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).

It is clarified how alternatives become relevant in group decisions by Borda`s method. Great importance has been attached to Plott`s example regarding manipulation by Borda`s method. It is shown that the alternative, in his example, which is relevant for his conclusion is irrelevant for each participant and can therefore be excluded from the settlement.

What chess people manage despite the theory

In the theory of group decisions Arrow´s condition that irrelevant alternatives should not affect the ordering of the other alternatives has become a central theme (Arrow, 1951). On the basis of the analogy between Borda´s voting method and the ranking of chess people in a chess tournament one gains a new insight into how alternatives become relevant in the context of social choice. By Borda´s method the voters order the alternatives by giving the alternative that is considered best the top rank, the second best alternative being given the second top rank, and so forth. If two alternatives are considered equally good, they are given the same rank. The settlement is a simple calculation and can be considered as a settlement in a chess tournament. The top alternative is given one point for every alternative it has defeated. The second rank is then given one point less, and so forth, the bottom rank receiving no points. If two alternatives have the same rank, each of them receives one point for every alternative that is ranked below them and additionally half a point for a draw between themselves.

These calculations are made for each individual voting paper, which can be interpreted as the results of a single chess tournament. Then the points for each alternative from all the voting papers are summed up. (It should be analogous to the summing up of points from equally many chess tournaments with the same players in all the tournaments). The alternative that gains most points is the winner.

All the players or alternatives count in the calculation of points. A player who is just mediocre can have an influence on who wins the first place. Chess people have no objections to that. Assume a chess tournament which is nearing its finish and that the situation is that A is at the top with half a point over B. They have only got left to play against J, whose performance has been rather poor so far. These last games end with a victory for B and a defeat for A, so that B is the winner of the whole chess tournament. The audience may discuss as to how far the two last games gave a just result. But they realize that more chess games could be argued in the same way. Chess people have therefore, in principle, no objections to average players affecting who wins the top place.

It is, however, different with the established social choice theory. The fact that an average alternative affects the ordering of the two-three top alternatives, as exemplified by point calculation of alternatives by Borda´s method, goes contrary to a principle determined by Arrow (1951). I have more thoroughly accounted for this contrast between chess people´s way of thinking and Arrow´s principle in a previous article (1982).

I also maintained that Arrow´s 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).

Two letters from Arrow

I asked Arrow for a commentary on my article, especially on the arguments and the conclusion on pp. 438-39. He replied with a letter of 19th June 1985. For the subsequent discussion I quote by his permission, marking the passages with (a) and (b):

(a) On pages 438-9, you draw an analogy between social choice and the ranking of chess players in a tournament. The analogy, while interesting, cannot be pushed too far. In social choice, the problem is that there are many orderings (one for each voter) which have to be aggregated into a single ordering. In chess tournaments (or other forms of contests of skill), there is nothing parallel to the multiplicity of voters.

One can look at chess rankings in different ways. One is to assume that there really is a fundamental ordering of ability, but that in any individual play there is room for chance. Hence the problem is a statistical one, to infer from the actual outcomes of the plays, which do not yield an ordering, an underlying "true" ordering. The underlying ordering would be revealed if there were enough plays between each pair of contestants that statistical fluctuations would be averaged out. In this case, "irrelevant alternatives" are helpful for yielding additional statistical information when there are too few plays between each pair.

Alternatively, one could simply accept that chess comparisons are not transitive. Suppose that A is in some sense clearly superior, in that he can almost always defeat C, D, and so forth. However, for some peculiarities of style or of psychology, B can usually defeat A even though B is usually defeated by C, D, and the others. In this case, the irrelevant alternatives are really irrelevant, but there is no paradox because transitivity is not required. In neither case is the situation really analogous to that of social choice.

(b) You largely defend Borda´s method, and indeed it is appealing. It does not take account of preference intensities, an individual who puts A over B by several ranks may nevertheless have only a "small"degree of preference (the alternatives may be close to being tied in his or her view), while an individual who prefers A to B with no intervening alternatives may nevertheless have a strong preference. Still, one could argue, as Laplace did, that in the absence of preference intensity information, equal intervals are a best guess. But in that case, why not introduce additional infeasible alternatives into the comparison. 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 last point, underlined by me, that alternatives which are not ranked among the top ones should be allowed to affect ordering among the top ones signifies that Arrow has changed his opinion, as this goes contrary to the third of his famous conditions (No. 2 in my article from 1982). — The question of intensity dealt with by Arrow was discussed in previous articles (Stefansson, 1974a, 1974b). On the occasion of Arrow´s discussion under (a) about the analogy between social choice and the ranking of chess players in a chess tournament I pointed out the following in a letter to him: Please note that I view every voter ordering as one chess tournament. The aggregating of the individual orderings is then compared with the aggregating of just as many chess tournaments between the same players but with varying results.

