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Optimal Item Pricing in Online Combinatorial Auctions

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Part of the Lecture Notes in Computer Science book series (LNCS,volume 13265)

Abstract

We consider a fundamental pricing problem in combinatorial auctions. We are given a set of indivisible items and a set of buyers with randomly drawn monotone valuations over subsets of items. A decision maker sets item prices and then the buyers make sequential purchasing decisions, taking their favorite set among the remaining items. We parametrize an instance by d, the size of the largest set a buyer may want. Our main result asserts that there exist prices such that the expected (over the random valuations) welfare of the allocation they induce is at least a factor \(1/(d+1)\) times the expected optimal welfare in hindsight. Moreover we prove that this bound is tight. Thus, our result not only improves upon the \(1/(4d-2)\) bound of Dütting et al., but also settles the approximation that can be achieved by using item prices. We further show how to compute our prices in polynomial time. We provide additional results for the special case when buyers’ valuations are known (but a posted-price mechanism is still desired).

Keywords

  • Combinatorial Auctions
  • Online allocations

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Fig. 1.
Fig. 2.

Notes

  1. 1.

    That is, for all sets A with \(|A|>d\), \(v(A):=\max _{B \subset A, |B| \le d} \{v(B)\}\).

  2. 2.

    And quite hard to approximate [22].

  3. 3.

    That is, each buyer has a fixed set T, and values all sets S at \(v(S)\,:=\,I(T \subseteq S) \cdot v(T)\).

  4. 4.

    Throughout the paper M is actually a set and refers to the set of different items.

  5. 5.

    Note that different copies of the same item need to get the same price.

  6. 6.

    In some of the constructions in Sect. 4 we break ties conveniently but all the results hold by slightly tweaking the instances.

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Correa, J., Cristi, A., Fielbaum, A., Pollner, T., Weinberg, S.M. (2022). Optimal Item Pricing in Online Combinatorial Auctions. In: Aardal, K., Sanità, L. (eds) Integer Programming and Combinatorial Optimization. IPCO 2022. Lecture Notes in Computer Science, vol 13265. Springer, Cham. https://doi.org/10.1007/978-3-031-06901-7_10

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