Of course, under certain conditions the vote is always perfect. Take for example, good old tyranny. A person decides everything on his own. In this case, there is no problem, there is no paradox and everyone is happy. In other words, if we exclude this pathological case and others of the same type, the vote can produce paradoxes. It is not the voting “system” that produces the paradox, it is the paradox that is contained in the vote itself.
The theorem on the impossibility of voting cannot be explained in two steps. The trick is that this theorem generalized an old paradox about votes, which is expressed more easily. This is the famous Condorcet paradox. Condorcet was a marquis specializing in political arithmetic who lived at the time of the French Revolution. Our Condorcet had developed his own paradox by studying the way of voting.
The secrets of the Brunelleschi dome in Florence
A hellish cycle
To fully understand the paradox, let’s take the example of the glacier. Imagine it’s a nice day and you’re heading to the ice cream shop. It offers vanilla, strawberry or chocolate. The trick is that you prefer vanilla to strawberry, strawberry to chocolate, and chocolate to vanilla. The choices form a cycle: vanilla > strawberry > chocolate > vanilla. It’s the infernal cycle, the absurd loop, the snake biting its own tail. Whatever flavor you choose, you are pale blue chocolate because there is always one that you prefer.
Now imagine that you have a group of a few people vote on which flavor of ice cream to buy for the group. A majority prefers vanilla to strawberry, an “other” majority strawberry to chocolate and yet another “other” majority chocolate to vanilla. They are different majorities each time, but the fact remains that you are in an absurd cycle of preferences. We speak of a transitive cycle because there is transitivity from the first to the last choice (we transfer a relationship from one element to a second, and from this second to a third, between the first and the third, or even 1 eats 2 eats 3 eat 1). Another way of saying that the loop closes on itself. This is the Condorcet paradox.
How to resolve this paradox? Condorcet’s paradox is too powerful a paradox. It doesn’t resolve. He walks around. To return to the ice cream parlor example, you could remove a flavor from the choice. For example, the strawberry tray is almost empty, the ice cream looks sad. You remove strawberry from the options. In this case, vanilla and chocolate remain and the paradox disappears because you know which one you prefer between the two remaining flavors. You need at least three choices to have a cycle. This is what is happening in France, for example. In presidential elections, they keep the two best candidates in the second round to avoid the cycle.
Galileo, the preventer of thinking in circles
Magnificent manipulation of votes
The Condorcet paradox also allows a magnificent manipulation of votes, as long as we grasp the trick during the debates. The trick had been used by Pliny the Elder in the 1st century AD. This is a typical example of anticipatory plagiarism. Good old Pliny defended before the Roman Senate a slave accused of having killed his master (but who claimed that his master had asked him to kill him, a sort of suicide by slave). The penalty incurred was the death penalty. The senators were divided into three groups: a first group wanted death, another group leaned towards exile and the last group wanted absolution.
Pliny noted that he was in Condorcet’s paradox (or the glacier paradox), with majorities that were certainly different but each preferred one option over another in a transitive cycle. The cycle in Pliny’s story was as follows: death beats absolution, absolution beats exile, and exile beats death (death > absolution > exile > death). The astute Pliny had subtly manipulated the vote by imposing the order of operations. He initially suggested voting between death or exile. Exile had taken him away. Then he weighed exile or absolution, and absolution won. His slave had been acquitted. To win, you have to offer options two by two, saving yours (the favorite) for last. For the record, Pliny was as proud as Artabanus of his trick and he recounted it in a book.
Paul Magnette doesn’t like others taking over his party, Bart De Wever is uncomfortable
Sin of gluttony
Condorcet also played the politician. The Revolution was a dangerous time for politicians because majorities were fragile. When his party was in trouble, instead of tearing down the walls, Condorcet had done his smartest thing and took the cart to the scaffold. The trickster had managed to escape, but he had been caught in the stomach, a sin of gluttony. Here he is in an inn ordering an omelette, the popular dish par excellence – the idea was to go more or less unnoticed. The innkeeper asks him how many eggs? The guy answers twelve! It was the answer not to do. What common man would eat a twelve-egg omelette?
Condorcet, however, managed to keep his head on his shoulders, in return he had to ingest poison in horse doses. He left us one last paradox for the road. In 1989, his remains were transferred to the Pantheon, but in truth his remains remained where they remained, that is to say in Bourg-la-Reine. The coffin at the Pantheon is an empty coffin. In short, the Pantheon has the non-remains of Condorcet.