He replied with a letter of 9th November 1987 which he has granted his permission to quote (I mark the passages with the letters (c)-(h)):

(c) I do not see the exact analogy you draw between chess tournaments and voting. In your paper, you spoke of "a" chess tournament. Now you consider that there are many chess tournaments with varying results. There is obviously some parallel between repeated chess tournaments and many voters, in that both create relations between alternatives (chess players in one case, issues or candidates in the other). However, there is one significant difference. We usually assume that each voter is rational, that is, that each voter ranks the candidates in an ordering. Presumably even one chess tournament may fail to satisfy transitivity.

(d) Before I go into details, let me make one further remark. Social choice theory is a critical analysis of social decision procedures, in that it compares actual procedures with ideal criteria. Similarly, I am perfectly prepared to say that the method of ranking chess players by the number of their wins does not yield a "best" player in the proper sense of the word, "best." The fact that actual tournaments are scored according to a Borda system is not, to me, evidence in favor of Borda counts, any more than the use of plurality voting to elect candidates for various offices in the United States is any evidence that it is always a good system (when there are three or more significant candidates, it very frequently chooses the "wrong" candidate by everyday observation).

(e) Let me start by taking the simplest situation for a tournament: any two players will have the same outcome whenever they play. In that case, one chess tournament has all the information. If one player defeats every other, he/she is clearly the best. Otherwise, what can one say? Suppose the Borda count gives A top rank, but B defeats A. I find it hard to ascribe any significant meaning to the statement that A is best. B is clearly better than A. In a rigorous sense, there is no best player. I note, by the way, that with regard to a World Championship, if Karpov defeats Kasparov, he is champion, even if Kasparov defeated a number of players who defeated Karpov. To retain the World title, the holder must defeat any (qualified) challenger.

(f) The case where repeated plays between a pair of players leads to varying results is really, in my judgement, significantly different from multiplicity of voters. For any pair, we assume that there is a probability that one will defeat the other. Repeated plays are repeated drawings of a binary random variable. In those cases, one possible approach is to proceed as follows. Assume that there really is an underlying index of ability for each player, the indices forming an ordered set (possibly even real numbers). The probability that one player will defeat the other in a given play is a function of the abilities of those two. In any given tournament, there are intransitivities due to the chance draws. Then with repeated play, one might hope to infer the true abilities and therefore order the players according to those abilities.

(g) It is possible (I have not looked into this) that under certain assumptions a Borda count will give a good statistical estimate of the true ranking of the players. But note the situation is fundamentally different from that of voters. In the above formulation of the problem of ranking chess players, it is assumed that there is a true ordering, of which each tournament supplies an estimate. Voters, on the contrary, have different values, they judge the candidates (or issues) by different criteria. Hence there is no real parallelism between voters´ preferences and the outcomes of several tournaments.

(h) You also ask what I take to be a different question, would I retain my insistence on independence of irrelevant alternatives "in the context of a theory of demand." Your remarks in your paper (p. 439) are so brief that I do not grasp the point being made. I assume you are referring to the theory of demand as usually presented in economics. Rational choice by a single individual always satisfies the independence of irrelevant alternatives. In the case of the theory of demand, the independence condition says simply that a shift in the indifference map in the region outside of the budget set but not within the budget set does not affect choice. That is certainly true of the economic theory of demand. Perhaps you have something else in mind. I did not intend my remarks about chess tournaments to apply to your statement about demand, since the two situations appear quite different.

I hope these remarks are clear.

To this discussion I wish to add the following remarks:

(1) As to the last paragraph under (c) regarding the comparison of many orderings: The ranking of chess players in a chess tournament is transitive. The outcome of an individual chess game is only part of the estimate of the performance of each player in the actual chess tournament. Playing ability is a many-sided quality. It depends, among other things, on the opponent´s gifts what abilities are put to the test.

