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The independence of irrelevant alternatives (IIA), also known as binary independence or the independence axiom, is an axiom of decision theory and various social sciences. The term is used with different meanings in different contexts; although they all attempt to provide a rational account of individual behavior or aggregation of individual preferences, the exact formulations differ from context to context. In individual choice theory, IIA sometimes refers to Chernoff's condition or Sen's property α (alpha): if an alternative ''x'' is chosen from a set ''T'', and ''x'' is also an element of a subset ''S'' of ''T'', then ''x'' must be chosen from ''S''.〔Sen, 1970, page 17.〕 That is, eliminating some of the unchosen alternatives shouldn't affect the selection of ''x'' as the best option. In social choice theory, Arrow's IIA is one of the conditions in Arrow's impossibility theorem, which states that it is impossible to aggregate individual rank-order preferences ("votes") satisfying IIA in addition to certain other reasonable conditions. Arrow defines IIA thus: :The social preferences between alternatives ''x'' and ''y'' depend only on the individual preferences between ''x'' and ''y''. Another expression of the principle: :If ''A'' is preferred to ''B'' out of the choice set , introducing a third option ''X'', expanding the choice set to , must not make ''B'' preferable to ''A''. In other words, preferences for ''A'' or ''B'' should not be changed by the inclusion of ''X'', i.e., ''X'' is irrelevant to the choice between ''A'' and ''B''. This formulation appears in bargaining theory, theories of individual choice, and voting theory. Some theorists find it too strict an axiom; experiments by Amos Tversky, Daniel Kahneman, and others have shown that human behavior rarely adheres to this axiom. In social choice theory, IIA is also defined as: :If ''A'' is selected over ''B'' out of the choice set by a voting rule for given voter preferences of ''A'', ''B'', and an unavailable third alternative ''X'', then if only preferences for ''X'' change, the voting rule must not lead to ''Bs being selected over ''A''. In other words, whether ''A'' or ''B'' is selected should not be affected by a change in the vote for an unavailable ''X'', which is irrelevant to the choice between ''A'' and ''B''. ==Voting theory== In voting systems, independence of irrelevant alternatives is often interpreted as, if one candidate (''X'') would win an election, and if a new candidate (''Y'') were added to the ballot, then either ''X'' or ''Y'' would win the election. Approval voting, range voting, and majority judgment satisfy the IIA criterion if it is assumed that voters rate candidates individually and independently of knowing the available alternatives in the election, using their own absolute scale. This assumption implies that some voters having meaningful preferences in an election with only two alternatives will necessarily cast a vote which has little or no voting power, or necessarily abstain. If it is assumed to be at least possible that any voter having preferences might not abstain, or vote their favorite and least favorite candidates at the top and bottom ratings respectively, then these systems fail IIA. Allowing either of these conditions alone causes failure. Another cardinal system, cumulative voting, does not satisfy the criterion regardless of either assumption. An anecdote that illustrates a violation of IIA has been attributed to Sidney Morgenbesser: :After finishing dinner, Sidney Morgenbesser decides to order dessert. The waitress tells him he has two choices: apple pie and blueberry pie. Sidney orders the apple pie. After a few minutes the waitress returns and says that they also have cherry pie at which point Morgenbesser says "In that case I'll have the blueberry pie." All voting systems have some degree of inherent susceptibility to strategic nomination considerations. Some regard these considerations as less serious unless the voting system fails the easier-to-satisfy independence of clones criterion. 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「Independence of irrelevant alternatives」の詳細全文を読む スポンサード リンク
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