On Deterministic Optimal Mechanisms in a Two-Item Setting for Distributions with Nondecreasing Density

Consider the problem of designing a revenue-optimal auction mechanism when two heterogeneous items are sold to a single buyer having independent valuations over the items. The distributions of the buyer's valuation for the items are assumed to have densities that are positive, nondecreasing, and continuously differentiable on their support sets $[c_i,c_i+b_i]$ in the positive axis. I prove that the optimal mechanism is deterministic if at least one of the minimum valuations (i.e., either $c_1$ or $c_2$) is sufficiently high. I provide a method to calculate the threshold of $(c_1,c_2)$ beyond which the optimal mechanism is deterministic. I also provide a sufficient condition on the distributions of buyer's valuations for which the individual sale mechanism is optimal. I show that when $c_1$ is low and $c_2$ is high, it is optimal for the seller to sell item $2$ at the minimum valuation $c_2$, thus effectively reducing the problem to finding the optimal mechanism in the one-dimensional setting only for item $1$. I conjecture with promising preliminary results that this result can be extended to the three-item setting. Specifically, I conjecture that when $c_1$ and $c_2$ are low but $c_3$ is high, it is optimal for the seller to sell item $3$ at the minimum valuation $c_3$, thus effectively reducing the problem to finding the optimal mechanism in the two-dimensional setting for items $1$ and $2$.