Democracy Center - The Arrow impossibility theorem

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The Arrow impossibility theorem. Eric Maskin and Amartya Sen with Kenneth J. Arrow, Partha Dasgupta, Prasanta K. Pattanaik and Joseph E. Stiglitz. (Kenneth J. Arrow lecture series) Columbia University Press, 2014. A discussion

 

Abstract: In three expressions in the literature, in 1972, 1991 and 2014, Arrow does not support the condition of independence of irrelevant alternatives. As Arrow´s theorem is based on that condition as much as the other three conditions, Arrow did not support “The Arrow Possibility Theorem” in this period.

 

This volume has three Nobel Memorial Laureates in Economics participating among others (the first time the Prize in Economics was awarded was in 1969, Arrow received it in 1972, Sen in 1998 and Stiglitz in 2001). Therefore it gives an excellent occasion to discuss the position of the theorem. We are concerned with the axiom of independence of irrelevant alternatives. In a commentary (pp. 57-63) Arrow expresses the following (p. 59):

The organizers of the lecture and this volume have asked me to adress a few other items. One concerns what aspects of social choice theory I would be interested in pursuing today. From a technical point of view, I would like to see more research on the condition that I find, in a way, to be the most problematic: the independence of irrelevant alternatives. In other words, if this condition is relaxed is there a decision mechanism that will satisfy the other restrictions? I might, for instance, put a few candidates on the ballott who were not actually available. They could serve, in a way, to give some measurement to the others. Now I do not exactly know how do do this in a way that is going to be consistent. But by dropping the independence of irrelevant alternatives condition, you could then employ a Borda count, the method proposes by Balinski and Laraki (“A Theory of Measuring, Electing, and Ranking,” Proceedings of the National Academy of Sciences 104, 21 (2007): 8720-8725), or perhaps some other alternative. Regardless, what emerges is a consistent ranking, which satisfies the other conditions, not based on cardinal utilities but instead on the rankings. Of course if a candidate drops out, different results emerge. Whether this is a good decision mechanism or not, I am not prepared to say. Regardless, it is an interesting area for furthger exploration.

This is not the first time Arrow expresses problems with the independence of irrelevant alternatives condition, as will be shown.

 

Arrow´s opinion in Stockholm 1972

When Arrow received the Bank of Sweden´s prize for economic science he stated his opinion on what the condition of the independence of irrelevant alternatives means. After mentioning three other conditions which a reasonable method for collective choice should fulfil, he said the following about the fourth (p. 229 in “General economic equilibrium: purpose, analytic techniques, collective choice.” Nobel Memorial lecture in Les Prix Nobel en 1972.):

The fourth condition which I have suggested, that of the Independence of Irrelevant Alternatives, is more disputable, though I would argue that it has strong pragmatic justification: the social choice made from any set of alternatives will depend on only the orderings of individuals among alternatives in that set. To see what is at stake, suppose that a society has to make a choice among some alternatives and does so. After the decision is made, an alternative which has not previously been thought of is mentioned as a logical possibility, although it is not feasible. The individuals can expand their preference orderings to place this new alternative in its place on their ranking; but should this preference information about an alternative which could not be chosen in any case affect the previous decision?

Any form of voting certainly satisfies the condition of Independence of Irrelevant Alternatives; the preferences of voters as between candidates and non-candidates or as between non-candidates, are of course, never asked for or taken into account.

Explanations in the literature on the condition of independence of irrelevant alternatives have never referred to this expression.

 

Correspondence with Arrow

A statement in a letter from Arrow:

If there are three alternative policies possible, why not have the voters rank these with one or more additional alternatives which are not feasible? These additional comparisons will provide still more information on the comparisons among the feasible alternatives.

The statement is published in my article “On irrelevant and infeasible alternatives”, Quality & Quantity 25 (1991), which has the following abstract:

In a letter to the author Arrow makes an important recognition regarding the question of irrelevant alternatives by expressing his view that alternatives which are not among the superior ones can, in fact, affect the choice of the best alternative (it is a question of choosing the best chess player).

 

In these three quotations Arrow does not support the condition of independence of irrelevant alternatives. As the theorem is based on all the four conditions Arrow did not support “The Arrow Possibility Theorem” in this period.

 

The correspondence with Arrow started in 1985. I had published the article “Group choice between three or more alternatives” in Quality & Quantity 16 (1982) where I among other things maintained that the condition concerning irrelevant alternatives went contrary to the basic assumption in the economic theory of demand that choice (demand) depends on the circumstances, that it is the alternatives that determine the circumstances and that it must show in each individual case what alternatives are relevant for the conclusion (the choice). Furthermore I discussed how meaning inherent in a voting rule would qualify the method. After a while I asked Arrow for a comment on the article, cf. the article in 1991.

 

In the present volume, The Arrow Impossibility Theorem, meaning inherent in a voting rule is not discussed. I wanted to refer to a more comprehensive discussion on the theme and on Arrow´s view on the importance of irrelevant alternatives in my book, published in 2013, Democracy with sequential choice and fund voting, Chapter II.D.4, «A more sensible method». The sensible method is sequential choice. In a sequential choice settlement, every single calculation has a choice meaning conforming with the meaning in every other single calculation. No voting paradox comes into existence. It shows how relevant every alternative is for the choice.

 

On this occasion, I find it convenient to quote the first and the last sentence in the abstract in my article in Quality & Quantity in 1995, “On the fundamental thought behind voting rules”: «The theme is Arrow´s requirement in his theorem of 1951 on methods for group choice, that the choice be independent of irrelevant alternatives. … The author expresses his surprise that a theoretical conclusion based on an arbitrary fundament has been admired so long.»