We consider an economic environment with one buyer and one seller. For a bundle $(t,q)\in [0,\infty[\times [0,1]=\mathbb{Z}$, $q$ refers to the winning probability of an object, and $t$ denotes the payment that the buyer makes. We consider continuous and monotone preferences on $\mathbb{Z}$ as the primitives of the buyer. These preferences can incorporate both quasilinear and non-quasilinear preferences, and multidimensional pay-off relevant parameters. We define rich single-crossing subsets of this class and characterize strategy-proof mechanisms by using monotonicity of the mechanisms and continuity of the indirect preference correspondences. We also provide a computationally tractable optimization program to compute the optimal mechanism for mechanisms with finite range. We do not use revenue equivalence and virtual valuations as tools in our proofs. Our proof techniques bring out the geometric interaction between the single-crossing property and the positions of bundles $(t,q)$s in the space $\mathbb{Z}$. We also provide an extension of our analysis to an $n-$buyer environment, and to the situation where $q$ is a qualitative variable.

翻译：我们考虑一个包含一个买方和一个卖方的经济环境。对于捆绑物$(t,q)\in [0,\infty[\times [0,1]=\mathbb{Z}$，其中$q$表示获得物品的获胜概率，$t$表示买方支付的金额。我们将买方在$\mathbb{Z}$上的连续单调偏好作为基本假设。这些偏好可以同时包含拟线性和非拟线性偏好，以及多维支付相关参数。我们定义了该类偏好中的丰富单交叉子集，并通过机制的单调性和间接偏好对应的连续性来刻画防策略机制。我们还提供了一个计算可行的优化程序，用于计算有限值域机制的最优机制。在证明中我们没有使用收益等价定理和虚拟估值作为工具。我们的证明技术揭示了单交叉性质与捆绑物$(t,q)$在空间$\mathbb{Z}$中位置之间的几何交互作用。我们还将分析扩展到$n-$买方环境，以及$q$为定性变量的情形。