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  "body": "\n## Statement\n\nOne 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 $a_i/t_i \u003e a_j / t_j$ then agent $i$ is favored, and if $a_i/t_i \u003c a_j / t_j$ 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.\n\n- $|a_i/t_i - a_j / t_j|$. 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).\n- $a_i - (t_i/t_j)a_j$ for $a_i/t_i \\geq a_j/t_j$ This leads to Adams's method.\n- $a_i(t_j/t_i) - a_j$ for $a_i/t_i \\geq a_j/t_j$. This leads to Jefferson's method.\n- $|t_i/a_i - t_j /a_j|$. This leads to Dean's method.\n- $\\left|\\frac{a_i/t_i}{a_j/t_j} - 1\\right|$. This leads to the Huntington-Hill method.\n\nThis 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.\n",
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