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How to Solve the Arrow Paradox
Content:
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
Notes
***
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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Notes:
(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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Last review 08.06.2007 / Eberhard Wesche
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