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'=

[Observations are made on the sources of cross-equation constraints and the ways they can be used as identification aids, on possible simplifying transformations of linear homogeneous cross-equation constraints, on a rank condition for identification of a block of equations under such linear constraints, and on a strategy for using these constraints for single-equation identification.]

[Assuming each assignment of strong preference orderings to individuals is equally likely, we examine how the probability of social intransitivity (under asimple majority vote decision rule) changes with changes in the number of alternatives and the number of voters. A similar study is made of violation of quasi-transitivity and failure of existence of a maximal alternative.]

1. TO BE CLEAR from the beginning, this note does not provide practical aids to identification; the information is very unlikely to be available. Rather, this paper serves an expository purpose, that of seeking to clarify two ideas: first, multiple-identification, by presenting and analyzing an example within a linear system, and second, Working's and Fisher's ideas on relative variances as identification aids, by showing the consequences of knowing that one variance is a specific large multiple of another. Along the way we provide a correction to the econometrics literature.

A social welfare function f is Arrowian if it has transitive values and satisfies Arrow's independence axiom. For any fraction t and any Arrowian f, either there will be some individual who dictates on a subset containing at least the fraction t of outcomes, or at least the fraction 1 minus t of the pairs of outcomes have their social ranking fixed independently of individual preference. And for any Arrowian f, there is a set containing a large fraction of the citizens whose preferences are never consulted in determining the social ranking of a large fraction of the pairs of alternatives.