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    <p>The problem you're trying to solve is generally known as an apportionment problem or vote allocation problem. This is the same problem as assigning the number of seats in the US House of Representatives to each state.</p> <p>The problem of robustness that your approach (known as Hamilton's method or the method of largest remainder) fails to have is known as the <a href="http://en.wikipedia.org/wiki/Apportionment_paradox" rel="nofollow">Alabama Paradox</a>. From the Wikipedia article, "The Alabama paradox was discovered in 1880, when it was found that increasing the total number of seats would decrease Alabama's share from 8 to 7."</p> <p>Historically, at least four different methods have been used in US: Jefferson's method, Hamilton's method, Webster's method and the current <a href="http://en.wikipedia.org/wiki/Huntington%E2%80%93Hill_method" rel="nofollow">Huntington-Hill's method</a> used since 1941.</p> <p>The idea behind these latter methods is the following. Let <code>D = N/k</code>, the total population divided by the number of seats/chaperones. Then let <code>d = D</code>, and modify <code>d</code> until the rounding <code>k_i = round(G_i/d)</code> adds up to the correct number of seats, i.e.</p> <p>   round(G_1/d) + round(G_2/d) + ... + round(G_m/d) = k</p> <p>The catch is in how the function <code>round</code> works. Webster's approach rounds in the usual sense: weakly above .5 go up and strictly below .5 go down, which is rather much like using an arithmetic mean. The Huntington-Hill method is based on the idea of using a geometric mean instead. There is a nice summary of these methods <a href="http://www.ctl.ua.edu/math103/apportionment/appmeth.htm" rel="nofollow">here</a>. Note that all of these divisor algorithms are flawed in that they violate the Quota Rule: a state is not guaranteed to get at least <code>floor(G_m/D)</code> representatives.</p> <p>If you want to play around with this more, there is an excellent article about this on <a href="http://www.cut-the-knot.org/ctk/Democracy.shtml" rel="nofollow">Cut The Knot</a> complete with history, equations, and fun applets.</p>
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    1. COThose are great links. Thanks.
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    2. COhttp://en.wikipedia.org/wiki/Pigeonhole_principle
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