Strategy-Proofness and Social Choice Functions without Singlevaluedness

Abstract

FOLLOWING ON SOME informal conjectures by Dummett and Farquharson [3] and Vickery [20] we now have independent proofs by Gibbard [7] and Satterthwaite [17 and 18] that no collective choice rule exists whose social choice functions are singlevalued, strategy-proof, nondictatorial and have a range containing at least three alternatives. Because strategy-proof ness seems desirable and because it is closely related to mainstream economic theory issues of evaluating resource allocation institutions with respect to incentive compatibility (cf. Hurwicz [9]), their theorem has excited considerable attention [5, 6, 10, 11, 12, 13, 14, 15, 16, 19, and 21]. In this paper, the requirement of singlevaluedness is dropped and explorations are made of the consequences this has on the Gibbard-Satterthwaite results. Let E be the set of all alternatives (which must, by assumption, be mutually incompatible) and N= {1, 2,.. ., n} be the set of individuals. A nonempty subset, v, of E (i.e., an element of 2E _-0}) is an agenda. RE is the set of all complete and transitive binary relations on E; RE is the n-fold Cartesian product of RE. An element, u, of RE is called a profile and if u = (R1, R2,... , Rn), we say that R, is the preference ordering for individual i in u. In the usual way, we use Ri to define strict preference, Pi, and indifference, Ii: xPiy if and only if xRiy and not yRix; xIiy if and only if xRiy and yRix. A social choice function (on V) is a function, C, on Vc2E _{0} into 2E _{0} satisfying C(v) c v. Here V is the set of admissible agenda. The set of all social choice functions on V is called ST. A collective choice rule (on V, U) is a function, F, on UcRE into c6. Here U is the set of admissible profiles. The first constraints on the social choice function in the Gibbard-Satterthwaite theorem are domain restrictions. They admit only one agenda, V= {E} and then require the collective choice rule to work for all societies, U = RE. The most important constraint they use is singlevaluedness: for each v in V, C(v) contains exactly one element. Of course, there is only one V, namely E, in the Gibbard-Satterthwaite theorem. The importance of this constraint stems from its use in all the rest of the problem; singlevaluedness is used in their method of formalizing both nondictatorship and strategy-proofness. Let us deal first with nondictatorship. Using singlevaluedness, let C(v) be the unique member of C(v). Then a collective choice rule, F, is nondictatorial if for no i, i = 1, ... , n, is it true that for all (R1,. . . , Rn) =uE U and for all x C(v) in the range of C = F(u), C(v)Pix. Finally, we turn to strategy-proofness. A collective choice rule is strategy-proof at (v, u) if it is not manipulable at (v, u). F is manipulable at (v, u) if, when u = (R1, R2, ... , Rn), there is a u'=