Never Say Never Optimal Exclusion and Reserve Prices with Expectations-Based Loss-Averse Buyers Benjamin BalzerAntonio Rosato

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Never Say Never: Optimal Exclusion and Reserve Prices

with Expectations-Based Loss-Averse Buyers∗

Benjamin Balzer†Antonio Rosato‡

June 16, 2023

Abstract

We study reserve prices in auctions with independent private values when bidders are expectations-

based loss averse. We ﬁnd that the optimal public reserve price excludes fewer bidder types

than under risk neutrality. Moreover, we show that public reserve prices are not optimal as

the seller can earn a higher revenue with mechanisms that better leverage the “attachment

eﬀect”. We discuss two such mechanisms: i) an auction with a secrete and random reserve

price, and ii) a two-stage mechanism where an auction with a public reserve price is followed

by a negotiation if the reserve price is not met. Both of these mechanisms expose more

bidders to the attachment eﬀect, thereby increasing bids and ultimately revenue.

JEL classiﬁcation: D44, D81, D82.

Keywords: Reference-Dependent Preferences; Loss Aversion; Reserve Price; First-Price

Auction; Second-Price Auction; Personal Equilibrium.

∗For helpful comments, we thank Zachary Breig, Jeﬀ Ely, Leslie Marx and audiences at the 2021 Australasian

Economic Theory Workshop at the University of Sydney, UTS, the 2021 INFORMS Annual Meeting in Anaheim,

the 12th Conference on Economic Design in Padova, the 2022 Asia-Paciﬁc Industrial Organization Conference in

Sydney, and the 2023 Workshop on Preferences and Bounded Rationality at the BSE Summer Forum. We are also

grateful to Andrew Barr, Gladys Berejiklian and, especially, Scott Morrison for the extended lockdowns that vastly

improved our productivity. Rosato gratefully acknowledges ﬁnancial support from the Australian Research Council

(ARC) through the ARC Discovery Early Career Researcher Award DE180100190.

†UTS Business School, University of Technology Sydney (Benjamin.Balzer@uts.edu.au).

‡University of Queensland, Universit`a di Napoli Federico II and CSEF (a.rosato@uq.edu.au).

arXiv:2210.10938v2 [econ.TH] 14 Jun 2023

1 Introduction

Reserve prices are a prevalent tool auctioneers use to raise their expected revenue. A reserve

price acts as an additional bid placed by the auctioneer since, in order to win, a buyer must also

outbid the reserve. Thus, a reserve price increases the competitiveness of an auction. Yet, this

comes at a cost for the auctioneer because trade does not happen if no buyer bids at least the

reserve price. Indeed, a reserve price excludes buyers with relatively low valuations from the auction

and reduces the overall probability of trade. Seminal theoretical contributions by Myerson (1981)

and Riley and Samuelson (1981) have characterized the revenue-maximizing reserve price as the

solution to this trade-oﬀ between decreasing the probability of trade and amplifying competitive

pressure. In particular, they show that with risk-neutral bidders having independent private values,

the optimal reserve price coincides with the classical monopoly price; hence, it is (i) deterministic

and public, (ii) always higher than the seller’s own value, and (iii) under mild conditions on the

distribution of bidders’ values, independent of the number of bidders.

However, these features are not always empirically observed. For instance, in real-world auc-

tions sellers often use secret reserve prices. A prime example is that of real-estate auctions in the

Australian state of Queensland, where prospective buyers are allowed to know whether the seller

set a reserve price, but not its exact value.1Moreover, some studies show that empirical reserve

prices are often signiﬁcantly lower than what the classical models predict; see Paarsch (1997) and

Haile and Tamer (2003). There is also evidence of reserve prices that vary with the number of

bidders and auctions with no reserve price at all; see Davis et al. (2011) and Gon¸calves (2013).

Overall, this evidence suggests that sellers may face additional trade-oﬀs not captured by the

classical risk-neutral and/or risk-averse model.2

In this paper, we analyze reserve prices in ﬁrst-price auctions (FPA) and second-price auctions

(SPA) where symmetric bidders have independent private values (IPV) and are expectations-

based loss averse ´a la K˝oszegi and Rabin (2006, 2007, 2009). We derive the revenue-maximizing

reserve price for each format and highlight how loss aversion modiﬁes the seller’s trade-oﬀ between

increasing competitive pressure and reducing the probability of trade. In particular, we show that

loss aversion can rationalize reserve prices that (i) are secret, (ii) vary with the number of bidders,

and (iii) are lower than what the theory predicts for risk-neutral and risk-averse bidders.

Section 2 introduces the auction environment and bidders’ preferences, and describes the solu-

tion concept. Following K˝oszegi and Rabin (2006), we posit that, in addition to classical material

utility, a bidder also experiences “gain-loss utility” when comparing her material outcomes to a

reference point equal to her expectations regarding those same outcomes, with losses being more

1Secret reserve prices are also documented by Elyakime et al. (1994) and Li and Perrigne (2003) in timber

auctions, and by Bajari and Horta¸csu (2004) and Hasker and Sickles (2010) in internet auctions.

