English: How Gerrymandering can influence the result of elections in a non-proportional system. In each case there are 9 blue squares and 6 orange dots in total. Depending on how the 3 districts are designed, there can be a 3:0 or a 2:1 win for the blue party, or even a 2:1 win for the state-wide minority party (orange).

Example rationale

In an election with 2 parties, 3 is the smallest number of districts to produce a winning minority, if ties cannot favour either party (so must not happen). Obviously, 1 district can lead only to the stronger party winning. With 2 districts, the best the weaker party can do is to force a 1-1 draw.

I then calculated the fewest voters that 3 districts could be equally divided to yield 3-0 (a), 2-1 (b) and 1-2 (c) results. Let's call the stronger party S and the weaker one W. S has s voters and W has w voters, making a total of n = s + w voters. The critical cases to satisfy are extreme cases (a) and (c).

Case (a): In each district, S must beat W by the slimmest possible margin i.e. 1 vote (assuming no tie-breaks). Thus for 3 districts, S beats W by 3 votes in total, so s = w + 3.

Case (c): In one district, S must beat W by the greatest possible margin i.e. 100% votes, using up (n/3) S votes. In the 2 remaining districts, W must beat S by the slimmest possible margin i.e. 1 vote (i.e. 2 votes in total), hence w = (s - n/3) + 2. ("- n/3" is for the votes used up earlier.) Substituting n = s + w gives w = s - (s + w)/3 + 2 or 4w/3 = 2s/3 + 2. Multiplying both sides by 3/2 gives 2w = s + 3.

Solving the 2 simultaneous equations gives s = 9 and w = 6, as in the example. cmɢʟee☺ τaʟκ 00:21, 17 July 2011 (UTC)