I suppose this post is at least in part a book review. I recently finished reading *Numbers Rule: The Vexing Mathematics of Democracy, From Plato to the Present*, by George Szpiro. If you have an interest in government, not just in theory but in practical implementation, then I highly recommend this book. It contains a chronological progression of our attempts to govern ourselves, and the fascinating and frustrating problems that can arise… or in many cases, are *guaranteed* to arise.

The book addresses two main challenges, both of which have been discussed here before. The first challenge is how a group can collectively make decisions, elect leaders, etc. This story has an interesting cast of characters. For example, Lewis Carroll proposed a very interesting election method that has a lot of theoretical advantages in its favor… if not for the fact that actually *implementing* the method turns out to be NP-hard. (As a side benefit, I learned about a new complexity class, , or “parallel access to NP,” for which this problem is complete.)

The second challenge addressed in the book is the focus of this post: how to apportion representatives. Just last month, the U. S. Census released its updated state population data used to reapportion the 435 seats in the House of Representatives, as directed by the Constitution: “*Representatives shall be apportioned among the several States according to their respective numbers.*” Unfortunately, this is not easy to do; when trying to allocate integral numbers of representatives to those 435 seats, some states inevitably end up with greater or less voting power in Congress than other states. With this latest 2010 reapportionment, residents of Rhode Island will have the greatest per capita representation, about 1.88 times that of the most under-represented residents of Montana.

The history of this problem and its many proposed solutions are described with great detail and not a little humor in Szpiro’s book. (Before you think this is a small problem, keep in mind that there are debates and lawsuits almost every ten years; in the 1920s, Congress had such a hard time with this problem that they directly violated the Constitution by simply giving up and leaving the apportionment as it had been for the previous 10 years.)

But one potential solution is mentioned only in passing, one that I think deserves more attention: why must we stick to nice, round, whole numbers? Why not allow some– or even all– representatives to have a fractional, or at least non-integral, vote?

This is not as crazy, nor as complicated, as it sounds. Szpiro suggests simply adding a few “fractional” representatives, at most one for each state, each of whose vote corresponds to the fractional “left over” population that causes all of the paradoxes and problems with most of the apportionment methods. But I think a cleaner approach would be to let *all* of the representatives have a *non-integral* vote, in exact proportion to the actual population of the corresponding district.

As is usually the case with my ideas, this one is not new, being most recently proposed and described in some detail in this 2008 article by Temple law professor Jurij Toplak. The article suggests leaving the composition of the House the way it is, even using the current Huntington-Hill method of apportionment of warm bodies, although I don’t see why this is necessary. That is, states could even choose the *number* of their representatives, within certain limits, based on their desired “expressive power” of diversity of opinions within their state. Of course, this could present more interesting districting problems, as we have also discussed here before. One step at a time, I suppose.