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\title{Hyperbolic Fixed Points of  $\C^*$ Actions}
\author{JWR \& DS}

\date{Thursday 13 July 2006 2:50}

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The constitution says that apportionment of representatives
to a state should be proportional to its population. 
If taken literally, this would mean
giving fractional representatives, so some method
of converting  fractions to whole numbers must be used.
Over the years, the choice of  method
has led to  many battles in the U.S. House of Representatives, 
and there will doubtless be more.
There is a vast
academic literature on the subject; see e.g.\ the 
\href{http://www.ams.org/featurecolumn/archive/apportionII5.html}{
\underline{references section}}
of the websites~\cite{Apportionment},\cite{Apportionment II}.
The clearest exposition I have found so far is~\cite{BY-AMM};
this article also contains many amusing details on the
history of the problem. The notes at the end of the article give
the sources for the historical information in given here.
Some of the material which follows is copied 
verbatim from various websites as indicated  
in Appendix~\ref{app:notes}.


Various methods of apportionment have been used over the years.
Often the result seems unfair to many people and many bitter disputes
have resulted. Here is a list
of surprising things that happened
as a result of using one method or another. 

\para \jdef{The Alabama Paradox.}\label{AlabamaParadox}
An increase in the total number of seats to be apportioned can cause a state to lose a seat. 
The Alabama Paradox first surfaced after the 1870 census. With 270 members in the House of Representatives, Rhode Island got 2 representatives but when the House size was increased to 280, Rhode Island lost a seat. After the 1880 census, 
C. W. Seaton (chief clerk of U. S. Census Office) computed apportionments for all House sizes between 275 and 350 members. He then wrote a letter to Congress pointing out that if the House of Representatives had 299 seats, Alabama would get 8 seats but if the House of Representatives had 300 seats, Alabama would only get 7 seats.
The method used at the time was Hamilton's method explained below
in paragraph~\ref{}.
\arap

% Illustrating the Alabama Paradox

\para \jdef{The Population Paradox.}\label{PopulationParadox}
An increase in a state's population can cause it to lose a seat.
The Population Paradox was discovered around 1900, when it was shown that a state could lose seats in the House of Representatives as a result of an increase in its population. (Virginia was growing much faster than Maine,
but Virginia lost a seat in the House while Maine gained a seat.)
The method used at the time was Hamilton's method explained below
in paragraph~\ref{}.
\arap

\para \jdef{The New States Paradox.}\label{NewStatesParadox}
Adding a new state with its fair share of seats can affect the number of seats due other states.
The New States Paradox was discovered in 1907 when Oklahoma became a state. Before Oklahoma became a state, the House of Representatives had 386 seats. Comparing Oklahoma's population to other states, it was clear that Oklahoma should have 5 seats so the House size was increased by five to 391 seats. The intent was to leave the number of seats unchanged for the other states. 
However, when the apportionment was mathematically recalculated, 
Maine gained a seat (4 instead of 3) and New York lost a seat (from 38 to 37).
The method used at the time was Hamilton's method explained below
in paragraph~\ref{}.
\arap

\para \jdef{The Quota Paradox.} \label{QuotaParadox}
A state may receive a number of seats
which is smaller than its lower quota or larger than its upper quota.
This happened in the apportionment
based on the 1820 census when 
New York had a population of 1,368,775,
the total U.S. population was 8,969,878,
and the size of the house was 213.
New York's quota was thus 
$
q=\frac{1,368,775}{8,969,878}\times 213=32.503.
$
However the apportionment method used at the time
(Jefferson's method explained below in paragraph~\ref{}) awarded New York
34 seats. 
\arap

\begin{quote}\em
 There seems to be no known apportionment method which avoids
 all these paradoxes! See Remark~\ref{rmk:paradox} below.
 \end{quote}




