• A
  • A
  • A
  • ABC
  • ABC
  • ABC
  • А
  • А
  • А
  • А
  • А
Regular version of the site

Mathematics of Natural Choices

The Borda count and independence of irrelevant alternatives

© ISTOCK

Eric Maskin, a leading research associate at the HSE International Centre of Decision Choice and Analysis, Professor at Harvard University and a Nobel Prize laureate in economics, spoke at a recent research seminar held by the Centre in which he proposed a modified independence of irrelevant alternatives condition (IIA) and demonstrated that the voting rule satisfies modified IIA, Arrow’s other conditions, and May’s axioms if and only if it is the Borda count, i.e., rank-order voting. Fuad Aleskerov, Doctor of Sciences, Professor of HSE Faculty of Economic Sciences, and Director of the International Centre of Decision Choice and Analysis, discussed how this affected the collective choice theory, elaborating on various choice models, starting from the Condorcet paradox.

Fuad Aleskerov
Director of the HSE International Centre of Decision Choice and Analysis

As a sub-discipline of the social sciences, the collective choice theory was based on very good mathematics at the end of the 18th century and evolved from the work of Nicolas de Caritat, marquis de Condorcet, a prominent French scientist. He put forward an example, later called the Condorcet paradox, where he showed that if the number of alternatives (for instance, candidates in an election) and the number of voters are more than two, then the simple majority rule may not result in any decisions. Condorcet's model accounted for both the most preferable options and the whole order of alternatives.

The Condorcet paradox made a huge impression on his contemporaries and fascinated scientists of the 19th and early 20th centuries. Since the Classical Period, people had believed that the majority was always right and that its opinion reflected the will of higher forces. There was even a wonderful Roman proverb ‘Vox populi—Vox dei’, which means ‘The voice of the people is the voice of God’. Condorcet demonstrated that sometimes the vox populi may lead to contradictory results.

Throughout the 19th century and the first half of the 20th century, scientists tried to propose rules that would help avoid the "Condorcet paradox". However, it continued to manifest itself in one way or another.

At the same time as Condorcet, another great French scientist, Jean-Charles de Borda, proposed another decision-making rule, where alternatives were ranked, their ranks were summed up, and the alternative with the maximum rank was selected. This was essentially a direct generalization of the plurality voting rule that we apply today.

A century and a half later in 1951, Kenneth Arrow—one of the five greatest economists of all time—developed a different approach to rules formation. He was the first to introduce a system of axioms, external constraints that any reasonable rule should satisfy, and as a result, shaped up a corresponding rule from these axioms. 

For example, it included the Pareto condition: if all participants of the voting procedure believe that alternative x is better than alternative y, then the collective choice should be the same. The axioms included neutrality—the rule should be applied to the same extent to all alternatives (candidates), and anonymity—the same condition, but with regard to voters. 

Apart from the theories above, Arrow introduced another fundamental axiom that he called independence of irrelevant alternatives. This condition seems to be absolutely crucial in the Arrow model. It states that when making decisions on the mutual ordering of alternatives x and y, we do not need to consider the mutual relations between x and other alternatives, as well as between y and other alternatives.

In other words, collective preference between alternatives x and y depends exclusively on individual preferences between x and y. Moreover, if x is more preferable to y from the range {x, y}, then the introduction of the third option z and expansion of the range up to {x, y, z} should not make y more preferable to x. This means that the preference for x or y should not change with the introduction of z.

We called this axiom a locality condition in our works. I remember telling this for the first time to Professor Arrow of whom I still cherish the fondest memories. He thought a little and said: "Indeed, this is a locality. I have never even thought of it". 

The result of Arrow's work, called "Arrow Theorem" or "Arrow Paradox", was unfavourable in the sense that the only procedure that meets the conditions above is completely non-democratic—a dictatorship. In other words, one member is assigned, and the collective choice inevitably coincides with his individual decision.

Two decades later, back in 1977, the young scientist Eric Maskin proposed considering such correspondences of group choice that allow representing the collective decision in the form of Nash equilibria in a specific formation. Later, Maskin successfully solved this fundamental problem, for which he was awarded the Nobel Prize in Economics in 2007 (which he shared with Leonid Hurwicz and Roger Myerson).

With this problem, the locality condition is formed in the following way: we consider the relations between x and y so that the elements under the x variant (i.e., the elements that are worse than x) would be the same, but their order is irrelevant.

We also obtained fundamental results in this field. One of my works describes an axiomatic substantiation of the so-called collective choice correspondences, which, in particular, meets Maskin's monotonicity condition (if the alternative (candidate) x improves his or her position compared to y, then his or her position in the collective decision should also improve). 

Just recently, Professor Maskin has received some new results. He proposed evaluating the relationship between the alternatives x and y, and if the positions of the elements between x and y remains unchanged, the relationship between x and y in the collective decision does not change either. If there are other well-known constraints—neutrality, anonymity and monotony—the Borda count turns out to be the solution that satisfies this system of axioms.

In my opinion, this is an outstanding achievement in the study of non-local procedures for collective decision making, which will trigger a significant amount of work afterwards. This is the first axiomatics ever that generalizes the Arrow axiomatics and leads to one of the well-known positional rules—the Borda rule. A fundamental breakthrough has been made in a field that is more than 200 years old, and I am very pleased to know that the author of this breakthrough is a friend of mine and an employee of our Centre and HSE, Professor Eric Maskin.

IQ

Author: Fuad T. Aleskerov, August 10