Wednesday, November 21, 2007

Assigning Seats in an Assembly

[NOTE: Though I still consider the underlying arguments below sound, I have determined that the actual procedure described below for assigning seats is less than optimal. For the new method that produces an assembly of an exact size (800 individuals), please go to]

Dear citizens of the world,

A few years ago, I developed an algorithm for assigning seats in a global assembly as part of my effort to promote the creation of a Provisional Peoples' Assembly. Since then I have been refining the algorithm and made it more consistent with its basic principles.

Activities in any assembly need to be meet the following criteria:

  1. Clearness of Action: The decision process of an act needs to be as transparent as possible without invading the private sphere of the acting agent. Otherwise the intent of the agent cannot be clearly understood and agents cannot be held to account for their decisions.

  2. Speed of Action: Representation needs to be by competent and freely active agents that can quickly adjust their behaviour without direct consent from their entire constituencies.

  3. Appropriateness of Action: A representative's overall performance as a freely active agent must be evaluated by the entire constituency at regular intervals. To get an accurate evaluation, the measurement distance between representatives and their constituencies must be minimal (i.e. they must be directly accountable to them).

  4. Fairness of Action: An agents power to influence the assembly must be commensurate with their constituencies size and their contributions to the well-being of the commons over which the assembly has jurisdiction.

Point 1 and 2 simply implies a representational model in itself (i.e that we have agents acting for us rather than acting on our own). The merits of a representational democracy versus a direct democracy are extensive and have been expounded numerous times by many, many others authors.

Point 3 stands for the the democratic system by which we all vote for who best manages our commons, and a system where everyone's opinion counts. Determining what is right and wrong in society should not be reserved to a cadre of experts appointed outside the democratic process itself. The soundness of this is not as well-accepted as one might think. But to prove it is better to give Joe Commons and Professor Smart one vote each requires another lengthy post. If you are in doubt, for now, you will simply have to go along with my statement that this is the case.

Point 4 is my concern here in regards to above mentioned algorithm. What is fair? If we were not concerned with fairness, anarchy might be our best model, since anarchy is a means by which the strongest enforces their will on the others. Supposedly the most well adapted conquers. This isn't quite true of course, since anarchy always gives way to tyranny and thereby ends the state of anarchy from which it originated.

And tyranny eventually leads to the collapse of society as rebellion sooner or later ferments within pockets of the oppressed. At some point all roads lead to a consensus-oriented approach. It may take a long time. But when everyone is tired of the killings and the arbitrariness of oligarchies (even the oligarchs themselves, especially those who have fallen from grace), the path is laid to methods for building consensus where needed. And so we are back to determining what is fair.

Fair can be balanced between what someone needs and what they deserve. Someone may need an oil pipeline but not deserve it. Someone may deserve a pipeline but not need it. Need is an easy term on the surface. Deserve seems is a bit harder. The former seems to imply simple observation and the latter complex and subjective moral belief systems. But actually it's the reverse.

We can all agree that someone deserves something if they have made a contribution to something else. I give you oil, you give me a pipeline. I don't give you oil, you don't give me a pipeline. That's how simple it is. It's a straight forward economic principle that can be abstracted into monetary systems. Actually, need is much trickier and subjective than deserve.

Need is sometimes objective and easy to determine as well. A starving child in Asia on the verge of death needs food and drink. No one in their sane mind will argue against this. But when we go beyond anything but the basics of food and shelter, need becomes a highly subjective matter. Does New york need a lot of electricity? Yes, it does. But do we need New York? Hard to say. As a New Yorker, I'm quite convinced we need places like this great city. But I'm sure there are a lot of people who would live a perfectly happy lives without anything like New York.

As an aside I think this is the mayor flaw in the Marxist concept expressed in "from each according to his ability, to each according to his needs". Resources are not and never will be infinite. Who gets to use cellular bandwidth cannot be determined by some higher objective means. Socialism fails because of its inability to determine actual needs versus perceived needs. Specific needs can only be determined through a constant tug-of-war between the individual and their fellow human beings. The objective of any regulatory efforts must be to keep such tug-of-wars peaceful and within the confines of basic criminal law ("though shalt not murder", "though shalt not steal", etc.). Beyond supplying basic food and shelter (which, as mentioned before, is objectively determinable), regulatory activities should never be to actually allocate resources according to some misfounded principles.