\section{Notation and Terminology}


\para 
Assume that there are $n$ states numbered $i=1,2,\ldots,n$
and that $p_ i$ denotes the population of the $i$th state
so that
$$
 p=p_1+p_2+\cdots+p_n
$$
is the total population of the entire country.
We call a sequence $(p_1,p_2,\ldots,p_n)$
of nonnegative integers a \jdef{population vector}.
We suppose that $h$ denotes the total number of seats
to be allocated and call it the \jdef{hpuse size}.
An 
 \jdef{apportionment}  of length $n$ and house size $h$
 is a sequence $(a_1,a_2,\ldots,a_n)$ of whole numbers such that
$$
h = a_1+a_2+\cdots+a_n. 
$$
The number $a_i$ is the  number of representatives assigned to the $i$th state
by the apportionment. 
An \jdef{apportionment method} is a sequence of  functions%\footnote{
%The notation
%$x\mapsto$\ y$ indicates a function which produces the output $y$
%for input $x$.
%)
$$
 (h,p_1,p_2,\ldots,p_n)\mapsto (a_1,a_2,\ldots,a_n)
$$
(one for every value of $n$) which assign to every house size $h$
and every population vector $(p_1,p_2,\ldots,p_n)$ an apportionment
$(a_1,a_2,\ldots,a_n)$  of house size $h$.  
The table in Figure~\ref{fig:ap} shows the outputs given
by five different apportionment methods based on
the U.S. census data for 1990.  Here we have $n=50$,  $h=435$, and
$p=249,022,783$. Wisconsin is the 49th state in alphabetical order
so $p_ {49}=4,906,735$.  Four of the apportionment methods
award Wisconsin 9 representatives, but   Jefferson's Method awards
Wisconsin only 8.
\arap


\begin{figure}[p]
\label{fig:ap}
\caption{Apportionment of the U.S. House of Representatives}
\bigskip

\tiny
\input{1990}
\end{figure}




\para
%
The \jdef{quota}\footnote{Some authors call this the {\em standard quota},
others call it the {\em exact quota}.} for $i$th state   
is the fraction 
$$
 q_i = \frac{p_i}{p}\times h
$$
of the total number of seats a state would be entitled to if the seats were not indivisible. The \jdef{lower quota}  of a state is its quota rounded down,
and the \jdef{upper quota} is its quota rounded up.  In standard mathematical
notation the lower  and upper quota of the $i$th state are
denoted $\lfloor q_i\rfloor$ and $\lceil q_i\rceil$ respectively.
\arap

An apportionment which avoids this paradox is said to
satisfy the quota condition, i.e. an apportionment
$(a_1,a_2,\ldots,a_n)$  satisfies the  \jdef{quota condition} iff
$$
    \lfloor q_i\rfloor\le a_i\le \lceil q_i\rceil
$$
for $i=1,2,\ldots,n$. 

\section{Apportionment Methods}
\para\jdef{Hamilton's Method}. \label{Hamilton}
For each $i$ calculate the quota $q_i=p_i/h$ of the $i$th state. The sum 
	 $h'=\lfloor q_1\rfloor+\lfloor q_2\rfloor+\cdots+\lfloor q_n\rfloor$
	 will satisfy  $h-n\le h' <h$. (The equality $h=h'$ occurs only
	 in the unlikely event that all the quotas $q_i$ are whole numbers.)
        Award the $i$th state
        either its lower quota $\lfloor q_i\rfloor$ or its upper quota $\lceil q_i\rceil$. 
	The states which receive their upper quota are the $h-h'$ states
	with the larger   fractional parts $q_i-\lfloor q_i\rfloor$.
\arap

