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How to Solve the Arrow Paradox



Arrows General Impossibility Theorem
The individualistic approach
The need for "enlightened" individual preferences
The condition of "unrestricted domain" for the shape of the individual preferences
The condition of non-dictatorship and the Pareto-Principle
Are individual interests measurable only by a ranking of alternatives?
The necessity of interpersonal comparisons of utility



Arrows General Impossibility Theorem

For a normative theory of collective choice and for a normative theory of democracy Arrow's General Impossibility Theorem is especially important. He asks: How is it possible to reach a collective choice by aggregating individual values? Or expressing the question in a different way: How can the collective interest be derived by aggregating the individual interests?

Arrow's answer is negative. In his Impossibility Theorem he has proved, that there is no "constitution", by means of which the individual interests, expressed by a ranking of all alternatives, can always be aggregated into a collective ranking of the alternatives - not as long as a number of seemingly reasonable conditions are fulfilled, such as an unrestricted domain for the individual orderings, the absence of a dictatorial individual, the Pareto-principle, and the independence from irrelevant alternatives.1)

Since the internal logic of the Impossibility Theorem is sound, one has to look at the assumptions that Arrow has made, if one wants to overcome the negative conclusion he draws.

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The individualistic approach

One of the basic premises of the whole approach is, that the collective interest is derived from individual interests. Now, one could simply evade the whole problem of aggregation by deciding which alternative should be realized independently from the interests of the individuals. All positions which appeal directly to the will of a superindividual entity such as the state, the nation or anything else proceed in this way.

However, to separate social choice from individual interests in this way appears to be untenable in methodological terms in the same way that it would be untenable to try to determine the truth about facts independently from the perceptions of the individuals simply by appealing to a superindividual source of knowledge. To find further confirmation for this position, one has to go a bit deeper into the methodological questions of normative knowledge.2)

Normative knowledge seeks to provide universally valid answers to questions about what should be. If one claims universal validity for a certain answer, there must in principle be the possibility of reaching a consensus with everybody by means of arguments. Oherwise the claim for validity would be an unfounded claim for belief or obedience towards those individuals, for which no consensus by argument is possible.3)

Assertions about which no consensus by argument is possible are in conflict with the basic principle of the intersubjectivity of any universally valid scientific knowledge. This applies for instance to empirical assertions, where a consensus by argument and free of force is in principle possible, because every individual can persuade himself of its truth by his own observations. In an analogous way one has to search for a consensus by argument in the case of normative assertions, if one claims not only obedience for them but also that they are valid.

However, the essence of a normative dissent does not rest in differing perceptions of reality - though empirical dissent may play an important part in normative disputes, too. The essence of normative dissent lies in the existence of differing volitions with regard to reality. Those participating in a normative dispute want incompatible things to be realized.

Such a conflict of volitions seems to be solvable in a way that is "reasonable" and free of force, if every individual considers the interests of every other individual as if these were at the same time his own. Only by such a "principle of solidarity", as it may be called, a consensus by argument in normative questions seems possible.4) This principle demands that every individual mentally puts himself in the place of the other individuals and that he looks at things from the others' points of view, too, thus taking account of the others' interests in the same positive way as he does with his own.

The result of these reflections is that the universally valid determination of the collective interest presupposes the aggregation of all individual interests. Therefore positions, which try to determine the collective interest independently from the interests of the individual are untenable. When an individual is told: "In determining the collective interest your interest does not count", the rule to seek a consensus by argument is broken. Force is introduced into the relation and the conflict is shifted from the level of argument to that of power relation. Hence the problem of aggregation as it is posed by Arrow remains relevant; it cannot be evaded this way.

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The need for "enlightened" individual preferences

To avoid a possible misunderstanding, it should be stressed, that the individual interests cannot be determined simply from the preferences that the individuals in question in fact express. Such a "subjective" determination of interests may be unsound, for there is no doubt that an individual may err with regard to what his own interests are. This can be readily seen from the fact that an individual may regret his own former decisions, for instances if these were affected by erroneous information, by logical error, by external sanctions or by subconscious motives. The autonomous formulation of interests by individuals therefore can only be a practical procedure, the application of which may be justified, if the individuals to a sufficient degree know what their own interests are.