In an assembly of representatives, constituencies are constantly at war as their agents battle to supply them with basic needs (and very often, unfortunately, more than basic needs). Who gets to tap the Colorado River for water? Who gets to fish off the coast of Spain? Who gets to drill in the Arctic Sea? Their power to influence these issues should be commensurate with the needs of their constituencies.

If we consider only basic needs, ideally resources should be supplied equally amongst all. Therefore, the power of a agent in an assembly should be equal to the number of individuals in the representative's constituency. Just because Utah is upstream of the Colorado River does not mean that California should not have a say in how its waters are tapped further upstream. California is much more populous and should get its fair share of the waters. The muscle it should be able to exert in an assembly should be greater than Utah due to its greater population..

This leads to the first decisive factor in allocating seats (s) in an assembly: population size. We will call it simply p. A simple and old model of allocating seats would be to take a fraction of p and assign seats accordingly. For every 10,000 person, a constituency gets 1 seat:

s = p /10,000

However, you want to limit the total number of seats in an assembly to an amount that allows for allocating time resources during deliberations equally. Otherwise, the entire process of deliberation becomes on oblique process not dissimilar from direct democracy. Why would we then need representational democracy at all? Consider also that the amount an agent can spend in the limelight is an important factor in determining power within an assembly. The point of having an representational assembly is to meet the first two criteria of assemblies (clearness and speed of action). Anything beyond some given total number of seats (call it x) would defy criteria 1 and 2. What x should be we will get back to later.

What we get is something like this

s = x * p / SUM( p[1],...p[n])

Where SUM( p[1],...p[n]) is the sum of the population in all constituencies (n here stands for the number of constituencies). In the case of a truly global assembly, it would amount to the entire population of the world.

At first glance it seems fair to allocate seats this way. Most "lower" assemblies in the world(that is, assemblies that allocate seats by the size of a constituency) have some form of this basic algorithm. But as Lionel Penrose proved to us already back in the 1940'ies, it is not very fair at all.

The reason for that has to do with the idea that, if you represent 51% of the population, shouldn't you cast a decisive vote 51% of the time? Let's assume that constituencies can have several seats in an assembly but that every representative of the same constituency votes in a single block. If Representative Jane, John and Kate from Marble Land always vote the same and they hold over 51% of the seats, Marble Land will cast a decisive vote 100% of the time. Regardless of how slim their majority is, this occurs, which does not seem in any way or shape fair. In effect, they have become what we call a tyranny of the majority.

As I said before, it seems more fair that they should determine the outcome of an issue 51% of the time. This, as Lionel Penrose discovered, can be better achieved by giving Marble Land only the square root of its population. Someone once said to me that "the square root seems awfully complicated" and someone else that "the square root defies the sacred rule of one person, one vote". First, roots are not complicated concepts. Its not the integration of a function. The Greeks new about roots a long time ago and my 8 year old son understands why the square root of 64 is 8 (since 8 groups of 8 items gives us a total of 64 things). Compared to some of the obfuscating methods for "fairness" used for voting in some of today's assemblies, its really very simple and fair.

Secondly, the square root actually guarantees that the spirit of "one person, one vote" is better fulfilled. The reason is exemplified by the whole concept of the tyranny of the majority. I won't try to demonstrate the validity of Penrose's root right here by extensive lists of voting combinations. But you can do it yourself by assuming that all agents from the same constituency vote in a block (which, of course, isn't always true but does happen very often in resource allocation issues).

Just combine all the number of votes of each constituency into all possible binary YES/NO combinations (i.e. possible alliances with other constituencies in voting for or against a resolution). Then count the number of times each constituency is in a decisive alliance (i.e. one that wins the vote). Do this for both the standard way of allocating seats and then Penrose's way. You will immediately begin to see how Penrose's approach is more fair and puts a tighter lid on the tyranny of the majority.