\para\jdef{Divisor Methods}. \label{Divisor}
These methods all involve a notion of  ``rounding'' as explained below.
The idea is that the (exact) quota for each state can be determined by
dividing the population of each state by the number of persons
represented (ideally) by each representative, i.e.
$$
        q_i=\frac{p_i}{p}\times h = \frac{p_i}{\lambda_0}, \mbox{ where }
        \lambda_0= \frac{p}{h}.
$$
For a divisor $\lambda$ which is near $\lambda_0$ the numbers
$p_i/\lambda$ will be close to quotas $q_i=p_i/\lambda_0$
but rounding $p_i/\lambda$ might give a different whole
number than  rounding $p_i/\lambda_0$.
The various divisor methods differ in how they define ``rounding''.
The various notions of rounding involve the choice, for each
nonnegative integer $a$ of a number $\mu(a)$ between
$a$ and $a+1$. Then the result of rounding a number $q$
is 
$\lfloor q\rfloor$ if $\lfloor q\rfloor\le q <\mu(\lfloor q\rfloor)$
and 
$\lceil q \rceil$ if $\mu(\lfloor q\rfloor)\le q<\lceil q\rceil$.
Here are the most important divisor methods.
\renewcommand{\arraystretch}{1.5}
\begin{description}
\item\jdef{Jefferson's Method.}  
This method rounds down,
i.e. the result of rounding $q$ is $\lfloor q\rfloor$, the largest
integer less than or equal to $q$. For example,
$$
\lfloor 2.2\rfloor=\lfloor 2.9\rfloor=2=\lfloor 2\rfloor.
$$
For this method the function $\mu$ is $\mu_\mathrm{J}(a)=a+1$.

\item \jdef{Adams' Method.} 
This method rounds up,
i.e. the result of rounding $q$ is $\lceil q\rceil$, the smallest
integer greater than or equal to $q$. For example,
$$
\lceil 2.2 \rceil = \lceil 2.9 \rceil =3= \lceil 3 \rceil.
$$
For this method the function $\mu$ is $\mu_\mathrm{A}(a)=a$.

\item\jdef{Webster's Method.} This method rounds in the usual way,
i.e. the result of rounding $q$ is $\langle q\rangle$ is
defined by
$$
\langle q\rangle=
\left\{\begin{array}{ll}
\lfloor q\rfloor &\mbox{if } \lfloor q\rfloor \le q<A\\
\lceil q\rceil & \mbox{if } A\leq q <\lceil q\rceil
\end{array}
\right.
$$ 
where $A=( \lfloor q\rfloor +\lceil q\rceil)/2$ is the
arithmetic mean (average) of  $\lfloor q\rfloor $ and $\lceil q\rceil$.
For example, the average $A$ of $2$ and $3$ is $2.5$ so,
$$
\langle 2.4\rangle= 2=\langle 2\rangle,\qquad
\langle 2.5\rangle= \langle 2.7\rangle=3.
$$
The notation $\langle q\rangle$ is not standard.
For this method the function $\mu$ is $\mu_\mathrm{W}(a)=a+\frac12$.

\item \jdef{Huntington-Hill Method.}   This method rounds by
comparing $q$ with the geometric mean $G$ of $\lfloor q\rfloor$
and  $\lceil q\rceil$,
i.e. the result of rounding $q$ is $\langle\langle q\rangle\rangle$
defined by
$$
\langle\langle q\rangle\rangle=
\left\{\begin{array}{ll}
\lfloor q\rfloor &\mbox{if } \lfloor q\rfloor \le q< G\\
\lceil q\rceil & \mbox{if } G\leq q <\lceil q\rceil
\end{array}
\right.
$$
where $G=\sqrt{\lfloor q\rfloor \times  \lceil q\rceil}$.
For example, the geometric mean $G$ of $2$ and $3$ is 
$\sqrt{2\times 3}=\sqrt{6}=2.449$
so
$$
\langle\langle 2.43\rangle\rangle=2=\langle\langle 2\rangle\rangle,
\qquad
\langle\langle 2.7\rangle\rangle=3=\langle\langle \sqrt6\rangle\rangle.
$$
The notation
$\langle\langle q\rangle\rangle$ is not standard.
For this method the function $\mu$ is $\mu_\mathrm{H}(a)=\sqrt{a(a+1)}$.


