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2. Strategy-Proof Collective Choice
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Chapter 2 Strategy-Proof Collective Choice Positive Political Theory I is concerned with aggregating various profiles of individual preferences. However, as we emphasized in that book, there is no logical necessity for choice and preference to coincide; individuals might abstain or might choose strategically. Of course, given that individuals have preferences, we expect any such strategic behavior to be directed toward promoting these preferences. For instance, consider Example 2.1. Example 2.1 Let N = {I, 2, 3}, X = {w, x, y, z}, and assume preferences are aggregated with the Borda rule under which, for all a, b E X, aPb if and only if 2:N ri(a) X (section 1.1). When considering collective choice functions, we abuse notation somewhat and write !.p(p) = x, for instance, rather than !.p(p) = {x}. For any profile P = (Rl' ..., Rn) E nn and any subset L ~ N, let PL denote the restriction of the profile P to individuals of L and define P-L == PN\L. For example, suppose P = (R1, ... , R5) and L = {I, 3, 5} C N; then P can be written (PL, P-L) with PL = (Rl, R3, R5) and P-L = (R2, R4 ). A particularly important case is when L = {i} for some individual i E Nj then we write P-i = (R1, R2, ..., ~-l, ~+l, ... , Rn) and P = (~, P-i). The 2.1. STRATEGY-PROOFNESS ON FINITE SETS following concept is fundamental. 21 Definition 2.1 A collective choice function 'P is strategy-proof if and only if, for all pEnn and any i E N, 'P(R;., p-i)R;.'P(R~, P-i) for all R~ E n. Thus a collective choice function is strategy-proof if, for any i, given every individual other than i reports their true preferences, i can do no better than likewise report his or her true preferences. An equivalent and useful way of defining strategy-proofness is in terms of manipulability: Definition 2.2 A collective choice function 'P is manipulable at profile P = (R1, ... , Rn) if and only if there exists some i E N and some R~ =I- R;. such that 'P(~, P-i)Pi'P(R;., P-i)' Then'P is strategy-proof if it is not manipulable at any profile. In other words, strategy-proof collective choice functions provide no opportunities for any individual to manipulate the outcome profitably by misrepresenting his or her preferences, given all others tell the truth. Notice that if'P were not assumed to be resolute, then the definitions of manipulability and strategy-proofness would not be well-defined. The difficulty is that, in general, there is no unequivocal way of inducing individual preferences over X, the family of nonempty subsets of X, from information only on preferences over X. However, given that we are interested in collective choice functions, the concept of stategy-proofness is really quite appealing. In particular, if 'P is strategy-proof, then it is irrelevant whether or not any individual even knows the preferences of others because whatever they might happen to be, reporting one's own preferences truthfully is always a best thing to do. Unfortunately, the only strategy-proof collective choice functions capable of distinguishing between at least three alternatives are dictatorial. Let IIcp denote the range of 'P, IIcp == UpERn'P(p) S;; X. Definition 2.3 A collective choice function 'P is dictatorial if and only if there exists some i E N such that, for all pEnn and all y E IIcp, 'P(p)R;.y. So a dictatorial collective choice function invariably respects the preferences of exactly one individual in the society. Theorem 2.1 (Gibbard-Satterthwaite) If the range of a strategy-proof collective choice function contains at least three alternatives, then it is dictatorial . 22 CHAPTER 2. STRATEGY-PROOF COLLECTIVE CHOICE The classical argument supporting this result uses Arrow's Possibility Theorem [PPTI, Theorem 2.1]: Theorem 2.2 (Arrow) Suppose there are at least three alternatives. If an aggregation rule with unrestricted domain is transitive, weakly Paretian and independent of irrelevant alternatives, then it is dictatorial. Where, we recall (Definition 1.2), a transitive preference aggregation rule with unrestricted domain, f : nn --. n, is weakly Paretian if, for any distinct alternatives x, Y EX, if XPiY for all i E N then XPYi and it is independent of irrelevant alternatives if, for every p, p' E nn and for any x, y E X, p[{x,y} = p'l{x,y} implies f(p...