Discussion Papers In Economics
And Business

Optimal Mechanism Design with Resale: An Ex-Ante
Price Default Model

Weiye Cheny

Discussion Paper 18-24

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANOptimal Mechanism Design with Resale: An Ex-Ante
Price Default Model

Weiye Cheny

Discussion Paper 18-24

July 2018

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANOptimal Mechanism Design with Resale: An Ex-Ante
Price Default Model∗
Weiye Chen†

Abstract
This paper investigates the optimal ex-ante price mechanism design of sellinga single indivisible object in a market that comprises one public risky buyer and
one regular risky buyer under unlimited or limited liability where resale is allowed.
First, we propose an endogenous liquidation rule requiring that the public buyer
acquires the object in the liquidation stage. Next, we design an optimal bankruptcy
transfer to prevent the buyer’s strategic default. On the basis of this liquidation rule,the optimal ex-ante price mechanism is designed to achieve the seller’s upper boundrevenue under unlimited and limited liability when resale cannot be prohibited prior
to the liquidation stage. Comparing the two mechanisms, the results illustrate thatthe effect on the seller’s behavior and that revenue over the liability and information
change in each case. In other words, (i) when faced with limited liability buyers, theregular buyer will obtain the object in the initial market, whereas they will become
the loser under the unlimited liability case; (ii) the seller’s expected revenue underunlimited liability is weakly higher than that under limited liability; and (iii) when
faced with the risky buyer, the seller prefers the buyer’s resale behavior and is averseto the speculator only under the limited liability case.

JEL classification:D82; G33
Keywords:Mechanism design Bankruptcy Endogenous bankruptcy recovery Resale∗

I am deeply indebted to Ken Urai for his careful guidance and constructive criticism. I am similarly
grateful to Shingo Ishiguro, Kouhei Shiozawa and seminar participants at Osaka University, AMES 2017
at The Chinese University of Hong Kong, CMES 2017 at Wuhan University, and at the JAE Spring
Meeting 2017 at Ritsumekan University.
†
The Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka 5600043, Japan. E-mail: chenweiye198966@yahoo.co.jp

1

Introduction

Consider the situation in which consumers purchase resources through advance reservation. Suppliers will introduce advance reservation while paying attention to the resalemarket and cancellation option , and if the cancellation option is available after theadvance reservation, suppliers could encounter default when the worth is less than the
demander’s expected value. Therefore, suppliers that use advance reservation may prefer to prevent a resale that would, according to traditional perception, raise the default
risk. On the other hand, the presence of post-reservation uncertainty enhances consumers
who rely on the resale market. This is especially the case in the personal retail market
through online shopping websites, which have prospered along with the development of
the Internet. Once a consumer purchases a resource, its value will be observed after thedetails of the event information (e.g., available seats and participants) are revealed, andthis resource will be resold if it is more profitable. Thus, suppliers may also benefit fromthe resale market. Examples of this include the sale of entertainment event tickets, luxury
goods, etc.
Previous studies have investigated the mechanism design for applying resale or default that ensure ex-post efficiency. When resale cannot be prohibited, Zhang and Wang(2013) configure a direct optimal mechanism design and point out that regardless of thebargaining power in the resale market, the seller always benefits more from a buyer whose
relevant information is common knowledge. Even if all the buyers have private information, a similar result is also described in other studies such as Zheng (2002). Haile (2003)
introduces uncertainty in the model when faced with resale opportunities and concludes
that if the information distribution is symmetric, the revenue equivalence may fail due toinformation conveyance and auction format choice in the secondary market. In those standard frameworks, the resale encourages an anticipatory bid from agents and may impose
ex-post inefficiency when the ex-post value is lower than the bid. Once default is allowed,
Board (2007) extend the work from DeMarzo et al. (2005) and propose a mechanism design using the endogenous bankruptcy recovery liquidation rule. The key to the existenceof symmetry equilibrium in Board (2007) is that the public common shock makes bidders
bid truthfully in the initial market. In the case where shocks are asymmetric, bidders may
strategically misreport in order to obtain more profits during the liquidation stage. As aresult, inefficient allocation arises due to both the information asymmetry and benefit of
the liquidation stage.
These suggestions pose three questions to consider: (1) How to design an ex-ante
price direct mechanism including resale and default that guarantees ex-post efficiency inthe case of asymmetric information and uncertainty? (2) Does the seller reveal some
information to buyers via a direct mechanism? If so, what information will be conveyed
and how will the revealed information influence each player’s belief? (3) Under this direct
mechanism, can the seller benefit from resale even if the default is allowed?
As such, in this paper, we study a simple model that enables one risk-neutral seller
to sell an individual object to two risk-neutral buyers while specifically incorporating
resale and default opportunities. We assume that only the ”regular” buyer has private
information about all market participants, while the other is designated as ”public”. Onceany buyer accepts the sales contract offered by the seller, the shock will be revealed, andthen they can offer a take-it-or-leave-it resale contract. After the secondary market isclosed, the winner can strategically declare bankruptcy.
This paper studies sellers’ revenue maximization problem on the basis of buyers’ in-1

centive compatibility and default behavior. In the presence of resale and limited liability,
the seller will face buyers’ private behaviors in three markets: an initial market, a resalemarket, and a liquidation market. After describing the general condition of every buyer
in each market, we consider an optimal mechanism that maximizes the seller’s expected
revenue under unlimited liability and limited liability. In this analysis, the seller faces the
problem of hidden information and hidden action because the regular buyer has private
information and the winner’s action in the resale market is unobservable by the seller.
When the default is allowed, the regular winner also benefits from declaring bankruptcy
because he has private information and there are no penalties in liquidation stage, i.e.,the strategic default occurs. Moreover, the public winner will choose a strategic default
on the basis of the design of the bankruptcy transfer.
To prevent the seller from strategic default, we propose endogenous liquidation recovery as ”resale recovery” following the framework of Board (2007). In other words, thetransaction will be forced to become invalid if any winner declares bankruptcy; however,the seller can resell an object only to the public buyer at the liquidation stage. With
such a rule, bankruptcy transfer can be designed ignoring private information althougha winner can strategically choose to default. This suggestion allows the seller to describe
all liability in detail in an ex-ante direct revelation contract. In addition, the results on
optimal bankruptcy transfer design indicate that neither strategic default nor welfare loss
arise at the liquidation stage.
In Section 4, an optimal ex-ante price mechanism is designed on the basis of an analysis
of buyers’ resale behaviors. We first find that regardless of the level of the buyer’s liability,
the seller can extract more profit allowing for resale while not revealing any information
to buyers. It is no surprise that both parties benefit from resale regardless of the liability:
the resale can either reduce the welfare loss due to uncertainty or decrease the probabilityof default. In our private information environment, the regular buyer can benefit morefrom misreporting during resale. Regarding information disclosure, the seller never reveals
anything to the winner in order to separate the information in the initial market, where apooled recommended resale price offer from the winner is the best response for the seller.
From there, a direct mechanism is designed only from information reports.
When liability is unlimited, the seller will allocate the object to the public buyer inorder to extract the entire surplus in the resale market through the optimal mechanism;
In other words, the risk of uncertainty is translated to the public winner. As far as limited
liability exists, the public winner will get some information rent by offering higher resale
price strategically because either the ex-post efficiency holds or screening resale behavior
is costly. The seller will reduce the surplus gain from the resale market in order to decrease
the information rent paid. Therefore, we address an important issue that has hitherto
been neglected in the literature: the effect of liability and information change on sellers’
behaviors and revenue in each case. If post-resale defaulting is allowed, the seller prefers to
set an ex-ante price to protect at the least one buyer capable of staying solvent. Throughanalyzing the information distribution, we highlight the result that the seller prefers, andhas the ability to set, the ex-ante price in advance to prohibit resale when the resalemarket becomes certain. Intuitively, in the optimal mechanism under unlimited liability,
the seller’s expected revenue is weakly higher than under limited liability. Thus, in somesituations, the seller can prevent the resale between buyers according to ”resale recovery”
and the ex-post allocation will also achieve the Myerson allocation. These foregoing
results illustrate that ”resale recovery” and the optimal ex-ante mechanism under limited
liability can both effectively prevent the public speculator from emerging, also ensuring2

that when facing limited liability buyers, the seller’s revenue does not decrease in somecases. These phenomena have important implications for sellers’ preferences for resale orspeculators in real markets.
Finally, we analyze the relationship between the optimal ex-ante price mechanism andinformation structure and show that our results are robust in more general situations, but
the price may not be unique; for example, some properties of the hazard rate cannot hold.
On the other hand, if the information parameter is relevant, the seller can still achieve
maximum revenue through the optimal ex-ante price mechanism but never needs to beconcerned with the information correlation.