So, if we apply the Penrose root to a simple voting allocation method in the UN, we get:

s = SQRT( p /1,000,000 )

This was what Penrose originally proposed for weighting votes in the General Assembly. A country would have gotten a square root of its inhabitants divided by one million had Penrose gotten his way. For the curious, Lionel Penrose is the father of the contemporary mathematical physicist Roger Penrose who wrote "The Emperor's New Mind".

To accommodate our need for defining the actual size of the assembly, we need to bootstrap the appropriate fraction. Just arbitrarily taking 1,000,000 will cause our assembly to balloon as the world population grows. Let's assign the letter d to the unknown denominator by which we have to divide the population of the constituency. We resolve the equation:

SQRT( p / d ) = s
p / d = s ^2
1 / d = s ^2 / p
d = p / s^2

If we take the average population size of all the constituencies, we can get the denominator that will produce a parliament that has approximately x number of total seats.

d = ( SUM(p[1],...p[n]) / n ) / s^2

To be able to resolve d, we will have work from the hypothetical situation where each constituency is the same size (and therefore would each get the same number of seats). This is of course not the case. But it will produce an assembly of approximately the size we have determined is ideal if the population sizes of the constituencies are normally distributed. If x is the ideal size of the assembly:

d = ( SUM(p[1],...p[n]) / n ) / (x/n)^2


d = ( SUM(p[1],...p[n]) * n^2 ) / ( n * x^2 )

It might seem sufficient to have this algorithm. However, taking only population into consideration only addresses the rightfulness to exercise power according to ones (basic) needs. It may, in some sense be considered equitable. But I think it is legitimate to distribute power according to what someone deserves. If you don't participate in the exchange of resources, why should you have the capacity to participate in the appropriation of resources?

Contributions to the well being of the commons can be measured simply by the market value of resources the constituency puts at the disposal of the federation (the union of all the constituencies). For example, if Marble Land provides a militia of 10,000 individuals to the federation at a yearly cost of 1 billion dollars, this can be objectively counted as providing 1 billion units of well-being to the commons. Counting monetary expenses as units of well being may seem wrong to some. But there really is no other way to objectively measure health at the macro level than by looking at monetary flows.

The only other option to measure well-being would be by asking some set of questions in a general referendum. What questions would you ask? Would you ask how satisfied the people of Marble Land are with the cooperative nature of the people of Stone Island? Any such scheme is pure hair brained balderdash nonsense. The economy is a quantifiable and objectively measurable abstraction. Where there is a steady stream of activity, there is usually happiness, or at least contentment.

The means by which we calculate seats by contributions is identical to the one we used for the population factor. As a matter of fact you could do it for any given quantifiable factor. Assigning the letter e to the amount a constituency contributes to the commons:

d = ( SUM(e[1],...e[n]) * n^2 ) / ( n * x^2 )

Now we have to join these 2 assemblies into one. That is to say, x is not the same for each calculation. If it was, we would not get an assembly of a given ideal size but an assembly twice the ideal size. We have to decide how many of our seats in the assembly should be allotted based on contributions and how many based on population. We would have to do this for each factor by which we want to bestow power in the assembly.

I can't think of a clear cut a priori logical way of establishing the ideal distribution of economic versus popular seats. In such a case, lets just allott half the seats by population and half by contributions (i.e. there is maximum uncertainty as to which factor is more important). Where P is the number of popular seats and E the number of economic seats (and x the ideal size of the assembly):

x = P * x / 2 + E * x / 2

We will be able to adjust this later if we discover that it leads to power structures that seem unbalanced (for example, if in a global assembly it results in almost all seats going to the US, China and Europe). I'm sure how to allot these seats will become a major contention between what we traditionally regard as the political left versus the political right. At the 2 extremes of this political division, you will get individuals who think one or the other factor should be entirely eliminated and reduced to x = P or x = E.

