Theorem of impossibility of Arrow

The theorem of impossibility of Arrow , also called “paradox of Arrow”, is a mathematical confirmation under certain precise conditions of the paradox evoked by Condorcet according to which there would not exist function of social choice indisputable, allowing to incorporate individual preferences in social preferences. For Condorcet, there did not exist simple system ensuring this coherence. Arrow shows, subject to acceptance of its assumptions, which there would not exist system whole ensuring coherence, except that where a Dictateur only would impose his choices on all the remainder of the population.

It was established later that the paradox of Arrow disappeared if the preferences of the voters could materialize by a position on a single axis. The Pre-order on the result of the choice becomes a total Ordre then and the paradox disappears. This characteristic is sometimes recalled during the discussions on the tendencies to the Bipartisme in policy.

Origin

The author

This result is due to Kenneth Arrow, prize winner of the “Nobel Prize” of economy 1972, which showed in his thesis and published it in 1951 in its book social Choix and individual values ( in Social Choice and Individual Values ).

Preferences

For the mathematicians, which the economists call “preferences” corresponds to a complete pre-order or, with the limit, a complete order (one speaks then about “strict preferences”).

For the others, the “preferences” of an individual correspond to the order which an individual establishes between the options which are offered to him. These preferences are known as strict when the individual never classifies two options ex-æquo. So that the description of this concept is complete, it is supposed that the order that an individual establishes between the various existing options is not modified by the addition of additional options. In other words, if an individual having preferences classifies the option once has in front of the option B, it will always classify has in front of B whatever the other options present, which constitutes a manifestation of the coherence of its choice.

A Profil of preferences is the name given to a “group” of individual preferences. One names social Préférences preferences which are worth with the social status.

Problems

It is a question of incorporating a whole of individual preferences in a whole of social preferences, in other words a whole of individual natures in a social order .

Some examples:

election: Various candidates present themselves to an election. The voters have preferences as for the candidates. Starting from these preferences, one wants to classify the candidates.
championship of formula 1: Various stables are present. One has classifications by stable for each large-price. Starting from these classifications by large-price, one wants a classification final.
choice between projects of infrastructure: Various projects of infrastructure are presented. These projects can be classified according to various criteria (price, duration of the work and various quality standards). Starting from these classifications by criteria, one wants a classification total of the various projects.

One names function of social choice the operation of passage of the individual preferences towards a collective preference.

Simplified statement

The theorem of Arrow is known in the following form.

For at least 3 options of choice and two individuals, there does not exist Fonction of social choice satisfying the following properties:

Universality: the function of social choice must be defined for any profile of preferences logically possible
Not-dictatorship: no individual must be able to impose his preferences, independently of the preferences of the others;
Unanimity: when all the individuals have the same preferences, the function of social choice must associate these same preferences with the company.
Indifference of the Not-Relevant Options: the relative classification of two options should depend only on their relative position for the individuals and not on the classification of third options; if only one subset of options is considered, the function should not lead to another classification of this subset.

Are the properties of universality and IONP reasonable? Yes, with the direction which it would be reasonable them to accept . Not, with the direction which it would be unreasonable them to require : These two properties are neither basic, nor elementary.

In another version of the theorem, the unanimity can be replaced by the two following assumptions:

Monotony: an individual should not be able to make decrease the total classification of an option by classifying it higher.
Sovereignty: no social choice must be impossible a priori.

Principle of the demonstration and interpretation

The demonstration is very technical (see the external bonds on) and rests on several lemmas which one deduces from particular cases.

This theorem is not a positive test: it does not allow a systematic illustration, but notes that for not-binary choices, there will be always problematic situations. Thus, a function of social choice which presents the elementary properties stated higher will be often sensitive to the not-relevant options. Let us notice nevertheless that our own choices are sometimes also influenced them also by not-relevant options and that does not affect in the general case our effectiveness.

Relieving of the conditions

If this theorem does not obstruct the partisans of dictatorial modes (which are ready to trust a “strong man” to carry out the people reasonably), and gene little the liberals (who challenge the idea to transform individual preferences into collective preference), it is on the other hand a spine in the foot of the partisans of the Démocratie.

The theorem consequently caused an abundant reflection on the possible procedures and the less strong conditions which remain compatible between them.

While working by elimination, one conceives not to be able to admit that a procedure of vote can

remain indefinitely undecided,
to be dictatorial,
to be nonmonotonous,
to have a result predetermined

For this last point, one can conceive that the result is partially predetermined: for example, one can not admit the elimination of a political party even if a vote decided some. One then utilizes of the choices of a level different from that of the choice of a voting system: the theorem does not consider the moral side or acceptable of the options; it treats just procedure transforming the starting options into final decision. There thus remains nonacceptable that a procedure of vote can exclude a priori from the options.

Only the indifference with the nonrelevant options (IONP) could reasonably be weakened. Besides we do not practice it in general ourself in our individual choices, weakness known of the salesmen and exploited by them (methods known as of the bait and switch - one crams the barge with products at low prices… then one proposes another article of “better” quality with more important margin)

It amounts saying, in the case of the choice of a project according to several criteria, that to propose a new project in the existing classification of the other projects should not intervene. In the case of an election, it amounts saying that the appearance or the desistance from a candidate should not intervene on what we think of the others. We are accustomed to electoral systems which violate this condition, without that shocking too much. The voter is sometimes thorough with the “useful Vote”, which implies that it must itself guess which are the relevant options, and eliminate those which are not it.

In fact, this condition IONP results in the fact that, only, the option finally selected is relevant, the introduction or the removal of all the others being without effect on the end result. That appears a little strong.

It appeared that one could plan to replace this condition by a less strong condition: the indifference with the least relevant options (IOMP). One retains as relevant options the members of the smallest whole who beat, in the matches two to two, all the options out of the unit (together of Smith). Thus and for example, If has beats B, B beats C and C beats has, but which has, B and C beat all the other options, the whole of Smith consists of have, B and C, and represent the options which are likely to carry it. The other options constitute the least relevant options.

With this looser definition, some methods of Condorcet check the criterion.

Examples of application

System by cumulated points

In the case of the automobile races, each car gains points with each test according to its order of arrival. The greatest total gains the competition. This device passes from a classification to a classification; it is universal, sovereign, monotonous, but it is not indifferent to the not-relevant alternatives.

Two stables ( has and B ) of two cars each one ( A1 , A2 and B1 , B2 ) are at the end of a competition, the last race is about to be completed. The leader A1 leads B1 of two points to general classification, but it is behind him and does not hope any more to catch up with it. Far in front of them, A2 and B2 are alone. The points on arrival are allotted as follows:

1st 10
2^nd 9
3rd 6
4th 5
…

A1 should gain the competition, since B1 will have only one point of better. But, this moment, the director of the stable B can ask B2 to give up the race: B1 will be then second with this race, and at three points of better than has, will gain the competition! If it does that, one can even imagine that A2 is tempted to give up to make it possible A1 to arrive only at one point of B1 , respectively in second and first position.

This situation shows well that a system by cumulated points is not indifferent to the not-relevant alternatives.

Invitation to tender for the market public

In the same way, one can hope that the committees of expertise which adopt a project according to several criteria are monotonous and sovereign, indifferent to the not-relevant options:

If need be, the code of the markets provided that the selection criteria must be balanced and not only treated on a hierarchical basis.

See too

the Theorem of Gibbart-Satervaithe,
the paradox of Condorcet says also paradox of the vote.
Criteria of voting systems

Arrow

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