Related LiteratureThis work is associated with the framework of mechanism design with resale, which
encompasses studies by Zhang and Wang (2013), Zheng (2002), Haile (2003), Hafalir andKrishna (2008), Calzolari and Pavan (2006), Garratt and Tröger (2006), Garratt et al.
(2009), and Groes and Tranaes (1999). These studies have tended to indicate the optimalmechanism design with resale by using auction theory, which shows that for optimalmechanism, the seller prefers the first-price auction (FPA) over the second-price auction(SPA).
Zhang and Wang (2013) consider a mechanism design with a bargaining resale market.
They focus on the bargaining game in a resale market that will change the seller’s revenue
between Myerson’s revenue and efficiency revenue. They demonstrate that in optimalmechanism, the public buyer always obtains the object regardless of their bargaining
power in the resale market. Our result for unlimited liability is closely related to thisframework and extends also to more general cases, such as the uncertainty of the resalemarket; in other words, their work describes a special case, whereas we obtain moregeneral results. Our paper extends the work of Haile (2003), which contains uncertain
bidders who have only one dimension of information in the first stage.
Furthermore, the work presented herein is related to Board (2007) and Saral (2009),
both of which concern the optimal auction design to maximize the seller’s revenue. Board(2007) discusses the seller’s preference over the choice of auction with differing forms oflimited liability and designs the optimal auction format for the seller to achieve higher
revenue. In his work, an endogenous liquidation rule is configured in a way the seller
can resell the object over the loser when ex-post auction bankruptcy occurs, the outcome
being that the seller prefers the SPA under this liquidation rule. Saral (2009) analyzes thebidder’s behaviors and the seller’s revenue in the SPA with limited liability. His results
show that the winner’s resale option increases the seller’s revenue and affects the bidding
strategy. Thus, when using the SPA, the seller may prefer the speculator. For further
details on the topic of optimal auction design with limited liability, see DeMarzo et al.
(2005) and Gorbenko and Malenko (2011).
The analysis of strategic default behavior in this paper most closely relates to that in
Alary and Gollier (2004) and Rhodes-Kropf and Viswanathan (2000), both of which study
the relationship between strategic default behavior and the liquidation rule (penalties
payment). For pertinent studies concerning renegotiation, monitoring, and bankruptcy,
see Livshits and Kovrijnykh (2015) and Martimort et al. (2015).
Throughout this paper, by analyzing the surplus gain from each winner, we explain
the seller’s preference for speculators and resale. Cui et al. (2014) propose also a similarargument but on the basis of a stronger assumption that the speculator faces a lower3resale cost. The idea of the resale market in this paper is derived also from the financial intermediaries problem (see Park (2000) and Allen and Gale (2004) who show that
financial intermediaries increase social welfare in incomplete markets).

2

Model

Consider an economy where a seller (”she”) provides one indivisible object to two
risk-neutral buyers (”he”) facing the initial market and the secondary (resale) market.
The buyer whose information is commonly known to the market participants is defined
as a public buyer, namely, P . The regular agent, R, has private information. Thereafter,
the economy will be regarded as the model whereby the seller only faces one public buyer
and one regular buyer. A sales contract will be concluded between the seller and thebuyer (now the ”winner”) in the initial market. Transactions, i.e., the object of trade andpayments, will be then completed at the end of the secondary market. In addition, resaleis permitted between the winner and loser and is modeled as a monopoly problem; inother words, the winner will provide a resale contract to maximize their expected payoff.
We assume that the seller cannot reject the offered sale contract; therefore, the winner isable to resell the object although they will receive it after the resale contract is offered.
Each buyer i ∈ {P, R} has type (signal) θi ∈ [θ, θ̄] supported by an independently
identical distribution (i.i.d) function Fi (θ) and density fi (θ), and a shock si ∈ [s, s̄],which is independent of θi , supported with i.i.d Gi (s) and density gi (s). Assume thatthose distributions are common knowledge. Notably the public buyer’s information willcommonly be observed by all the market participants; thus, the public buyer’s type iscommon knowledge for all the player while his shock will be revealed by the regular buyer
in resale market and by the seller only in liquidation stage. Denote the post using avaluation function as ν : (θi , si ) 7→ R+ , which is twice differentiable, increasing over boththe parameters θ, s, and uniformly bound. After the shock si is revealed, the winner iwill choose to use the object or propose a take-it-or-leave-it resale offer to the loser. The
loser can choose to accept or reject this offer. If any resale price Pi ∈ R+ is provided by
the winner, the resale will occur if and only if ν(θj , sj ) ≥ Pi . In addition, we assume themonotonicity to be as follows:
Assumption 1.
ν(θi , si ) −∂ν(θi , si ) 1 − Fi (θi )>0∂θifi (θi )increases in θi for every si .
Note that we do not need to discuss the effect of information disclosure on buyer beliefs
if the inequality holds.
Taking into account the winner’s resale and default behavior, the seller will design
a mechanism to solve her revenue maximization problem. The timing of the unlimited
liability game proceeds as follows:(1) type θi is naturally observed for each buyer;(2) the seller provides a sales contract to a buyer on the basis of the information report;(3) shock si is revealed by each buyer after the trade occurs;4(4) the secondary (resale) market opens, the winner proposes a take-it-or-leave-it resaleoffer into the resale market, and the loser chooses to accept or not accept it;(5) after the secondary market closes, the winner receives the object, and then makes
the payment while reselling it when the resale offer is accepted.
The timing of limited liability game is the same as the unlimited liability game in periods(1)-(4). In period (5), instead, all the players will enter the liquidation stage when thewinner receives the object but declares bankruptcy; otherwise, the winner will do thesame as in an unlimited liability game. It is notable that once a sales contract is signed,
the seller can only participate again in the liquidation stage.
In the case of unlimited liability, shocks are treated as uncertainty as the seller cannot
commit it during the whole game. However, once the signals are revealed, the emergenceof shock can be predicted. Therefore, those signals allow the seller to convey the resale
price in every situation by designing a direct mechanism. On the other hand, when
any winner declares bankruptcy, the seller can observe the public buyer’s ex-post using
valuation after participation in the liquidation stage. Thus, the seller can provide thewinner direct mechanisms, including the bankruptcy transfer, based only on the publicbuyer’s valuation.
In this game, we assume that the winner will never receive the object in the initial
market but has resale opportunity in the secondary market; thus, the initial market also
can be called reservation market. All the transactions will be completed after secondary
market is closed. Therefore, the ex ante contract can be considered a promised contract,
and the default will become the winner’s strategy. Once the resale contract is accepted,
any buyer will never benefit by breaking it strategically. Because it will reduce eachbuyer’s ex post payoff, but never affect the acceptance of contract in the initial market.
On the basis of this game setting, we focus on perfect Bayesian equilibria in a threestage game, initial market stage, resale market stage and liquidation stage, using backward
induction, in which the unlimited liability case is characterized as a two-stage game without the liquidation stage. Accordingly, to derive the individual rationality and incentive
compatibility (IRi and ICi ) for each buyer in each stage, a direct mechanism will be
designed to maximize the seller’s expected revenue directly where the public buyer’s typeis given as θp∗ ∈ [θ, θ̄]. For simplification, a feasible ex-ante price mechanism is defined asfollows:
Definition 1. A mechanism, as quadruple parameters (ρi , Qi , Pi , Bi ), is designed by theseller to maximize her revenue, which depends on the buyers’ signal reports.
In general, the direct mechanism allows the seller to convey the information as muchas she wants, such as the allocation rule ρi (θp∗ , θr ), the monetary transfers Qi (θp∗ , θr ),the recommended resale price Pi (θp∗ , θr ), and the bankruptcy transfer Bi . Similarly, themechanism, as triple parameters (ρi , Qi , Pi ), will be offered to the buyers with unlimited
liability. As mentioned above, the seller is able to recommend a resale price depending on
the information report and shock prediction, but neither the resale action nor the resale
price choice can be committed. Assume that the buyers report the information (θp∗ , θr )truthfully; in that case, the seller can forecast the winner’s resale behavior against allavailable states of nature as {(θp∗ , θr , sp , sr )}(s ,s )∈[s,s̄]2 . In equilibrium, the seller believes
p rthat the winner will act with the incentive-compatible recommended resale price in eachstate of nature. Thus, the recommended resale price can be specified in an ex-ante direct
mechanism.
5

In the limited liability game, the winner can commonly declare bankruptcy regardlessof his ex-post payoff. If the winner i declares bankruptcy, he should pay the bankruptcy
transfers Bi during the liquidation stage. Additionally, we do not allow any bankruptcy
transfers greater than the social welfare during the liquidation stage. Accordingly, if thewinner stays solvent, their payment is equal to an ex-ante price Qi . The definition of
the liquidation rule and the design of bankruptcy transfers will be described in detail in
Section 3.
Thus, the seller’s optimization problem involves the transfer of two different cases,
one equivalent to the ex-ante price when the winner stays solvent and the other fromliquidation. In Section 3, we will describe the liquidation rule and design the optimalbankruptcy transfers that will facilitate the characterization of the seller’s optimization
problem. In Section 4, we analyze the resale behavior of each winner and design the
optimal mechanism under unlimited and limited liability.

3

Liquidation rule

When a bankruptcy is considered in an economy, the winner will declare bankruptcy
if they can get a higher payoff in an insolvent rather than solvent state. In other words,
strategic default arises in this economy. In this section, we will analyze the strategic
default behavior of each winner in the case of asymmetric information and the optimalbankruptcy transfer Bi will be designed.
Because the seller faces limited liability buyers, we will then address the liquidation
rule of ”resale recovery”, which is used in this paper as follows:(1) if any winner declares bankruptcy, the object will be compulsorily transacted to the
public buyer;(2) otherwise, the seller will receive the promised transfers as Qi .
This ”resale recovery” rule is different from that employed by Board (2007). As in part(1), we assume that the resale contract, if accepted, will be annulled if any winner declares
bankruptcy. Thus, the public buyer will receive the object in the liquidation stage whether
or not he is winner. In other words, the object will be resold if the winner is a regularbuyer; on the other hand, the public winner will hold the object compulsorily. According
to the above definition, this liquidation recovery is close to an endogenous recovery i.e., theseller has the ability to set bankruptcy transfers for each bankrupt without disclosing theregular buyer’s information. Because the public buyer’s information is common knowledge
during the liquidation stage, the seller can extract the entire surplus, but less than thesocial welfare, in the liquidation stage. Therefore, we have the following proposition:
Proposition 1. The bankruptcy transfer is optimal if and only if Bi ≡ {ν(θp∗ , sp )|sp ∈[s, s̄]}.
The bankruptcy transfer can be specified in an ex-ante contract after revealing the
public buyer’s signal even if the seller never receives any information report from theregular buyer. This is because the public buyer’s ex-post valuation in each state can beforecasted on the basis of the commonly known signal θp∗ and, in the liquidation stage,
it will be revealed by the seller. Using such common knowledge, the seller will be able6

