Abstract
Interdependent values make basic auction design tasks – in particular maximizing welfare truthfully in single-item auctions – quite challenging. Eden et al. recently established that if bidders’ valuation functions are submodular over their signals (a.k.a. SOS), a truthful 4-approximation to the optimal welfare exists. We show existence of a mechanism that is truthful and achieves a tight 2-approximation to the optimal welfare when signals are binary. Our mechanism is randomized and assigns bidders only 0 or \(\frac{1}{2}\) probabilities of winning the item. Our results utilize properties of submodular set functions, and extend to matroid settings.
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Notes
- 1.
The algorithm runs in time polynomial in its input size, which consists of set functions over n elements and so is exponential in n.
- 2.
Our algorithm has two iterations: At \(s_1=0\), an appropriate pair is not found and so the highest bidder (bidder 1) wins the item with probability \(\frac{1}{2}\), which is propagated forward to this bidder at \(s_1=1\). At \(s_1=1\), an appropriate pair is again not found and so the highest bidder (bidder 2) wins the item with probability \(\frac{1}{2}\).
- 3.
This notation is not to be confused with the value for a set of items S; in our model there is a single item, and a bidder’s interdependent value for it is determined by the set of signals, i.e., which subset of signals is “on”.
- 4.
As mentioned above, submodularity over signals is not to be confused with submodularity over items in combinatorial auctions.
- 5.
Note the difference from dominant-strategy IC, in which this guarantee should hold no matter how other bidders report.
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Amer, A., Talgam-Cohen, I. (2021). Auctions with Interdependence and SOS: Improved Approximation. In: Caragiannis, I., Hansen, K.A. (eds) Algorithmic Game Theory. SAGT 2021. Lecture Notes in Computer Science(), vol 12885. Springer, Cham. https://doi.org/10.1007/978-3-030-85947-3_3
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