The key facts

  • The Condorcet paradox (late 18th century) shows that the simple majority rule may fail to produce a winner with three candidates.
  • In 1951, Kenneth Arrow proved that all natural requirements for voting can be satisfied only with a 'dictator'.
  • Aleskerov and Aizerman's models, which develop Arrow's theorem, led to the same conclusion.

A perfect voting system does not exist — this has been mathematically proven

A perfect voting system does not exist and never will exist — this has been rigorously proven mathematically. This was told to Gazeta.Ru by the mathematician Fuad Aleskerov, director of HSE University's International Center of Decision Analysis and Choice and head of the Choice Theory Laboratory at the V.A. Trapeznikov Institute of Control Sciences of the Russian Academy of Sciences. According to him, the imperfection of majority rule was noted as far back as antiquity, and the earliest evidence of its danger was left by medieval chroniclers.

From Novgorod to Condorcet

As an early example, Aleskerov cited an episode from Veliky Novgorod in the 13th century: in an election for the city's administrator, opinions were divided, one of the candidates won by a narrow margin, and infighting broke out. 'As the chronicles say — I am almost quoting — two or three days later everything was stopped, the dead were buried, people sat down, began voting again, and chose the same person,' the scholar recounted.

The Condorcet paradox and the 'carousel'

The first rigorous proof appeared at the end of the 18th century, on the eve of the French Revolution: the Marquis de Condorcet showed that with three candidates and diverging voter opinions, even the simple majority rule fails. The paradox is easiest to demonstrate with a 'carousel': if for one third of voters A is better than B, and B better than C; for a second third B is better than C, and C better than A; and for the third C is better than A, and A better than B, then A beats B, B beats C, and C beats A — the circle closes, and there is no winner. 'Throughout the 19th century and half of the 20th century, people tried to devise procedures free of this paradox. They failed,' Aleskerov noted.

Arrow's theorem and its development

The final point was made in 1951 by the American Kenneth Arrow, a future Nobel laureate. He described voting with a set of minimal natural requirements and proved that all of them can be satisfied only by appointing a 'dictator' — a participant whose preferences automatically become the decision for everyone. The result entered science as the impossibility theorem. 'I knew Arrow, met him many times, and you would not believe what an impression he made on me,' Aleskerov remarked.

Checked again — the answer did not change

Aleskerov himself, together with his teacher Mark Aizerman, one of the founders of choice theory, built models that develop Arrow's theorem: the classical models were revised, but the conclusion remained the same. According to the scholar, Arrow respected, loved, and supported these models. 'Perfect models within the theory do not exist. If there are disagreements of opinion, whatever decision is made, some part of the voters will always be dissatisfied with it,' the mathematician concluded. The practical meaning of this conclusion is that debates about the 'best' way of voting have an objective limit: choosing any procedure means choosing which part of the participants' dissatisfaction to consider acceptable.