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For all permutations τ from N into itself, and every profile w, denote by uT the profile wT = ( w / T ) / e A f , where ul τ = wT(/), all / e N. Then S is anonymous if and only if for all τ and all i#, S(uT) = S(u). c. ) to discriminate: we call it the neutrality requirement. Social choice functions and correspondences 23 Definition 5. c. ) S we say that S is neutral if the following holds true. For every permutation σ of A into itself, denote by u° the profile ua = ( u ? ) i G N 9 where u°(a) = ui(o(a)), Then for all u and all σ, ^(ι*) 0 = σ " all / e TV and all a e Λ.

In problem 1 we explore the existence of an anonymous and neutral social choice function. f. }. Show that the converse implications do not hold. (2) Given A and JV with cardinalities/? and w, respectively, such that (ii) holds true, show that there exist no anonymous and neutral social choice function on L(A)N. f. 5. c. The Condorcet approach to the social choice problem is crucial to the strategic analysis developed in subsequent chapters. A Condorcet winner is The strategy of social choice 26 an outcome defeating every other outcome in binary contest where a majority decides: for instance, an alternative which is ranked first by a strict majority of agents is the Condorcet winner in every associated profile.

C. Fix two integers /c, 1 < k < p, and m, 1 < m < «. c: Smk(u) = {a ^ A/a is ranked at most h by at least m agents}. We can easily check that Sm only if: m^\(k/p)-n], h is not empty valued (for all profiles) if and 47 Monotonicity and the A rrow theorem where [z] denotes the smallest integer not inferior to z. Moreover, Smk clearly satisfies SPA. c. all are anonymous, neutral and strongly monotonie (notice that they do not satisfy efficiency when h ^ 2). c. c. We fix a particular outcome a0 that plays the role of a status quo.

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