[EM] Apportionment

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

Dear Elections List,

Some comments about apportionment.

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  

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.


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.



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:


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