Strategy-proofness, Pareto optimality and strictly convex norms
The author considers a voting situation where individual agents choose from a 2-dimensional real space of alternatives. The voting scheme selects a social choice compromise point in the plane. The space of alternatives is given by a metric derived from a strictly convex norm. Earlier work by the author, \textit{H. Peters} and \textit{T. Storcken} [Z. Oper. Res. 38, No. 1, 19-53 (1993; Zbl 0794.90031)] showed that in this situation a voting scheme satisfies strategy-proofness, Pareto optimality and anonymity (of the voters) if and only if it is a median scheme, that is, the compromise point has coordinates that are the medians of the coordinates of the individual points for some suitable orthogonal coordinate system. In this paper, the author works in the same setting, but relaxes the condition of anonymity. For three voters, the author introduces the notion of a shifted median scheme. In such a scheme, the compromise point is chosen by a median scheme, but the choice of median scheme depends on the particular arrangement of the individual choices. Additionally, for an arbitrary number of voters, the author introduces coalition dependent median schemes, which can be thought of as simple games over the family of winning coalitions. The main result of the paper is that a voting scheme satisfies strategy-proofness and Pareto optimality if and only if it is a coalition dependent median scheme or all individuals except three are dummies and with respect to these three special voters, the scheme is shifted median. The author also investigates the case of voting schemes satisfying strategy-proofness, Pareto optimality and a weakening of anonymity called equality of power. Some partial results and conjectures are given.
scientific article; zbMATH DE number 1452732