From the viewpoint of a normative methodology, however, not every subjectively formulated individual interest may justly claim recognition when determining the collective interest, for according to the principle of solidarity stated above all individual interests must be determinable interpersonally by "putting-oneself-in-the-other's-place". In a similar way in the positive sciences not all subjectively expressed perceptions of reality enter into the formulation of knowledge but only those which in principle can be made by everyone.

Arrow's Impossibility Theorem omits the question of the qualification of individual interests, but this is quite justified in the context of his problem. He is only dealing with the logical problem of aggregation, and this logical problem would exist even when it is conceded that the individual interests are determined correctly.

When, however, not the logic of collective decision-rules but the acceptability of real decision-procedures is at stake, the quality of individual articulations of interest is of great importance, for the collective decision cannot be better than the individual interests it is derived from.

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The condition of unrestricted domain for individual preferences

For some theorists the "condition of unrestricted domain" is the 'black sheep' among Arrow's assumptions. This condition postulates, that every logically possible individual preference ordering must be admitted. However, if one assumes special constellations of interests, there then exist some rules of aggregation, which lead to a consistent collective preference ordering in each case. For instance, if preferences are "single-peaked", the majority rule yields a consistent collective ordering of the alternatives.5)

Yet such a restriction of the domain of individual interests to those which pose no problem of aggregation appears to be doubtful from the point of view of a normative methodology. Either one has to prove that such "difficult" constellation of interests do in fact not occur - a proof which was not yet provided and which seems quite impossible - or one has to "correct" these constellations of preferences and exclude certain existing individual interests from recognition. However, then one can no longer talk about a recognition of interests according to the principle of solidarity, and consensus by argument becomes impossible. So in this respect, too, Arrow's "individualistic" premises resist its critics.

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The condition of non-dictatorship and the Pareto-Principle

Consequently the question remains, what other conditions should be abandoned, to achieve a consistent determination of the collective interest.

The "condition of non-dictatorship", which demands that no single individual's interest become collectively decisive independently from the interests of the other individuals, also seems sound according to the solidarity-principle.

The same is true for the Pareto-Principle which roughly states that if the alternative x is better than the alternative y for each individual, then it is better for the collective, too. For if one has to consider the interests of all individuals as if they were one's own, this also means that one has to consider them positively. Now if all individual interests unanimously point into the same direction, the collective interest must lie in this direction, too.

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Individual interests measurable only by a ranking of alternatives?

Then only two premises of the Impossibility Theorem remain that may possibly be debatable: first the condition of "collective rationality", which demands that from each possible constellation of individual orderings of the alternatives a collective ordering must be derivable. By elimination of this condition the whole problem would be canceled and one would accept the negative result.

Second there is the condition of "independence from irrelevant alternatives", which states that the collective decision regarding any two alternatives must be based solely on the individual rankings of just these two alternatives. Both conditions have in fact a premise in common, which seems highly problematical from the point of view of normative methodology. This is the assumption that the individual interests shall be measured merely on an ordinal scale, i.e. that they can only be represented by a ranking of the alternatives. Recognition of the utility distances - or the intensity of the preferences - between the alternatives is excluded. So it does not matter whether an alternative is very much or only slightly better for an individual than another alternative, because in both cases the same ordering will result.

Such a merely ordinal understanding of individual interests seems to be incompatible with the principle of solidarity formulated above. The individual often is able to express not only rankings of the alternatives but also the magnitudes of the differences in the utility of the alternatives. Statements like: "Alternative x is for me much better than alternative y, whereas alternative y is only slightly better than z" are in no way senseless.

This becomes especially clear, when two different decisions by an individual are combined to one. Take for example an individual who loves beef much more than pork but who likes potatoes only slightly better than rice.

Now if both decisions with regard to the kind of meat and to the kind of side-dish are combined into a single decision between the two dishes "beaf with rice" and "pork with potatoes" the individual will choose the first dish, because the greater difference of utility with respect to meat proves decisive (if any interdependence of utilities is excluded).

