By Jerry S. Kelly and Karl Shell (Auth.)
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Additional info for Arrow Impossibility Theorems
Example text
Therefore CUo({z,x}) = {x}. If x $ CUo({x,y}), then CUo({x,y}) = {y} and then CMo({y,z}) = {y} by base quasitransitivity. But then, by independence of irrelevant alternatives, yPDw*z, contrary to our choice of W. , k is a weak pairwise dictator and W is a pairwise oligarchy. □ Comparing this result with Arrow's second impossibility theorem (Theorem 4-3), we have weakened the regularity constraint from transitive rationality to base quasitransitivity at the cost of a strengthening from nondictatorship to absence of pairwise oligarchies.
Proof This is immediate from (Arrow's second impossibility) Theorem 4-3 and Theorem 3-15, which allow us to derive independence of irrelevant alternatives from /c-ary independence and |£| = k + 1 as holds here by (ii) and (iv). □ Of course this use of (iv) instead of IIA is just an apparent weakening: in the presence of (i)-(iii), Theorem 3-15 tells us that (iv) and IIA are logically equivalent. Clearly we can also develop this apparent weakening for Theorem 4-4 on quasi-transitive rationality or Theorem 4-8 on acyclic rationality.
Equivalently, they inform us of what is not chosen from supersets according 5 This result together with the proof of Theorem 8 shows that Property a is equivalent to PI* (cf. Parks [245]). 6 Since V contains all pairs, this is equivalent to Sen's theorem that C is normal if and only if Properties a and 7 are satisfied. 31 SIMPLICITY: CONDITIONS to what is not chosen from subsets. An expansion-consistency condition gives information on what is chosen from supersets from information on what is chosen from subsets or, equivalently, information on what is not chosen from subsets according to what is not chosen from supersets.
