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\noindent
 Eric   Rasmusen \\
January 19, 2021
 
\noindent
{  {\bf   ``How Should We Measure Polarization in Congress?''
}}
 
   
 Suppose we have an index of left-right ideology for all members of Congress, and they are in two parties, Democrat and Republican. What do we mean by saying  Congress is more polarized than ten years ago?

1.  The  average opinions diverge more: The average Democratic ideological position becomes further from the average Republican one. Call this ``Party Polarization''?   \\
2.  The clusters become  more dispersed and more overlapping: more Democrats are close to more Republicans and vice versa.  Call this ``Cluster Polarization''?   
 

  Suppose ideology is from left to right, with 0 in the center. Let the Democratic congressmen be  located at  circles $d_1, \ldots d_D$ and the Republicans at dots $r_1, \ldots r_R$, with mean values $\mu_d$ and $\mu_r$.
  \includegraphics[width=.3\linewidth]{polarization4.png}
\includegraphics[width=.3\linewidth]{polarization2.png}
\includegraphics[width=.3\linewidth]{polarization3.png}
\begin{equation} \label{e0}
   P^2  \equiv \frac{1}{2}\cdot \frac{ \sqrt{ \Sigma_{i=1}^D(d_i  - \mu_r)^2/D}}  { \sqrt{ \Sigma_{i=1}^D (d_i  - \mu_d)^2/D}} +\frac{1}{2} \cdot \frac{ \sqrt{ \Sigma_{i=1}^R(r_i  - \mu_d)^2/R}}  { \sqrt{ \Sigma_{i=1}^R (r_i  - \mu_r)^2/R}}   - 1
\end{equation}
\begin{equation} \label{e0}
   P^3  \equiv   1 - \frac{1}{ \frac{1}{2}\cdot \frac{ \sqrt{ \Sigma_{i=1}^D(d_i  - \mu_r)^2/D}}  { \sqrt{ \Sigma_{i=1}^D (d_i  - \mu_d)^2/D}} +\frac{1}{2} \cdot \frac{ \sqrt{ \Sigma_{i=1}^R(r_i  - \mu_d)^2/R}}  { \sqrt{ \Sigma_{i=1}^R (r_i  - \mu_r)^2/R}}  } 
\end{equation}
\begin{equation} \label{e0}
   P^3  =  1 - \frac{.5} {   \frac{ \sigma_{d-r}}  {  \sigma_d } +     \frac{ \sigma_{r-d} }  {  \sigma_r   } }
\end{equation}
 1.  Two parties with identical means have zero polarization. \\
 2.  Polarization rises with the distance between party means.\\
3.  Polarization goes to the maximum level ($\infty$ or 1) as the distance between means goes to $\infty$. \\
4.  Moving a party member closer to the party mean increases polarization (IS THIS TRUE?)\\
\begin{small}
\begin{tabular} { l|llll |l | lll | l|lll l  |l l }
    &   &    &    &  &   &    &    & && && &&  &  \\
Congressmen:     &  $d_1$ &  $d_2$  &  $d_3$ & $d_4$ &   &   $r_1$ &  $r_2$  &  $r_3$& & $\mu_d$ &$\mu_r$ & $\sigma_d$ & $\sigma_r$ &  $P^2$  & $P^3$  \\
    &   &    &    &  &   &    &    & && && &&& \\
\hline
    &   &    &    &  &   &    &    & && && &&& \\
Congress 1: Unpolarized                 &  -2 &  -1& \;1 & \;2 &   &   -3 &  \;0 &  3 &  &\;0 & 0 &\;2.5 &6&  0.0& .00 \\
Congress 2:  Polarized both ways  &  -4 & -3 & -3 & -2  &   &   \;2 & 3 &  4 &  & -3 & 3 &0.7 & 0.7&  7.0 & .97\\
Congress 3:  Parties differ  &  -6 & -6 & \;0 & \;4  &   &  \;0 & 2 &   7 &  & -2 & 3 & 4.2  & 2.9 & 0.8?? &.86\\
Congress 4:  Congressmen concentrated  &  -1 & -1 &\;0 & \;0  &   &   \;1 & 1 &  2&  & -0.5  & .5 &\;.5 &0.7& 2.9  & .94\\
\end{tabular}
\end{small}
\begin{quotation}
``1. The difference between the location of the median Democrat and the median
Republican. This measure gets at one aspect of inter-party heterogeneity.\\
2. The ratio of the standard deviation of ideal points in the Democratic party to that of
the full House, which indicates variation in intra-party homogeneity.\\
3. The proportion of overlap between the two parties’ distribution of ideal points
subtracted from one. Overlap is measured by the minimum number of ideal points
that would have to be changed to yield a complete separation of the two parties, with
all Democrats' ideal points being to the left of all Republicans’ ideal points.\\
4. The R2
 resulting from regressing the Member’s ideal point location on party
affiliation.'' (Aldrich-Rohde (1998))
\end{quotation}

 

 

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