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...
View Full Document
- Fall '10
- Sociology, Voting system, IIA, Social Preference, anti-dictator SWF satisﬁes, simple social preference