proj-liquidRepublic-derivationOfConstituencyLimitFormula

The formula for the member number limit is constituency_member_limit = 5^((1 + sqrt(1 + 8*(log(electorate_size/constituency_votes)/log(5))))/2), where electorate_size is the total number of voters in the organization, and constituency_votes is the total number of votes controlled by this constituency. Why?

First, why are there 5 people on the Board?

• there are at least 2 sides to every controversial issue
• having only two sides is not enough, because often there is more than two sides, or more than one way to break an issue into sides; so we want at least 3
• having only one representative per side is not enough, because due to noise (or sometimes even corruption), ppl whose constituencies would be on opposite sides will find themselves agreeing; so we need some redundancy; the first number after 3 is 4
• but we want an even number so there are not unbreakable tie votes; and we'd prefer a prime number so there cannot be tied votes when there are more than 2 choices. The first odd number (and the first prime) greater than 4 is 5
• we want the minimal number meeting our other criteria, because meetings are much more efficient with less people. So 5 it is.

Now, we considering an imaginary 'ideal' organization (we are just calling it that, we're not going to give any justification for this form being 'ideal') in which constituencies are organized in the following way:

• the CEO is a Delegate of the Board
• the Board is a constituency of 5 members (the Board members)
• each member of the Board is a Delegate of a lower-level Constituency with 5*5 members
• each member of those constituencies is a Delegate of a lower-level Constituency with 5*5*5 members
• ... etc

How many voters are there in total? It depends on how many levels down we go.

• If there are 3 levels, then there is the CEO, then the Board, and then a Constituency for each Board Member, each of which contains 5*5 voters. So there are 5*(5*5) voters.
• If there are 4 levels, then there is the CEO, then the Board, and then a Constituency for each Board Member, and a Constituency for each member of each of those constituencies, each of which contains 5*5*5 voters. So there are 5*(5*5)*(5*5*5) voters.

The general formula here is total_voters = 5^(x*(x-1)/2), where 'x' is the number of levels.

Now, what if we had 3 levels, but less than 5*(5*5) voters? We'd still have the CEO, then the Board, then a Constituency for each member. How many of those lowest-level Constituencies do we have? We have 5. So each of these 5 constituencies would have electorate_size/5 voters. And if we had 4 levels, we'd have 5*(5*5) lowest-level constituencies, so each one would have electorate_size/5^2 voters. The formula is electorate_size/5^(x*(x-1)/2). todo

Now, if we are given a number of voters in such a structure, compute the number of levels needed. That is, given a = electorate_size/5^(x*(x-1)/2), solve for x. The solution is: todo

x = (1 + sqrt(1 + 8*(log5(n/a))))/2

(todo explain why we set a = indirect_constituency_votes)

now member_limit = 5^x

(todo note that we've set our limit to the number of members in the 'ideal' structure)

so substituting in x, we get:

member_limit = 5((1. + sqrt(1. + 8.*(log(electorate_size/votes)/log(5.))))/2.)

To display the effect of this formula in Python:

electorate_size = 1000; largest_permitted = 5((-1. + sqrt(1. + 8.*log(float(electorate_size))/log(5.)))/2.); constituency_votes_array = arange(ceil(largest_permitted),electorate_size); figure(); plot(constituency_votes_array, 5((1. + sqrt(1. + 8.*(log(float(electorate_size)/constituency_votes_array)/log(5.))))/2.)); xlabel('constituency_votes'); ylabel('constituency_member_limit');

Note: largest_permitted there is the largest constituency you can have in which each member contributes at least 1 vote (with a voting strength of 1). This reaches 150, Dunbar's number, when electorate_size is about 30000. This is close to the ideal with 4 levels, CEO, Board, 5*5, 5*5*5, with about 15k members. That is to say, if you demand an organization of this form in which no constituency is larger than Dunbar's number, you'll be able to support on the order of 15k, with in which there is about one to two Delegates in between each member and the Board, and each Board Member reports to about 25 delegates.