Straightforward Elections, Unanimity and Phantom Voters
研究不可操纵的直接揭示社会选择函数，在选民偏好为可分离星形且存在全局最优时，若函数满足弱一致性条件，则仅依赖选民理想点，并可分解为一维机制，其中非独裁机制由连续性或幽灵选民刻画，但可微性导致独裁。
Non-manipulable direct revelation social choice functions are characterized for societies where the space of alternatives is a euclidean space and all voters have separable star-shaped preferences with a global optimum. If a non-manipulable choice function satisfies a weak unanmity-respecting condition (which is equivalent to having an unrestricted range) then it will depend only on voters' ideal points. Further, such a choice function will decompose into a product of one-dimensional mechanisms in the sense that each coordinate of the chosen point depends only on the respective coordinate of the voters' ideal points. Each coordinate function will also be non-manipulable and respect unanimity. Such one-dimensional mechanisms are uncompromising in the sense that voters cannot take an extreme position to influence the choice to their advantage. Two characterizations of uncompromising choice functions are presented. One is in terms of a continuity condition, the other in terms of “phantom voters” i.e. those points which are chosen which are not any voter's ideal point. There are many such mechanisms which are not dictatorial. However, if differentiability is required of the choice function, this forces it to be either constant or dictatorial. In the multidimensional case, non-separability of preferences leads to dictatorship, even if preferences are restricted to be quadratic.