to specify all the liability in each liquidation state. Therefore, the seller can offer an exante contract that conveys the bankruptcy payment but is never greater than the publicbuyer’s ex-post valuation.
To clarify the importance and practicability of this liquidation rule and the bankruptcy
transfers, we will explain the following aspects. Under the ”resale recovery” and optimalbankruptcy transfers defined in Proposition 1, the strategic default chosen by any winner
will reduce their profit. Assume that the regular buyer is the initial market winner; inthat case, in accordance with the definition, the regular winner will compulsorily resell
the object when they declare bankruptcy, and thus, the net payoff of regular bankrupt
equals to zero. As a result, there is no incentive to motivate the regular buyer to reject
the contract offered by the seller because his ex-post payoff is indifferent to whether hedeclares bankruptcy or becomes a loser. When the public buyer is the winner on theinitial market, the result is more complex. The seller cannot commit with the resale
behavior of the public buyer and thus a problem of hidden action, private resale behavior,arises. However, due to the liquidation, the resale offer will be mandatorily rejected.
Thus, this rule prevents the public winner from choosing the strategic default; in otherwords, the public winner has no incentive to resell the object but declare bankruptcy.
Furthermore, because the public bankrupt’s utility equals zero in the liquidation stage,
the optimal bankruptcy transfers Bi also effectively prevent any solvent public winner fromchoosing the strategic default. Therefore, the strategic default has never been a weakly
dominated strategy for each winner and welfare loss does not arise during the liquidation
stage. Then, the strategy of declaring bankruptcy will never affect the signal report in
the initial market even with no penalties; hence, a direct mechanism as in Definition 1
can be designed without another information report.
Intuitively, this result shows that the asymmetric information problem will not appear
in the liquidation stage. Therefore, the seller needs to observe neither any buyer’s shock
nor the resale behavior. Through ”resale recovery”, the seller can set an ex-ante pricedirectly depending on thetai . In this way, on the basis of ”resale recovery” and Proposition
1, we can design an ex-ante price direct mechanism to maximize the seller’s expected
revenue, subject to the individual rationality and incentive compatibility constraints ofeach buyer in the initial market.

4

Mechanism design with unlimited and limited liability

In this section, the optimal ex-ante price mechanism will be designed according to the
behavior of each buyer under unlimited and limited liability, which is useful to investigate
the effect of changes in buyer’s liability on the seller’s behavior and revenue. In subsection
4.1, on the basis of each buyer’s incentive compatibility and individual rationality in theunlimited liability case, a directly optimal ex-ante price mechanism will be designed toachieve the upper bound of the seller’s revenue. When facing the limited liability buyer,
the optimal ex-ante price mechanism can still be uniquely designed according to the”resale recovery” even if the resale cannot be prohibited. These results will be describedin subsection 4.2.

7

4.1

Ex-ante price mechanism design with unlimited liability

In the unlimited liability case, the ex-ante price mechanism contains parameters ρi , Qi ,and Pi . In addition, the resale price will maximize the winner’s surplus in the secondary
market, and this resale behavior, which cannot be observed by the seller, affects not only
the allocation rule but also the ex-ante price. Using backward intuition, the directly optimal mechanism will maximize the seller’s surplus in the two-stage game, which depends
on each buyer’s equilibrium path in two markets. In addition, because the regular buyer
has private information, we have an assumption as follows:
Assumption 2. For every interval Θr ∈ [θ, θ̄],EΘr

1 − H(ν|θr )h(ν|θr )is increasing in ν.
Note that H(ν|θi ) is denoted as a conditional probability distribution function and,
therefore, h(ν|θi ) is the density function. Assumption 2 also derives from the frameworkof Myerson (1981, 1983) and is useful for calculating the unique equilibrium path foreach buyer. Based on assumptions 1 and 2, this equilibrium path will be separately
discussed for two situations, i.e., each winner sets the resale price following the seller’s
recommendation if it satisfies incentive compatibility.
First, consider the case where the regular buyer is the initial market winner and offersa take-it-or-leave-it resale contract to the loser at price Pr . Because the public buyer’s
information is common knowledge, the optimal resale price Pr must satisfy the following:(ν(θr , sr ) < ν(θp∗ , sp )ν(θp∗ , sp ).
(1)Pr∗ (sp ) => ν(θp∗ , sp ) otherwise
It can be easily verified that this recommended price satisfies the public buyer’s incentive0
compatibility constraints. In addition, suppose that the regular winner reports θr insteadof the true signal θr . Then, the offer price is unchanged because it depends only on common knowledge for the regular buyer. Therefore, this price is incentive-compatible in theresale market, and the seller will recommend the resale price equal to Pr∗ (sp ). Furthermore, although the seller cannot observe the shock of each buyer, she will believe that thebuyer will accept the recommended resale price if it satisfies the incentive compatibility
constraint in the secondary market. Thus, we obtain the following result:Lemma 1. The resale price offer Pr∗ (sp ) is incentive-compatible if and only if it satisfies:(ν(θp∗ , sp )ν(θr , sr ) < ν(θp∗ , sp )∗Pr (sp ) =.
> ν(θp∗ , sp ) otherwise
Furthermore, the seller will recommend price Pr∗ (sp ) when she receives the signal reportas θr , and the regular winner will follow this recommendation.
Clearly, IRRr is an individual rationality constraint and ICRr is the incentive compatibility constraint for the regular loser in the resale market. Furthermore, because
the public winner follows the recommendation and the seller cannot observe the regular loser’s shock, the ICRr only contains the truthful type report. The final equation,8

0

σ(θr , θr ), shows that the resale occurs if and only if the regular buyer’s ex-post valuation
is greater than the resale price. According to the ICRr , the public winner has no incentive to reject the recommended price and set up an alternative resale price on the basisof the information report from the regular loser in the resale market. Furthermore, theICRr shows that this recommended price must be the global maximum. In that case, theoptimization problem in the resale market can be rewritten as follows:
max Eθr (ν(θp∗ , sp )H(Pp (sp )|θr ) + Pp (sp )[1 − H(Pp (sp )|θr )])Pp (sp )Solving this optimization problem, we have the next result:Lemma 2. The resale price offer Pp∗ (sp ) that satisfies the following:
Z min{s̄,s̄∗r }[ν(θr∗ |ξ) − ν(θp∗ , sp )]fr (θr∗ |ξ)(− (1 − Fr (θr∗ |ξ))dGr (ξ)∂ν(θ∗ |ξ)r

max{s,s∗r }= 1 − Gr (min{s̄, s̄∗r })∂θr∗is incentive-compatible, is the global maximum, and is increasing in sp whereν(θr∗ |ξ) = Pp∗ (sp ). Furthermore, the seller will recommend a price Pp∗ (sp ) regardless of theinformation report she receives and the public winner will follow this recommendation.
Proof. Appendix
Note that the resale occurs if and only if the regular loser’s valuation is greater than
the resale price Pp∗ (sp ). As a result, the public winner has no incentive to reject therecommended price. If min{s̄, s̄∗r } = s̄∗r , the resale price will balance two parts: theinformation rent and the utility of the lowest type regular loser. In other words, if resale
price increases, the information rent will increase while the lowest type regular loser’s
utility will decrease.
On the basis of the resale behavior and price Pp∗ (sp ), Pr∗ (sp ) of each winner on theequilibrium path, the seller will choose ρi , Qi to maximize her expected revenue subject
to the individual rationality IRi and incentive compatibility ICi for all buyers i = p, r
in the initial market. Because the public buyer has no private information, he does nothave an incentive compatibility constraint. Suppose that the public buyer is the winner,
if the seller receives information report as θr , the individual rationality constraint of the
public winner can be written as follows:
Esp (ν(θp∗ , sp )H(Pp (sp )|θr ) + Pp (sp )[1 − H(Pp (sp )|θr )]) − Qp (θp∗ , θr ) ≥ 0 IRIp
As Lemma 1 shows, the public loser’s individual rationality constraint in the resale market
is as follows:ν(θp∗ , sp ) − Pr (sp ) ≥ 0 IRRp0

Next, for the regular buyer, suppose he reports the type θr instead of θr . Then, theex-post payoff of the regular winner is as follows:
Z sr
Z s̄
0
∗
E sp [
ν(θr , sp )dGr (ξ) +ν(θr , ξ)dGr (ξ)] − Qr (θr )ssrwhere sr satisfies ν(θr , sr ) = ν(θp∗ , sp ). For the regular loser, his payoff is as follows:00

Esr [ν(θr , sr ) − Pp (θr |sp )]σ(θr , θr |sp )9

Furthermore, the information report will affect the allocation rule, i.e., the regular buyer’s
expected payoff is calculated as follows:
0

0

0

0

Ur (θr , θr ) = ρp (θr )Esr [ν(θZr , sr ) − Pp (θr |sp )]σ(θrZ, θr |sp )
sr

0

s̄

ν(θr∗ , sp )dGr (ξ) ++ρr (θr )[Esps0ν(θr , ξ)dGr (ξ) − Qr (θr )]srThus, the incentive compatibility constraint of the regular buyer in the initial market canbe written as follows:0

Ur (θr , θr ) ≥ Ur (θr , θr ) ICr
Moreover, in each market, the individual rationality constraint of the regular buyer canbe simplified as follows:
Z sr
Z s̄
∗
Esp [
ν(θr , ξ)dGr (ξ)] − Qr (θr ) ≥ 0 IRIr
ν(θr , sp )dGr (ξ) +ssr

