The Non-Existence of Equilibrium in Sequential Auctions When Bids Are Revealed

Article excerpt


Sequential auctions of homogeneous objects are common in public and private marketplaces. Weber derived equilibrium results for what is now a classic model of sequential auctions. However, Weber's results are derived in the context of two particular price quote assumptions. In this paper, we examine a model of sequential auctions based on online auctions, in which, after each auction, all bids are revealed. We show that a pure-strategic, symmetric equilibrium does not exist, regardless of whether the auctions are first- or second-price, if all bids are revealed at the end of each auction.

Keywords: sequential auctions; online auctions; e-commerce; Nash equilibrium

(ProQuest-CSA LLC: ... denotes formulae omitted.)

1. Introduction

A sequential auction consists of a sequence of individual auctions. Those auctions may be first-price sealed-bid (FPSB) auctions, second-price sealed-bid (Vickrey) auctions, English auctions, or reverse (Dutch) auctions. According to Krishna (2002), a single English auction is strategically equivalent to a Vickrey auction; and a single Dutch (reverse) auction is strategically equivalent to an FPSB auction.

Online auctions that encourage sniping by having fixed deadlines are considered equivalent to sealed-bid auctions. As shown by Roth and Ockenfels (2002), bidders tend to bid at the last minute, which makes them strategically distinct from traditional English auctions. According to Bajari and Hortacsu (2003), "more than 50% of final bids are submitted after 90% of the auction duration has passed." To help bidders bid at the last minute, some companies (e.g. provide software to submit bids just before the end of the auction. However, according to Lucking-Reiley (2000), sniping "destroys the English auction's attractive feature that bidders have a dominant strategy to bid up to their maximum willingness to pay." To restore this desirable feature, some online auctions, such as eBay, introduce a proxy bidding policy to auctions, in which bidders submit their maximum bids and a proxy agent will automatically outbid other competitors until reaching the maximum bid. The winner pays an amount of the second highest valuation plus a minimum increase. When the survey was done in 2000, 65 out of 142 online auction sites had adopted proxy bidding (Lucking-Reiley 2000). With proxy bidding, online auctions, such as those on eBay, are again equivalent to the Vickrey auction (Ockenfels and Roth 2005). Without the proxy bidding, in online auctions such as and, the winner pays what he/she bids. As Lucking-Reiley (2000) points out, "if all bidders were to follow a strategy of bidding only at the last minute, the game would become equivalent to a first-price sealed-bid auction."

In this paper, we study sequences of first-price sealed-bid auctions and second-price sealed-bid (Vickrey) auctions. It is quite common in practice to see identical or nearly-identical items sold in a sequence of auctions. Examples include auctions for satellite broadcast licenses, art, wine, fish, flowers, mineral rights, government debts, and many others (Gale and Stegeman 2001). Among those reported in the academic literature are the sequential sale of 120 identical cases of wine in 1990 at Christie's of Chicago (McAfee and Vincent 1993) and the sale of pelts on the Seattle Fur Exchange (Lambson and Thurston 2003). eBay, the world's largest electronic auction, can be viewed as an unending series of auctions for hundreds of thousands of nearly identical items. The practical importance of studying sequential auctions can also be partially supported by Caillau et al. (2002). According to Caillau et al., "many goods, services and contracts are allocated in sequential auctions, sometimes with quite long time periods between two consecutive auctions, sometimes with several auctions almost in a row. As documented in the literature, estate, cattle, fish, vegetables, timber and wine are often allocated in comparable lots at sequential auctions, to a quite well-established and limited group of potential buyers. …