Limits of Efficiency in Sequential Auctions

TL;DR

This paper demonstrates that sequential first-price auctions can have linearly inefficient equilibria even with simple valuation types, highlighting fundamental limitations in auction efficiency.

Contribution

It proves linear inefficiency in equilibria for auctions with additive or unit-demand valuations, filling a key gap in understanding auction performance.

Findings

Inefficiency grows linearly with the number of items or bidders.

Inefficient equilibria persist after elimination of dominated strategies.

Results apply to various valuation models including gross substitute and budget constraints.

Abstract

We study the efficiency of sequential first-price item auctions at (subgame perfect) equilibrium. This auction format has recently attracted much attention, with previous work establishing positive results for unit-demand valuations and negative results for submodular valuations. This leaves a large gap in our understanding between these valuation classes. In this work we resolve this gap on the negative side. In particular, we show that even in the very restricted case in which each bidder has either an additive valuation or a unit-demand valuation, there exist instances in which the inefficiency at equilibrium grows linearly with the minimum of the number of items and the number of bidders. Moreover, these inefficient equilibria persist even under iterated elimination of weakly dominated strategies. Our main result implies linear inefficiency for many natural settings, including auctions with gross substitute valuations, capacitated valuations, budget-additive valuations, and additive valuations with hard budget constraints on the payments. Another implication is that the inefficiency in sequential auctions is driven by the maximum number of items contained in any player's optimal set, and this is tight. For capacitated valuations, our results imply a lower bound that equals the maximum capacity of any bidder, which is tight following the upper-bound technique established by Paes Leme et al. \cite{PaesLeme2012}.