On the benefits of dynamic bidding when participation is costly

Abstract

Consider a second-price auction with costly bidding in which bidders with i.i.d. private values have multiple opportunities to bid. If bids are observable, the resulting dynamic-bidding game generates greater expected total welfare than if bids were sealed, for any given reserve price. Making early bids observable allows high-value bidders to signal their strength and deter others from entering the auction. Nonetheless, as long as the seller can commit to a reserve price, expected revenue is higher when bids are observable than when they are sealed.

Introduction

Bids in sealed-bid auctions may arrive at different times but, since they are sealed, equilibrium play is the same as if bids were simultaneous. This paper considers the welfare and revenue implications of an alternative policy of publicly revealing all bids as they arrive, prior to an otherwise standard second-price auction with costly bidding in which bidders have i.i.d. private values. In particular, I consider a dynamic-bidding game that extends Samuelson's [27] costly-bidding model to a setting with multiple “bidding rounds” in which bids can be simultaneously submitted, with bids made in each round automatically revealed prior to the next round.

Bidders with higher values submit earlier bids in equilibrium, allowing them to deter lower-value bidders from competing. Such entry deterrence benefits higher-value bidders, by allowing them to obtain the object at a lower price, while also benefiting lower-value bidders as they avoid entering auction contests they would otherwise lose. Consequently, bidders' interim expected surplus is higher under “dynamic bidding” when there are multiple bidding rounds than under “sealed bidding” when there is only one bidding round, for any given reserve price.

Equilibrium entry is not efficient under dynamic bidding, even when the reserve price is set to zero. The reason is that each bidder's private benefit from deterring others' entry, that he can win the object at the reserve price rather than the second-highest bidder value, differs from the social benefit of entry deterrence, that others do not incur the cost of bidding. This contrasts with the well-known result that equilibrium entry is efficient under sealed bidding with a zero reserve price; see e.g. Stegeman [28].

Example 1 Inefficient equilibrium entry

Two bidders have i.i.d. private values uniformly distributed on $[0,1]$. The cost of bidding $c=\frac{1}{10}$ and there are two bidding rounds. The efficient symmetric entry thresholds in this example are $\frac{5}{8}$ in the first round and $\frac{1}{4}$ in the second round, while the equilibrium entry thresholds are $\frac{2}{5}$ in the first round and $\frac{1}{5}$ in the second round. (See the online supplementary material for details.) Note that the object is more likely to be sold in equilibrium than is efficient ($\frac{1}{5}<\frac{1}{4}$) and bidders are more likely to enter early in equilibrium than is efficient ($\frac{2}{5}<\frac{5}{8}$).

Although equilibrium entry is inefficient under dynamic bidding, equilibrium expected total welfare is strictly higher under dynamic bidding than when bids are sealed, for any given reserve price (Theorem 1). For an intuition, note that allowing multiple bidding rounds has two sorts of effects on equilibrium play, each of which tends to increase expected total welfare. First, dynamic bidding facilitates welfare-enhancing entry deterrence, as bidders who would have entered but lost in a sealed-bid auction now avoid incurring the cost of bidding. Second, dynamic bidding reveals information about others' values to those who choose not to enter early, encouraging some bidders who would have chosen not to participate in a sealed-bid auction to enter in a later round of the dynamic-bidding game.

What about expected revenue? For any given reserve price, the effect of dynamic bidding on seller expected revenue is ambiguous, as the revenue from new sales to lower-value bidders may or may not dominate the lost revenue from selling to higher-value bidders at lower prices. If the seller is able to commit to a reserve price,¹ however, expected revenue is higher under dynamic bidding than under sealed bidding. Intuitively, the reason is that since dynamic bidding makes the auction more attractive to bidders at any given reserve price, the seller can raise the reserve without losing sales. Indeed, raising the reserve price allows the seller to extract all the welfare gains associated with better bidder coordination in the form of greater expected revenue.

The paper focuses on a setting in which bidding is costly and bids are publicly observable, but the analysis carries over to an alternative setting in which bidding is costless and unobservable but there are costs associated with participating in the auction and other bidders can observe when these costs are incurred. The notion that participation can be costly is well-accepted in the auction literature,² but the analysis here also depends on the notion that (i) these costs can be incurred prior to the auction and (ii) the act of incurring these costs is observable to other bidders. For instance, the cost of traveling to an auction site would not fit within the framework studied here, since bidders cannot observe who has traveled to the auction site until they have already incurred the travel cost.

