# Arrow Impossibility Theorems by Jerry S. Kelly and Karl Shell (Auth.)

By Jerry S. Kelly and Karl Shell (Auth.)

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Sample text

X#z by x G Q Q I ^ ) u C(i;2)) and zKy by z G C(Î; 1 ). By transitivity, xRy. Therefore x e C(vl u v2) and C ( C K ) u C y ) ç C K u 4 *PI and PI* combine to give path independence. (*PI) □ Important for the position we have shown Plott to have taken is that a total choice function may satisfy path independence but not rationality. The following example is from Plott [266] : C({x,y}) = {x,y}9 C({y,z}) = {y,z}, C({x,y,z}) = {x,y}. R-best in {x, y, z}. But z <£ C({x, y, z}); thus C is not rational.

Theorem 4-6 [45] There is no collective choice rule / satisfying (i) base quasitransitivity, (ii) the weak pairwise Pareto condition, (iii) independence of irrelevant alternatives, (iv) the standard domain restriction, (v) there is no pairwise oligarchy. Proof As before, we can invoke Lemma 4-1, allowing us to conclude that a set is pairwise decisive if it is weakly, locally pairwise decisive for at least one alternative against another. Let W be a smallest pairwise decisive set. The weak pairwise Pareto condition ensures that such a W exists and is nonempty.

Suppose / satisfies neutrality and {/} is globally decisive for x against y; then i is a dictator. Theorem 2 shows a conflict between a weakened liberalism requirement and a strengthened no-weak-dictator constraint. ) Theorem 5-2 There is no collective choice rule / satisfying (i) the standard domain constraint (2), (ii) independence of irrelevant alternatives, (iii) triple acyclicity of the base relation, (iv) no individual is locally pairwise semidecisive between all pairs, (v) neutrality, (vi) there exists a set S of two individuals and a pair of alternatives x, y such that xPDsy.