(2) Concerning ideal criteria and voting forms in North America - under (d): I added an ideal criterion, namely, that all alternatives are taken for consideration. I referred to a method for estimating them, i.e. point calculation. In that way it is calculated what alternatives are relevant when choosing the best alternatives. - The logical drawbacks of voting forms in the United States of America are well-known. They are applied in lack of something better, whereas chess people do not see any logical flaws in their way of ranking chess players in a tournament where everybody plays against everybody.

(3) As to (e) that nobody is best, in a rigorous sense: Rather than saying that the question is of who is the best it is a question of who has performed best in a concrete case. B is better than A, if it is only considered how they performed against each other. In a bigger group many performances (many chess games) will count, which may place B behind A. Chess people regard it as meaningful, among other things, because it varies what qualities of the opponent are put to the test by individual chess players. The playing chances in chess are influenced by the opening. The different openings are called variants. Each chess player can to some extent influence the choice of a variant by his opening moves. One chess player can be clever in one variant, while another is clever in another variant. Thus, a chess player who is clever in a particular variant and only average in other variants can win a chess game against one who is only average strong in the opponent´s favourite variant, but who otherwise is clever. Thus, it is a question of what qualities are tested. The greater the number of players, the clearer conception one gets of the playing qualities in general. But the ranking of chess players only applies to the performances actually rendered by them, it does not count as an absolute estimate corresponding to the performances of discus throwers who are ranked according to metres and centimetres.

It is determined by the International Chess Federation how one qualifies for a match against the World champion. The extent of the establishment makes it impossible to make everybody play against everybody. The Federation divides the world into zones. Those who win in their zonal tournament (e.g. in Scandinavia which is a single zone) move on to their interzonal tournament. At the interzonal tournament, in which Scandinavia took part 1987, there were 18 participants. Everybody played against everybody, and the two players who gained most points, Salov and Johann Hjartarson, became World Championship candidates, despite Salov losing his game against Todorcevic, while Johann was defeated by Adorjan.

My aim of showing how chess people rank chess players transitively was to analyze their logic in order to do away with arguments that social choice theorists had raised against a method, which, after all, had won their recognition compared with other forms, namely, Borda´s method, and which besides possessed interesting qualities with regard to the social organization it seemed to be forming.

(4) On repeated chess tournaments under (f) and (g): Here I have unfortunately given an incomplete formulation. I compared aggregation of individual preference orderings with aggregation of equally many chess tournaments with the same participants but with varying results, assuming the common knowledge that nothing of the kind existed in the world of chess. By this comparison I wanted to point out that the analogy with chess could not be drawn out to that extent. Chess people argue that their ranking, where A can lose his game against B, but still get a higher place than B in a bigger group, does not pre-suppose that A´s defeat is due to random variation.

As regards the aggregation of points from many preference orderings it then seemed to me quite reasonable that it was made by adding up the points for each alternative. For that purpose I had neither the practical experience nor the judgement of chess people to refer to. Thus, it is not necessary in this context to analyze the variation in the performance of chess players in repeated tournaments, as done by Arrow, but it can, however, give an insight into how it can vary what qualities are tested, depending on the alternatives under trial. Variable performance can be due to playing ability in general, luck or bad luck, disposition of players and the types of variants played.

(5) As to the question of the theory of demand under (h): As the outcome by Borda´s method can be dependent on what alternatives (possibilities) are to be chosen among, I drew attention to an axiom in studies on the theory of demand that the demand (choice) may depend on the situation, and I pointed out that the situation is, among other things, determined by the existing possibilities. I shall clarify this by a fictitious example of a visit to a restaurant. A man used to frequent a restaurant where there were only two dishes on the menu, a cheap one (A), which he used to choose, and a very expensive dish (B). One day the restaurant celebrated its 10 years in business by offering 5 dishes on the menu in addition to the usual 2 dishes. The man was surprised, but regarded the celebration as an occasion to indulge himself. He chose the dish B, ignoring the dish A. The new situation with more alternatives made him evaluate different qualities in the previous alternatives.

I shall clarify by another example how important it can be, as far as Borda´s method is concerned, that the participants take a new alternative for consideration. A society is to choose a piece of sculpture as a present. Many sketches are made. The members of the society are to choose among these. Roughly speaking, they may have criteria, such as colour, form, material, size and price. Thus a new alternative, h, which is, apart from the colour, in every respect identical with a previous alternative f and the same as a former alternative g, except for the size, will be able to disarrange the ordering of the participants in different ways. Some of those who have ordered f before g will place h between them. This will increase the significance of f in relation to g. Some of those who have ordered g before f will place h between them, which will make gmore significant in relation to f. To others the new colour will mean quite new standpoints, so that h comes before both fand g. The fact that h is included can therefore give a clearer picture for the comparison of the original alternatives f and g.

