|
In social choice theory, Arrow’s impossibility theorem, the General Possibility Theorem or Arrow’s paradox states that, when voters have three or more distinct alternatives (options), no rank order voting system can convert the ranked preferences of individuals into a community-wide (complete and transitive) ranking while also meeting a pre-specified set of criteria. These pre-specified criteria are called ''unrestricted domain'', ''non-dictatorship'', ''Pareto efficiency'', and ''independence of irrelevant alternatives''. The theorem is often cited in discussions of election theory as it is further interpreted by the Gibbard–Satterthwaite theorem. The theorem is named after economist Kenneth Arrow, who demonstrated the theorem in his doctoral thesis and popularized it in his 1951 book ''Social Choice and Individual Values''. The original paper was titled "A Difficulty in the Concept of Social Welfare". In short, the theorem states that no rank-order voting system can be designed that always satisfies these three "fairness" criteria: * If every voter prefers alternative X over alternative Y, then the group prefers X over Y. * If every voter's preference between X and Y remains unchanged, then the group's preference between X and Y will also remain unchanged (even if voters' preferences between other pairs like X and Z, Y and Z, or Z and W change). * There is no "dictator": no single voter possesses the power to always determine the group's preference. Voting systems that use cardinal utility (which conveys more information than rank orders; see the subsection discussing the cardinal utility approach to overcoming the negative conclusion) are not covered by the theorem.〔(Interview with Dr. Kenneth Arrow ): "''CES:'' Now, you mention that your theorem applies to preferential systems or ranking systems. ''Dr. Arrow:'' Yes ''CES:'' But the system that you’re just referring to, Approval Voting, falls within a class called cardinal systems. So not within ranking systems. ''Dr. Arrow:'' And as I said, that in effect implies more information.〕 The theorem can also be sidestepped by weakening the notion of independence. Arrow rejected cardinal utility as a meaningful tool for expressing social welfare,〔"Modern economic theory has insisted on the ordinal concept of utility; that is, only orderings can be observed, and therefore no measurement of utility independent of these orderings has any significance. In the field of consumer's demand theory the ordinalist position turned out to create no problems; cardinal utility had no explanatory power above and beyond ordinal. Leibniz' Principle of the Identity of the Indiscernables demanded then the excision of cardinal utility from our thought patterns." Arrow (1967), as quoted on (p. 33 ) by .〕 and so focused his theorem on preference rankings. The axiomatic approach Arrow adopted can treat all conceivable rules (that are based on preferences) within one unified framework. In that sense, the approach is qualitatively different from the earlier one in voting theory, in which rules were investigated one by one. One can therefore say that the contemporary paradigm of social choice theory started from this theorem. == Statement of the theorem == The need to aggregate preferences occurs in many disciplines: in welfare economics, where one attempts to find an economic outcome which would be acceptable and stable; in decision theory, where a person has to make a rational choice based on several criteria; and most naturally in voting systems, which are mechanisms for extracting a decision from a multitude of voters' preferences. The framework for Arrow's theorem assumes that we need to extract a preference order on a given set of options (outcomes). Each individual in the society (or equivalently, each decision criterion) gives a particular order of preferences on the set of outcomes. We are searching for a ranked voting system, called a ''social welfare function'' (''preference aggregation rule''), which transforms the set of preferences (''profile'' of preferences) into a single global societal preference order. The theorem considers the following properties, assumed to be reasonable requirements of a fair voting method: ; Non-dictatorship: The social welfare function should account for the wishes of multiple voters. It cannot simply mimic the preferences of a single voter. ; Unrestricted domain: (or universality) For any set of individual voter preferences, the social welfare function should yield a unique and complete ranking of societal choices. Thus: : * It must do so in a manner that results in a complete ranking of preferences for society. : * It must deterministically provide the same ranking each time voters' preferences are presented the same way. ; Independence of irrelevant alternatives (IIA): The social preference between x and y should depend only on the individual preferences between x and y (''Pairwise Independence''). More generally, changes in individuals' rankings of ''irrelevant'' alternatives (ones outside a certain subset) should have no impact on the societal ranking of the subset. For example, the introduction of a third candidate to a two-candidate election should not affect the outcome of the election unless the third candidate wins. (See Remarks below.) ; Positive association of social and individual values: (or monotonicity) If any individual modifies his or her preference order by promoting a certain option, then the societal preference order should respond only by promoting that same option or not changing, never by placing it lower than before. An individual should not be able to hurt an option by ranking it ''higher''. ; Non-imposition: (or citizen sovereignty) Every possible societal preference order should be achievable by some set of individual preference orders. This means that the social welfare function is surjective: It has an unrestricted target space. Arrow's theorem says that if the decision-making body has at least two members and at least three options to decide among, then it is impossible to design a social welfare function that satisfies all these conditions at once. A later (1963) version of Arrow's theorem can be obtained by replacing the monotonicity and non-imposition criteria with: ; Pareto efficiency: (or unanimity) If every individual prefers a certain option to another, then so must the resulting societal preference order. This, again, is a demand that the social welfare function will be minimally sensitive to the preference profile. The later version of this theorem is stronger—has weaker conditions—since monotonicity, non-imposition, and independence of irrelevant alternatives together imply Pareto efficiency, whereas Pareto efficiency and independence of irrelevant alternatives together do not imply monotonicity. (Incidentally, Pareto efficiency on its own implies non-imposition.) Remarks on IIA * The IIA condition can be justified for three reasons (Mas-Colell, Whinston, and Green, 1995, page 794): (i) normative (irrelevant alternatives should not matter), (ii) practical (use of minimal information), and (iii) strategic (providing the right incentives for the truthful revelation of individual preferences). Though the strategic property is conceptually different from IIA, it is closely related. * Arrow's death-of-a-candidate example (1963, page 26) suggests that the agenda (the set of feasible alternatives) shrinks from, say, X = to S = because of the death of candidate c. This example is misleading since it can give the reader an impression that IIA is a condition involving ''two'' agenda and ''one'' profile. The fact is that IIA involves just ''one'' agendum ( in case of Pairwise Independence) but ''two'' profiles. If the condition is applied to this confusing example, it requires this: Suppose an aggregation rule satisfying IIA chooses b from the agenda when the profile is given by (cab, cba), that is, individual 1 prefers c to a to b, 2 prefers c to b to a. Then, it must still choose b from if the profile were, say, (abc, bac) or (acb, bca) or (acb, cba) or (abc, cba). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Arrow's impossibility theorem」の詳細全文を読む スポンサード リンク
|