We provide algorithms that learn simple auctions whose revenue is approximately optimal in multi-item multi-bidder settings, for a wide range of valuations including unit-demand, additive, constrained additive, XOS, and subadditive. We obtain our learning results in two settings. The first is the commonly studied setting where sample access to the bidders' distributions over valuations is given, for both regular distributions and arbitrary distributions with bounded support. Our algorithms require polynomially many samples in the number of items and bidders. The second is a more general max-min learning setting that we introduce, where we are given "approximate distributions," and we seek to compute an auction whose revenue is approximately optimal simultaneously for all "true distributions" that are close to the given ones. These results are more general in that they imply the sample-based results, and are also applicable in settings where we have no sample access to the underlying distributions but have estimated them indirectly via market research or by observation of previously run, potentially non-truthful auctions.
Our results hold for valuation distributions satisfying the standard (and necessary) independence-across-items property. They also generalize and improve upon recent works, which have provided algorithms that learn approximately optimal auctions in more restricted settings with additive, subadditive and unit-demand valuations using sample access to distributions. We generalize these results to the complete unit-demand, additive, and XOS setting, to i.i.d. subadditive bidders, and to the max-min setting.
Our results are enabled by new uniform convergence bounds for hypotheses classes under product measures. Our bounds result in exponential savings in sample complexity compared to bounds derived by bounding the VC dimension, and are of independent interest.
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The menu-complexity of ``one-and-a-half'' dimensional mechanism design
We study the menu complexity of optimal and approximately-optimal auctions in the context of the ``FedEx'' problem, a so-called ``one-and-a-half-dimensional'' setting where a single bidder has both a value and a deadline for receiving an item [FGKK 16]. The menu complexity of an auction is equal to the number of distinct (allocation, price) pairs that a bidder might receive [HN 13]. We show the following when the bidder has $n$ possible deadlines:
1) Exponential menu complexity is necessary to be exactly optimal: There exist instances where the optimal mechanism has menu complexity $\geq 2^n-1$. This matches exactly the upper bound provided by Fiat et al.'s algorithm, and resolves one of their open questions [FGKK 16].
2) Fully polynomial menu complexity is necessary and sufficient for approximation: For all instances, there exists a mechanism guaranteeing a multiplicative $(1-\epsilon)$-approximation to the optimal revenue with menu complexity $O(n^{3/2}\sqrt{\frac{\min\{n/\epsilon,\ln(v_{\max})\}}{\epsilon}}) = O(n^2/\epsilon)$, where $v_{\max}$ denotes the largest value in the support of integral distributions.
\item There exist instances where any mechanism guaranteeing a multiplicative $(1-O(1/n^2))$-approximation to the optimal revenue requires menu complexity $\Omega(n^2)$.
Our main technique is the polygon approximation of concave functions [Rote 91], and our results here should be of independent interest. We further show how our techniques can be used to resolve an open question of [DW 17] on the menu complexity of optimal auctions for a budget-constrained buyer.
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- Award ID(s):
- 1717899
- NSF-PAR ID:
- 10069454
- Date Published:
- Journal Name:
- Symposium on Discrete Algorithms
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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