Condorcet voting method

At Open Agora, Condorcet voting system is like our Philosopher’s stone. A source of development and progress for Humankind which we try to promote through our values and products.

This article will present this voting method using a concrete example. Using another example, I will demonstrate some advantages of Condorcet system when a consensus, i.e. a decision which satisfy most of the voters, is needed.

This voting system was created by the French mathematician Nicolas de Condorcet at the end of the 18th century. In this earlier article, I already spoke about Nicolas de Condorcet and his scientific theories. In July, I will write another article about his political and philosophical commitment.

What is it?

Before continuing, it is necessary to know that Condorcet voting system is relevant when there are at least 3 possible choices. Indeed, having only 2 choices to rank will lead to the same result as majority vote.

Now, let us start explaining this voting system with a very simple example. Assume there are 3 proposals: A, B and C. Each voter will rank these choices by preference, for example:

  1. A
  2. B
  3. C

Nothing innovative yet, since other systems also use lists of preferences, like the Instant-Runoff voting. Let’s have a look at the global result.

Assume there are 75 voters for these 3 choices. In the table below, we can read that 35 voters have chosen A > B > C; 25 voters have chosen B > C > A; 15 voters have chosen C > B > A.

VotersFirstSecondThird
35ABC
25BCA
15CBA
I voluntarily reduced the list of rankings in order to simplify explanations. There are three other rankings namely : A > C > B, B > A > C or C > A > B.

But how to compute the winner out of this table? This is where the Condorcet voting system innovates with a consensual method → voting analysis is based on duels between choices.

Explanations!

Each choice faces each other in duels. A duel is won by a choice when it is ranked above another one in a given vote → In the “A > B > C” list, A wins one duel versus B and versus C, furthermore, B wins versus C and C does not win any duel.

Here are the duels between the choices, extracted from the table:

  • 35 voters prefer A > B and 40 (25 + 15) B > A (B wins against A)
  • 35 voters prefer A > C and 40 (25 + 15) C > A (C wins against A)
  • 60 (35 + 25) voters prefer B > C and 15 C > B (B wins against C)

To conclude, B is the Condorcet winner because it wins its duels against every other candidate.

Why use Condorcet ?

Condorcet analysis with duels between choices produces a more consensual result. Indeed, the Condorcet winner is the one which would win any 1 versus 1 confrontation facing any other candidate. The Condorcet winner is not the choice which got more votes but the choice which won versus every other choice. You can observe why this system facilitates consensus and approbation by a majority of voters.

Let us take a more concrete example demonstrating the interest of this system, in order to satisfy a majority of voters. Imagine four cities (A, B, C and D) want to decide the location of a common hospital.

This schema represents the geographical distribution of cities, and their respective populations. It is assumed that voters are selfish, they all want the hospital to be as close as possible to their own city.

With the majority vote, city A wins because its population is more important. But, looking at the schema, we can say it is not the best choice. Similarly and without recalling its methodology, Instant-Runnoff would find city D as the winner, which does not seem to be the best solution for the location of an hospital shared between the cities.

Here is the table of ranked votes:

VotersFirstSecondThirdFourth
42kABCD
26kBCDA
17kCDBA
15kDCBA

We can see A choice is the most approved but it’s also the most rejected choice. A situation which is not as paradoxical as it may seem, especially for people who followed the recent presidential elections in the USA.

Here are the duels results:

  • 42k voters prefer A > B and 58k B > A (B wins against A)
  • 42k voters prefer A > C and 58k C > A (C wins against A)
  • 42k voters prefer A > D and 58k D > A (D wins against A)
  • 68k voters prefer B > C and 32k C > B (B wins against C)
  • 68k voters prefer B > D and 32k B > D (B wins against D)
  • 83k voters prefer C > D and 17k D > C (C wins against D)

The Condorcet winner is B. This solution seems to be the most logical one given the geographical configuration of the four cities.

To conclude

There are other features of Condorcet method that we will discuss later on. I think, for example, to methods for “tie breaking” that Open Agora uses when, occasionally, the ballot does not directly designate a winner.

I hope this presentation of this method was clear enough. Feel free to comment at the bottom of the page.

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