ν(θr , sr ) − Pp (θr |sp ) ≥ 0

IRRr

Therefore, the seller’s optimization problem is subject to individual rationality, incentive compatibility, and the allocation rule of each buyer in each market, which is clarified
by the following equation :max(ρp ,ρr ),(Qp ,Qr )subject toEθr [ρp (θr )Qp (θr ) + ρr (θr )Qr (θr )]IRIp , IRIr , IRRp , IRRr , ICr ,ρp (θr ) + ρr (θr ) ≤ 1

Generally, if there is an upper bound of expected revenue for all feasible mechanisms,
intuitively, an optimal mechanism exists to maximize the seller’s expected revenue. By
this approach, the optimal allocation rule can be directly determined and the optimal
ex-ante price can be derived according to the optimal allocation rule and the equilibrium
path of the winner. On the basis of lemmas 1 and 2, it transpires that the optimalmechanism always exists and is formulated as follows:
Theorem 1. The mechanism that maximizes the seller’s revenue is given as follows:(1) Allocation rule: (ρ∗p , ρ∗r ) = (1, 0);(2) Resale price: (Pp∗ (sp ), Pr∗ (sp )) satisfies lemmas 1 and 2 for every θr ;(3) Ex-ante price for buyer i ∈ {p, r}:Q∗i (θr ) = EθZr ,sp [ν(θp∗ , sp )Gr (min{s∗r (sp ), s̄})s̄∂ν(θr , ξ) 1 − Fr (θr ))dGr (ξ)] − Ur (θ, θ)+(ν(θr , ξ) −∂θrfr (θr )min{s∗r (sp ),s̄}
Proof. Appendix
As the result shows, the public buyer will acquire the object even if ν(θp∗ , sp ) = 0 for all
sp (speculators) and, in the optimal mechanism, the optimal ex-ante price will be uniquelydetermined. This result has further properties as discussed in what follows. First, fromthe optimal resale price and lemmas 1 and 2, we obtain the following result:
10

Corollary 1. In any optimal mechanism identified in Theorem 1, the seller will neverreveal any information of the regular buyer to the resale market, which will reduce her
expected revenue.
From the lemmas 1 and 2, it can be easily verified that the resale price recommended
from the seller to the public buyer satisfies incentive compatibility if and only if it is apooling price. Consider if this recommended price is a separate price; in that case, theregular loser will always report false information to the market in order to obtain thelowest resale price so that price is not incentive-compatible. Furthermore, the regularloser’s type can be conjectured if an inverse one-to-one function of price and type exists.
In this case, the public winner will have an incentive to reject the recommended price but
will require the regular loser to report the shock sr , which may increase the public winner’s
payoff. However, because an ex-ante price mechanism that relies on the recommended
price is designed, according to the optimal allocation rule, this feasible mechanism cannot
achieve the upper bound revenue and, therefore, is not the optimal mechanism. Hence, inthe optimal mechanism satisfying Theorem 1, the optimal recommended resale price forthe public winner must be the pooling price, i.e., the seller never reveals any information
to the public winner.
This corollary shows that in any optimal mechanism satisfying Theorem 1, the seller
does not disclose any additional information to the resale market, and the feasible recommended resale price for the public winner is the pooling price. Then, we have the nextcorollary:
Corollary 2. The optimal ex-ante price in Theorem 1 may be greater than the winner’s
highest ex-post valuation, i.e., the public speculator will certainly be the winner and willachieve the upper bound revenue.
This result has been already discussed in numerous extant frameworks of mechanism
design with resale, such as Zhang and Wang (2013) and Garratt and Tröger (2006). In thebasic model herein, we assume that for any information parameter, the distribution mustbe mutually independent. Therefore, under the optimal ex-ante price mechanism, a lower
type public buyer may face an ex-ante price higher than all of their ex-post valuations.
Moreover, if we suppose that the distribution of the public buyer’s shock is Gp (s) = 1 andassume ν(θ, s) = 0, the optimal ex-ante price mechanism in Theorem 1, the public buyer
with (θ, s) must be the winner, but he is merely a pure speculator.
Through this analysis, we find that even if some post-contract uncertainty exists, theseller always prefers the resale market as a feasible mechanism that will achieve higher
revenue. Moreover, in the optimal mechanism, if the speculator discloses true information
to the market, he will be the winner. As Theorem 1 shows, Zhang and Wang (2013)’s
result, which does not consider the bargaining power in the resale market, can exist asa simple case of our result. In addition, according to a comparison of the revenue rankin each market, the results show that asymmetric information in the initial market has
more of an effect on the seller’s expected revenue. This result was not discussed by Haile(2003) in their exploration of certainty and uncertainty in the resale market.

4.2

Ex-ante price mechanism design under limited liability

Now we characterize the seller’s optimization problem when facing a limited liability
buyer. The optimal mechanism for limited liability will be designed using the ”resale
11

recovery” and a similar analysis as per the unlimited liability case. By comparing the unlimited and limited liability mechanisms, the effects of changes in distribution or liability
on seller preference and behavior are obtained. To solve the optimization problem, we
invoke also a similar assumption to the previous subsection, but stronger:
Assumption 3. For every interval Θr ∈ [θ, θ̄] and for all sp ∈ [s, s̄],EΘr

1 − H(ν|θr )H(ν|θp )h(ν|θr )H(ν|θp ) + H(ν|θr )h(ν|θp )is increasing in ν.
Note that this assumption implies that Assumption 2 and, hence, the argument, which
will be discussed in the proof of Lemma 3, is satisfied as well.
In general, a feasible mechanism in limited liability, which has been defined in Section
2, contains quadruple parameters ρi , Qi , Pi , and Bi . In Section 3, buyers’ default strategies
were analyzed and the unique bankruptcy transfers Bi∗ by ”resale recovery” were outlined.
Therefore, the seller will choose ρi , Qi , and Pi , depending on the individual rationality
and incentive compatibility of each buyer affected by Bi∗ , to maximize her revenue.
Suppose the regular buyer is the initial market winner and receives an ex-ante price Qrfrom the seller. With a similar approach to the unlimited liability case, following Lemma
1, the uniquely optimal resale price satisfies:(ν(θp∗ , s∗p )max{ν(θr , sr ), Qr } < ν(θp∗ , s∗p ).
(2)
Pr (sp |Qr ) => ν(θp∗ , s∗p ) otherwise
If the regular winner’s ex-post valuation is no less than the ex-ante price, i.e., ν(θr , sr ) ≥Qr , then he must stay solvent from Definition 1 and Proposition 1. Therefore, Pr (sp |Qr ) is
the optimal resale price and satisfies incentive compatibility and uniqueness. On the other
hand, if there are no buyers, neither a winner nor a loser, whose ex-post valuation is higher
than the ex-ante price Qr , then the regular winner will truthfully declare bankruptcy. In
this case, no matter what the recommended resale price Pr (sp ) ≤ Qr , the expected revenueof the seller is indifferent. As a result, the seller prefers to recommend the resale price
above the ex-ante price, i.e., Pr (sp ) ≥ Qr . Note that if ν(θr , sr ) < Qr ≤ ν(θp∗ , sp ), thensimilarly under Lemma 1, the incentive-compatible recommended resale price must satisfy
Pr (sp |Qr ) = ν(θp∗ , sp ).
Next, suppose the initial market winner is the public buyer with θp∗ and the ex-ante
price is Qp . If the public winner’s ex-post valuation is higher than Qp , the public winner
will participate in the resale market only when he can benefit from the resale market,
which can be expressed as follows:ν(θp∗ , sp )H(Pp (sp , Qp )|θr ) + Pp (sp |Qp )[1 − H(Pp (sp , Qp )|θr )] − Qp≥ ν(θp∗ , sp ) − Qp .
Because ν(θp∗ , sp ) ≥ Qp implies Pp (sp |Qp ) ≥ Qp , the default never occurs in this case. On
the other hand, the public winner will declare bankruptcy if their ex-post valuation is
below Qp and they cannot resell the object. From Definition 1 and Proposition 1, thepublic bankrupt’s payoff becomes zero. Thus, the individual rationality constraint of the
public winner who faces the lower ex-post valuation in the resale market can be written
as follows:(Pp (sp |Qp ) − Qp )[1 − H(Pp (sp |Qp )|θr )] ≥ 0
12Because 1 − H(Pp (sp |Qp )|θr ) ≥ 0, this equation can be simplified to Pp (sp |Qp ) ≥ Qp . In
that case, the optimization problem for the public winner in the resale market can be
written as follows:max[Pp (sp |Qp (θr )) − Qp (θr )][1 − H(Pp (sp |Qp (θr ))|θr )]subject toPp (sp |Qp (θr )) ≥ Qp (θr )Pp (sp |Qp (θr ))if the seller receives θr . On the basis of the behavior of each winner and the optimal
recommended resale price, the equilibrium path can be uniquely determined and, hence,
the result can be summarized as follows:
Lemma 3.

