Remedies to the paradox: Borda Method

Another method to circumvent Condorcet’s paradox is that was proposed by Chevalier de Borda in 1770, which was in contrast with Condorcet on this topic. It is an ancient method, dating back to the Roman Senate, and is still used in different fields. It consists essentially of generating a ranking of points of the various candidates in which each voter assigns an order of preference. Let’s take an example. Suppose the candidates are 6, Aldo, Bruno, Carlo, Dario Ezio and Fabio. We still have 3 voters and their preferences are:

1 => A>B>C>D>E>F
2 => D>B>A>C>E>F
3 => D>C>A>F>B>E

Let’s put them in a table where we assign, for each voter, 6 points to the most preferred one, 5 points to the second, and so on.

1 6 5 4 3 2 1
2 4 5 3 6 2 1
3 4 2 5 6 1 3
Tot: 14 12 12 15 5 5

From the maximum score we see that the winner should be Dario.
Let’s now consider the possibility that, before the vote, voter 1, who would prefer to be elected Aldo, looks at the result of a survey and hence feel that its best candidate was not the favorite. He could modify its preferences as follows:

1 6 3 2 1 5 4
2 4 5 3 6 2 1
3 4 2 5 6 1 3
Tot: 14 10 10 13 8 8

So voter 1, even though he did not modify his highest preference, could even bring it to victory by intervening on other weight. A similar result could also occur if, in a set of candidates, with their relative weights, were to be added another candidate. This is because the Borda method lacks an important feature that is defined as “independence to irrelevant alternatives that says that the choice between two options should be determined solely by the relation of preference between them and not by the other options in the game.

Remedies to the paradox: Majority

This is a solomonic solution to the problem: the candidate who scores more votes wins. In this case, you renounce the opportunity to reward a possible “Condorcet winner” . For example, let’s consider the following table:

Number of voters per alternative
2 3 4

In this case, the winner is C who has scored 4 votes, but if we do Condorcet’s analysis, we note that A>B with 6 votes against 3, A>C with 5 votes against 4 and B>C with 5 votes against 4. The method certainly satisfies Carlo’s supporters, but everyone else would find a winner who is at the last position of their preferences. We have a report of type A>B>C then we could apply the Condorcet method and have Aldo as a winner. Bruno and Carlo’s supporters would still have a winner having the second place in their rankings. Unfortunately, the majority rule does not catch this opportunity.

Remedies to the paradox: Double Turn

There is a second vote, having as alternatives the two candidates who scored more votes in the first round. So considering the situation in the previous paragraph Aldo would still be defeated despite being the “winner according to Condorcet” but his votes probably go to Bruno given the preference chain A>B>C. Given the previous scheme, despite Carlo being the candidate with the highest relative share and Aldo being the second best choice for everyone, Bruno would win anyway! With this system we not only do not take the opportunity to elect the “Condorcet winner”, but we also displease the majority of the voters who are the supporters of Carlo.