A different look at elections

[This is an automatic translation of the original post in Spanish and has not been edited yet.]

The image that heads this post: Can you guess what it is? When I see it, its texture and volume make me think that it is some type of stone, not exactly spherical but rather with the figure that one recites at school to describe the earth: flattened at the poles and bulging at the ends . Or maybe it’s a fruit, like a pumpkin …

Let me give you a hint: the photo shows the side view of an object carved in wood. And the next two images show the front and top views of the same object.


¿Can you get a better idea of ​​what the object is? If you connect the three images you can see that it is actually a bowl.

Now which of the three photos is correct? The question doesn’t make much sense since all three capture a real side of the object we want to understand. The front view may be the one with the most information, but we need the other two to reconstruct the three dimensions in our minds. The first image I showed in this post – the side view – is the one with the least information, exaggerating the height but confusing ourselves with the length. And in any case, from just these photos it is difficult to know how deep the inside of the bowl is.

A few days ago I wrote an entry summarizing the results of a preference survey of candidates for this year’s elections in Colombia. I received a response from 252 people who organized the five candidates who ran in the first round, plus the blank vote, from highest to lowest preference. Each chain of preference granted a determined number of votes to each candidate depending on their position. To obtain the collective vision, all the votes given to the different candidates by the 252 respondents were added together.

Depending on how many votes are assigned to each position one gets different electoral systems, and the uncomfortable observation is that their results can be totally different from each other. For example, from the survey I obtained the following orders of preference using five different systems (using the nomenclature “So-and-so> Mengano” to say that so-and-so over Mengano is preferred):

Duque> Petro> Fajardo> Street> Vargas Lleras> Blank

Duque> Fajardo> Petro> Street> Vargas Lleras> Blank

Fajardo> Duque> Petro> Street> Vargas Lleras> Blank

Fajardo> Street> Vargas Lleras> Duque> Blank> Petro

Vargas Lleras> Blank> Street> Fajardo> Petro> Duque

Clearly, the five results seem to contradict each other. The first chain, in particular, is the one obtained using the plurality system, which is the one we use in our elections, and in which we only take into account the candidate that we prefer the most and reject the rest.

¿Which of all these results is correct? Well, each of those results corresponds to a photo of what the group of respondents thought. They are all correct, insofar as each one shows a different angle of the object that we want to understand. But as with the photos of the bowl, a single photo does not necessarily give us enough information to correctly reconstruct the object. The imposition of using a plurality system to choose our rulers is no more arbitrary than making the intention of only taking photos from side views.

The question remains: Is there an electoral method that is better than the rest? The analogy with the photos of the wooden bowl I think still applies. Look at the next photo.

Taking the photo diagonally and at a certain angle helps solve many problems. Likewise, the Borda Recount or the Condorcet method are electoral systems that tend to introduce fewer distortions.

However, we can forget about having a perfect electoral system someday, that we can always use and with the peace of mind that we are not registering a deceptive image. And we know this with mathematical rigor: Kenneth Arrow, whom I had already mentioned in a previous post, accomplished the feat of demonstrating that any electoral system that one can invent will always have a blind spot that manifests itself in the form of electoral paradoxes. Translated into the world of photos, Arrow’s impossibility theorem tells us that if we always point our camera at the same angle – and no matter what angle it is – there will always be objects that we will not be able to correctly represent.

We, the electors, may have free will and choose the candidate we want. The shocking thing is to discover that destiny is encoded in the particular electoral system that has been imposed on us.

Can we design a democratic system for choosing rulers in which we incorporate more than a single rigid and incomplete photo of what the people say?