We consider one buyer and one seller. For a bundle $(t,q)\in [0,\infty[\times [0,1]=\mathbb{Z}$, $q$ either refers to the wining probability of an object or a share of a good, and $t$ denotes the payment that the buyer makes. We define classical and restricted classical preferences of the buyer on $\mathbb{Z}$; they incorporate quasilinear, non-quasilinear, risk averse preferences with multidimensional pay-off relevant parameters. We define rich single-crossing subsets of the two classes, 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. 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. Our proofs are simple and provide computationally tractable optimization program to compute the optimal mechanism. The extension of the optimization program to the $n-$ buyer environment is immediate.

翻译：我们考虑一个买家和一个卖家的情形。对于捆绑商品$(t,q)\in [0,\infty[\times [0,1]=\mathbb{Z}$，其中$q$指代物品的获胜概率或商品的份额，$t$表示买家支付的金额。我们在$\mathbb{Z}$上定义了买家的经典偏好与受限经典偏好；这些偏好涵盖了具有多维支付相关参数的拟线性、非拟线性及风险规避偏好。我们定义了两类偏好中丰富的单交叉子集，并通过机制单调性与间接偏好对应关系的连续性来刻画策略性证明机制。我们还提供了一个计算可行的优化程序来计算最优机制。在证明中我们未使用收益等价定理或虚拟估值作为工具。我们的证明技术揭示了单交叉属性与捆绑商品$(t,q)$位置之间的几何相互作用。证明过程简洁，并为计算最优机制提供了计算可行的优化程序。该优化程序可直接扩展至$n-$买家环境。