# [EM] Apportionment and minimum desirable legislature size?

Alex Small alex_small2002 at yahoo.com
Mon Apr 18 17:03:44 PDT 2011

```I've been following some of the apportionment threads, and wanted to pose a related question:  Is there an optimal legislature size from the standpoint of apportionment?

Let's say that you decide to apportion seats among constituencies.  We'll leave aside the issue of whether those constituencies will elect their representatives from districts within the constituencies or proportionally in an at-large election or whatever.  We'll just stipulate that each constituency gets at least one seat, and that the number they get is roughly linear in their population.

The common way to do this is you start off with each constituency having 1 seat, and then you hand out the remaining seats one-at-a-time based on some formula.  Different people have proposed different formulas, either based on who is most under-represented at the moment, or who would be least over-represented with an extra seat, or whatever, but regardless of what formula you use, let's just agree that the method goes one-at-a-time.

The first few seats will mostly go to the largest constituencies.  So, if you have 50 states with a broad distribution of populations, and you have a legislative chamber with 60 seats, most of your states will be under-represented.  If you expand to 1000 seats or whatever, your biggest disproportionality problem will probably be at the low end of the scale, i.e. adding or removing 1 seat from a state with 5 representatives changes the ratio of reps to population by 20%, while adding or removing 1 seat from a state with 100 reps changes the ratio of reps to population by 1%.

If your only concern is numerical fairness, of course, you should just keep adding more seats, until you reach something like the square root of the population.  But if you want to keep your legislature size low for practical reasons (i.e. in the 3 digit range), is there any good quantitative criterion for minimum size?

My intuition tells me that either there is no fundamental criterion here, but a practical one is to keep adding until most of the changes are in the median state.  Or keep adding until the ratio of population to reps for the median state equals the ratio of population to reps for the largest state, plus or minus some small tolerance that you decide on.

Anybody have ideas for formalizing this issue?

Alex Small
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