Abstract. 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 attention is drawn to (1) his own explanation of this requirement in 1972, which is a quite different understanding than has been discussed in the voluminous literature on the theorem, (2) that Arrow, in fact, in 1985 showed an understanding for how "irrelevant alternatives" might in a meaningful way influence the group choice, (3) that admittedly the border-line between irrelevant and relevant alternatives in Arrows original statement is arbitrary, and (4) that Arrow, if he had observed the final thought in the origin of the group theory by Borda, which he admittedly did not, might have realized that Borda´s method stringently estimates the relevance of each alternative for the result. The author expresses his surprise that a theoretical conclusion based on an arbitrary fundament has been admired so long.

In the theory of collective choice Arrow´s contribution has been of central importance for more than 40 years (Arrow, 1951). In 1963 he added a further theoretical discussion (Arrow, 1963; 92-120). In 1972 he accounted for the problem on 2 pages in a lecture in memory of Alfred Nobel (Arrow, 1972; 228-30). Beyond that, his only published contribution to the discussion is in a letter to me which I published with his permission Stefansson, (1991b). The letter gives rise to a thorough evaluation of his contribution.

The discussion was started in France by Borda and Condorcet more than 200 years ago.1 Borda devised his voting method for solving a practical problem when electing new members in the French Academy of which he was a member. The problem was that a cyclical majority could arise where there were three on choice and there was voting between them in pairs. Borda´s solution was that the voters rank the alternatives. Then the voting committee calculated points for each alternative by comparing all alternatives in pairs in the order for each voter. The first alternative of the two receives one point. The alternative with most points is the choice of the group. When calculating the points it shows which of the alternatives are relevant for the result. There are no requirements for the alternatives in this respect, they can all count in the calculation.2 Condorcet, on the other hand, favoured the principle to vote in pairs between the alternatives. The alternative that wins is the group´s choice. In the case of a cyclical majority he proposed special rules.

As far as I know, no arguments have been presented that either one of these principles be better than the other, but nevertheless Condorcet usually serves as a starting-point. A recent example is that Newenhizen came to the conclusion when evaluating voting rules that Borda´s method satisfies Condorcet´s principle best (Newenhizen, 1992). There should be at least as good a reason to study how other voting methods follow Borda-choice. As a matter of fact Borda´s method has the advantage over Condorcet´s principle that one need no special rules for exceptional cases. It is, in fact, valid under all circumstances.

Arrow did not apply Borda´s practical starting-point. Instead he formulated some fundamental requirements for methods for group choice. One of them was that the choice must not depend on irrelevant alternatives. He concluded that it was impossible to satisfy all this requirements simultaneously (Arrow, 1951; V). What he first thought as a possibility theorem became an impossibility theorem. The scientific world accepted his way of presenting the problem, the requirements he made on the method, and the conclusion. In 1972 he was awarded the prize for economic science in memory of Alfred Nobel by the Bank of Sweden, among other things for the above contribution (Bentzel, 1972). The following words of praise can be regarded as an another example of how great importance has been attached to his contribution (MacKay, 1980; 5): "It is a rigorous demonstration of an entirely unobvious, surprising, paradoxical result. Like all the great paradoxes, Arrow´s result on collective rationality resists understanding and resolution equally as it demands the attempt."

Here, MacKay admires Arrow for being rigorous, although he finds it difficult to understand him. Much has been written about how Arrow should be understood. One could rather have asked the instigator to explain. That was not done, and he did not explain himself by any further important contribution to the discussion. Nevertheless, in his lecture when he received the Bank of Sweden´s prize for economic science he states 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 says the following about the fourth (Arrow, 1972; 229):

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. It turns out (Arrow [1951b, 1963b])3 that these four reasonable-sounding requirements are contradictory. That is, if we devise any constitution, then it is always possible to find a set of individual orderings which will cause the constitution to violate one of these conditions.

It is surprising that Arrow should on such an occasion explain the independence of irrelevant alternatives in this way. The discussion about it has by no means focussed on such an understanding, no matter which author one reads. I cite here the interpretation given by Hansson: If there are two different choice situations which are such that the preferences between all alternatives in a set of alternatives are the same for each of the participants in both situations, the common order in this set shall be the same for both situations. It shall not influence the order in the common set of alternatives how alternatives outside the set are preferred, this shall, as already mentioned, be "irrelevant" (Hansson, 1970). — In the article from 1982 I, in fact, criticized Arrow´s demand for independence of irrelevant alternatives referring to conditions in the theory of demand in economics. 4

My evaluation of methods of collective choice was not based on how they fulfilled ideal requirements, but what the method would mean for the presentation of issues, their treatment and the organization around them. When I happened to discover that in chess tournaments a method is applied to measure who is the best player, which gives the same result as Borda´s method, I assessed it in the light of it (Stefansson, 1982). Of course our form of chess tournaments was not known in Borda´s times. Then I discovered that Borda considered the method in the same way as I by analysing the method for ranking players in chess tournaments (Stefansson, 1991a). This insight has, in fact, been lacking among the collective theoreticians that have appeared on the scene after Borda.5 They have, in fact, evaluated Borda´s method as if it was merely a calculation of points without any fundamental thought and discussed it together with methods that look similar to Borda´s method as a method for calculating points but lack the same fundamental principle, some of them without any principle at all. 6

In a letter from the year 1985 Arrow presented a new evaluation of the question on alternatives that are relevant for the choice, where he presented the question, as follows (Stefansson, 1991b; 299): "If there are three alternative policies possible, why not have the voters rank these with one or more additional alternatives which are not feasible?" He considered these additional comparisons would provide still more information about the comparisons among the feasible alternatives. — To take account of such additional information is obviously a rejection of what was called the condition for the independence of irrelevant alternatives.