\item \jdef{Dean's Method.}\footnote{James Dean was a mathematician from the
University of Vermont and Dartmouth in the nineteenth century.}  
This method rounds by
comparing $q$ with the harmonic mean $H$ of $\lfloor q\rfloor$
and  $\lceil q\rceil$,
i.e. the result of rounding $q$ is $[[q]]$
defined by
$$
[[ q]]=
\left\{\begin{array}{ll}
\lfloor q\rfloor &\mbox{if } \lfloor q\rfloor \le q<H\\
\lceil q\rceil & \mbox{if }H \leq q <\lceil q\rceil
\end{array}
\right.
$$
where 
$\ds
H=
\frac{2\times\lfloor q\rfloor \times  \lceil q\rceil}{\lfloor q\rfloor +  \lceil q\rceil}.
$
For example, the harmonic mean $H$ of $2$ and $3$ is 
$12/5=2.4$
so
$$
[[ 2.23]]=2=[[2]],
\qquad
[[ 2.7]]=3=[[ 2.4]].
$$
This method is not mentioned in the text~\cite{COMAP} and the notation
$[[q]]$ is not standard.
For this method the function $\mu$ is $\mu_\mathrm{D}(a)=2a(a+1)/(2a+1)$.
\end{description}
\arap


%\rmk Other names for these methods are often used  in the academic literature.
% Hamilton's Method is also called 
% the  Method of \jdef{Largest Remainders}. 
%It was championed by Samuel F. Vinton was a member of the House of Representatives from Ohio in the middle of the 19th century.
%Jefferson's Method   is also called the
%the  Method of  \jdef{Greatest divisors}.
%In Europe it is named after Victor d'Hondt, a Belgian lawyer who lived in the 19th century.
%Adams' Method is also called the
%the  Method of  \jdef{Smallest Divisors}.
%John Quincy Adams was the 6th president and later
%served as  a representative of Massachusetts in the House.
%Webster's Method  is also called the
%the  Method of  \jdef{Major Fractions}.
%In Europe it is named after Andr\'e Sainte-Lagu\"e, a French mathematician.
%In 1911 Joseph Hill (chief statistician of the Census Bureau) proposed the 
%method  which eventually became known as the Huntington-Hill method.  After the First World War a Harvard professor of mathematics,
%Edward Vermilye Huntington (1874-1952),
%revised the ideas of Hill to obtain an apportionment method, which he referred to as the Method of \jdef{Equal Proportions}. 
%It is also called the Method of the  \jdef{Geometric Mean}.  
%Dean's Method is
%also called the Method of the  \jdef{Harmonic Mean}.
%It is named after James Dean who a mathematician at the University of Vermont
%and Dartmouth in the nineteenth  century. 
%\kmr

\section{Fairness} 
Let  $a_i$ and $p_ i$ be the number of seats for state $i$ and its population, respectively. The  fraction $a_i/p_ i$ represents the number of representatives
per person  in state $i$. 
Ideally the persons of state $i$ have the same number of representatives
per person as those of state $j$, i.e.
$$
  \frac{a_i}{p_ i}=\frac{a_j}{p_ j}
$$
but of course this generally won't happen because the numbers $a_i$
must be whole numbers. If 
$$
  \frac{a_i}{p_ i}>\frac{a_j}{p_ j} \eqno(*)
$$
the people in $i$th state have more representatives per person
than those of the $j$th state and are  thus ``better represented''.
A function $T(p_i,a_i,p_j,a_j)$ such that
$$
   T(p_i,a_i,p_j,a_j)>0\iff  \frac{a_i}{p_ i}>\frac{a_j}{p_ j} 
$$
is called a \jdef{fairness measure}. The idea is that the bigger
$T(p_i,a_i,p_j,a_j)$ is the more unfair is the gap
between the representation of $i$th state and the $j$th state.  
An allocation is called \jdef{stable}  for a given fairness measure 
iff  for each pair of states, switching a representative
from the better represented state $i$ to the other state $j$ makes
state $j$ better represented and the measure larger, i.e. iff
$$
T(p_j,a_j+1, p_i,a_i-1)\ge T(p_i,a_i,p_j,a_j)
$$
for all pairs of states.
%%

