We provide novel simple representations of strategy-proof voting rules when voters have uni-dimensional single-peaked preferences (as well as multi-dimensional separable preferences). The analysis recovers, links and unifies existing results in the literature such as Moulin's classic characterization in terms of phantom voters and Barber\`a, Gul and Stacchetti's in terms of winning coalitions ("generalized median voter schemes"). First, we compare the computational properties of the various representations and show that the grading curve representation is superior in terms of computational complexity. Moreover, the new approach allows us to obtain new characterizations when strategy-proofness is combined with other desirable properties such as anonymity, responsiveness, ordinality, participation, consistency, or proportionality. In the anonymous case, two methods are single out: the -- well know -- ordinal median and the -- most recent -- linear median.

中文翻译：

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单峰下策略验证的新特征

我们提供的防策略投票规则新颖简单的陈述时，选民有一维的单峰偏好（以及多维分离的喜好）。该分析恢复，链接并统一了文献中的现有结果，例如穆林根据幻影选民的经典特征以及巴伯，古尔和斯塔切蒂的赢得联盟的思想（“广义中位数选民计划”）。首先，我们比较了各种表示形式的计算属性，并表明渐变曲线表示形式在计算复杂度方面是优越的。此外，当将策略验证与其他期望的属性（例如匿名性，响应性，常规性，参与性，一致性或相称性）结合使用时，新方法可使我们获得新的特征。