Elsevier

Games and Economic Behavior

Volume 117, September 2019, Pages 289-315
Games and Economic Behavior

An experimental study on sequential auctions with privately known capacities

https://doi.org/10.1016/j.geb.2019.06.005Get rights and content

Abstract

We experimentally study sequential procurement auctions where bidders' capacity constraints are private information. Our experiment involves two first-price auctions with a belief elicitation stage at the end of the first. Our results show that (i) observed behavior in the second auction is overall consistent with sequential rationality; (ii) first auction bids are decreasing in the capacity of the bidder, but (iii) stated beliefs are inconsistent with the actual play. Hence, subjects seem to be aware of the opportunity cost of early bids (which leads capacity constrained bidders to bid more cautiously than unconstrained ones); on the other hand, since they do not recognize the informative content of bids, the potential signaling cost associated with early bids does not come into play.

Introduction

Several goods and services are procured or sold through auctions that are run in sequence. The most prominent example is represented by electricity markets: in these markets, the delivery of electricity is usually procured by means of auctions, the day-ahead market and the intra-day market, where sellers commit to deliver a certain amount of power in a specific time interval the next day or in the same day, followed by a real-time auction, the balancing market, meant to secure real time balance between actual demand and supply. Other examples of sequential auctions include the sale of spectrum rights, oil and gas leases, greenhouse gas emission permits and treasury bonds.

The distinguishing feature of sequential auctions is that the outcome of one auction may alter the setting in which the following auction takes place and/or may convey additional, though imperfect, information on some relevant elements of the environment. This introduces a strategic linkage between the auctions, as rational bidders should anticipate that their behavior in one auction will, directly or indirectly, affect their payoffs in the next.

In procurement contexts, this strategic linkage is certainly relevant when sellers have capacity constraints. This has been documented empirically by Jofre-Bonet and Pesendorfer, 2000, Jofre-Bonet and Pesendorfer, 2003 and De Silva (2005), who show that, in auctions for road construction contracts, firms that won previous auctions typically participate less and bid less aggressively in later auctions. The idea is that, given that completing a project requires several months whereas new contracts are auctioned off at high frequency, a firm that is awarded a contract, having more committed capacity, may not have the necessary resources to carry out future projects, or can obtain them only at a relatively high cost. In other words, when firms have capacity constraints, winning an early auction entails an opportunity cost, as the firm will lose the opportunity to effectively compete in the next, where market conditions may possibly be more favorable. Inspired by these findings, Brosig and Reiß (2007) and Saini and Suter (2015) have then tested in the lab whether subjects do indeed properly account for this opportunity cost of early bids. However, these papers assume that bidders' capacities are common knowledge, thus potentially missing other concurring strategic effects. In particular, when bidders may have limited capacities – but this information is privately held – the opportunity cost of winning an early auction for capacity constrained bidders interplays with a signaling cost: bidders that are far from their capacity limits should anticipate that their bids might signal their actual capacity, thereby affecting the intensity of competition in future auctions. This adds complexity to the bidders' tasks and makes the analysis of observed bids more challenging.

To investigate how bidders react to these strategic forces, we design an experiment with two sequential first-price auctions, each involving a single unit, and two sellers, who may have one or two units to sell: hence, the winner of the first auction will bid in the second only if her initial capacity was two units. Whereas the information on capacity is privately held, costs are common knowledge and, for simplicity, normalized to zero. At the end of the first auction, the outcome and the winning bid are disclosed, and bidders reveal – through an incentive compatible mechanism – their beliefs on the opponent's (initial) capacity. To match real-world situations where firms are unsure about the demand or even the occurrence of future auctions, we also introduce (exogenous) uncertainty about the realization of the second auction. We consider four treatments, which differ in the ex ante probability distribution of bidders' capacities and in the degree of uncertainty about the second auction's implementation.

Our preliminary theoretical analysis shows that, while the opportunity cost of early bids pushes toward a separation of bids in the early auction – all else equal, bidders with only one unit to sell have an incentive to bid more cautiously than those with a two-unit capacity – the signaling cost is so strong that, in equilibrium, bids are, at least partially, pooled: a bidder with a capacity of two units prefers to significantly reduce her chance of winning the first auction than to fully reveal her type.

Our experimental results can be summarized as follows. Bidding behavior in the second auction matches the main predictions associated with the Perfect Bayesian Equilibrium of the game: in particular, a subject who lost the early auction tends to (rationally) bid the more aggressively, the higher the likelihood she attaches to her opponent having an initial capacity of two units. This implies that, for a bidder with two units to sell, revealing her own capacity would be costly, as this would intensify competition in the later auction. However, the joint analysis of the behavior in the first auction and the belief elicitation phase, shows that this signaling cost is only potential, as bidders seem to be unable to appreciate the informative content of bids. In fact, although first auction bids are significantly and negatively affected by capacity, beliefs do not respond accordingly. In particular, beliefs, net of a partial reversion to the 50/50 odds, are, on average, aligned with prior probabilities. Therefore, our evidence suggests that, while subjects seem to properly account for the opportunity cost of early bids – capacity constrained bidders bid more cautiously than those who are not – the signaling cost of early bids, which should countervail the opportunity cost, is essentially absent.

