While I was rummaging through the science and philosophy magazines literally hanging off the sides of the bookshelves in my bedroom/office, I found this old gem of article from an Italian popular science magazine called “Le Scienze” (The Sciences) which is actually nothing more than the Italian version of Scientific American, containing mostly translations of SciAm articles from the English, but with the occasional outstanding contribution from some of the top matematicians and scientists working here in Italy.
In this particular case, the writer, Piergiorgio Odifreddi, is one of the top mathematical logicans in Italy. But he is also one of those great popularizers of science, ala Steven Jay Gould, Oliver Sacks and Steven Pincker, who know how to make complex ideas accesible to a popular audience without compromising the substance.

The title of the article is, “The Paradox of Democracy” and I think it should definitely interest even the non mathematically-inclined folks out there.

my translation from the Italian

Let’s begin with the problem of electoral districts. Since it never occurs in real life, as it does in Borge’s short story “the Parlaiment”, that every voter gets elected to office, the number of voters is always greater than the number of seats being contested, and the ratio bewteen the two is rarely an integral number.

In order to be fair, the distribution of seats should above all satisfy a principle of proportionality :
the number of seats assigned to a district on the basis of the size of the population, or to a party on the basis of the number of votes, should be one of the two approximations of that ratio. For example, if the seats to be assigned are ten, a party that wins 1/3 of the vote should receive three or four.

A second obvious condition that should be satsified is monotonicity : if a party wins more votes than another, it shouldn’t receive less seats. And this should apply not only synchronically, in single elections, but diacronically, in different elections, with respect to the ratio between the percentages of votes.

But in 1982 two US mathematicians, Michael Balinsky and Peyton Young, proved a surpising theorem: there is no general method for distributing seats in a manner that will satisfy the two aforementioned conditions at the same time. As soon as there are, in fact, at least four parties or districts, and seven seats being contested, there exist distributions of votes that make the thing impossible.

For example, let’s suppose that in a first election the four parties A,B,C and D obtain, respectively, the following distributions of votes relative to the seven seats: 5.01 – 0.67 – 0.67 – 0.65. And that, in a second election, the distributions become: 3.99 – 2.00 – 0.50 – 0.51. Synchronic proportionality and monotonicity would require that in the first election the distribution of seats be: five to A, one each to B and C, and none to D. In the second: four to A, two to B, none to C, and one to D. But in that way diachronic monotonicity is violated, because A loses a seat and D gains one, even though A increased his number of votes with respect to D from 7.5 to almost 8 times.

Naturally, the principle of proportionality is rejected by the majority based electoral systems, which however have their own nice little problems. For example, it’s possible that a party with slightly less than 50% of the national vote not obtain a single seat, and that every seat goes instead to parties with a minumum national representation. It’s sufficient in fact that in each district one and the same national party obtain 50% – 1 of the votes, and that a small local party obtain 50% + 1, in order for the seat to go to the latter.

Another problem of majoritarian systems is that they require an aggregation of policital forces in two counterpoised blocks. The deleterious consequences become evident immediatly by comparing the electoral campaigns of the two parties to the selling of icecream on a linear beach about a chilomter long: the optimal disposition of the two icecrem vendors is 250 from the etxremes, becasuse in this way no bather has to travel more than 250 meters to buy an icecream. But if one of the vendors moves slightly toward the center, he won’t lose the the buyers at the extreme and he will take some from his competitor: since the phenomenon works symmetrically, the reciprocal movements of the vendors toward the center will leave them both at the center, leaving everyone else underserved and those in the center spoiled.

Now isn’t that one of the best illustrations you’ve ever read of the fundamental problem with majoritarianism? Brilliant!!!

The rest of the article discusses Kenneth Arrow’s impossibility theorem and illustrates why democracy, when carefully and precisely defined, is mathematically impossible!!

Arrow’s theorem of 1951 demonstrates in fact that there do not exist systems which satify the minimal conditions which are normally required for democracy: the fact, that is, that each voter has the right to freely choose the candidate he wants (freedom of choice); that if all voters prefer one candidate over another, the latter must never be elected (unanimity); and that the choice of the winner depend only on the votes expressed and not on other factors (indepdence from irrelevant alternative).

In reality, things are even worse than they seem. A theorem of Amartya Sen from 1970 poves in fact that the first two conditions are already incompatible wiht the possibility that , in one society, more than one person can have rights. This is evident in the case of rights regarding the same alternative: if in fact two individuals each have a right regarding the choice between A and B, it is sufficne that one choose A and the other B in order to obtain a contradiction.

From Sen’s theorem it is easy to derive Arrow’s. Lets’ consider two alterantives A and B not indifferent to society. If noone had rights on them, evey individual could prefer to choose the alternative contrary to that of the whole society, by the freedom of choice. But if everyone beahved in the same manner, the independnce from irrelevnat alteranatives and unanimity would contrain society to choose the alternative whch everyone prefers, contary to the hypothesis that that choice is in fact the other.

Therfore, for every alternative not indifferent for society, there must exist an individual who has a right WRT it. By Sen’s theorem, this individual must always be the same, becaseu at the most only one person can have rights. But one persaon who has rights overall the altrenatives is a dictator…….

I’ll leave you all to tease this one out for a while.