  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.
