# [EM] Apportionment and minimum desirable legislature size?

Raph Frank raphfrk at gmail.com
Mon Apr 18 17:45:04 PDT 2011

```On Tue, Apr 19, 2011 at 1:03 AM, Alex Small <alex_small2002 at yahoo.com> wrote:
> 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.

What will happen is that lots of states would get zero seats.  The US
compensates for this by requiring that all states get 1 seat.  This
favours smaller states.

> 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%.

Right, the error is reduced as you add more seats.

> 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?

I would suggest that you set a minimum number of seats per state and a
minimum number of total seats.

If every state has 3 seats, then the max difference in reps per seat

This is much better than a state with 0.1 seats worth of population
getting a full seat.

Effectively, rather than giving states "free" seats, just keep adding
seats until all states get at least 3.

Ofc, this means that the size of the house would depend on the
population of the smallest state (roughly, 3 * (national population) /
(state population) )

Wyoming is the smallest state with a population of 563k, while the US
population for 50 states is 308 million.  This gives a house size of
roughly 1641 required for Wyoming to get 3 seats.

> 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.

Most of the seats will go to the larger states since they have more
population.  If a state gets 20 of the first 100 seats, it will get
roughly 20 of the 2nd 100 too.

```