\begin{theorem}[Huntington]\label{thm: Huntington}
Assume the apportionment $(a_1,a_2,\ldots,a_n)$
results from one of the aforementioned divisor methods.
Then
%%
\begin{itemize}
\item  For   
$\ds
 T(p_i,a_i,p_j,a_j)  =a_j-a_i(p_j/p_i)
$,
 Adams' Method  is stable.
\item For 
$\ds
T(p_i,a_i,p_j,a_j)  =p_i/a_i-p_j/a_j
$,
Dean's Method   is stable. 
\item For   
$\ds
 T(p_i,a_i,p_j,a_j)  =(p_ja_i/p_ia_j)-1
$,
 Huntington-Hill's Method is stable.
\item For   
$\ds
 T(p_i,a_i,p_j,a_j) =a_j/p_j-a_i/p_i
$,
 Webster's Method is stable. 
\item For   
$\ds
 T (p_i,a_i,p_j,a_j)  =a_i(p_j/p_i)-a_i
$,
 Jefferson's Method is stable.
\end{itemize} 
%%
(In all definitions the $i$th state is better represented,
i.e. $p_i/a_i\ge p_j/a_j$.)
\end{theorem}


\rmk It is easy to see the appeal of Webster's Method;
it is the method which attempts to make the
number of representatives per person the same
in all states by minimizing the differences
$p_i/a_i-p_j/a_j$. Similarly Dean's Method
attempts to minimize the differences 
$a_j/p_j- a_i/p_i$ in the average district size
for the various states. The Huntington-Hill Method
attempts to minimize the relative differences
of these quantities as we now explain.

The  \jdef{relative difference} of two numbers
is the result of subtracting the larger from the smaller
and then dividing by the smaller. 
Now the inequality~$(*)$ can be written in four ways:
$$
 \frac{a_i}{p_ i}>\frac{a_j}{p_ j}\iff
 \frac{p_j}{a_j}>\frac{p_i}{a_i} \iff
 a_i>a_j\frac{p_i}{p_ j}\iff 
 a_i\frac{p_j}{p_i}>a_j.
$$
We can get four of the five fairness measures in 
Theorem~\ref{thm: Huntington} from these four ways
by subtracting the right side from the left.
The differences  of the two sides in each of these ways.
The relative difference between the two sides
is the same for all four ways, i.e.
$$
\frac{\frac{a_i}{p_ i}-\frac{a_j}{p_ j}}{\frac{a_j}{p_ j}}=
 \frac{\frac{p_j}{a_j}-\frac{p_i}{a_i}}{\frac{p_i}{a_i}}=
 \frac{a_i-a_j\frac{p_i}{p_ j}}{\frac{p_i}{p_ j}}=
 \frac{a_i\frac{p_j}{p_i}-a_j}{a_j}=\frac{p_ja_i}{p_ia_j}-1.
 $$
 The last expression is the fairness measure for
 the Huntington-Hill Method. Perhaps this is why  Huntington thought
 his  fairness measure was the best, but which measure is the
 best is a value judgment, not a mathematical question.
\kmr


\rmk\label{rmk:paradox}
In~\cite{BY-AMM} Balinski and Young 
devised an ingenious method which they
call the  \jdef{Quota Method}, which 
avoids both 
the Quota Paradox
and the Alabama paradox.% (see paragraph~\ref{AlabamaParadox}). 
However, their Quota Method does not avoid the 
Population Paradox. % (see paragraph~\ref{PopulationParadox}).
In their book~\cite{BY} they show that

