york.cuny.edu

web page:

http://www.york.cuny.edu/~malk

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