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For half a century, the analysis of the size of national assemblies was dominated by the famous cube-root relation with the population. However, a revisitation of that historical work with a physicist’s approach reveals basic conceptual problems that fatally undermine its conclusions. Furthermore, the assembly size evaluation exceeds the accuracy of all power equations, which cannot be reliably used for political analysis.

Could the “optimal” size for the national assembly of a country be evaluated with methods similar to physics research? This is a timely question: the debate about insufficient representation at the federal and state levels is raging in the USA. On the other side, there were recent initiatives to reduce the number of representatives in the national parliaments of many countries, including France, Hungary, Ireland, Japan, Mexico, the Netherlands, Portugal, Romania and the United Kingdom. And Italy just emerged from a referendum on this issue.

The classic reference is the 1972 work of Taagepera [_{o}, the population:

Alternate approaches were later presented [

The relation between _{o} size must be appreciated in a more general context [

Furthermore, cube-root scaling laws have alternate mathematical explanations [

Taagepera’s work [

The cube-root law was not derived from its data and the corresponding fit was arbitrarily forced.

The theoretical steps that were used to derive

The model assumed that each representative spends on the average equal times for communications inside and outside the parliament, an arbitrary hypothesis that has unrealistic consequences.

No evaluation of the “optimal” size based on a power law, including the cube-root one, can reach a meaningful accuracy.

Concerning the first problem, the original article [

However, it surprisingly argued against using it to fit the data: “_{o}
^{n} …

We analyzed the consequences of the above argument by applying the same fitting procedure as Ref. [

The exponent

Log-log plot of the original data of Taagepera [

If one forces the same data set to be fitted by a cube-root law, the result is:

The corresponding fit (

To present the second and third of the problems affecting Ref. [

Two kinds of channels were considered: first, those between each parliament member and his/her active constituency. The average number of such channels per member is:_{o} is the fraction of the population that is politically involved.

The second type of communication channels connects different members of the assembly, to discuss and implement the measures identified by the first type of channels. While communicating between them, two assembly members share the same channel, and it was argued in [

Which, except for unrealistically small assemblies, can be approximated as:

What is the relation between _{C} and _{A}? Ref. [_{C} = _{A}, leading to:

That is, to the cube-root law of ^{1/3}.

However, this logic frame is affected by two conceptual problems. First,

Which, assuming again that _{C} = _{A}, leads to:

Not a cube-root law but a square-root law [

To better understand why

The other flaw in the above logic frame is that there is absolutely no evidence supporting its hypothesis that _{C} = _{A}. On the contrary, this assumption causes problems. In the original work of Ref. [

The balance between different types of communications can actually change from country to country and evolves with time. For example, modern communication instruments can reduce _{C}. Symmetrically, effective negotiators can decrease _{A}. Thus, assuming _{C} = _{A} is arbitrary.

Supposing instead that _{C}/_{A} =

In both cases, the multiplication factor is a combination of

The difficulties in evaluating

Note that Ref. [

Hypothetically, one could try to extract an “optimal” value by using a subset of “good” countries, perhaps those with low indexes for corruption and bureaucratic ineffectiveness. However, not even filtering could solve the fourth problem affecting Ref. [

(d_{o}
^{n} ln(_{o}) = _{o}) ,

_{o}is large, an uncertainty d

_{o}) and produces a large relative uncertainty d

In short, accurately evaluating the “optimal” size of a national assembly is illusory. And trying to inject additional factors besides the population cannot solve the above problems.

At most, this kind of approach can identify the countries that strongly deviate from the “average”, as Ref. [

In conclusion, we surprisingly found that the historical and very influential work of Taagepera [

Publicly available datasets were analyzed in this study. This data can be found here: Reference 1 in the article.

The author confirms being the sole contributor of this work and has approved it for publication.

Work supported in kind by the Ecole Polytechnique Fédérale de Lausanne (EPFL).

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.