2With private values, Hu et al. (2010) show that risk aversion can explain low reserve prices in the FPA but not

in the SPA; if in addition bidders have interdependent values, Hu et al. (2019) show that risk aversion can explain

low reserve prices also in the SPA.

painful than equal-size gains are pleasant.

We apply the solution concept of “unacclimating personal equilibrium” (UPE) introduced by

K˝oszegi and Rabin (2007). According to this concept, bidders choose the strategy that maximizes

their payoﬀ keeping expectations ﬁxed, and the distribution of outcomes so generated must coincide

with the expectations; hence, when deviating from her equilibrium bid, a bidder holds her reference

point ﬁxed.3As there might be multiple UPEs, we assume bidders select their preferred personal

equilibrium (PPE) — the one that maximizes their utility from an ex-ante perspective.

We begin our analysis in Section 3 by deriving the revenue-maximizing public reserve price

in the FPA.4First, we show that the “attachment eﬀect” (K˝oszegi and Rabin, 2006) is the main

driving force behind the bidding behavior of loss-averse buyers. In particular, the higher the

probability with which a bidder, in equilibrium, expects to win the auction, the bigger the loss

she endures if she ends up losing it. Hence, a bidder has an incentive to increase her bid, so

as to win more often and avoid experiencing the loss. Thus, the attachment eﬀect induces an

upward pressure on the equilibrium bidding strategy, ensuring that a bidder’s cost from an upward

deviation is higher than the beneﬁt. Importantly, the attachment eﬀect only aﬀects the incentives

of a bidder who expects to win the auction with strictly positive probability — however small —

and is thus exposed to potential losses in equilibrium; yet, it does not aﬀect those bidders who

abstain from the auction since they do not incur a loss when not winning it.

The fact that bidders who do not expect to win are not exposed to the attachment eﬀect

has several implications for the characterization of the revenue-maximizing public reserve price.

First, it puts downward pressure on the optimal reserve price. Indeed, by increasing the reserve

price, the seller excludes a larger set of bidder types from the auction. As a bidder’s attachment

increases in her type, the higher the marginally excluded type, the larger the attachment eﬀect

that the seller forgoes. Thus, with expectations-based loss aversion, increasing the reserve price

is more costly for the seller compared to a situation where the attachment eﬀect is not present;

e.g., with risk-neutral bidders. This ﬁnding is especially relevant for those empirical studies that

ﬁrst estimate the distribution of bidders’ values and then use the theoretical insights of Myerson

(1981) and Riley and Samuelson (1981) to estimate the revenue-maximizing reserve price as the

minimum bid that excludes all bidders with a “virtual value” lower than the seller’s value. For

instance, Paarsch (1997) and Haile and Tamer (2003) ﬁnd that sellers exclude fewer types than

their estimation predicts as revenue maximizing. Yet, as our ﬁrst result implies, it is theoretically

optimal to include in the auction bidders with virtual values lower than the seller’s own value

because of the attachment eﬀect. While risk aversion can rationalize such low reserve prices in the

FPA (see Hu et al., 2010), it cannot explain it for the SPA.5

3For other applications of UPE see, for instance, Heidhues and K˝oszegi (2008, 2014) Karle and Peitz (2014,

2017), Karle and M¨oller (2020), Karle and Schumacher (2017) and Rosato (2016).

4As shown by Balzer and Rosato (2021), under UPE the FPA and SPA are revenue equivalent; therefore, our

results on the reserve price for the FPA carry over to the SPA.

5However, beyond risk aversion, several diﬀerent explanations for low reserve prices in both the SPA and FPA

Furthermore, the optimal public reserve price varies with the number of bidders. Indeed, the

more bidders are present, the less optimistic each of them is about her chances of winning, which

reduces their attachment. Yet, this does not imply that the reserve price always increases in the

number of bidders. Indeed, when raising the reserve price, the seller forgoes the attachment eﬀect

of the marginal type. However, with already many bidders participating, adding an extra one

reduces this cost, leading to more exclusion.6With risk aversion, instead, the optimal reserve

price (in the FPA) is decreasing in the number of bidders; see Vasserman and Watt (2021).

The fact that the attachment eﬀect does not operate on those bidders excluded by a public

reserve price suggests that a seller could raise an even higher revenue by exposing more bidders

to this eﬀect. In Section 4 we show that this intuition is correct. In particular, we characterize

two tactics whereby a seller can expose almost all bidders to the attachment eﬀect, resulting in a

strictly larger revenue than an auction with a public reserve price.

In Subsection 4.1 we show that secret and random reserve prices are revenue superior to public

and deterministic ones. To see why, notice that with a secret reserve price each bidder type expects

to win the auction with strictly positive — albeit potentially arbitrarily small — probability. In

such an auction, therefore, every bidder is exposed to potential losses and thus has an incentive

to bid more aggressively in order to avoid them. Hence, by transforming the public reserve price

into a secret one, the seller can ensure that every bidder experiences the attachment eﬀect, which

enhances revenue. By doing so, however, the seller also reduces the competitive pressure on the

buyers’ side, which could potentially harm revenue since those low-type bidders excluded under a

public reserve would be participating now. Yet, the seller can choose a distribution for the (secret)

reserve price that puts large probability mass on relatively high prices and arbitrarily small mass on

low ones. Such a distribution ensures that, while the seller exposes every bidder to the attachment

eﬀect, the competitive pressure is almost the same as under a public reserve price.

