The People at
www.RangeVoting.com have an explanation for the cube root rule (in addition to an interesting voting method idea):
Suppose 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 do 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 which minimizes total communication to make something that Legislator wants, get done. This 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 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 each Legislator typically must walk a distance proportionate to L to reach a random target Legislator. So 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·(L
2), by choice of L, now getting the cube-root law, L
∝ P
(1/3).
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·L
(1.5), by choice of L, now getting the two-fifths power law, L
∝ P
(2/5).
So it seems as though some power law is the "right answer" – although perhaps it is now not so clear what the correct power is! (I do not see any good argument for L∝log(P).)
