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Revenue and Reserve Prices in a Probabilistic Single Item Auction

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Abstract

We investigate the effect of limiting the number of reserve prices on the revenue in a probabilistic single item auction. In the model considered, bidders compete for an impression drawn from a known distribution of possible types. The auction mechanism sets up to \(\ell \) reserve prices, and each impression type is assigned the highest reserve price lower than the valuation of some bidder for it. The bidder proposing the highest bid for an arriving impression gets it provided his bid is at least the corresponding reserve price, and pays the maximum between the reserve price and the second highest bid. Since the number of impression types may be huge, we consider the revenue \(R_{\ell }\) that can be ensured using only \(\ell \) reserve prices. Our main results are tight lower bounds on \(R_{\ell }\) for the cases where the impressions are drawn from the uniform or a general probability distribution.

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Notes

  1. Real life ad exchanges have access to a vast log history, which allows for a good estimate of the bidders’ values based on historical bids and other, less direct, means such as the conversion rate associated with the bidder and impression at hand.

  2. An instance of an auction consists of the possible impression types, as well as the probability associated with each type. The exact type of the impression that arrives in practice is not considered part of the instance. A formal definition of the auctions we consider is given in Sect. 2.

  3. If the choice of reserve prices is based on historical bids, it might be beneficial for a bidder to lie in the current auction round, and probably lose some value, in order to affect the reserve prices set in future auctions. We leave such complex (and probably unpractical) plots outside our scope.

  4. Technically, when removing impressions from an auction, we get an invalid auction where the total probability of all impression types is less than 1. We assume throughout this section that if \(\sum _{i = n - n' + 1}^{n} p_i < 1\), then with probability \(1 - \sum _{i = n - n' + 1}^{n} p_i\) no impression arrives, and the revenue is 0.

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Authors and Affiliations

  1. Microsoft Research, 46725, Herzliya, Israel

    Noga Alon

  2. Tel Aviv University, 69978, Tel Aviv, Israel

    Noga Alon

  3. EPFL, 1015, Lausanne, Switzerland

    Moran Feldman

  4. Technion-IIT, 32000, Haifa, Israel

    Moshe Tennenholtz

Authors
  1. Noga Alon
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  2. Moran Feldman
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  3. Moshe Tennenholtz
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Correspondence to Moran Feldman.

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Alon, N., Feldman, M. & Tennenholtz, M. Revenue and Reserve Prices in a Probabilistic Single Item Auction. Algorithmica 77, 1–15 (2017). https://doi.org/10.1007/s00453-015-0055-1

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  • DOI: https://doi.org/10.1007/s00453-015-0055-1

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