\begin{itemize}
\item 
Divisor methods avoid the Alabama Paradox, the  Population Paradox,
and the New State Paradox, % (see paragraph~\ref{NewStatesParadox}),
but
%
\item No divisor method  can avoid the Quota Paradox. In other words,
for any divisor method there exists $(p_1,p_2,\ldots,p_n)$ and $h$
such that the method assigns $(a_1,a_2,\ldots,a_n)$
where either $a_i<\lfloor q_i\rfloor$ for some $i$ or
$\lceil q_i\rceil<q_i$ for some $i$. (Note however,
that the Jefferson Method always gives 
$\lfloor q_i\rfloor\le a_i$ and the Adams Method  always
gives $a_i\le\lceil q_i\rceil$.)
%
\end{itemize}
%
\kmr

 \para Figure~\ref{fig:ap} shows
 the apportionment of the U.S. House of Representatives using various apportionment methods and the 1990 census data.
 Each of the apportionment methods has been modified, where necessary, so that every state gets at least one seat as provided for under the Constitution.
\arap


\appendix



\section{U.S. Constitution}


The way the U.S. House of Representatives is to be constituted is specified in Article 1, Section 2 of the U.S. Constitution.

\begin{quote}
\em 
The House of Representatives shall be composed of members chosen every second year by the people of the several states... No person shall be a representative who shall not have attained to the age of twenty-five years, and been seven years a citizen of the United States, and who shall not, when elected, be an inhabitant of that state in which he shall be chosen.

Representatives and direct taxes shall be apportioned among the several states which may be included within this Union, according to their respective numbers... The actual enumeration shall be made within three years after the first meeting of the Congress of the United States, and within every subsequent term of ten years in such manner as they shall by law direct. The number of representatives shall not exceed one for every thirty thousand, but each state shall have at least one representative; and until such enumeration shall be made, the state of New Hampshire shall be entitled to choose three, Massachusetts eight, Rhode Island and Providence Plantations one, Connecticut five, New-York six, New- Jersey four, Pennsylvania eight, Delaware one, Maryland six, Virginia ten, North-Carolina five, South-Carolina five, and Georgia three.
\end{quote}


