In the United States, it is often a raw point discussing the issue of redistricting, or so unfavorably called, "Gerrymandering."

Background: In the United States House of Representatives, each State is allotted a number of representatives based on the State's population. Within the state, each of its representatives are assigned a geographical "district." The point of contention comes with the fact that a states Governor may periodically get to redraw the districting lines.

My question is: mathematically, probabilistically, can this work? My intuition tells me that if one line is moved over to improve the chances of one side winning one district, won't that just hurt the chances of winning the other?

  • $\begingroup$ As long as you were going to lose that other district, it doesn't matter. $\endgroup$ – Thomas Andrews Oct 9 '13 at 1:33
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    $\begingroup$ Terrible intuition, unfortunately. Here is a well thought-out alternative. $\endgroup$ – Andrés E. Caicedo Oct 9 '13 at 1:33

Imagine a state with 12 million inhabitants, and 6 districts. The voters are equally split with 6 million Democrats and 6 million Republicans. Note that each district contains two million voters, and it seems fair that there should be an even split in representation.

However, the districts are gerrymandered so that district one contains only 2 million Democrats and no Republicans. The other 4 million Democrats and 6 million Republicans are evenly distributed through the other districts.

Now there are five districts, each 60% Republican and 40% Democrat - as a result, we expect one Democratic district and 5 Republican districts. So gerrymandering significantly changed the result.

This is an extreme case, but if we were to even set it up so that one district contains 1.5 million Democrats with the remaining Democrats evenly distributed, we'd find that the remaining five districts have moderate Republican advantages, even though the first district is overwhelmingly Democratic.

  • $\begingroup$ Ok sounds good. Is there away to come about an "upper bound" or a "maximum gerrymander"? (Don't have permission to accept your answer for another few min.) $\endgroup$ – Mr. AM Oct 9 '13 at 1:43
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    $\begingroup$ @Mr.AM The maximum amount of Gerrymandering would be to put ALL of the members of one party into a single district and then to control all of the others. Of course, this is completely impractical given the geographic dispersal of voters. $\endgroup$ – Kevin Driscoll Oct 9 '13 at 1:59

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