Abstract
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 the main open question from this literature of extending the buy-many constraint to multiple buyer settings and developing an approximation.
We propose a new revenue benchmark for multi-buyer mechanisms via an ex-ante relaxation that captures several different ways of extending the buy-many constraint to the multi-buyer setting. Our main result is that a simple sequential item pricing mechanism with buyer-specific prices can achieve an \(O(\log m)\) approximation to this revenue benchmark when all buyers have unit-demand or additive 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.
From a technical viewpoint we make two novel contributions. First, we develop a supply-constrained version of buy-many approximation for a single buyer. Second, we develop a multi-dimensional online contention resolution scheme for unit-demand buyers that may be of independent interest in mechanism design.
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Notes
- 1.
Consider, for example, multiple interactions of a buyer with the mechanism interleaved by purchases made by other buyers. As the item supply changes, the mechanism can update the prices on its menu, and no longer necessarily needs to satisfy a subadditivity constraint on the final pricing observed by the buyer. In fact, by exploiting this supply-based pricing approach, a multi-buyer buy-many mechanism can simulate any single-agent non-buy-many mechanism, inheriting the unbounded simple-versus-optimal revenue gaps of the latter setting. Sybil-proofness or false-name-proofness is even easier to achieve in principle, unless some symmetry-type restrictions are placed on the mechanism (as in [18, 25], for example), as the mechanism can simply refuse to make any allocations unless the number of agents is exactly n.
- 2.
For formal definitions, see [10].
- 3.
In fact, the function \(\textsc {Rev}(x)\) defined as maximum revenue from any restricted set of mechanism with allocations at most x is concave. This is because for any two allocations x and y and coefficient \(1\ge \alpha \ge 0\), one can consider mechanisms \(\textsc {M}(x)\) and \(\textsc {M}(y)\) that define \(\textsc {Rev}(x)\) and \(\textsc {Rev}(y)\), and run the former with probability \(\alpha \) and the latter with probability \((1-\alpha )\). Then \(\textsc {Rev}(\alpha x+(1-\alpha )y)\ge \alpha \textsc {Rev}(x)+(1-\alpha )\textsc {Rev}(y)\).
- 4.
For any buyer i, it is possible that p has allocation \(y\le x^o\), with \(\textsc {SRev}(y)=\textsc {SRev}(x^o)\). However, for any item j such that \(y_{j}<x^o_j\), since \(\textsc {SRev}(y)=\textsc {SRev}(x^o)\), the gradient \(\nabla \textsc {SRev}(x^o)\) has value \(c_j=0\) on the jth component. Thus the profit of item pricing p for the buyer with production costs \(c\) is still \(\textsc {SRev}(x^o)-c\cdot x^o\), although the actual allocation is less than \(x^o\).
- 5.
We can reduce the price of \(\lambda ^{(j)}\) by some small \(\epsilon >0\) to make each buyer type’s utility be strictly positive.
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Chawla, S., Rezvan, R., Teng, Y., Tzamos, C. (2024). Buy-Many Mechanisms for Many Unit-Demand Buyers. In: Garg, J., Klimm, M., Kong, Y. (eds) Web and Internet Economics. WINE 2023. Lecture Notes in Computer Science, vol 14413. Springer, Cham. https://doi.org/10.1007/978-3-031-48974-7_2
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