(1) the resale price offer Pp∗ (sp |Qp (θr )) that satisfies

Z min{s̄,s̄∗r }((ν(θr∗ |ξ) − max{ν(θp∗ , sp ), Qp })fr (θr∗ |ξ)max{s,s∗r }= 1 − Gr (min{s̄, s̄∗r })∂ν(θr∗ |ξ)∂θr∗− [1 − Fr (θr∗ |ξ)])dGr (ξ)

is incentive-compatible where ν(θr∗ |ξ) = Pp∗ (sp |Qp (θr ));(2) The resale price offered, Pr∗ (sp |Qr (θr )), is incentive-compatible if and only if it satisfies the following equation:(max{ν(θr , sr ), Qr } < ν(θp∗ , s∗p )ν(θp∗ , s∗p );Pr∗ (sp , Qr ) => ν(θp∗ , s∗p ) otherwise(3) the seller will recommend the resale price (Pp∗ (sp |Qp (θr )), Pr∗ (sp |Qr (θr ))) to eachbuyer when she receives the signal report as θr and sets the ex-ante price as (Qp (θr ), Qr (θr ));each buyer will follow the recommendation.
Proof. Appendix
This lemma can be directly derived from lemmas 1 and 2 and intuitively shows thatthe optimal resale price is the pooling price and is affected by an ex-ante price. As aresult, the winner may declare bankruptcy only when there is no resale. Lemma 3 thendeduces the equilibrium path of the incentive-compatible resale price and the behavior ofeach winner in a solvent state.
Now, consider that the winner of the initial market is a public buyer with θp∗ , if
the regular buyer reports θr and the seller offers Qp (θr ). In such a case, the individual
rationality constraint of the public winner can be calculated as follows:
Up (θr ) = max{ν(θp∗ , sp ), Qp (θr )}H(Pp∗ (sp |Qp (θr ))|θr )+Pp∗ (sp |Qp (θr ))[1 − H(Pp∗ (sp |Qp (θr ))|θr )] − Qp (θr )
≥ 0.
IRIp
Note: Lemma 3 shows that formulas such asPp∗ (sp |Qp (θr )) ≥ max{ν(θp∗ , sp ), Qp (θr )}Pp∗ (sp |Qp (θr )) = max{ν(θp∗ , sp ), Qp (θr )}hold only if Qp = ν(θ̄, s̄) is established. Therefore, IRIp is binding only if the seller sets
an ex-ante price that prevents any resale possibility. On the other hand, if the public
13

buyer is the loser in the initial market, they will participate in the resale market if andonly if the following holds:
Up (θr , sp ) = ν(θp∗ , sp ) − Pr∗ (sp |Qr (θr )) ≥ 0, IRRp
when the seller offers an ex-ante price Qr (θr ) to the regular winner. Similar to the caseof unlimited liability, not only the public buyer has no private information, but also theseller cannot monitor the public winner’s resale behavior in the solvent state and hence
the public buyer has no incentive compatibility constraints.
Now, suppose that the regular buyer with type thetar is the initial market winner andwill truthfully report. In that case, the individual rationality constraint of the regularwinner in the initial market is as follows:
Ur (θr , θr ) = max{ν(θr , sr ), Qr (θr )}H(Pr∗ (sp |Qr (θr ))|θr )+Pr∗ (sp |Qr (θr ))[1 − H(Pr∗ (sp |Qr (θr ))|θr )] − Qr (θr )
≥ 0.
IRIr
where the ex-ante price is given as Qr (θr ). If the regular buyer is the initial market loser,
based on the result in Lemma 3, his individual rationality constraint in the resale market
can be calculated as follows:
Ur (θr , θr ) = max{ν(θr , sr ) − Pp∗ (sp |Qp (θr )), 0}[1 − H(Pp∗ (sp |Qp (θr ))|θr )]
≥ 0,
IRRr
0

where Qp (θr ) is the ex-ante price for the public winner. If the regular buyer reports θrinstead of θr , his payoff is equivalent to the following:
0

0

0

0

Ur (θr , θr ) = ρp (θr ) max{ν(θr , sr ) − Pp∗ (sp |Qp (θr )), 0}[1 − H(Pp∗ (sp |Qp (θr ))|θr )]000+ρr (θr )[max{ν(θr , sr ), Qr (θr )}H(Pr∗ (sp |Qr (θr ))|θr )
0
0
0
(3)
+Pr∗ (sp |Qr (θr ))(1 − H(Pr∗ (sp |Qr (θr ))|θr )] − Qr (θr )]For simplification, the regular buyer’s incentive compatibility constraint resolves to thefollowing:0

Ur (θr , θr ) ≥ Ur (θr , θr ) ICr
According to each buyer’s individual rationality and incentive compatibility constraint,
the seller’s optimization problem in the limited liability case can be specified as per theunlimited liability case. However, the limited liability and the liquidation rule show thatin insolvent states, the seller’s surplus gains must be equivalent to the public buyer’svaluation. Thus, in the initial market, the seller’s optimization problem is an expectedfunction that comprises two parts-the maximization problem in the solvent state and thebankruptcy state. Using Proposition 1, the seller’s optimization problem is characterized
as follows:maxEθr ,sp [ρp (θr )(max{Qp (θr ), ν(θp∗ , sp )}H(Pp∗ (sp |Qp (θr ))|θr )(ρp ,ρr ),(Qp ,Qr )subject to+Qp (θr )[1 − H(Pp∗ (sp |Qp (θr ))|θr )])0+ρr (θr )(max{Qr (θr ), ν(θp∗ , sp )}H(Pr∗ (sp |Qr (θr ))|θr )0+Qr (θr )[1 − H(Pr∗ (sp |Qr (θr ))|θr )])]
IRIp , IRIr , IRRp , IRRr , ICr ,ρp (θr ) + ρr (θr ) ≤ 1

Using the same logic as Theorem 1 and lemmas 1-3, the optimal ex-ante price mechanism, which will achieve the upper bound revenue, will be designed. The results are
summarized below:
14

Theorem 2. The mechanism that maximizes the seller’s revenue is given as follows:(1) allocation rule: (ρ∗p , ρ∗r ) = (0.1);(2) Liquidation monetary transfers: Bi∗ = ν(θp∗ , sp ) for all i ∈ {p, r}(3) resale price: (Pp∗ (θr |sp ), Pr∗ (sp )) that satisfies Lemma 3 for every θr ;(4) ex-ante price: Q∗p (θr ) = ν(θp∗ , s∗p ) that satisfies
Esr ν̂(θr |sr , sp ) = 1 − Gp (s∗p )Gr (s∗r ) where Pp∗ (Q∗p ) = ν(θ, s∗r ) and

=

ν̂(θ
r)
Rr |s∗ [Q∗ −ν(θ ∗ ,ξ)]f (θ |s ) ∂P ∗
s
r r r
p
p
 p ( p ∂ν(θ∗ − [1 − Fr (θr |sr )])dGp (ξ)|s )Pp∗ ∈ [ν(θr , s), ν(θr , s̄)]0otherwise

s

r r
∂θr

∂Qp

∗∗
(5) ex-ante price: Q∗r = ν(θp∗ , s∗∗p ) = ν(θ, sr ) that satisfies Esr ν̃(θr |sr , sp ) = 1 −∗∗∗∗Gp (sp )Gr (sr ) where

=

ν̃(θ
r)
Rr |s∗∗ [Q∗ −ν(θ ∗ ,ξ)]f (θ |s )
s
r r r
p
 p ( r ∂ν(θ− [1 − Fr (θr |sr )])dGp (ξ)|s )Q∗r ∈ [ν(θr , s), ν(θr , s̄)]0otherwise

s

r r
∂θr

Proof. Appendix
As Theorem 2 shows, if the ex-ante price is equivalent in each winner’s case, thesolvent winner is more likely to appear when the regular buyer is allocated the object in
the initial market. Intuitively, if the probability of the solvent winner occurring is thesame in each case, the optimal ex-ante price is higher when the regular buyer becomes theinitial market winner. This shows that resale reduces the probability of the seller facing
bankruptcy. In addition, the optimal ex-ante price mechanism, defined by Theorem 2,
has more important properties than the result under the unlimited liability case and isable to explain the behaviors of sellers in the real market. These aspects will be described
below.
It can be easily verified that in the optimal ex-ante price mechanism defined by Theorem 2, the result also implies Corollary 1. Intuitively, the reason behind this is that
Lemma 3 is a direct extension of lemmas 1 and 2, and because the result of Corollary 1 isbased on the regular loser’s incentive compatibility constraint in the resale market. Thus,
if the seller reveals any regular buyer’s information to the public winner, although it candecrease the probability of default, the public winner will reject the recommended resale
price and propose a new optimal resale price depending on the additional informationthat will reduce the seller’s revenue. In addition, from Lemma 3, the next property is
intuitive:
Corollary 3. The seller cannot extract the entire surplus from the public winner if resale
and limited liability are allowed simultaneously.

15

In the resale market, because the seller cannot monitor the winner’s resale behavior,
the action of the public winner becomes private under the limited liability case. As aresult, the public winner has more incentive to hide their resale action to gain more
surpluses from the seller. Therefore, the seller cannot extract the entire surplus from thepublic winner through the optimal ex-ante price mechanism. This intuition shows that
including endogenous bankruptcy recovery, the public winner becomes a regular winner
to the seller and, through the unmonitored resale action, acquires some information rent
from the seller. Therefore, even if the final allocation in solvency states achieves theMyerson allocation when the public buyer becomes the winner of the initial market, theseller will need to pay more information rent to the market.
If resale is prohibited in the solvent state, efficient or Myerson allocation may not be
achievable. When resale is allowed, it will be effective in not only improving social welfare
but also reducing the possibility of buyer bankruptcy. Therefore, Myerson allocation canbe achieved only if it satisfies the efficient allocation in solvent states. Further, the next
result will be derived by comparing the seller’s revenue upper bound under the optimalmechanism of unlimited liability and limited liability:
Corollary 4. The upper bound of the seller’s revenue in the optimal mechanism is weakly
higher under unlimited liability compared with that under limited liability.
Proof. Appendix
In general, it is easy to verify that given the ex-ante price Q∗r , defined by Theorem 2,
for every feasible shock sp , the seller’s expected revenue is weakly lower under the limited
liability case. With the expectation of the revenue under all shocks, the upper bound of
the seller’s revenue will also be weakly lower. As confirmed by the proof, the expected
Myerson allocation is achievable in the optimal mechanism defined in Theorem 2 when
the resale market becomes certain because, in this case, the allocation satisfies efficiency
in the solvent state, and social welfare losses are caused only by bankruptcy.
As the preceding properties show, the ex-ante price mechanism by Theorem 2 will
maximize also the seller’s revenue, which is equivalent to the revenue in the case of unlimited liability. As Corollary 4 shows, in such a mechanism, the seller prefers to prohibit
the resale market through the mechanism and the revenue upper bound under unlimited
liability is achievable by using endogenous recovery. Per this fact, the seller is averseto the resale in the case of limited liability. On the other hand, when only the ex-ante
price mechanism is available, the seller prefers the resale market when she faces uncertain
buyers because it will reduce the seller’s loss in the event of the buyer’s bankruptcy. Thisconclusion can be summed up as follows:
Corollary 5. The seller prefers resale that is correlated with the distribution of shock in
the limited liability case but is averse to the public speculator in all situations.
Our conclusions show that the seller prefers to set up an ex-ante price to prohibit theresale market when the resale market becomes certain. Therefore, in such a case, theseller is averse to the speculator whether he is public or regular. When resale cannot be
prohibited, in our results, the regular speculator may emerge but social welfare loss isprevented in solvent states. Moreover, the emergence of the regular speculator will also
reduce the probability of bankruptcy. In addition, the probability of the occurrence ofthe regular speculator is higher when higher θp∗ is observed. However, when the secondary
market becomes certain, the seller will also prevent the emergence of the regular speculator through the optimal ex-ante price mechanism. This conclusion is deduced from16

