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The Declining Price Anomaly Is Not Universal in Multi-Buyer Sequential Auctions (but almost is)

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Abstract

The declining price anomaly states that the price weakly decreases when multiple copies of an item are sold sequentially over time. The anomaly has been observed in a plethora of practical applications. On the theoretical side, Gale and Stegeman (Games and Economic Behavior, 36(1), 74–103, 2001) proved that the anomaly is guaranteed to hold in full-information sequential auctions with exactly two buyers when one item is sold in each time period. We prove that the declining price anomaly is not guaranteed in full-information sequential auctions with three or more buyers. This result applies to both first-price and second-price sequential auctions. Moreover, it applies regardless of the tie-breaking rule used to generate equilibria in these sequential auctions. To prove this result we provide a refined treatment of subgame perfect equilibria that survive the iterative deletion of weakly dominated strategies and use this framework to experimentally generate a very large number of random sequential auction instances. In particular, our experiments produce an instance with three bidders and eight items that, for a specific tie-breaking rule, induces a non-monotonic price trajectory. Theoretical analyses are then applied to show that this instance can be used to prove that for every possible tie-breaking rule there is a sequential auction on which it induces a non-monotonic price trajectory. On the other hand, our experiments show that non-monotonic price trajectories are extremely rare. In over eighteen million experiments only a 0.000183 proportion of the instances violated the declining price anomaly.

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Notes

  1. If the values are affiliated then prices can have an upwards drift.

  2. A source is a vertex v with in-degree zero; that is, there no arcs pointing into v.

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Authors and Affiliations

  1. McGill University, Montreal, Canada

    Vishnu V. Narayan & Adrian Vetta

  2. École normale supérieure de Lyon, Lyon, France

    Enguerrand Prebet

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  1. Vishnu V. Narayan
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  2. Enguerrand Prebet
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Correspondence to Vishnu V. Narayan.

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This article belongs to the Topical Collection: Special Issue on Algorithmic Game Theory (SAGT 2019) Guest Editors: Dimitris Fotakis and Vangelis Markakis

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Narayan, V.V., Prebet, E. & Vetta, A. The Declining Price Anomaly Is Not Universal in Multi-Buyer Sequential Auctions (but almost is). Theory Comput Syst 66, 546–580 (2022). https://doi.org/10.1007/s00224-021-10059-4

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