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

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 proved that the anomaly is guaranteed to hold in full information sequential auctions with exactly two buyers. 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. Theoretic 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 six million experiments only a 0.000183 proportion of the instances violated the declining price anomaly

The Price of Anarchy of Two-Buyer Sequential Multiunit Auctions

We study the efficiency of sequential multiunit auctions with two-buyers and complete information. For general valuation functions, we show that the price of anarchy is exactly $1/T$ for auctions with $T$ items for sale. For concave valuation functions, we show that the price of anarchy is bounded below by $1-1/e\simeq 0.632$. This bound is asymptotically tight as the number of items sold tends to infinity.Comment: 20 pages, 1 figur

Two-Buyer Sequential Multiunit Auctions with No Overbidding

We study equilibria in two-buyer sequential second-price (or first-price) auctions for identical goods. Buyers have weakly decreasing incremental values, and we make a behavioural no-overbidding assumption: the buyers do not bid above their incremental values. Structurally, we show equilibria are intrinsically linked to a greedy bidding strategy. We then prove three results. First, any equilibrium consists of three phases: a competitive phase, a competition reduction phase and a monopsony phase. In particular, there is a time after which one buyer exhibits monopsonistic behaviours. Second, the declining price anomaly holds: prices weakly decrease over time at any equilibrium in the no-overbidding game, a fact previously known for equilibria with overbidding. Third, the price of anarchy of the sequential auction is exactly $1 - 1/e$.Comment: 30 pages, 3 figures. Submitted to SAGT'2