That said, there are several sorts of potentially substantial participation costs that must be incurred before an auction and which therefore have the potential, if observable by other bidders, to influence their decisions whether to participate.

Bidders in auctions of high-value assets are often required to post bond and/or to receive third-party certification of their ability to pay. Similarly, bidders in complex procurement auctions must often first establish that they are capable of delivering the desired products or services.³

When competing to provide expert services, law firms, consulting firms, and others who coordinate such services may first need to secure a qualified expert. This may entail substantial cost even if the bid is unsuccessful, if the experts' services are needed to prepare the bid.

In a corporate acquisition, top managers of each potential acquirer may need to meet at length with the target firm's management to communicate their plans and build personal trust.⁴

Some of these “pre-participation costs” could potentially be kept secret. For instance, when a corporate acquisition target and potential suitor's management teams meet, both sides could choose to keep quiet about it. Within the context of this paper's model, however, the acquisition target benefits by committing to a transparent policy of revealing whenever such meetings take place (Theorem 2).

Other times, the seller may be unable to observe when bidders incur the costs of participating in the auction. In such cases, however, each bidder has an incentive to reveal to other bidders when it has incurred such costs, so as to deter others from participating in the auction. For instance, consider the example in which bidders need to secure an expert's services in order to prepare a bid. In a “small world” with only a few qualified experts, each of whom is in regular contact with all of the bidders, each bidder would naturally be able to observe whenever anyone else secured an expert's services, as that expert would then be unavailable. On the other hand, the seller might not be able to observe anything about which bidders have secured an expert until the auction itself.

The paper fits into the costly-bidding literature following Samuelson [27],⁵ the novel feature here being that bidders have multiple opportunities to enter the auction. The only other paper I am aware of that allows for multiple bidding rounds in an auction with costly bidding is Li and Conitzer [18], who characterize the optimal reserve-price path in a K-round second-price auction in which the seller can change the reserve price over time. The seller's expected revenue is obviously higher under dynamic bidding with an optimal reserve-price path than under sealed bidding with an optimal reserve, since the seller could always replicate the sealed-bid environment by committing to an infinite reserve after the first round. This paper shows that, in fact, the seller's expected revenue is higher under dynamic bidding even if the seller can only commit to an unchanging reserve.⁶

Another closely related paper is Levin and Peck [17] (hereafter “LP”), who consider a dynamic-entry game that can in some cases be interpreted as a second-price auction with zero reserve.⁷ The basic threshold structure of equilibrium entry is similar here and in LP, but the papers take different (and complementary) analytical approaches. For instance, whereas LP use a contraction argument to establish equilibrium uniqueness, I provide a direct proof and a simple algorithmic method to compute the equilibrium.

The paper also touches more indirectly on several other literatures:

A key feature of equilibrium bidding in this paper's model is that bidders are more likely (ex ante) to enter in earlier bidding rounds (Corollary 1 to Proposition 1 in Appendix A), because earlier entry allows bidders to deter subsequent entry. As such, the paper is similar in spirit to the jump-bidding literature (e.g. Avery [1] and Horner and Sahuguet [16]) while also relating, more indirectly, to the broader literature on preemptive bidding (see e.g. Fishman [9] and Hirshleifer and Png [15]) and the diverse literature on bidder-to-bidder signaling in auctions.⁸

A key result here is that, when bidding is costly, restricting bidding to occur at a single moment makes auctions less efficient. The paper therefore relates, at least in spirit, to others that consider when markets should be opened or closed to maximize the expected gains from trade. See e.g. Ockenfels and Roth [23] on the benefits of not committing to a hard deadline in eBay-style auctions and Fuchs and Skrzypacz [10] on the benefits of sometimes imposing a “lock-up period” (in which no trade is allowed) in a continuous-time lemons market.

This literature identifies optimal disclosure policies, as well as optimal allocation rules, in multi-round sales mechanisms such as when a seller faces a sequence of potential buyers (e.g. Pancs [25]) or when the winner of an auction may then resell the good (e.g. Calzolari and Pavan [5], Zhang and Wang [29]). This paper touches on the related issue of bid disclosure in a second-price auction, comparing two bid-disclosure regimes: never revealing any bids vs. always revealing all bids.

This literature characterizes optimal mechanisms in a wide variety of dynamic environments with changing bidder preferences and/or random bidder arrivals. (See Bergemann and Said [3] for an excellent survey.⁹) This paper touches on this literature by analyzing a particular dynamic mechanism – the second-price auction with multiple bidding rounds – but differs by considering a static environment in which dynamics are driven by the presence of a participation cost.