How the number of alternatives is limited

William H. Riker, a well-known social choice theorist, has in a letter of 23 January 1985 to me valued the possibilities, which are rendered by the Borda method of dealing with many alternatives, by the following argumentation:

In any event, however, the main problem resulting from the abandonment of the Independence condition is that then there is no control over the number of irrelevant alternatives admitted to consideration. The admission of additional alternatives is the easiest method of manipulating some kinds of voting, e.g., the Borda method. (See Peter Fishburn, “Paradoxes of Voting”, American Political Science Review, 68 (June 1974), pp. 537-46). Thus, Independence is often a guard against manipulation, although, of course, all methods are manipulable. (See Allen Gibber, “Manipulation of Voting Schemes”, Econometrics 45 (July 1973), pp. 587-601).

Fishburn (1974) presented an example with preferences, where the outcome by Borda´s point calculation depends on whether an alternative is included or not. Riker also refers to Plot (1976) who on pp. 516-17 makes use of Fishburn´s example. Apart from the outcome they do not discuss what new insight this single alternative renders. The outcome is striking, but it is not beyond criticism. Since great importance has been attached to it I feel I have to trespass on the readers´ time with my criticism. I present the example in my own way for discussion.

There are 3 alternatives to choose among 7 voters (Table 1). a is given most points, followed by b and c.

Table 1. Borda choice. Example with 3 alternatives and 7 participants

Now the alternative d, which all the voters rank next to c (Table 2), is added.

Table 2. Borda choice. Example with 4 alternatives and 7 participants

This has changed the sequence. In the calculation for the participants 2, 3, 5 and 6 d pushes c upwards in relation to a or both a and b. The introduction of d has not, however, rendered any insight into the relation among a, b and c, since it is always ordered after c. d is to none of the participants an alternative to c and need not therefore be weighed against other alternatives. This possibility of manipulation can therefore on good grounds be avoided, if alternatives, which all participants order closest behind or at side of another alternative, are eliminated before the calculation of points. At the end of my article (1982), unaware of Fishburn´s example, I stress the importance “of considering all actual possibilities, i.e. all the possibilities that he who is entitled to make proposals prefers to some of the other alternatives”.

In his letter Riker called my attention to Plott´s contribution regarding the question of irrelevant alternatives, as follows:

The issue about Independence from Irrelevant Alternatives is complicated by a misstatement in Arrow´s example (see Charles Plott, “Axiomatic Social Choice Theory”, American Journal of Political Science 20 (Aug. 1976), pp. 511-96).

Plott (originally in 1971) wants to correct Arrow´s mistake by using the concept “infeasible” alternative instead of irrelevant. Since I have already explained the way Borda´s method shows formally and objectively, how relevant and feasible individual alternatives are, Plott´s contribution will be irrelevant, as far as this method is concerned.

Acknowledgements

The Icelandic Science Fund has allocated a grant for the work. Professor Knut Midgaard read a draft and gave constructive advice for the set-up of the article. It has been important to be able to discuss Arrow´s letters with cand. real. Trausti Jonsson.

References

• Arrow, K.J. (1951). Social Choice and Individual Values. New York: John Wiley. (2nd edn., 1963).
• Fishburn, P.C. (1974). “Paradoxes of voting”, The American Political Science Review 68: 537-46.
• Plott, C.R. (1971): “Recent results in the theory of voting”, pp. 109-27 in M. D. Intriligator (ed.): Frontiers of Quantitative Economics. Amsterdam: North-Holland.
• Plott, C.R. (1976): “Axiomatic social choice theory: an overview and interpretation”, American Journal of Political Science 20: 511-96.
• Stefansson, B.S. (1974a). “Mathematical aggregation of preferences”, Quality and Quantity 8: 121-38.
• Stefansson, B.S. (1974b). “Public choice through vote funds. A model for transactions”, Scandinavian Political Studies 9: 51-74.
• Stefansson, B.S. (1982). “Group choice between three or more alternatives”, Quality and Quantity 16: 433-54.

Quality & Quantity 25 (1991) 297-306