In an attempt to shed light on the behavioral reasons behind the imperfect belief updating process by subjects, we also show that winners of the first auction (who are not informed of their opponent's bid) seem to follow an irrational gambler's fallacy heuristic. On the other hand, losers, who observe the opponent's (winning) bid, are not affected by the gambler's fallacy; yet, they do not seem to recognize the link between capacity and bids, as if the information conveyed by the opponent's bid were not strong and clear enough to prompt a belief revision.

We believe, that, beyond its theoretical interest, investigating the consequences on behavior of private information about capacity constraints has also practical relevance. Hortaçsu and Puller (2008) claim that, in electricity markets, it is realistic to assume that generation costs are common knowledge across firms. Instead, it is less likely that firms have accurate information regarding each other's available capacities at the time of bidding. This is because generating firms that participate in the auctions usually also trade electricity through bilateral forward contracts with electricity users: it is unrealistic to believe that firms perfectly know the exact contract positions of their rivals at the time of bidding. Similar considerations are likely to be applicable also in other procurement markets, especially those in which firms, beyond competing for contracts tendered by public authorities, also operate in the private sector. In these contexts, our paper suggests that the information that is endogenously generated in the market may trigger significant behavioral effects. In this sense, we offer novel insights into the possible determinants of the deviations from equilibrium bidding that have been documented by the empirical literature on electricity and recurring procurement auctions (see Section 2 below).

Notice, in addition, that the strategic incentives faced by sellers in our setting are similar to those confronted by financially constrained buyers in a standard sequential auction: on the one hand, a bidder with small budget cannot effectively bid in later auctions if she depletes her budget early on (this is the opportunity cost); on the other hand, a bidder with large budget should anticipate that, should the other bidders learned that her budget is indeed large, competition will be fiercer later on (this is the signaling cost). Hence, the insights from our paper easily extend to those auction markets where buyers' budgets are limited and privately known, an assumption that is realistic in many circumstances, as argued by several authors (see, e.g., Salant, 1997, Ghosh, 2013; and Ghosh and Liu, 2019).

The rest of the article is organized as follows. Section 2 reviews the related literature. Section 3 presents the experimental design and the theoretical predictions under standard bidders' preferences and equilibrium behavior. Section 4 analyzes the experimental results. Section 5 discusses the results, focusing on the belief updating process by subjects. Finally, Section 6 concludes.

Section snippets

Related literature

In the benchmark model of sequential auctions (see Milgrom and Weber, 2000), all bidders are assumed to have unit demand and private valuations.1

Baseline game and treatments

Our experimental setup consists of two sequential first-price auctions with incomplete information. Two sellers participate in two consecutive auctions, A1 and A2: in each auction, they compete to sell one unit of a homogeneous good to a hypothetical buyer, and the buyer buys the unit from the seller posting the lowest price, provided this price does not exceed a commonly known reserve price R, that was set equal to 120. Sellers do not bear any cost for producing the units they sell. However,

Experimental results

In line with the structure of the theoretical part, we will first analyze bids in A2, studying how they depend on the outcome of the first auction and on the beliefs elicited between the two auctions. We will then focus our attention on behavior in A1, identifying its main determinants. Lastly, we will examine beliefs and how they respond to the information provided at the end of the first auction. In the next section, we will comment on the results, suggesting a possible explanation of the

Discussion

A direct comparison between our experimental evidence, summarized in results R1-R11 in Section 4, and the predictions implied by the equilibrium theory (presented in Section 3.2), leads us to draw mixed conclusions: whereas bidding behavior in A2 basically conforms with sequential rationality, behavior in A1 departs from equilibrium requirements in that beliefs are clearly inconsistent with actual play.29

Conclusion

This article reported the results of an experiment involving two sequential first-price procurement auctions, where sellers may be capacity constrained, and this information is privately held.

That capacity constraints may crucially affect behavior in repeated procurement auctions is a well documented empirical fact. However, we are not aware of other experimental papers that study sequential auctions where bidders' capacities are private information.

We designed four treatments that differ in

Acknowledgements

We thank for their comments: Alexey Belianin, Federico Boffa, Arturo Lorenzoni, Fausto Pacicco, Giancarlo Spagnolo, Steve Tadelis, Mirco Tonin, Luigi Vena, Andrea Venegoni and participants in the 2018 RES Annual Conference, International Workshop on Competition, Regulation and Procurement in Moscow (May 2018), 2018 ESA World Meeting, 2018 European Winter Meeting of the Econometric Society and department seminar at the Free University of Bozen. We are also grateful to 2 anonymous Referees, an

References (32)

  • J.D. Donna et al.

    Complements and substitutes in sequential auctions: the case of water auctions

    Rand J. Econ.

    (2018)
  • P. Février et al.

    Buy or wait, that is the option: the buyer's option in sequential laboratory auctions

    Rand J. Econ.

    (2007)
  • U. Fischbacher

    Z-Tree: Zurich toolbox for ready-made economic experiments

    Exp. Econ.

    (2007)
  • Ghosh, G., 2013. Simultaneous first-price auctions with budget-constrained bidders....
  • T. Henckel et al.

    Belief Adjustment: A Double Hurdle Model and Experimental Evidence

    (2018)
  • A. Hortaçsu et al.

    Understanding strategic bidding in multi-unit auctions: a case study of the texas electricity spot market

    Rand J. Econ.

    (2008)
  • View full text