Overview of some findings on democracy

The dictum is to have one subject per letter. This paradise is no longer possible when time passes and letters and subjects accumulate. Let me take stock of some findings on democracy.

Economic theory needs a stronger defence against unwise application of mathematics. Mathematicians are trained for abstract thought and not for empirical science. Their contribution can wreak havoc, for example in education with real life pupils and students, in finance by neglecting real world risks that contribute to a world crisis, or in voting theory where they don’t understand democracy.

Nowadays, though, I am also wary of students from the Humanities who rely upon legal views (their version of mathematics) instead of empirical understanding.

For the following, distinguish single seat elections (president, prime minister) and multiple seats elections (parliament). There is also a key distinction between Equal Proportional Representation (EPR) with proper elections and District Representation (DR) that has contests rather than proper elections.

Key findings

(1) Montesquieu’s Trias Politica of the separation of powers is failing, and we need the separation of a fourth power, an Economic Supreme Court, based upon science, with a position in the constitution at the same level as the Executive, Legislative and Judiciary. The current setup allows too much room for politicians to manipulate the information for policy making. This need for separation can also be proven logically in a model using stylised facts, see the book DRGTPE. A short discussion on the 2007+ European crisis is here.

(2) Kenneth Arrow in his Impossibility Theorem has a correct deduction (there is an impossibility) but a wrong interpretation. He confuses voting and deciding. For this debunking of Arrow’s Theorem, see Chapter 9.2 of Voting Theory for Democracy (p239-251). Sheets of a presentation in June 2018 are here.

(3) A voting method that many might find interesting is the Borda Fixed Point method. See the counterfactual example of selecting a Prime Minister for Holland.

(4) Political science on electoral systems is no science yet but still locked in the Humanities, and comparable to astrology, alchemy and homeopathy. People in the USA, UK and France still have taxation without representation.

(4a) The key paper is One woman, one vote. Though not in the USA, UK and France.

(4b) A supportive paper develops the SDID distance measure for votes and seats.

(4c) This paper reviews the role of statistics for the latter measure. Sheets of a presentation in June 2018 are here.

(4d) An earlier comparison of Holland and the UK in 2010 (update 2015) contains a major stepping stone, but is not as critical as (4a). This analysis resulted in a short paper for Mathematics Teaching 222 (May 2011) at the time of the UK referendum on Alternative Vote.

Minor results because these lead to dead ends

(5) There are some supplementary findings, that I do not regard as major, but as roads that you might need to walk in order to discover that they do not lead far.

(5a) There are Two conditions for the application of Lorenz curve and Gini coefficient to voting and allocated seats. The Lorenz curve is a neat way to graphically show the disproportionality and inequality of votes and seats. The Gini is its associated measure. However, above measure SDID is to be preferred, since it is symmetric and doesn’t require sorting, has a relation to the R-squared and the Weber-Fechner law.

(5b) We can compare votes and seats but also use a policy distance. A crucial question is who determines the distance between policies ? When we have a distance, how do we process it ? I am not convinced by the method, but a discussion is here.

(5c) The Aitchison geometry might present a challenge to SDID. This paper provides an evaluation and finds this geometry less relevant for votes and seats. Votes and seats satisfy only two of seven criteria for application of the Aitchison distance.

(5d) This paper tries to understand the approach by Nicolaus Tideman and compares it with the distinction between voting and deciding.

(5e) Mathematician Markus Schulze was asked to review VTFD but did not check his draft review with me, which caused needless confusion, see here and here. PM. Schulze now has this 2017 paper, but doesn’t refer to Borda Fixed Point, perhaps thinking that he understands it, but he apparently is not open to the diagnosis that his “review” is no proper review.


For the above, it is pleasant that a distinction can be made between key results and findings about dead ends. I listed my debunking of Arrow’s Theorem as a key result, but it also identifies this theorem as a dead end. Thus, it is also a matter of perspective. When you are at the dead end, and turn around, the whole road is open again.

PM. Earlier weblog entries on democracy are here.


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