Buy-Many Mechanisms for Many Unit-Demand Buyers

A recent line of research has established a novel desideratum for designing approximately-revenue-optimal multi-item mechanisms, namely the buy-many constraint. Under this constraint, prices for different allocations made by the mechanism must be subadditive implying that the price of a bundle cannot exceed the sum of prices of individual items it contains. This natural constraint has enabled several positive results in multi-item mechanism design bypassing well-established impossibility results. Our work addresses a main open question from this literature involving the design of buy-many mechanisms for multiple buyers. Our main result is that a simple sequential item pricing mechanism with buyer-specific prices can achieve an $O(\log m)$ approximation to the revenue of any buy-many mechanism when all buyers have unit-demand preferences over $m$ items. This is the best possible as it directly matches the previous results for the single-buyer setting where no simple mechanism can obtain a better approximation. Our result applies in full generality: even though there are many alternative ways buy-many mechanisms can be defined for multi-buyer settings, our result captures all of them at the same time. We achieve this by directly competing with a more permissive upper-bound on the buy-many revenue, obtained via an ex-ante relaxation.