Arrow did not turn to the origin of the theory when working with the impossibility theorem (Arrow, 1963; 93). By considering the final words of Borda´s paper, he could have realized that his method estimates the relevance of each alternative for the result. 7 If he had given attention to this under his earlier reflections on collective choice there would not have been any reason to present the condition about the independence of irrelevant alternatives. It could of course have become an "Arrow´s theorem" without that condition, but it would have been quite different from the one we know.

It is common to draw the conclusion from Arrow´s theorem that the methods entail that collective choice must be arbitrary. An authoritative reference is the speech by Bentzel in 1972 on behalf of the Swedish Academy when the Bank of Sweden´s prize for economic science was presented to Arrow. His concluding words were, as follows (Bentzel, 1972; 203):

In his doctoral thesis, which was published in 1951, Arrow put the following question. Let us assume that in a society one has a number of alternative conditions to choose between and that each individual in the society can rank all these alternatives in order of desirability. Is it, in this case, possible to find ethically acceptable, democratic rules, for making a collective (or social) ranking of the different alternatives in order of desirability? Arrow showed that that question must be answered in the negative. It is in principle impossible to find such rules. This conclusion, which is a rather discouraging one, as regards the dream of a perfect democracy, conflicted with the previously established welfare theory, which had long employed the concept of a social-welfare function. However, this concept is nothing but an expression of a social ranking order of desirability such as Arrow had shown that it was in principle impossible to make.

MacKay´s footnote to the above words of praise is a more recent example of the same evaluation (MacKay, 1980; 5): "Omitted here, obviously, are more mundane considerations of "importance": that Arrow´s Theorem allegedly sounds the death knell of democratic theory, welfare economics, and various other good things." In this way, Arrow´s theorem has been of importance in political discussion. This has been particularly obvious in the "public choice" school.

Riker has carried out a widely known argumentation about the importance of Arrow´s theorem (Riker, 1982). It is not easy to see what would have been left of such an argumentation, if Arrow had started his studies at their origin and found there a clear method of measuring how relevant the alternatives are for the result. There is a special reason to be surprised at the great respect for Arrow´s conclusion, as the border-line between irrelevant and relevant alternatives, on which the whole depends, is admittedly arbitrary.8 It is a normal task for economists to reveal contradictory ends and means for individuals and society. Arrow´s message was of that kind, calling attention to apparently reasonable considerations and examining how far they could be satisfied simultaneously. This is clear as far as it goes. The problem is only that one of the considerations turns out to be arbitrary and that in addition it is contrary to a more reasonable consideration, namely being open to any evaluation among the participants and then seeking a logical method to assess the importance of every evaluation. This was, in fact, involved in Borda´s method.

Quality & Quantity 29 (1995) 433-38

References

• Arrow, Kenneth J.: “General economic equilibrium: purpose, analytic techniques, collective choice.” Nobel Memorial Lecture, December 12, 1972. pp. 209-31 in Les Prix Nobel en 1972.
• Arrow, Kenneth J.: Social choice and individual values, John Wiley & Sons, New York, 1951.
• Arrow, Kenneth J.: Social choice and individual values, second edition, Yale University Press, New Haven, 1963.
• Bentzel, Ragnar: “The prize for economic science, in memory of Alfred Nobel.” pp. 202-3 in Les Prix Nobel en 1972.
• Björn S. Stefansson: “Group choice between three or more alternatives,” in Quality & Quantity vol. 16, 1982: 433-54.
• Björn S. Stefansson “Borda´s method applied - The right to make a proposal. A historical note,” in Quality & Quantityvol. 25, 1991: 389-92.
• Björn S. Stefansson: “On irrelevant and infeasible alternatives,” in Quality & Quantity, vol. 25, 1991: 297-306.
• Hansson, Bengt: “Valsystem och beslutsprocesser,” in Minerva´s kvartalsskrift vol. 14, 1970: 190-198.
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• MacKay, Alfred: Arrow´s theorem. The paradox of social choice.. A case study in the philosophy of economics, Yale University Press, New Haven, 1980.
• Newenhizen, Jill Van: “The Borda method is most likely to respect the Condorcet principle,” Economic theory, vol. 2, 1992: 69-83.
• Nurmi, Hannu: “Preferences, choices, tournaments: alternative foundations for the evaluation of voting schemes,” Quality & Quantity, vol. 25, 1991: 393-405.
• University of Oxford. Faculty of social studies. Social studies faculty centre, working paper 1/89. The political theory of Condorcet. 1989.
• Riker, William H.: Liberalism against populism. A confrontation between the theory of democracy and the theory of social choice, W. H. Freeman and Company, San Francisco, 1982.
• Sen, Amartya: “Social choice theory: A re-examination,” Econometrica, vol. 45 1977: 53-89.