\section{Apportionment Timeline}

\begin{description}
\item{1787} Constitution drafted by the Constitutional Convention
\item{1790} First Census
\item{1791} After much debate, Congress approved a bill for a 120 member House and Alexander Hamilton's method to apportion seats among the states. Hamilton's method won out over Thomas Jefferson's method. Hamilton's method was supported by the Federalists while Jefferson's method was supported by the Republicans.
 President Washington vetoes the above bill (first veto in U.S. history!).
The House, unable to override the veto, passed a new bill for a 105 member House and Jefferson's method to apportion seats among the states.
(This method was used until 1840.)
\item{1822} Rep.\ William Lowndes (SC) proposed an apportionment method now known as the Lowndes's method. It never passed.
\item{1832} John Quincy Adams (former President and, at this time, a representative from Massachusetts) proposes the Adams's method for apportionment. It fails.
 Senator Daniel Webster (MA) proposes Webster's method. It fails.
 Congress passes a bill that retains Jefferson's method but changes the size of the House to 240.
\item{1842} Webster's method is adopted and the size of the House is reduced to 223.
\item{1852} Rep.\ Samuel Vinton (OH) proposed a bill adopting Hamilton's method with a House size of 233. Congress passes this bill with a change to a House size of 234, a size for which Hamilton's and Webster's methods give the same apportionment.
\item{1872} A very confusing year! First the House size was chosen to be 283 so that Hamilton's and Webster's methods would again agree. After much political infighting, 9 more seats were added and the final apportionment did not agree with either method.
\item{1876} Rutherford B. Hayes became president based on the botched apportionment of 1872. The electoral college vote was 185 for Hayes and 184 for Tilden. Tilden would have won if the correct apportionment as required by law had been used.
\item{1880} The Alabama Paradox surfaced as a major flaw of Hamilton's method.
\item{1882} Concerns continued over the flaws in Hamilton's method. Congress passed a bill that kept Hamilton's method but changed the House size to 325 so that Hamilton's method gave the same apportionment as Webster's.
\item{1901} The Census Bureau gave Congress tables showing apportionments based on Hamilton's method for all House sizes between 350 and 400.
 For all House sizes in this range (except for 357) Colorado would get 3 seats. For 357, Colorado would get 2 seats. Rep.\ Albert Hopkins (IL), chairman of the House Committee on Apportionment, submitted a bill using a House size of 357--causing an uproar.
 Congress defeated Hopkins' bill and instead adopted Webster's method with a House size of 386.
\item{1907} Oklahoma joined the union and the New States Paradox was discovered as a result.
\item{1911} Webster's method was readopted with a House size of 433. A provision was made to give Arizona and New Mexico each 1 seat if they were admitted to the union.
\item{1921} No reapportionment was done after the 1920 census.
\item{1929} The Speaker of the House, Nicholas Longworth, suggested that the National Academy of Sciences make an objective study of the problem. A committee of mathematicians consisting of G.A. Bliss (1876-1951), E.W. Brown (1866-1938), L. P. Eisenhart (1876-1965) and Raymond Pearl (1879-1940) was formed to investigate the situation. 
Their report indicated support for Huntington's method. (in 1949  an even more prestigious group of mathematicians provided a report to the President of the National Academy of Sciences on apportionment methods. The report was signed by Harold Marston Morse (1892-1977), John Von Neumann (1903-1957), and Luther Eisenhart.
Their report again supported the Huntington-Hill method.) 
\item{1931} Webster's method was adopted with a House size of 435.
\item{1941} Senator Vandenburg of Michigan suggested that Arkansas prefered
the Huntington-Hill Method over the Webster Method in order to gain
a single seat. The only difference between these methods for the
1940 census was that the former gave Arkansas 7 and Michigan 17
while the latter  gave Arkansas 6 and Michigan 18.
The Huntington-Hill method was adopted with a House size of 435.
\item{1990} The U.S. Census Bureau, for only the second time since 1900, allocated Defense Department overseas employees for apportionment purposes. This resulted in Massachusetts losing a seat to Washington. Massachusetts filed suit.
\item{1992} Overruling a U. S. district court decision, the U. S. Supreme Court ruled against Massachusetts on technical grounds involving ``the separation of powers and the unique constitutional position of the President.'' (The President is charged with calculating and transmitting the apportionment to Congress.)
\item{1992} Montana challenged the constitutionality of the Huntington-Hill method (Montana v. U.S. Dept.\ of Commerce). The Supreme Court upheld the method. Montana was upset because it lost a seat to Washington based on the results of the 1990 census.
\end{description}

\section{Notes}\label{app:notes}
I downloaded the time line,  Figure~\ref{fig:ap}, and
the material in pargraphs~\ref{AlabamaParadox}-\ref{NewStatesParadox}
from~\cite{Bowen}.
The material in paragraph~\ref{QuotaParadox} comes from~\cite{COMAP}. 



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\begin{thebibliography}{99}
\bibitem{Bowen} Larry Bowen:
MATH 103: Introduction to Contemporary Mathematics,
\url{http://www.ctl.ua.edu/math103}
\bibitem{Apportionment} 
\url{http://www.ams.org/featurecolumn/archive/apportion1.html}
\bibitem{Apportionment II} 
\url{http://www.ams.org/featurecolumn/archive/apportionII1.html}\
\bibitem{BY-AMM}  Michel Balinski and H. Peyton Young:
The quota method of apportionment, Amer. Math. Monthly 82 (1975) 701-730.
\bibitem{BY}  Michel Balinski and H. Peyton Young:
{\em Fair Representation}, Yale University Press, New Haven, 1982, (2nd. ed.), Brookings Institution, 2001.
\bibitem{COMAP} Comap:
{\em For all Practical Purposes}, 7th Edition, 
W.H. Freeman, 2005.
\end{thebibliography}