Abstract
Competitive equilibrium, the central equilibrium notion in markets with indivisible goods, is based on pricing each good such that the demand for goods equals their supply and the market clears. This equilibrium notion is not guaranteed to exist beyond the narrow case of substitute goods, might result in zero revenue even when consumers value the goods highly, and overlooks the widespread practice of pricing bundles rather than individual goods. Alternative equilibrium notions proposed to address these shortcomings have either made a strong assumption on the ability to withhold supply in equilibrium, or have allowed an exponential number of prices.
In this paper we study the notion of competitive bundling equilibrium – a competitive equilibrium over the market induced by partitioning the goods into bundles. Such an equilibrium is guaranteed to exist, is succinct, and satisfies the fundamental economic condition of market clearance. We establish positive welfare and revenue guarantees for this solution concept: For welfare we show that in markets with homogeneous goods, there always exists a competitive bundling equilibrium that achieves a logarithmic fraction of the optimal welfare. We also extend this result to establish nontrivial welfare guarantees for markets with heterogeneous goods. For revenue we show that in a natural class of markets for which competitive equilibrium does not guarantee positive revenue, there always exists a competitive bundling equilibrium that extracts as revenue a logarithmic fraction of the optimal welfare. Both results are tight.
S. Dobzinski—Incumbent of the Lilian and George Lyttle Career Development Chair. Supported in part by the I-CORE program of the planning and budgeting committee and the Israel Science Foundation 4/11 and by EU CIG grant 618128.
M. Feldman—partially supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC grant agreement number 337122.
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Notes
- 1.
In the full version available on arXiv.org we use linear programming to show that our solution concept actually applies more widely than this one.
- 2.
To emphasize this distinction we propose a different name – competitive bundling equilibrium – for the solution concept we focus on in this paper.
- 3.
The classes addressed in [9] are a strict subclass of budget-additive valuations, and superadditive valuations. We provide a general treatment of budget-additive valuations in the full version of this paper [6], and re-derive the result for super-additive valuations as a corollary of a more general argument in the full version: It was observed in [19] that the linear program introduced by [3] has an integrality gap of 1 for superadditive valuations, guaranteeing the existence of non-linear supporting prices; we show that an integrality gap of 1 also implies the existence of a competitive bundling equilibrium with optimal welfare.
- 4.
An interesting future direction is trying to better understand the market processes leading to such bundling.
- 5.
This shows that even for slightly more complicated valuations than those considered by Vickrey, not only is it the case that a standard competitive equilibrium may not exist, but also no competitive bundling equilibrium necessarily provides a constant fraction of the optimal welfare. In the full version we also show that randomization – in the form of correlated lotteries – does not improve the logarithmic approximation factor. This tight and robust bound can be seen as a kind of price of stability result, which establishes a clear separation between the optimal welfare and the optimal stable welfare in markets with indivisible items.
- 6.
i.e., a coarser bundling of the original m items.
- 7.
A tractable algorithm for multi-unit market runs in time \({\text {poly}}(\log m,n)\) and not \({\text {poly}}(m,n)\) – see [7].
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Dobzinski, S., Feldman, M., Talgam-Cohen, I., Weinstein, O. (2015). Welfare and Revenue Guarantees for Competitive Bundling Equilibrium. In: Markakis, E., Schäfer, G. (eds) Web and Internet Economics. WINE 2015. Lecture Notes in Computer Science(), vol 9470. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48995-6_22
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