Corollary 4, i.e., the seller will prefer to set up an ex-ante price that is higher than every
available resale price in order to prohibit the speculator’s occurrence. This important
corollary illustrates the reason why the seller is averse to the buyer’s resale behaviors or
the speculator’s emergence in reality; and without the speculator, regardless of the limited
liability or unlimited responsibility, resale will always enhance the seller’s expect revenue.

5

Discussion and Conclusion

In this paper, a basic model has been posited under a symmetric information range andvaluation function, where an optimal ex-ante price mechanism exists under each liability
situation. Furthermore, optimal mechanism has been uniquely determined on the basis of
assumptions 1-3. In general, these assumptions can be extended to more relaxed contexts.
The first extension is the relaxation of the virtual valuation assumption in assumption 1,
i.e., virtual utility may be negative. Notably, in our framework, only the regular buyer
has virtual valuation and may be less than his valuation. Therefore, the seller will believethat the regular buyer, whose virtual valuation is negative, will never participate in any
market. In addition, in the resale market, the public buyer will believe that only theregular buyer with a positive virtual valuation will accept a resale offer. On the other
hand, although the seller’s belief will be updated, the hazard rate never changes because
the virtual valuation is irrelevant to it. Therefore, according to lemmas 1-3, the conclusion
remains unchanged and optimal mechanism is uniquely determined in each liability caseby assumption of the hazard rate. Next, supposing that assumptions 2 and 3 do not hold,it is easy to verify that the uniqueness of the optimal ex-ante price and resale price may
not hold because the concavity of the maximization function may be satisfied. However,
optimal mechanism is determined on the basis of the upper bound of the seller’s revenue.
Then, the optimal ex-ante price also uniquely exists in the unlimited liability situation
because the revenue’s upper bound is unique. On the other hand, although the allocation
rule in Theorem 2 is uniquely determined, the ex-ante price may not be unique. Moreover,
the result reveals that all the properties contained in each mechanism, as explained, still
hold and that the optimal ex-ante price mechanism can be implemented in a more generalsituation.
The next extension is the relevance of each buyer’s information. In the case of unlimited liability, the seller does not disclose any information to the buyer, which is based on
Lemma 2 and Corollary 1. In other words, the recommended resale price is independentof the information reports but depends on the joint probability distribution. Hence, even
if there is a non-trivial correlation between the buyer’s type and shock, the resale price
uniquely exists depending on Assumption 2. Furthermore, because the upper bound of theseller’s revenue is only related to the virtual valuation of the buyer, the optimal ex-ante
price mechanism will still achieve its upper bound. These arguments show that the resultof Theorem 1 is independent of the correlation of buyer information. Similarly, in the caseof limited liability, when Assumption 3 holds, the resale occurs only in the solvent states
and thus the resale price is also directly depends on the joint probability distribution.
When any winner declares bankruptcy, from Proposition 1, the bankruptcy transfers justdepends on the ex-post valuation of the public buyer. Therefore, the optimal ex-ante
price mechanism uniquely exists even if there is a correlation between buyer information.
This article considered an environment in which the seller is unable to prohibit reselling by unlimited or limited liability buyers. In each situation, the optimal ex-ante
price mechanism is designed to maximize the expected revenue of the seller. In the un17

limited liability case, the optimal ex-ante price mechanism designed uniquely achieves
revenue maximization for the seller. Thus, resale will ensure the efficiency and increase
the seller’s revenue in the optimal mechanism. In the case of limited liability, we propose
an endogenous recovery, i.e., ”resale recovery”, to prevent the seller from the strategic
default. When resale cannot be prohibited in any feasible ex-ante price mechanism, thepublic buyer becomes a private buyer whose surplus cannot be extracted fully by theseller. Through the analysis of each buyer’s incentive-compatible equilibrium path in the
optimal mechanism of Theorem 2, the seller prefers to allocate the object to the regularbuyer to obtain higher revenue while the resale reduces the bankruptcy loss. In addition,these results show that when the resale market tends to be certain, the seller prefers
to prohibit the winner’s resale behavior through endogenous recovery and the ex-ante
price mechanism, and can achieve the same revenue as in the case of unlimited liability.
Through these analyses, this paper shows that when faced with a limited liability buyer,
the seller is averse to the speculator rather than the resale.

Appendix
Proof of Lemma 2
Proof. First, let the recommended resale price sequence defined as{Pp∗ (θr |sp )}θ ∈[θ,θ̄] , and suppose, in this sequence, there are at least two different elements,r

00that is, Pp∗ (θr |sp ) 6= Pp∗ (θr |sp ) for any θr 6= θr . The above inequalities are divided into two00cases, that is, Pp∗ (θr |sp ) > Pp∗ (θr |sp ) and Pp∗ (θr |sp ) < Pp∗ (θr |sp ). Then for the first case,there is following inequality:00[ν(θr , sr ) − Pp∗ (θr |sp )][1 − H(Pp (θr |sp )|θr )]> [ν(θr , sr ) − Pp∗ (θr |sp )][1 − H(Pp (θr |sp )|θr )].
And the second case:00[ν(θr , sr ) − Pp∗ (θr |sp )][1 − H(Pp (θr |sp )|θr )]00> [ν(θr , sr ) − Pp∗ (θr |sp )][1 − H(Pp (θr |sp )|θr )].
It is easy to verify that both inequalities are contrary to the IC condition. Therefore,
the recommended resale price sequence satisfies incentive compatibility if and only if
00Pp∗ (θr |sp ) = Pp∗ (θr |sp ) for any θr 6= θr , that is, the optimal recommended resale price isindependent with θr .
Next, consider the recommended resale price Pp∗ (sp ). Define the inverse functions
as sr = νθ−1
(Pp (sp )) and θr = νs−1
(Pp (sp )) if ν(θr , sr ) = Pp (sp ), then the optimizationrr
problem becomes to:Z ν −1 (Pp (sp ))
Z s̄
θ̄
∗
max
ν(θp , sp )dGr (ξ) +Pp (sp )dGr (ξ)
Pp (sp )νθ−1 (Pp (sp ))s

Z ν −1 (Pp (sp ))θ+νθ̄−1 (Pp (sp ))(ν(θp∗ , sp )Fr (νξ−1 (Pp (sp ))|ξ)+Pp (sp )[1 − Fr (νξ−1 (Pp (sp ))|ξ)])dGr (ξ)
Differentiate this function with respect to Pp , thenν(θp∗ , sp )Gr (s∗r )∂ν(θ̄,s∗r )∂s∗r+Pp∗ (sp )Gr (s̄∗r )∂ν(θ,s̄∗r )
∂s̄∗r

18

−

ν(θp∗ , sp )Gr (s∗r )∂ν(θ̄,s∗r )∂s∗r

Z s̄∗r(+ν(θp∗ , sp )fr (θr |ξ)∂ν(θr ,ξ)∂θrs∗r+[1 − Gr (s̄∗r )] −=−Pp∗ (sp )fr (θr |ξ)Pp∗ (sp )Gr (s̄∗r )∂ν(θr ,ξ)∂θr

+ [1 − Fr (θr |ξ)])dGr (ξ)∂ν(θ,s̄∗r )∂s̄∗r
Z s̄∗r∗ν(θp , sp )fr (θr |ξ) Pp∗ (sp )fr (θr |ξ)(−+ [1 − Fr (θr |ξ)])dGr (ξ)∂ν(θr ,ξ)∂ν(θr ,ξ)
s∗r
∂θr
∂θr
+[1 − Gr (s̄∗r )]

= 0.

