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By Gaertner W.

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Due to our inferences in step 4, a must be ranked above c due to condition I , and c is Pareto-preferred to b. Therefore, by transitivity of the social relation, a is preferred to b, and this holds whenever person m orders a above b. R1 c · · b a . . Rm−1 c · · b a Rm a c b · · Rm+1 c · · b a . . 4. If we now permute alternatives b and c in the arguments above, we obtain the same qualitative result. The ranking of a is above alternative c when person 28 ARROW’S IMPOSSIBILITY RESULT m orders a above that alternative.

Since R ∗ is an ordering, three ways of ranking points in region II against u¯ are possible: the points in II could be preferred, indifferent, or worse. In our argument above, we had postulated a strict preference against u. ¯ We could also have started by assuming u¯ to be preferable to all points in II. 6. a2 32 ARROW’S IMPOSSIBILITY RESULT would have been completely analogous. However, indifference between points in II and u¯ would lead to a contradiction. We would, for example, have aI ∗ u¯ and bI ∗ u.

The simple majority rule is clearly non-dictatorial, from May’s theorem we know that this rule satisfies unrestricted domain and the independence condition (remember that the 42 MAJORITY DECISION UNDER RESTRICTED DOMAINS latter is weaker than neutrality), and one can easily see that the method of majority rule fulfils the weak Pareto principle (actually, neutrality, and positive responsiveness together imply the Pareto rule). Where lies the problem? The problem comes from the fact that simple majority voting can yield a social relation that is not transitive.

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