GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No. 307

Auctions Versus Negotiations: The Role of Price
Discrimination

Chia-Hui Chen and Junichiro Ishida

May 2013

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Auctions Versus Negotiations: The Role of Price
Discrimination
Chia-Hui Chen∗ and Junichiro Ishida†
May 2, 2013

Abstract
Auctions are a popular and prevalent form of trading mechanism, despite the restriction that the seller cannot price-discriminate among potential buyers. To understand
why this is the case, we consider an auction-like environment in which a seller with anindivisible object negotiates with two asymmetric buyers to determine who obtains theobject and at what price. The trading process resembles the Dutch auction, except thatthe seller is allowed to oﬀer diﬀerent prices to diﬀerent buyers. We show that when theseller can commit to a price path in advance, the optimal outcome can generally be implemented. When the seller lacks such commitment power, however, there instead exists
an equilibrium in which the seller’s expected payoﬀ is driven down to the second-priceauction level. Our analysis suggests that having the discretion to price discriminate is notnecessarily beneﬁcial for the seller, and even harmful under plausible conditions, which
could explain the pervasive use of auctions in practice.

JEL Classiﬁcation Number: D44, D82.
Keywords: Dutch auction, second-price auction, negotiation, commitment, price discrimination, asymmetric buyers.
∗†

Institute of Economics, Academia Sinica. Email: chchen@econ.sinica.edu.tw
Institute of Social and Economic Research, Osaka University. Email: jishida@iser.osaka-u.ac.jp

1

Introduction

Auctions, in various formats, are a very popular and prevalent form of trading mechanism
when a seller has an indivisible object up for sale. At a glance, however, the reason for their
popularity is not so obvious because it is well known that standard auctions fail to realize
the optimal (revenue-maximizing) outcome in many instances. The case in point is whenbuyers are ex ante asymmetric with respect to their valuations for the object: in a case like
this, the seller must bias against “strong buyers” and level the playing ﬁeld to achieve theoptimal outcome (Myerson, 1981).1 Since buyers in most trading environments are in fact
heterogenous in terms of readily observable characteristics, such as age, gender, occupationand so on, it remains to be seen why sellers usually prefer auctions to more ﬂexible forms ofnegotiation where they can apparently retain a greater degree of discretion in price setting,including the ability to price discriminate among potential buyers.
To address this issue, this paper considers a less structured negotiation process in which
a seller with an indivisible object negotiates with two buyers to determine who obtains theobject and at what price. The two buyers are ex ante asymmetric in that their private
valuations for the object are drawn independently from diﬀerent distributions. The selleroﬀers a pair of prices at the beginning and gradually lowers them over time until one of
the buyers accepts or the seller terminates the negotiation without selling the object. The
trading environment resembles the Dutch auction (and in fact encompasses it as a specialcase), except that the seller is allowed to oﬀer diﬀerent prices to diﬀerent buyers at any point
in time. Within this environment, we evaluate the value of price discrimination, i.e., theextent to which the seller can beneﬁt from having the discretion to oﬀer diﬀerent prices todiﬀerent buyers.
We largely obtain two results in this setup. First, we show that the seller can implement
any individually rational and incentive compatible mechanism when she can commit to apair of price sequences, or simply a “price path,” in advance, even though she is restricted tooﬀer weakly descending price sequences (Theorem 1). This result immediately implies thatthe seller can implement Myerson’s optimal outcome with full commitment power, suggestingthat the value of price discrimination cannot be negative in general and is strictly positive in
1

Between two buyers, a buyer is weaker if his value distribution is ﬁrst-order stochastically dominated by
the other buyer’s.

1the face of asymmetric buyers. The optimal outcome necessarily involves price discrimination
between the two buyers, and is not eﬃcient in that the seller might not sell the object to thebuyer with the highest valuation. By carefully tailoring a price path, the seller can improveupon the Dutch auction in which she is required to call out a single price for both of thebuyers.
In a typical negotiation environment, however, it is usually prohibitively costly, or simply
infeasible, for the seller to fully commit to a particular price path in advance. We thus shift
our attention to the case where the seller lacks such commitment power, and all the priceoﬀers must be sequentially rational. We show that, in the case without commitment, there
instead exists an equilibrium whose allocation coincides with that of the second-price auction(Theorem 2). Since the seller’s payoﬀ in the second-price auction is lower than in the Dutch
auction under many plausible circumstances (Vickrey (1961) and Maskin and Riley (2000)),
this result implies that the value of price discrimination can even be negative in the absenceof commitment power. Moreover, we also show that with some reasonable restrictions on
the strategy space and the buyers’ type distributions, this outcome is the unique equilibriumin this environment (Theorem 3). In light of these results, we argue that the seller would
lose very little, and even gain under plausible conditions, by giving up the discretion to pricediscriminate, which could explain the pervasive use of auctions in practice.
To see why the lack of commitment may drive the equilibrium payoﬀ down to the secondprice auction level, recall that when there are two asymmetric buyers, Myerson’s optimalmechanism must be designed in a way that even when the weak buyer’s realized valuation isslightly lower than the strong buyer’s, the object is still allocated to the weak buyer. The
optimal mechanism is, however, vulnerable to the commitment problem because the seller
inevitably gains additional information about the buyers’ valuations from their rejections towhich she cannot resist reacting. The key here is that after each rejection, the buyers become
more and more “symmetric” from the seller’s point of view, which diminishes the seller’s
incentive to favor the weak buyer and forces the seller to deviate from the initially intended
optimal path. Knowing this, the buyers no longer bid as aggressively as they would under
the optimal mechanism. We show that in the limit case where the seller can incorporate new
information and revise the price oﬀers continuously, the equilibrium eventually converges tothe second-price auction outcome in which the buyer with the higher valuation always obtains
the object.
2