(4)

where s∗r = νθ̄−1 (Pp∗ (sp )) and s̄∗r = νθ−1 (Pp∗ (sp )). Rewrite equation (4), thenZ s̄∗r([Pp∗ (sp ) − ν(θp∗ , sp )]fr (θr |ξ)∂ν(θr ,ξ)∂θrs∗r

− [1 − Fr (θr |ξ)])dGr (ξ) = 1 − Gr (s̄∗r )Since Pp∗ (sp ) is independent with θr , then Pp∗ (sp ) is incentive compatible and the global
maximum.
Finally, Athey (2001) shows that a pure strategy Nash equilibrium exists in nondecreasing strategies if Up (sp , Pp (sp )) is log-supermodular in (sp , Pp (sp )). Differentiating
the ICRr with respect to θr and applying the envelope theorem, the regular loser’s payoffin the secondary market becomes to:
Z s−1∗
r (νr )∗Up (sp , νr ) = Eθr [ν(θp∗ , sp )dGr (sr |θr )Z s̄s∂ν(θ, ξ)[ν(θr , ξ) −+]dGr (sr |θr )] − Ur (θ, θ|νr∗ ),−1 ∗∂θrsr (νr )where νr∗ = Pp∗ (sp ) and s−1r (·) : max{ν(θr , s), min{·, ν(θr , s̄)}} 7→ sr . Suppose that these
0
0
∗
∗ 0
inequalities, νr > νr = Pp (sp ) and sp < sp , is hold, then since νr∗ is the global maximum
for sp and given Assumption 1, we have:
Z s−1∗
r (νr )
∂ν(θ, ξ)0∗Up (νr ) − Up (νr ) = Eθr(ν(θp∗ , sp ) − [ν(θr , ξ) −)])dGr (sr |θr )
0
∂θr
s−1
r (νr )0
−Ur (θ, θ|νr∗ ) + Ur (θ, θ|νr )
> 0.
0

0

0

0

On the other hand, Up (sp , νr∗ ) − Up (sp , νr ) must be positive because ν(θp∗ , sp ) < ν(θp∗ , sp ).
000Hence this is contrary to the fact that νr = Pp∗ (sp ) is the global maximum when sp is
p (sp )
observed. Therefore, Pp∗ (sp ) is increasing in sp and ∂P∂s
> 0. Intuitively, Up (sp , Pp∗ (sp ))p
is also increasing in sp .
Using these foregoing results, we can show that Up (sp , Pp (sp )) is log-supermodular in(sp , Pp (sp )). By redefining the payoff function as:
Up (sp , x) − ν(θp∗ , sp ) = Eθr

Z s−1∗r (νr )

[ν(θp∗ , sp ) − ν(θr , ξ) +s̄

∂ν(θ, ξ)]dGr (sr |θr ),∂θrwhere x = ν(θp∗ , sp ) − Pp (sp ). Taking the cross-partial verifies that the following equation:[Up (sp , x) − ν(θp∗ , sp )]19∂ 2 Up∂Up ∂Up−∂x∂sp∂sp ∂x(5)

is negative. First, take the boundary point P̄p (sp ) such that Up (sp , x) − ν(θp∗ , sp ), thenp ∂Upequation (5) equals − ∂U
, which is negative. Therefore, taking the derivative at x
∂sp ∂xin equation (5) when it equals 0, is just needed to confirm this differential equation isnegative. Substituting for Up (sp , x) − ν(θp∗ , sp ) at equation (5) equals to 0, this equation’ssign must equivalent to:∂[Up (sp , x) − ν(θp∗ , sp )] ∂Up ∂ 3 Up
∂ 2 Up ∂ 2 Up
(
−
).
∂sp
∂x ∂ 2 x∂sp ∂x∂sp ∂ 2 x
p
From the definition of hazard rate in Assumption 2, it is easy to verify that ∂U
is log∂x
submodular in (θr , x), and therefore:p3∂ 2 ln ∂U∂ 2 Up ∂ 2 Up∂x sgn ∂Up ∂ Up:=−<0
∂θr ∂x∂x ∂θr ∂ 2 x ∂θr ∂x ∂ 2 xp
Furthermore, since ∂P> 0, then∂sppp∂ 2 ln ∂U∂ 2 ln ∂U
∂x
∂x ∂ν
=
< 0.
∂sp ∂x∂ν∂x ∂sp

Thus, Up (sp , Pp (sp )) is log-supermodular in (sp , Pp (sp )), and, hence, Pp∗ (sp ) is uniquelydetermined by sp .

Proof of Theorem 1
Proof. Differentiate the ICr with respect to θr :∂ν(ι, sr )dUr (θr , ι)
= Esp (ρp (ι)[[1 − H(Pp∗ (sp )|θr )] − Qr (ι)]
dθr∂θr
∂ν(ι, sr )+ρr (ι)[1 − H(Pr∗ (sp )|θr )]).
∂θr
Apply the envelope theorem:Z θr∂ν(ι, sr )Ur (θr , θr ) = Esp([ρp (ι)[1 − H(Pp∗ (sp )|θr )] − Qr (ι)]∂θ
rθ
∂ν(ι, sr )+ρr (ι)[1 − H(Pr∗ (sp )|θr )])Fr (ι)dι + Ur (θ, θ).
∂θr
Therefore, the seller’s optimization problem can be rewritten as follows:
max Eθr ,sp [ρp (θr )(ν(θp∗ , sp )H(Pp∗ (sp )|θr ) + Pp∗ (sp )[1 − H(Pp∗ (sp )|θr )])ρp ,ρr+ρr (θr )[ν(θp∗ , sp ) − Pr∗ (sp )]H(Pr∗ (sp )|θr )+ρp (θr )[ν(θr , sr ) − Pp∗ (sp )][1 − H(Pp∗ (sp )|θr )]+ρr (θr )(Pr∗ (sp )H(Pr∗ (sp )|θr ) + ν(θr , sr )[1 − H(Pr∗ (sp )|θr )])Z θr∂ν(ι, sr )(ρp (ι)[1 − H(Pp∗ (sp )|θr )]−∂θrθ∂ν(ι, sr )+ρr (ι)[1 − H(Pr∗ (sp )|θr )])dFr (ι)] − Ur (θ, θ)∂θr= Eθr ,sp [ρp (θr )(ν(θp∗ , sp )H(Pp∗ (sp )|θr )20∂ν(θ, sr )][1 − H(Pp∗ (sp )|θr )])∂θr+ρr (θr )(ν(θp∗ , sp )H(Pr∗ (sp )|θr )∂ν(θ, sr )+[ν(θr , sr ) −][1 − H(Pr∗ (sp )|θr )])] − Ur (θ, θ)∂θr+[ν(θr , sr ) −Since Pp∗ (sp ) is the global maximum and the virtual utility increases in θr , then for every
sp , there has:∂ν(θ, sr )][1 − H(Pp∗ (sp )|θr )]∂θr∂ν(θ, sr )> ν(θp∗ , sp )H(Pr∗ (sp )|θr ) + [ν(θr , sr ) −][1 − H(Pr∗ (sp )|θr )]∂θr
ν(θp∗ , sp )H(Pp∗ (sp )|θr ) + [ν(θr , sr ) −Hence (ρ∗p (θr ), ρ∗r (θr )) = (1, 0). Finally, since each winner has unlimited liability, theoptimal ex ante price must be equivalent to the upper bound of the seller’s expected
revenue. That is, the unique optimal ex ante price (Q∗p (θr ), Q∗r (θr )) satisfies:
Q∗i (θr ) = EθZr ,sp [ν(θp∗ , sp )Gr (min{s∗r (sp ), s̄})s̄∂ν(θr , ξ) 1 − Fr (θr )+(ν(θr , ξ) −)dGr (ξ)] − Ur (θ, θ).
∂θrfr (θr )min{s∗r (sp ),s̄}
Hence this proof is completed.

Proof of Lemma 3
Proof. Note that the second part of this lemma is intuitive, that is, the optimal resale
price must be higher than the ex ante price and the public loser has no private information.
To prove the part one, let us to divide the optimization problem of the public winner into
two cases. The first case is that, when the public winner’s ex post valuation is higher
than Qp , the optimization problem becames:max ν(θp∗ , sp )H(Pp (sp , Qp )|θr ) + Pp (sp , Qp )[1 − H(Pp (sp , Qp )|θr )].

Pp (sp ,Qp )Equation (1) depends on the fact that Pp∗ (sp , Qp ) ≥ ν(θp∗ , sp ) ≥ Qp . Thus from Lemma 2,
the result is optimal and incentive compatible. The next case isν(θp∗ , sp ) < Qp , then:max (Pp (sp , Qp ) − Qp )[1 − H(Pp (sp , Qp )|θr )].

Pp (sp ,Qp )By Proposition 1, the bankrupt’s payoff is zero, then this optimization problem can be
rewritten as:
max Qp H(Pp (sp , Qp )|θr ) + Pp (sp , Qp )[1 − H(Pp (sp , Qp )|θr )].

Pp (sp ,Qp )and, therefore, from Lemma 2, the optimal recommendation price must satisfy:
Z min{s̄,s̄∗r }(ν(θr∗ |ξ) − max{ν(θp∗ , sp ), Qp })fr (θr∗ |ξ)(− [1 − Fr (θr∗ |ξ)])dGr (ξ)∂ν(θ∗ |ξ)
max{s,s∗r }= 1 − Gr (min{s̄, s̄∗r }).

r

∂θr∗

This proof is completed.
21

Proof of Theorem 2
Proof. Differentiate the equation (3) with respect to θr , and apply the envelope theorem:Z θr∂ν(ι, sr )(ρp (ι)[1 − H(Pp∗ (sp |Qp (ι))|ι)]Ur (θr , θr ) = Esp∂θrθ∂ν(ι, sr )+ρr (ι)[1 − H(Pr∗ (sp |Qr (ι))|ι)]) + Ur (θ, θ)(6)∂θr
Equation (6) shows that the optimal ex ante price for each winner must be the poolingprice.
To show the optimal ex ante price is uniquely determined, it needs to be verified thatthe seller’s expected payoff is log-supermodular in (θp , Qi ). From Proposition 1, the seller’s
payoff is equivalent to the public buyer’s valuation if any winner declares bankruptcy. The
seller’s payoff loss in insolvent states is Eθr ,(sp ,sr ) [Qi − ν(θp , sp )], and the seller’s payoff is:
Us (θp , Qi ) = Qi − Eθr ,(sp ,sr ) [Qi − ν(θp , sp )]DeMarzo et al. (2005) show that an increasing ex ante price function exists if and only if∂2
log Us (θp , Qi )∂θp ∂Qi(7)2i
is negative at theis log-supermodular. Assumption 3 shows that ∂ ∂2 Qi log Us (θp , Qi ) ∂Q∂θp2

point of ∂θp∂∂Qi log Us (θp , Qi ) = 0, that is, equation (7) is log-supermodular using the same
logic in the proof of Lemma 2. Hence, there exists an increasing ex ante price function
depending on θp .
Since the optimal ex ante price is determined uniquely, the price must solve the maximization problem of the seller. Now, suppose that the public buyer is the initial market
winner, then by Proposition 1 and Lemma 3, the seller’s optimization problem can be
rewritten as
Z ν −1 (Pp−1 (Qp ))Z s̄
Z ν −1∗ (Qp )
θp
θ̄
∗
max
[
ν(θp , ι)dGr (ξ) +Qp dGr (ξ)
Qpνθ−1 (Pp−1 (Qp ))ss