If the two interests of the individual are measured only by an ordinal scale, no decision is possible. With regard to the criterion meat the dish 'beaf with rice' is preferred, and with regard to the criterion side-dish 'pork with potatoes' is preferred, and there is no possibility of weighing the degrees of superiority in both cases against one another.

This demonstrates that whenever an individual splits a decision into a number of partial decisions according to different criteria and when it values the alternatives on an ordinal scale only then there exists the possibility of an incomplete ordering of the alternatives as a result of aggregating the partial decisions into one total decision.

The problem here is analogous to the problem of aggregating ordinal measured individual interests into a collective decision. Neglect of individual utility differences is especially important because without taking them into account there is no way to compare the utility differences of different individuals. In determining the collective interest the different urgency of individual interests is neglected. The exclusion of such interpersonal comparisons of utility differences, however, seems incompatible with the Principle of Solidarity stated above.

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Interpersonal comparisons of utility

Suppose one has to decide whether alternative x or y is more in the collective interest. Now for individual A alternative x is only slightly better than y, whereas for B the choice of y is of outmost importance. With a consideration of all interests as if they were at the same time one's own, alternative "y" had to be chosen, whereas with a merely ordinal measurement of individual interests, x and y are incomparable. However, if different degrees of urgency of interests are measured by means of utility differences between the alternatives, an incomplete collective preference scale can no longer be found. For example, with the summation of cardinal individual utilities no inconsistency with regard to collective utility will occur.

As the result of the foregoing reflections one can say that Arrow's Impossibility Theorem presents no principal problem for the concept that collective interest is derived from individual interests. The problem only occurs when the measurement of individual interests is restricted to ordinal, interpersonally incomparable preferences.

Arrow's own reflections about removing the Impossibility Theorem also point in the direction of weakening the condition of "independence from irrelevant alternatives". This condition is mainly responsible for the merely ordinal and subjective comprehension of individual interests.6) Surely the difficulties of such an interpersonal comparison of utility differences are considerable. But in principle one can get that information about the external situation, the personal attributes, and the emotions and attitudes of the individuals, that is needed to reach a consensus based on the interpersonal weighting of their interests. In everyday life such comparisons of the different urgencies of interests are frequently made, and they may in principle be attacked or defended with arguments. Of course, there are still unsolved problems in connection with this interpersonal comparison of utilities, which need further clarification.

Yet even if it could be made sufficiently clear what is meant by recognizing interests in accordance with the Principle of Solidarity, probably no concrete procedure of collective decision-making would result that could be applied directly to normative disputes. For instance the costs of reaching a consensual determination of individual interests would be immense, especially for larger collectives. Nevertheless an epistemological clarification of the interpersonal comparison of utility can deliver a theoretical standard to criticize the acceptability of different procedures for making decisions, for instance in assessing the majority system or the property-contract-system of an exchange economy. These procedures must be understood as "rules of thumb" methods which under certain conditions may yield acceptable approximations of the collective interest according to the Principle of Solidarity.

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(1) Kenneth J. ARROW, Social Choice and Individual Values, 2nd ed., New Haven - London 1963
(2) The following, rather sketchy passage is based on a larger text "Tauschprinzip - Mehrheitsprinzip - Gesamtinteresse" published in 1979 by the Klett-Cotta Verlag.
(3) This 'consensus-theory of truth' was elaborated by Habermas in his theory of the "herrschaftsfreier Diskurs". Cf. J. Habermas, Legitimationsprobleme im Spätkapitalismus, Frankfurt a.M. 1973, pp.147ff.
(4) Similar ethical principles are postulated by authors like L. Nelson, S.I. Benn, R.M. Hare or J.C. Harsanyi
(5) Cf. A. Sen, Collective Choice and Social Welfare, San Francisco 1970, pp.l73ff.
(6) Cf. Arrow, loc.cit. p.114.


Also compare the following similar pages of the Ethics-Workshop:

    Eine Lösung für das Arrow-Problem
Einzelinteresse und Gesamtinteresse, § 37.3


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