(Rodrigo Peñaloza, March 3rd, 2015)

Here I show that a democratically elected candidate by majority voting in two rounds can also be a candidate who is rejected by an outstanding majority of voters. For the wining candidate in a democratic election, it is not wise to take for granted the political support of the population. Victory in the polls does not preclude him or her from the need to maintain perennial talk with the opposition, not only to assure agreement on important issues, but also to truly convince people.

This is one virtue of Democracy that some presidents, prime ministers and winning parties seem not to understand. The reason for this error is their not noticing that the electorate’s preference orderings are built on the whole list of candidates in the first round, not only the two most voted candidates that compete in the second round, should it by chance occur. A case in point is Brazilian President Dilma Rousseff and her Workers’ Party. She is known for her incapability to talk with the opposition, not to mention her inability to spontaneously deliver any intelligible speech.

In the Theory of Social Choice, this is a mere instance of Arrow’s Impossibility Theorem, which says that no voting system simultaneously satisfies a set of reasonable properties known as Arrow’s axioms. One of these properties, the one which is violated by the example I will provide, is the Axiom of the Independence of Irrelevant Alternatives, according to which the collective decision with respect to a pair of candidates should depend only on voters’ private choice with regard to these two alternatives only. In simple words, the decision on a pair of candidates should not depend on the existence of a third candidate.

Suppose there are 3 candidates (A, B e C) and 100 voters, whose preference orderings are as follows:

A ≻ C ≻ B (26 voters)
B ≻ C ≻ A (49 voters)
C ≻ A ≻ B (25 voters)

The preference ordering A ≻ C ≻ B means that, for the voters in the first group above: “A is preferred to C”, “C is preferred to B” and, by transitivity, “A is preferred to B”. The number of voters is 100 for obvious reasons. I want to give it the meaning of percentage of voters. Thus 26% of the voters (the voters in the first group above) have preference orderings described by the following sequence of decisions: if elections were run between candidates A and C, they would vote for A; if between C and B, they would vote for C; if between A and B, they would vote for A. Similar reasoning of course applies to the other preference orderings as well.

The rule of election is “majority decision with two rounds”: if no candidate gets the absolute majority of votes (that is, at least 50% plus 1 vote), then the most voted two candidates will compete in the second round, again by majority voting. The winning candidate in this case is the one who gets the majority of votes in the second round.

In the example, the result of polls is:

Candidate A: 26 votes in 100
Candidate B: 49 votes in 100
Candidate C: 25 votes in 100

Since no candidate got the absolute majority of votes, the less voted candidate, C, is eliminated, thus candidates A and B go to the second round. Removing candidate C from the preference orderings given above, we get the following orderings regarding candidates A and B only:

A ≻ B (26 voters)
B ≻ A (49 voters)
A ≻ B (25 voters)

In the second round, candidate A gets 51% of approval against 49% for candidate B. Therefore candidate A wins the election.

Notice, however, that according to the original preference orderings with all 3 candidates, C is preferred to A by 74% of the electorate. Indeed, for the reader’s comfort, let me reproduce the original preference orderings, to make it simpler for the reader to check that out:

A ≻ C ≻ B (26 voters)
B ≻ C ≻ A (49 voters)
C ≻ A ≻ B (25 voters)

If the election were to be run with candidates A and C only, absent B, then candidate C would be elected with 74% of approval (the 49 voters in the second group plus the 25 voters of the third group). In other words, candidate A, notwithstanding the victory in the polls, is still rejected by 3/4 of the voters!

Letter A can stand for Dilma Rousseff. Notice that regarding candidates B and C, candidate B’s index of rejection is also 75%, also 3/4 of the voters. It does not matter if candidate B stands for Aécio Neves. He is not the president. Dilma is. This explains, by the way, why the opposers to Dilma do not go for Aécio either. In addition, candidate C, which may stand for Marina Silva, will never win, because 75% of voters prefer either Aécio or Dilma! The accidental distribution of preferences among the three candidates is the sole source of all this conundrum. Here is where Democracy takes place: talks, agreements and checks-&-balances. If a president is unable to understand that, and the only self-protection he or she can devise against opposition is the rhetoric and ridiculous point of majority election, then he or she is definitely not fit to be even a mediocre politician, let alone a president.

The lesson is very simple, though unfortunately not evident. The unwise winning candidate — say the elected candidate A is of one this type — normally draws attention only to the fact that he or she got elected by majority voting, but totally forgets the fact that the election in two rounds hides the electorate’s preference ordering regarding candidate C as well, the one who was kicked out in the first round. The winner tends to focus on the manifested poll decisions, not on the voters’ whole preference orderings.

If voters do not change their preference orderings in their minds, they will still reject the elected candidate A in comparison with C. The unwise elected candidate A may believe that his having winning over B by majority suffices to give him the necessary approval to rule the country. The winner wrongly believes to have 51% of approval. However, in the mind of 74% of the voters, the elected candidate A will be always seen with suspicion and rejection, unless people change their minds and start to prefer A over C after election. Acceptance is not enough: people have to truly change their minds. If that does not happen, it will likely be the case that voters for C will join voters for B to oppose A’s government and form a coalition with 74% of the people which will then do whatever it can to block the ruling party and oppose the ruler.

The key of the problem is to realize that voters for C ended up voting for A in the second round only because they had to choose between A and B, not between A and C. This does not mean that in their minds they truly approve A. This mechanical aspect of the voting system cannot be mistaken for actual approval. In addition to importance of persuasion, the example highlights the importance of also acquiring majority in the Parliament.

This is what is going on now with respect to President Dilma Rousseff and her Workers’ Party. Her fiercest supporters insist on the surreal argument that she was elected by the majority of the people and they keep repeating the mantra that her opposers undemocratically do not accept the final results of the polls. Nothing far from the truth!

The good Stateman, or Statewoman for that matter, should understand that in a true Democracy election is actually an ongoing process, not just a temporal point in the course of events. The elected candidate should bear in mind that talking with the opposition is a matter of mentally convincing people that he or she is doing things right. It is not only majority voting that defines Democracy: it is its constant need for talking and persuasion.