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Explicitly Simple Near-Tie Auctions

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Algorithmic Game Theory (SAGT 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13584))

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Abstract

We consider the problem of truthfully auctioning a single item, that can be either fractionally or probabilistically divided among several winners when their bids are sufficiently close to a tie.

While Myerson’s Lemma states that any monotone allocation rule can be implemented, truthful payments are computed by integrating over each profile, and may be difficult to comprehend and explain. We look for payment rules that are given explicitly as a simple function of the allocated fraction and the others’ bids. For two agents, this simply coincides with (non-negative) Myerson’s payments. For three agents or more, we characterize the near-tie allocation rules that admit such explicit payments, and provide an iterative algorithm to compute them. In particular we show that any such payment rule must require positive payments to some of the bidders.

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Notes

  1. 1.

    For a PSA, \(x_{\underline{k}}\) is a step function. In the general case Myerson’s lemma uses an integral over \(x_{\underline{k}}\) rather than a sum.

  2. 2.

    This can be formally implemented, for example, if each agent j reports (in addition to \(b_j\)) a rational number \(r_j\). In case of a tie, we rank the tied agents according to \(R_j:=r_j\cdot \sqrt{\psi _j}\) where \(\psi _j\) is the j’th prime. Note that \(R_j,R_{j'}\) are never tied (since \(\sqrt{\psi _j}\) are linearly independent over the rationals [1]), unbounded on both sides, and that for any \(r_{j'}<r_{j''}\) and j there is \(r_j\) s.t. \(R_{j'}<R_j<R_{j''}\).

  3. 3.

    We do not allow agents for which \(\alpha _j=0\), as they would change the allocation without being affected.

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Acknowledgement

The authors thank Okke Schijvers and Ella Segev for fruitful discussions.

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Authors and Affiliations

  1. Meta, Core Data Science, London, UK

    Reshef Meir & Riccardo Colini-Baldeschi

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  1. Reshef Meir
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  2. Riccardo Colini-Baldeschi
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Correspondence to Reshef Meir .

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Editors and Affiliations

  1. University of Essex, Colchester, UK

    Panagiotis Kanellopoulos

  2. University of Essex, Colchester, UK

    Maria Kyropoulou

  3. University of Essex, Colchester, UK

    Alexandros Voudouris

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Meir, R., Colini-Baldeschi, R. (2022). Explicitly Simple Near-Tie Auctions. In: Kanellopoulos, P., Kyropoulou, M., Voudouris, A. (eds) Algorithmic Game Theory. SAGT 2022. Lecture Notes in Computer Science, vol 13584. Springer, Cham. https://doi.org/10.1007/978-3-031-15714-1_7

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  • DOI: https://doi.org/10.1007/978-3-031-15714-1_7

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  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-031-15714-1

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