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By Gaertner W.

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The social ordering is assumed to be a weak order. We pick any two distinct alternatives a and b from X . In step 1, alternative a is ranked highest and alternative b lowest by every person i ∈ {1, . . , n}. Condition P then requires that a is strictly at the top of the social ordering. Imagine now that alternative b is raised, step by step or rank by rank, to the top of individual 1’s ordering, while the ranking of all other alternatives is left unchanged. Due to the independence condition, a either remains at the top of the social ordering or is replaced by b.

Transitivity of the social relation now yields xPz. But then the single member of V1 would be almost decisive, in contradiction to what we have assumed at the outset. The impossibility result now follows from the lemma. The reader should note two points. The first refers to the profile that we have used above. It has the structure of the so-called paradox of voting to which we shall come back in the next chapter. The second refers to the fact that the preferences of the individuals in V3 are not needed in our argumentation.

Consider a small society of seven voters where two individuals prefer x to y, two individuals prefer y to x and the rest is indifferent between x and y. e. xIy. Now assume that one of the hitherto indifferent voters declares a strict preference for x over y, with everyone else’s preference between x and y remaining the same. Positive responsiveness requires that xPy socially which is what the simple majority rule but not the absolute majority rule brings about. Thus simple majority voting is more sensitive to changes in the preference profile of society.

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