Note: this document is under construction! See the introduction.

It is not even a finished draft yet, just parts of a draft interspered with notes.

Motivation for the size of the senate, the number of layers, and the minimal vote thresholds for each layer

First, I decided what I thought the size of the senate should be. I wanted a formula that took as input the size of the electorate, and yielded the size of the Senate. I wanted this formula to be scalable, that is, to give reasonable answers for moderately small groups and also for groups the size of the entire population of Earth, and larger. In numerical terms, I require a rule that scales for groups of about 1000 up to groups of about 9 billion. If possible, it would be nice if the rule would scale down to about 15 and up to about 1e12.

Some approximate example group sizes to help get a handle on these numbers:

Size of representative bodies in this system with those numbers, and 8,15 and 1e18 thrown in just to see what happens with really small and with ridiculously large groups4

:

By the reasoning given elsewhere, I felt that the current size of the U.S. Senate, 100, is significantly too large. The Senate, I feel, should be as small as possible, a tiny body that is small enough to sit down together and have serious discussions; the only reason it needs to grow with population at all is to allow all of the major political viewpoints to have a voice (and to allow each major viewpoint to be represented by more than one member so that the personal preferences of one person doesn't overly dominate the expression of the will of their faction).

Let me emphasize that: I see the proper role of the Senate as a guide for public discourse and a focus for attention. So, the number of Senators should grow in proportion to the number of points of view that the public should be made to focus upon. This is in opposition to an idea in which the number of Senators should grow roughly in proportion to the size of the population itself, in order to ensure each faction of some fixed size at least one representative in the Senate.

Now, the simple fact is that, no matter how much you would like the public to devote sufficient time to be able to understand all of the subtleties of politics, that people will refuse (and rightly so) to spend too much time on this, and so the media will simplify the world into a small number of factions. In the United States today, by looking at how the media talks about political factions, I guess they divide domestic politics into about 6 factions: (religious Republicans, other Republicans, anti-corporate Democrats, other Democrats, greens, "other"). So let's say that the magic number is between 5 and 7. Since we will allow each of these factions (except "other") to field 2 reps, we have a body of at least 9 to 13. Now, if even the small factions such as the greens have 2 reps, surely some of the larger ones will have more. So we want a Senate of about 13 to 24 members.

Nth-root formulas are one alterantive, but n must be 6 or more in order to bring the Senate below 50 for 350,000,000 people. I worry that this scheme would not scale too well if the population continues to increase exponentially. For example, for 18 trillion people, the 6-th root rule yields about 162 senators, which seems like a lot to me. Does this fit with out intuitions? This would mean that with a population 3000 times larger than the current population of the entire Earth, the media would pretend that there is about 40-56 factions. This seems like too many to me.

On the other end of the scale, this formula gives Senate sizes that I feel are too slightly too low for small groups. For example, the 6-th root rule would give 3 senators for a group of 500, which seems like too few; would a 500 employee company have a board of directors of only 3 people?

In addition, the 6 seems like a kludge.

All in all a 6-th root rule would not be too bad, but log seems better. Log gives answers that seem better for very small and very large groups.

To start with, I tried natural log. log(350000000) \approx 19.7, which seems like a reasonable size for the Senate of the United States. log(25e9) \approx 24, so even with a population of more than twice current projections for the maximum world population within the forseeable future, this rule yields a suitably small Senate size. log(18e12) is only 30.5. Does this fit with out intuitions? This would mean that even with a population 3000 times larger than the current population of the entire Earth, the media would pretend that there is only about 7-11 factions. This seems too small to me, but I think it's closer than the guess of 40.

On the other end of the scale, for a 30 person group, which is about the size of the computational neurobiology graduate subprogram that until recently i was part of, we get a senate of 3.4, which seems about right. For a 65 person group, about the size of my class in elementary school we get a senate of 4.2, which also seems reasonable. For a

One model that would justify this is if we assume that the cognitive cost of holding in mind n factions is exponential in the number of factions, and the benefits of adding a new faction is directly proportional to the number of people who will be represented by that faction.

In this case, if we have P people and F factions, then that must be because the benefits of creating the last faction outweighed the costs. The benefits are proportional to P/F. The costs are proportional to exp(F) - exp(F-1). We have

exp(F) - exp(F-1) = P/F

F*(exp(F) - exp(F-1)) = P

F*(F e^{F-1} - e^{F-1}) = P

F*e^{F-1}*(F - 1) = P

ln F*e^{F-1}*(F - 1) = ln P

ln (F*(F-1)) + ln(e^{F-1}) = ln P

ln (F*(F-1)) + F-1 = ln P

F = O(ln P)

Natural log seems to wor

For example, according to http://physics.ucsd.edu/was-sdphul/dept/pr/gintro.html, UCSD currently has on the order of 2500 graduate students. The Senate would correspond to the size of the Graduate Student Council, our student government.

\begin{comment} (log_S P -1 - L)/(\sum_{j=1}^L j) = k

log_S P = 1 + L + k*(\sum_{j=1}^L j)

log_S P = log_S S^{1 + L + k*(\sum_{j=1}^L j)}

P = S^{1 + L + k*(\sum_{j=1}^L j)}

P = S^{1 + L + k + 2*k + .... + ik + .... + L*K}

P = S^{1 + (1 + k) + (1 + 2*k) + .... + (1 + ik) + .... + (1 + L*K)}

P = \prod_{i=0}^L S^1 S^{1 + k} S^{1 + 2*k} * ... * S^{1 + ik} * .... * S^{1 + L*K}

P = \prod_{i=0}^L S^{1+k i}

x = \prod_{i=0}^{ceil(log(ceil(log(x))))} (ceil(log(x))^{1+k i}

If each individual in the electorate

5 \end{comment}

Glossary

Corporation: An abstract entity representing a group of people in some capacity which is not itself capable of consciousness.

Person: Sentient being.

Renew: A Renewable bill may be Renewed by being repassed, exactly as before, except with dates advanced to the current time period.6