Rubinstein2005-page135

# Rubinstein2005-page135 - between a and b be applied to any...

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October 21, 2005 12:18 master Sheet number 133 Page number 117 Social Choice 117 The IIA condition requires that if two proﬁles agree on the relative rankings of two particular alternatives, then the social preferences attached to the two proﬁles also agree in their relative ranking of the two alternatives. Notice that IIA allows an SWF to apply one criterion when com- paring a to b and another when comparing c to d . For example, the simple social preference between a and b can be determined ac- cording to majority rule while that between c and d requires a 2 / 3 majority. Condition IIA is sufﬁcient for Arrow’s theorem. However, for the sake of simplifying the proof in this presentation, we will make do with a stronger requirement: Condition I (Independence of Irrelevant Alternatives + Neutrality): For all a , b , c , d X , and for any proﬁles ± and ± ² if for all i , a ± i b iff c ± ² i d , then a ± b iff c ± ² d . In other words, in addition to what is required by IIA , condition I requires that the criterion that determines the social preference
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Unformatted text preview: between a and b be applied to any pair of alternatives. Arrow’s Impossibility Theorem Theorem (Arrow): If | X | ≥ 3, then any SWF F that satisﬁes conditions Par and I ∗ is dic-tatorial, that is, there is some i ∗ such that F ( ± 1 , . . . , ± n ) ≡± i ∗ . We can break the theorem’s assumptions into four: Par , I ∗ , Transi-tivity (of the social ordering), and | X | ≥ 3. Before we move on to the proof, let us show that the assumptions are independent . Namely, for each of the four assumptions, we give an example of a nondictatorial SWF, demonstrating the theorem would not hold if that assumption were omitted. • Par : An anti-dictator SWF satisﬁes I ∗ but not Par . • I ∗ : Consider the Borda Rule. Let w ( 1 ) > w ( 2 ) > . . . > w ( | X | ) be a ﬁxed proﬁle of weights. Say that i assigns to x the score...
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