Minimizing pairwise inequality

Statement

One way to evaluate apportionment methods is by whether they minimize the amount of inequality between pairs of agents. Clearly, inequality should take into account the different entitlements: if $a_i/t_i = a_j / t_j$ then the agents are treated "equally" (w.r.t. their entitlements); otherwise, if $a_i/t_i > a_j / t_j$ then agent $i$ is favored, and if $a_i/t_i < a_j / t_j$ then agent $j$ is favored. However, since there are 16 ways to rearrange the equality $a_i/t_i = a_j / t_j$, there are correspondingly many ways by which inequality can be defined.

• $|a_i/t_i - a_j / t_j|$. Webster's method is the unique apportionment method in which, for each pair of agents $i$ and $j$, this difference is minimized (that is, moving a seat from $i$ to $j$ or vice versa would not make the difference smaller).
• $a_i - (t_i/t_j)a_j$ for $a_i/t_i \geq a_j/t_j$ This leads to Adams's method.
• $a_i(t_j/t_i) - a_j$ for $a_i/t_i \geq a_j/t_j$. This leads to Jefferson's method.
• $|t_i/a_i - t_j /a_j|$. This leads to Dean's method.
• $\left|\frac{a_i/t_i}{a_j/t_j} - 1\right|$. This leads to the Huntington-Hill method.

This analysis was done by Huntington in the 1920s. Some of the possibilities do not lead to a stable solution. For example, if we define inequality as $|a_i/a_j - t_i/t_j|$, then there are instances in which, for any allocation, moving a seat from one agent to another might decrease their pairwise inequality. There is an example with 3 states with populations (737,534,329) and 16 seats.