Z ν −1 (Pp−1 (Qp ))θ+νθ̄−1 (Pp−1 (Qp ))(ν(θp∗ , ι)Fr (νξ−1 (Pp−1 (Qp ))|ξ)+Qp [1 − Fr (νξ−1 (Pp−1 (Qp ))|ξ)])dGr (ξ)]dGp (ι) +

Z s̄
νθ−1
∗ (Qp )Qp dGp (ι)

pDifferentiate equation (8) with respect to Qp , thenZ s∗p Z s̄∗r
[Q∗p (θr ) − ν(θp∗ , ι)]fr (θr |ξ) ∂Pp∗
− [1 − Fr (θr |ξ)])dGr (ξ)[(∂ν(θr ,ξ)
∂Q∗p
s∗r
s
∂θr
−1 + Gr (s̄∗r )]dGp (ι)= 1 − Gp (Sp∗ )(8)

Similarly, if the regular buyer is the initial market winner, the seller’s optimization problem is:Z ν −1Z ν −1 (Qr )Z s̄∗ (Qr )
θp
θ̄
∗
max
[
ν(θp , ι)dGr (ξ) +Qr dGr (ξ)
Qrsνθ−1 (Qr )s22Z ν −1 (Qr )θ+νθ̄−1 (Qr )(ν(θp∗ , ι)Fr (νξ−1 (Qr |ξ))+Qr [1 − Fr (νξ−1 (Qr |ξ)])dGr (ξ)]dGp (ι) +

Z s̄
νθ−1
∗ (Qr )

Qr dGp (ι)(9)

pDifferentiate equation (9) with respect to Qr , thenZ s̄∗r
Z s∗∗p[Q∗r (θr ) − ν(θp∗ , ι)]fr (θr |ξ)− [1 − Fr (θr |ξ)])dGr (ξ)[(∂ν(θr ,ξ)ss∗r

−1 + Gr (s̄∗r )]dGp (ι)= 1 − Gp (s∗∗
p )

∂θr

(10)

Therefore, there exists a unique optimal ex ante price Q∗i (θr ) if and only if equations (8)and (10) hold.
Finally, to varify the optimal allocation rule, the expected revenue rank of upper boundunder the optimal ex ante price in each case of different winners needs to be confirmed.
Suppose that the optimal ex ante price (Q∗p , Q∗r ) satisfies equations (8) and (10), yields
from the ICr , the seller’s optimization problem becomes:
max Eθr [ρp (θr )(ν(θp∗ , s∗p )H(Pp∗ (sp )|θr )ρp ,ρr∂ν(θ, sr )][1 − H(Pp∗ (sp )|θr )])∂θr+ρr (θr )(ν(θp∗ , s∗p )H(Q∗r |θr )∂ν(θ, sr )][1 − H(Q∗r |θr )])] − Ur (θ, θ)+[ν(θr , sr ) −∂θr+[ν(θr , sr ) −Consider that there is a price Qr such that Qr = Pp∗ > Q∗p . Since Pp∗ > Q∗p , for each
sp , the difference of expected revenue from each winner is equivalent to:Z s̄∗r∂ν(θ, ξ)Eθr [(ν(θp∗ , sp )H(Pp∗ |θr ) + [ν(θr , ξ) −][1 − H(Pp∗ |θr )]
∂θ
∗
r
sr
∂ν(θ, ξ)∗][1 − H(Qr |θr )])dGr (ξ)−ν(θp , sp )H(Qr |θr ) − [ν(θr , ξ) −∂θrZ s̄∂ν(θ, ξ)∂ν(θ, ξ)+(ν(θr , ξ) −− ν(θr , ξ) +)dGr (ξ)
∂θr
∂θr
s̄∗r
+[ν(θp∗ , sp ) − ν(θp∗ , sp )]Gr (s∗r )] − Ur∗ (θ, θ) + Ur (θ, θ)= −Ur∗ (θ, θ) + Ur (θ, θ)≤ 0
The final inequality depends on the fact that Ur∗ (θ, θ) ≥ Ur (θ, θ) since Q∗p < Qr . That is,Ur∗ (θ, θ) = 0 implies Ur (θ, θ) = 0. Since Q∗r is the global maximum, then (ρ∗p , ρ∗r ) = (0, 1),
and this proof is completed.

Proof of Corollary 5
Proof. To shows this corollary, we need to verify that the upper bound of the expected
revenue is lower under the limited liability case and at least, a distribution pair (Gi , gi )exists such that the revenue is equivalent in each optimal mechanism. The first part is
intuitive. The reason is that, under the unlimited liability case, the expected revenue
23

upper bound in each sp must be achieved, but it may not be achievable for every sp in
the limited liability case. To show the second part, now, suppose the emergence of publicbuyer’s shock certainty, that is, there exists s∗p such that(0 sp < s∗p.
Gp (sp ) =1 sp ≥ s∗p
From Theorem 1, the upper bound revenue in the unlimited liability case is equivalentto:Eθr (ν(θp∗ , s∗p )H(Pp∗ (s∗p )|θr )∂ν(θr , sr ) 1 − F (θr )+[ν(θr , sr ) −][1 − H(Pp∗ (s∗p )|θr )]) − Ur (θ, θ).
∂θr
f (θr )

(11)

On the other hand, the upper bound revenue in the limited liability case is equivalentto:Eθr (ν(θp∗ , s∗p )H(Q∗r (s∗p )|θr )∂ν(θr , sr ) 1 − F (θr )][1 − H(Q∗r (s∗p )|θr )]) − Ur (θ, θ).
+[ν(θr , sr ) −∂θrf (θr )

(12)

Therefore, in this case, equations (11) and (12) are equivalent since Q∗r (s∗p ) = Pp∗ (s∗p ).
Hence this proof is completed.

References
Alary, D. and Gollier, C. (2004). Debt contract, strategic default, and optimal penalties with judgement errors. Annals of
Economics and Finance, 5(2):357–372.
Allen, F. and Gale, D. (2004). Financial intermediaries and markets. Econometrica, 72(4):1023–1061.
Athey, S. (2001). Single crossing properties and the existence of pure strategy equilibria in games of incomplete information.
Econometrica, 69(4):861–889.
Board, S. (2007). Bidding into the red: A model of post-auction bankruptcy. The Journal of Finance, 62(6):2695–2723.
Calzolari, G. and Pavan, A. (2006). Monopoly with resale. RAND Journal of Economics, 37(2):362–375.
Cui, Y., Duenyas, I., and zge ahin (2014). Should Event Organizers Prevent Resale of Tickets?
60(9):2160–2179.

Management Science,

DeMarzo, P. M., Kremer, I., and Skrzypacz, A. (2005). Bidding with securities: Auctions and security design. American
Economic Review, 95(4):936–959.
Garratt, R. and Tröger, T. (2006). Speculation in standard auctions with resale. Econometrica, 74(3):753–769.
Garratt, R. J., Tröger, T., and Zheng, C. Z. (2009). Collusion via resale. Econometrica, 77(4):1095–1136.
Gorbenko, A. S. and Malenko, A. (2011). Competition among sellers in securities auctions. American Economic Review,101(5):1806–41.
Groes, E. and Tranaes, T. (1999). On the efficiency of decentralized exchange with resale possibilities. International
Economic Review, 40(2):423–38.
Hafalir, I. and Krishna, V. (2008). Asymmetric auctions with resale. American Economic Review, 98(1):87–112.
Haile, P. A. (2003). Auctions with private uncertainty and resale opportunities. Journal of Economic Theory, 108(1):72–110.
Livshits, I. and Kovrijnykh, N. (2015). Screening as a unified theory of delinquency, renegotiation, and bankruptcy.
Martimort, D., Menezes, F., Wooders, M., MENEZES, F., and RYAN, M. (2015). Default and renegotiation in public-private
partnership auctions. Journal of Public Economic Theory, 17(1):49–77.

24

Myerson, R. B. (1981). Optimal auction design. Mathematics of Operations Research, 6(1):58–73.
Myerson, R. B. (1983). Mechanism design by an informed principal. Econometrica, 51(6):1767–1797.
Park, C. (2000). Monitoring and structure of debt contracts. The Journal of Finance, 55(5):2157–2195.
Rhodes-Kropf, M. and Viswanathan, S. (2000). Corporate reorganizations and non-cash auctions. Journal of Finance,
55(4):1807–1854.
Saral, K. J. (2009). An analysis of market-based and statutory limited liability in second price auctions. (25143).
Zhang, J. and Wang, R. (2013). Optimal mechanism design with resale via bargaining. Journal of Economic Theory,
148(5):2096–2123.
Zheng, C. Z. (2002). Optimal auction with resale. Econometrica, 70(6):2197–2224.

25