Condorcet Voting Method

Condorcet voting method

Do you know the Condorcet voting method? At Open Agora, we consider this system like our Philosopher’s stone. This innovative voting method is very important to us and we strive to promote it through our values and products.

To demonstrate some of it advantages, we will, in this article, present the principles and methodology of this technique through a concrete example of the Condorcet vote.

Condorcet Voting Method

First, although this method may seem innovative, it is not new. Indeed, it was developed by the French mathematician Nicolas de Condorcet at the end of the 18th century. In a previous article, I already discussed about Nicolas de Condorcet and his various scientific works on social mathematics. In particular, he conducted research to establish a voting theory which would reflect more consistently the will of the people. It states that “if there is a candidate who, when confronted with any other candidate, is preferred, by a majority of voters, over that other candidate, then that candidate is the one of all that the people prefer“.

Thus, the goal of this method is to achieve consensus, in other words, the choice that satisfies the majority of voters.

How Does it Work?

Before continuing, it is necessary to recall that Condorcet voting system is only 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 by explaining this voting system on 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
35 votersABC
25 votersBCA
15 votersCBA

But how to compute the winner out of this table? This is where the Condorcet voting method innovates with a consensual method → voting analysis is based on duels between pairs of 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 Should you Prefer the Condorcet Voting Method?

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’s take a Concrete Example

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.

Condorcet voting method - a concrete example

This map 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 map, we can say it is probably not the best choice.

Let’s now try with the Condorcet voting method:

VotersFirstSecondThirdFourth
42kABCD
26kBCDA
17kDCBA
15kCDBA

We can see that choice A is the most approved but it’s also the most rejected choice (it is the choice which gets the most first ranks… and the most last ranks).

Here Are the Duels Results:

DuelsWinsDefeated
A vs B42k58k
A vs C42k58k
A vs D42k58k
B vs C68k32k
B vs D68k32k
C vs D15k85k

In this table, we can read the results of all the one-vs-one confrontations. For example, in the first line we see that A wins 42,000 times but loses 58,000 against B, but we can also read the reverse: B wins 58,000 times and loses 42,000 times against A. Overall, we see that the city B wins against each of the other cities. City B is therefore the Condorcet winner of this election.

This solution seems the most logical and appropriate given the geographical configuration of the 4 cities. It can also be noted that with these data, the result of an Instant-Runoff vote is city D! Indeed, in this method, in the “first round” (virtual), city C is eliminated because it arrives last. The 15,000 voters who placed it first are added to D (since it is their second choices), which is 32,000 voters in the second virtual round. B is then eliminated, and his voters are added to D who wins in the third round.

How to Use the Condorcet Voting Method?

Thanks to Open Agora, it is possible to make a Condorcet poll in Slack, very easily. By installing our application, you can carry out as many polls as you want with more consensual results.

Our poll application allows the expression of balanced opinions using the Condorcet voting method. Participants rank proposals according to their preferences, our system computes the winner based on the results of the duels and highlights the choice that satisfies the most people.

Feel free to test Open Agora for Slack now:

We are currently working on Instant Agora, a tool that will soon allow you to create surveys using the Condorcet voting method and other innovative voting and polling systems. We will very soon launch a new version of the tool, completely transformed with many new features.


To discover more about Nicolas Condorcet and this work, you can read our previous article:

Read also on our blog

Nicolas de Condorcet, a scientist serving the public interest

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