Physics challenges the optimal size of parliaments

What is the best size of a parliament? That is a question at the center of many countries today, including the 2020 referendum in Italy where almost 70% of voters selected to slash the number of members of parliament by about a third. Among others, the complex issue involves matters of governing efficiency, logistics, and financial costs.

But one thing many people might not realize is that there is a science behind all this. In 1972, political scientist Rein Taagepera published a seminal paper proposing that the ideal size of a parliament corresponds to the cube root of the country’s population: A=αP_o^1/3, where A is the parliament size, P_o is the population size, and α is a constant. In general terms, the bigger a country’s population, the bigger its parliament ought to be.

Taagepera’s famous “cube-root law” was quickly taken up by governments, but hasn’t been without critics: In 2007 and 2012, researchers used empirical data to come up with a square-root relationship rather than a cube-root, while another paper in 2019 questioned the actual cause-effect sequence that lies at the foundation of Taagepera’s law. So while all agree that a bigger parliament would be needed for a bigger country, the exact relation has remained a matter of dispute.

Now, physicist Giorgio Margaritondo, Professor Emeritus with EPFL’s School of Basic Sciences, has published a paper analyzing Taagepera’s model. Published in Frontiers in Physics, his findings challenge the math behind the paper and the accuracy of its predictions, and raises concerns about the way the data was used.

“I was astonished,” says Margaritondo. “The law has been, – and still is – widely used, but the paper’s flaws have gone undetected for half a century.” He points out that Taagepera’s original paper actually evaluates the actual size of a parliament not its “optimal” size.

Four fatal flaws

“The original derivation of the ‘cube-root law’ is affected by fatal mistakes, unnoticed for half a century,” says Margaritondo. Taking a physicist’s view, he analyzed the 1972 paper, and discovered four flaws.

• First, that the cube-root law was not derived from the paper’s data and that the corresponding trend that led to the formula was “arbitrarily forced”.
• Second, that the theoretical steps used to derive the formula incorrectly evaluated one of its key factors.
• The third flaw has to do with real-world politics: Taagepera’s model assumes that each member of parliament spends on average an equal amount of time communicating inside and outside parliament, which Margaritondo describes as a “an arbitrary hypothesis that has unrealistic consequences.”
• Finally, there is generally no evaluation of “optimal” size based on a power law that can reach meaningful accuracy, and that includes the cube-root law.

Square, not cube

Margaritondo’s suspicion was that the paper’s data could better fit a broader, more “general” formula that the cube-root law. But in the paper, Taagepera challenged this notion by claiming that the it would be a “dead end” and that it would be “more fruitful to look for a plausible theoretical model which would fit the observed general trend”.

“This argument is fundamentally flawed from a physicist’s point of view,” writes Margaritondo. “It considers only one hypothesis, renouncing a priori to demonstrate its superiority with respect to others.” Spurred on, Margaritondo used the paper’s original data and applied to them the same statistical fitting method that Taagepera had in 1972. Except that here, he used a similar, but more general formula to fit the data: A=αP_oⁿ. Here the cube root changes and the exponent is n.

Applying this equation to the 1972 data, Margaritondo discovered that n equals 0.45 ± 0.03. This is actually closer to a square-root law, already proposed in 2012 by the researchers Emmanuelle Auriol Robert J. Gary-Bobo. “Even the original data did not support the cube-root law,” says Margaritondo. “The fit was arbitrarily forced.”

Beyond math

In short, it seems that, at least for now, physics cannot decide what the optimal parliament size is based on math alone, but it might be time to abandon a cherished rule that has governed governments for almost half a century.

“To think that we can accurately derive the ‘optimal’ parliament size with a power law is an illusion,” says Margaritondo. “It’s akin to fake news: the wrong use of ‘scientific’ arguments to propagate political notions.”