## Answer to Puzzle #63: Optimum size for legislature

Puzzle (ACE):
Consider the amount of communication a congressman has to do with both his constituency and with his fellow legislators.

1. Argue that it is minimized if the size of the legislature is proportional to the square root of the country's population P.
2. But can you argue for other "optimal legislature size" functions such as P1/3 or P2/5 or log(P)?

Some constant fraction of the constituency probably wants to communicate with the legislator, which is c·P/L communication with him where c is a constant, P is the population of the country, and L is the cardinality of the legislature.

Meanwhile each legislator needs to communicate with all the others (or anyhow a constant fraction of them) to get things done (e.g. convince them to enact something he wants). That's about k·L communication for each legislator per thing he wants to do (where k is another constant).

If we now minimize

c·P/L + k·L
by choice of L, we get the square-root law
L = (P · c/k)1/2,    i.e.    L ∝ √P
which is the "optimum" legislature size that minimizes total communication to make something that legislator wants, get done. The above formula is "optimum" if we assume each legislator aims for some constant number of goals per (fixed length) term.

Suppose the communication with the constituents is by mail or e-email or telephone; but the communication with fellow legislators is face-to-face 1-on-1 meetings in random order. Further, all the legislators are along one long corridor. Then Joe Legislator typically must walk distance L to reach a random target legislator. Then the difficulty of communication with the L-1 others is then not proportional to L, but rather to its square. In that case we need instead to optimize by minimizing

c·P/L + k·L2
by choice of L, now getting the cube-root law
L ∝ P1/3.
(One can also get the cube-root law if we assume each legislator monitors communications between all pairs of other legislators, although that does not seem realistic to me.) If instead of one corridor, they sit in a 2-dimensional grid, then the typical walk-distance is proportional to √L. In that case optimizing is instead to minimize
c·P/L + k·L1.5
by choice of L, now getting the two-fifths power law
L ∝ P2/5.

Similarly if we had assumed a three-dimensional grid we would have found L∝P3/7. If we had assumed his constituents had to travel distances proportional to (P/L)1/2 to meet with the legislator, then...

So it seems that, no matter which assumption we make, some power law is the "right answer" – albeit it perhaps now is less clear what the correct power is! The best power seems fairly clearly to lie within the real interval [1/4, 1/2].

[I do not see any good argument for L∝log(P).]

ACE claims they did a survey of cross-country data, finding a good fit of L to the cube root of 2·P·W·A where P=population, W=fraction of it of "working age", and A=literacy fraction. "Very few countries have assemblies that are larger than twice the size predicted by this equation and only a few have assemblies that are smaller than half [of it]."