So what we now have to do is divide x by half when calculating the denominator of each factor. Instead of e and p, lets just use f as a stand-in for either factor:

d = ( SUM(f[1],...f[n]) * n^2 ) / ( n * ( x / 2 )^2 )
d = 4 ( SUM(f[1],...f[n]) * n^2 ) / ( n * x^2 )

Since we have the same amount of seats for each factor, this works fine. If we had fewer seat for either of the factors, we would have a slightly different formula for each one. That is to say the x/2 would be different. If we had only one third economic seats, we we would replace x/2 with x/3 for economic seats and for popular seats we would use 2x/3.

So what should x be? What approximately is the maximum number of seats you can have in an assembly to have clearness and speed of action (criteria 1 and 2 of assemblies)? The number of agents for each constituency will depend on its population. Each agent will not represent the same number of people due to the Penrose root, which curbs the power of populous constituencies in order to prevent the unjust tyranny of even the slightest majority (51%). Given the Penrose root, as a block the agents of a each constituency will represent the power of the constituency in appropriate proportions, if and only if each agent is allotted the same resources to deliberate on issues.

It seems fair that each agent should be given at least one opportunity during a week to be the main focus of attention. It should be sufficient to allow them 3 minutes of time: 1 minute of setup, 1 minute of elaboration and 1 minute to draw a conclusion. Agents are given the "podium" not to expound their views of an issue in full. They are given the podium merely as an opportunity to draw everyone's attention to their views, which can be explored in further detail by tuning to the agent's dedicated channels (such as personal web sites, radio stations, television shows etc.).

We can assume 40 hours of "podium" time every week. There may be 24 hours in the day. And it might be tempting for a global assembly to fill all the hours of the day as the earth rotates through the sun's benevolent light. But if agents should have a chance to broadcast themselves to all individuals in all constituencies in a global assembly, we will have to replay their deliberations as the light cone of the sun slides through the time zones we have invented. So we can apply the same criteria for deliberations in a global assembly as we can apply to any assembly.

40 hours is 2,400 minutes:

2400 / 3 = 800

Therefore, 800 agents seems to be the maximum number of individuals an assembly can consist of in order to not become too unwieldy and obscure the processes of common deliberation as well as the speed at which it can be done. We are getting closer to an actual equation for calculating seats based on 2 equal factors (population and contributions) in a maximally large assembly. Let's plug 800 into the equation in place of x:

d = 4 ( SUM(f[1],...f[n]) * n^2 ) / ( n * 800^2 )
d = 4 ( SUM(f[1],...f[n]) * n^2 ) / ( n * 640000 )

For a single factor assembly the equation would be:

d = ( SUM(f[1],...f[n]) * n^2 ) / ( n * 640000 )

When the factors are all of equal importance (and there are the same total number of seats for each factor), the coefficient is simply the square of the number of factors. For a three factor assembly, the equation would be:

d = 9 ( SUM(f[1],...f[n]) * n^2 ) / ( n * 640000 )

These equations are not, of course, complete. They do not calculate a constituency's actual seats. they only calculate the denominator of the Penrose root equation (the "millions"of taking the square root of the number of millions of inhabitants). So, lets replace "millions" with the equation for a 2 factor assembly:

s = SQRT( f / ( 4 ( SUM(f[1],...f[n]) * n^2 ) / ( 640000 * n ) ) )
s = SQRT( ( 640000 * f * n) / 4 ( SUM(f[1],...f[n]) * n^2 ) )
s = SQRT( 160000 f n / SUM(f[1],...f[n]) n^2 )

This is the full equation. Once it is resolved for each factor of a constituency and the totals of each factor added up, the number of seats a constituency can fill is known. Note that this equations (or its variations with different number of factors and total number of seats) is applicable to any assembly, not only those that are global in nature. Below is the full equation for an assembly that equally considers population and contribution to the commons:

s =


SQRT( 160000 p n / SUM(p[1],...p[n]) n^2 )


SQRT( 160000 e n / SUM(e[1],...e[n]) n^2 )


The CEIL part stands for "ceiling", which means we round the result up to its closest integer. This guarantees that all constituencies that are part of the federation will have at least one agent representing them in the assembly.

In my next post, I will explore the actual power distribution in a global assembly based on this equation.

No comments: