Bayesian Combinatorial Auctions: Expanding Single Buyer Mechanisms to Many Buyers

For Bayesian combinatorial auctions, we present a general framework for approximately reducing the mechanism design problem for multiple buyers to the mechanism design problem for each individual buyer. Our framework can be applied to any setting which roughly satisfies the following assumptions: (i) The buyer's types must be distributed independently (not necessarily identically). (ii) The objective function must be linearly separable over the set of buyers (iii) The supply constraints must be the only constraints involving more than one buyer. Our framework is general in the sense that it makes no explicit assumption about any of the following: (i) The buyer's valuations (e.g., sub modular, additive, etc). (ii) The distribution of types for each buyer. (iii) The other constraints involving individual buyers (e.g., budget constraints, etc). We present two generic $n$-buyer mechanisms that use $1$-buyer mechanisms as black boxes. Assuming that we have an$\alpha$-approximate $1$-buyer mechanism for each buyer\footnote{Note that we can use different $1$-buyer mechanisms to accommodate different classes of buyers.} and assuming that no buyer ever needs more than $\frac{1}{k}$ of all copies of each item for some integer $k \ge 1$, then our generic $n$-buyer mechanisms are $\gamma_k\cdot\alpha$-approximation of the optimal$n$-buyer mechanism, in which $\gamma_k$ is a constant which is at least $1-\frac{1}{\sqrt{k+3}}$. Observe that $\gamma_k$ is at least $\frac{1}{2}$ (for $k=1$) and approaches $1$ as $k$ increases. As a byproduct of our construction, we improve a generalization of prophet inequalities. Furthermore, as applications of our main theorem, we improve several results from the literature.