# Scaling and Universality in Proportional Elections

Somebody (I can’t remember who) pointed me to this paper by Claudio Castellano and Santo Fortunato. Here’s the abstract:

A most debated topic of the last years is whether simple statistical physics models can explain collective features of social dynamics. A necessary step in this line of endeavor is to find regularities in data referring to large-scale social phenomena, such as scaling and universality. We show that, in proportional elections, the distribution of the number of votes received by candidates is a universal scaling function, identical in different countries and years. This finding reveals the existence in the voting process of a general microscopic dynamics that does not depend on the historical, political, and/or economical context where voters operate. A simple dynamical model for the behavior of voters, similar to a branching process, reproduces the universal distribution.

I wrote to them, doing the traditional academic thing of referring to my own papers:

You may be interested in our paper, The mathematics and statistics of voting power, which considers some mathematical models for correlations among votes. Further empirical analysis is in our paper, Standard voting power indexes don’t work: an empirical analysis.

I do not think it is necessary to assume contact by word of mouth (as you say in your paper) as the correlations can occur for other reasons, including similarity of interests among nearby voters, and also news media effects.

Fortunato and Castellano replied,

We have finally had the time to read the two papers you recommended to us. We find them interesting. However we think that the problem we deal with in our paper is rather different from what is discussed in yours.

In particular, we consider proportional elections with multiple-seat constituencies and open lists and we focus on the performance of candidates within each party list, trying to factor out the choice of the party and the issues associated to it.

Our main result is that the empirical analysis reveals striking similarities for the histograms of the relative performance of candidates in different years and in different countries. This result is very remarkable to us.

In order to explain these findings we have devised an elementary model, based on the idea of word-of-mouth influence in the network of voters, that allows to reproduce the empirical evidence. We do not claim that word-of-mouth is a necessary ingredient for understanding the phenomenology uncovered. Nor we assert that our model is the only way to explain the data. The model is an attempt in this direction. It is likely that other plausible modelizations of the correlations among voters, as those you mention, could reproduce our results. Additional empirical evidence is needed to allow for a more stringent test of the model(s).

We have tried to approach the social/political science community in order to have feedback on our work and find relevant related literature.

And so I posted this discussion here.

## 2 thoughts on “Scaling and Universality in Proportional Elections”

1. Interesting. Who would have thought Italian elections would be orderly?

There are a lot of constraints in here, and it's possible their model might be a function of these rather than some underlying dynamic.

2. I was playing with number during the Ukrainian election, which like Germany has a minimum threshold for representation that is more than the number of votes required to get a single delegate.

It is really remarkably how profoundly this slight quirk from the platonic ideal system the papers cited discuss impacts the dynamics of the entire system.

One of the main reasons to go to a multi-party list system proportional representation voting plan is to end the spoiler effects of marginal third parties, which largely eliminates the need to vote strategically or have some sort of primary system. But, even a very modest cutoff produces impacts very similar to the Nader effects in the Bush v. Gore race.

If a small party with say 5% support fails to hold together and the cutoff is say 4% of the vote, a split into two parties, one with 3.8% of the vote, and the other with 1.2% of the vote, for example, dramatically damages that coalition that both of the daughter parties would otherwise be a part of in a government.

Similar effects arise if any faction smaller than the cutoff splits from the biggest party in a coalition, typically in the 30s in terms of percent support in that kind of election.