Vote for Brain for world domination.

(Rodrigo Peñaloza, March 4th, 2016)

Pindorama, the “Kingdom of Heaven on Earth”, is a democratic republic in which the run for presidency is characterized by absolute majority voting. If the most voted candidate gets 50% of the votes plus one, that is, absolute majority, then he is elected and will take office. The polls of course allow Pindoramans to count the sum total of votes to each candidate. In the end, it is possible to rank them all from first to last according to the share of votes. If no candidate gets absolute majority of votes, then the two most voted candidates will compete in a second round, again under majority voting. No voting system is perfect. Majority decision is no exception. Just Arrow’s Impossibility Theorem at work.

Eventually, Pindoramans were fortunate to elect their president already in a single round. “Candidate X rocks!”, people said. Actually, X is already president but is now running for his reelection. That year there were 4 candidates (X, A, B and C). There were 15 voters, each with his or her own preference ordering on the set of candidates:

X ≻ C ≻ B ≻ A (6 voters)
X ≻ C ≻ A ≻ B (2 voters)
X ≻ B ≻ C ≻ A (1 voter)
X ≻ B ≻ A ≻ C (1 voter)
A ≻ X ≻ C ≻ B (1 voter)
A ≻ B ≻ X ≻ C (2 voters)
B ≻ A ≻ X ≻ C (2 voters)

Take, for instance, the first array above. It says there are 6 voters with the same preference ordering X ≻ C ≻ B ≻ A. For them, “X is preferred to C”, “C is preferred to B”, and “B is preferred to A”. The other arrays are interpreted correspondingly. Of course the number of voters, 15, does not mean necessarily that only 15 people vote. We can say, for example, that “X≻C≻B≻A” is the preference ordering of 6/15 or 40% of the electorate, and so on. All can be taken in a distributional sense.

It is now easy to count the number of votes to each candidate:

candidate X: 10 votes out of 15 (67% of the votes)
candidate A: 3 votes out of 15 (20% of the votes)
candidate B: 2 votes out of 15 (13% of the votes)
candidate C: no votes (0% of the votes)

Thus the ranking of candidates is:

X > A > B > C

I use the symbol > (which differs from ≻) to rank candidates according to the number of votes. This is the aggregate order relation. Candidate X is then elected with absolute majority, approximately 67% of the votes. An overwhelming majority indeed. No need of a second round. Candidate A got the second place, B is third, and C is last. Poor C, he got no votes. Total humiliation.

Now imagine that, some time after X was sworn in and took office, people find out that X mismanaged the finances of his campaign. He used his influence to deviate money from public enterprises in order to finance the campaigns of his political allies and to perpetuate his party in power. In addition, the country’s economy is in a mess because of his heterodox economic measures. Unable to cope with such scandals, X resigns.

What happens now? When I ask students this question, they all agree that candidate A, who was ranked second in the election, should come in and take office. Indeed, he was the second and it seems fair that, once the winner is out, the prize should go to the next to first. From the original ranking, X>A>B>C, if we remove X, we get:

A > B > C

Simple as that.

Wrong! Economic wisdom teaches us that when one parameter is changed, the whole system should be put to work again in order to compare two optima, not just two states of the world. What happens if there are new elections with all the candidates but X? I will repeat below the original arrays of preference orderings without candidate X:

C ≻ B ≻ A (6 voters)
C ≻ A ≻ B (2 voters)
B ≻ C ≻ A (1 voter)
B ≻ A ≻ C (1 voter)
A ≻ C ≻ B (1 voter)
A ≻ B ≻ C (2 voters)
B ≻ A ≻ C (2 voters)

Candidate X has been removed, everything else is constant. This means that voters didn’t change in their minds the way they rank candidates. They just don’t have candidate X any more. Assume then that we have new elections under the same rules.

If we count the number of votes to each of the remaining candidates, we have:

candidate A: 3 votes out of 15 (20% of the votes)
candidate B: 4 votes out of 15 (27% of the votes)
candidate C: 8 votes (53% of the votes)

Thus the ranking of candidates is:

C > B > A

If there were new elections, candidate C would be elected in a single round with absolute majority. Remember that C got no vote in the first election and was placed last! Besides, compare the ranking of the new election (C>B>A) with the ranking of the first election with X removed (A>B>C). It is the reverse ordering!

The example shows that, if the winner X resigns and A takes office, then it is likely the case that A will face a strong opposition and popular aversion. To the naïve eyes, this would be strange and unexpected. Indeed, if A was second, how come he is now so rejected? Did voters suddenly change mind? Of course not. On the other hand, candidate C, who got no vote in the original election, would be the winner if new elections occurred. The purpose of this text is pretty obvious. I want to show that Democracy has many problems people don’t even think about and highlight the importance of Social Choice Theory and Arrow’s Impossibility Theorem.

This is the meaning of Mathew 20:16, “So the last shall be first, and the first last; for many be called, but few chosen”.

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