Thus, expectations-based loss aversion provides a novel rationale for secret and random reserve

prices.7This result is reminiscent of those in Heidhues and K˝oszegi (2014) and Hancart (2022),

who characterize the optimal pricing strategy for a monopolist selling to an expectations-based

loss-averse buyer. In line with the ﬁndings of Azevedo and Gottlieb (2012), who showed that risk-

neutral sellers beneﬁt from oﬀering gambles to consumers exhibiting prospect-theory preferences,

these papers ﬁnd that the monopolist beneﬁts from using random prices. In particular, Heidhues

and K˝oszegi (2014) show that if the seller has suﬃcient commitment power, a stochastic pricing

have been proposed. These include correlated types (Levin and Smith, 1996), interdependent values (Quint, 2017;

Hu et al., 2019), endogenous entry (McAfee, 1993; Levin and Smith, 1994; Peters and Severinov, 1997), bidders’

selection neglect when sellers are privately informed about the quality of the objects they sell (Jehiel and Lamy,

2015), level-k bidders (Crawford et al., 2009) and taste projection (Gagnon-Bartsch et al., 2021).

6Menicucci (2021) obtains a similar result in the classical IPV risk-neutral model when the bidders’ virtual values

are not monotone; in contrast, our result holds also for the regular case of increasing virtual values.

7An indirect way of implementing a secret and random reserve price is via “shill bidding”, a prominent albeit

often illegal practice in real-world auctions whereby a dummy buyer submits pre-speciﬁed bids on behalf of the

seller; see Ashenfelter (1989).

scheme featuring low, variable sale prices and a high, sticky regular price yields more revenue than

posting a single price. Our secret and random reserve price scheme has similar features, but diﬀers

from their characterization since we consider an environment with multiple, privately-informed

buyers. Nonetheless, we are able to draw a connection between the optimal reserve price and the

optimal monopoly pricing scheme with expectations-based loss-averse buyers that is analogous to

the well-known one for risk-neutral buyers.

In Subsection 4.2 we show that the seller can achieve a higher revenue than what is achievable

with a public reserve price by employing a simple two-stage mechanism. In this mechanism, the

seller ﬁrst runs an auction with a public reserve price; then, if the reserve price is not met, with

some probability, the seller posts a price that would be accepted by those bidders whose types are

below the marginally excluded one. In this way, the seller exposes also the bidder type marginally

excluded from the initial auction to the attachment eﬀect; this, in turn, pushes the marginal type

to bid more aggressively, thereby increasing the overall revenue.

Section 5 concludes the paper by summarizing the results of our model and discussing some

further implications. All proofs are relegated to Appendix A.

2 The Model

In this section, we introduce the auction environment and bidders’ preferences, and provide a

formal deﬁnition of our solution concept (UPE) in the context of sealed-bid auctions.

2.1 Environment

A seller auctions oﬀ an item to N≥2 bidders via a sealed-bid auction. Each bidder i∈

{1,2, ..., N}has a private value tiindependently drawn from the support t, t, with t > t = 0,

according to the same cumulative distribution function F.8We assume that Fis continuously

diﬀerentiable, with strictly positive density fon its support. Further, we impose the standard

assumption that Fhas a monotone hazard rate; i.e, f(x)

1−F(x)is increasing for all x∈t, t. This, in

turn, implies that bidders’ “virtual values” are increasing; i.e., V(ti)≡ti−1−F(ti)

f(ti)is increasing in

ti. The seller has a commonly-known value tS∈[0,¯

t).

We consider two canonical selling mechanisms: the ﬁrst-price sealed-bid auction (FPA) and the

second-price sealed-bid auction (SPA). We restrict attention to symmetric equilibria in increasing

strategies; in such equilibria, the bidder with the highest type wins the auction, conditional on

placing a bid above the reserve price.9Let F1denote the cumulative distribution function of the

highest order statistic among N−1 draws, and denote by f1its corresponding density. Finally,

8We normalize t= 0 to simplify the exposition. Moreover, under this assumption, a seller facing risk-neutral

bidders would always choose a non-trivial reserve price; i.e., there are no corner solutions.

9Throughout the paper, we restrict attention to symmetric (i.e., non discriminatory) auction mechanisms; for a

recent analysis of asymmetric auctions with expectations-based loss-averse bidders, see Muramoto and Sogo (2022).

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NeverSayNever:OptimalExclusionandReservePriceswithExpectations-BasedLoss-AverseBuyers∗BenjaminBalzer†AntonioRosato‡June16,2023AbstractWestudyreservepricesinauctionswithindependentprivatevalueswhenbiddersareexpectations-basedlossaverse.Wefindthattheoptimalpublicreservepriceexcludesfewerbiddertypestha...

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Never Say Never Optimal Exclusion and Reserve Prices with Expectations-Based Loss-Averse Buyers Benjamin BalzerAntonio Rosato

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