Related Literature: The paper is in spirit most closely related to the literature which compares the performances of auctions and other trading mechanisms. Wang (1993) compares
standard auctions with posted-price selling in an environment where buyers arrive stochastically over time and shows that auctions are generally superior absent any auctioning costs.
Manelli and Vincent (1995) consider a sequential oﬀer process in which the order of buyers
with whom the seller negotiates and the prices oﬀered are determined in advance, and theseller receives, at most, one chance to negotiate with each buyer.2 They then ﬁnd that thenegotiation of this form outperforms the second-price auction from the seller’s perspective
under certain conditions.3 Bulow and Klemperer (1996) compare an English auction withno reserve price and an optimally-structured negotiation with one less bidder and show thatthe auction is always preferable under plausible assumptions. Bulow and Klemperer (2009)
consider a sequential negotiation process in which potential buyers in turn decide whether
to enter the bidding and compare this with a standard English auction. They ﬁnd that
although the sequential negotiation is always more eﬃcient, the auction usually generates
higher revenue because it is more conducive to entry. The current paper also addresses asimilar question, comparing an (asymmetric) auction with a particular form of negotiation,
but approaches from a diﬀerent perspective with emphasis on the role of price discrimination.
Since our model describes a dynamic trading process without commitment, it also has
an inherent connection with the literature on durable goods monopoly. There is a critical
diﬀerence, however, between the durable-goods problem and the current setup. In the prototype durable-goods problem, the monopolist has unlimited supply of the good, making it
impossible to stop at the right moment. In contrast, an important feature of our model isthat the seller only has limited supply (one unit) compared to potential demand (two buyers),
which works as a strong commitment device just like in auction settings. Due to this feature,
the seller can have the buyers compete against one another and extract rents from them evenwhen they are inﬁnitely patient as we assume here.
In the sense that the seller has only limited supply, the paper is more closely related to the2

To avoid confusion, we consistently apply the terms as speciﬁed in our model by referring to the informed
party as the buyers and the uninformed party as the seller.
3
In this paper, we show that with independent private value, if the number of chances to negotiate isunlimited and the price path is determined in advance, Myerson’s optimal outcome can always be implemented,
meaning that the negotiation always outperforms the second-price auction. This result thus indicates thatthe key driving force behind their work is the limited number of chances to negotiate in the independent
private-value setting.

3

so-called revenue management problem which examines the optimal pricing strategy when theseller has ﬁnitely many goods to sell before a deadline. While much of the literature assumes
perfect commitment power, several recent works (Horner and Samuelson, 2011; Chen, 2012;
Dilme and Li, 2012) analyze this problem when the seller lacks the ability to commit toany price path in advance. Aside from the fact that our model has no exogenously imposed
deadline, the critical departure from this strand of literature is that potential buyers are ex
ante symmetric in those previous works, so that there is no inherent need to price discriminate
among the buyers.4 In contrast, the two buyers in our setup are ex ante asymmetric, where
their valuations (or their “types”) are possibly drawn from diﬀerent distributions, so that theability to price discriminate is supposed to be highly valuable.
On the more technical side, to formalize the negotiation environment of our interest, we
employ a continuous-time model in which the seller continuously adjusts the prices oﬀered
to the two buyers. The technique adopted to analyze the model is related to the theory of
diﬀerential games originated by Isaacs (1954). Most applications of this technique involve
complete-information games, on issues such as oligopoly games with dynamic prices, R&D
competition, and capital accumulation (e.g., Dockner et al., 2000). Recently, the technique
has also been applied to settings under incomplete information as in ours (see, e.g., Boltonand Harris, 1999; Bergemann and Valimaki, 1997, 2000, 2002; Decamps and Mariotti, 2004).
The remainder of the paper is organized as follows. Section 2 describes the model. Section
3 characterizes the optimal outcome that the seller can achieve when commitment is possible.
Section 4 characterizes the equilibrium without commitment and proves that this equilibriumis unique under some mild conditions. Section 5 concludes the paper. All the proofs arerelegated to Appendix A.

2

The Model

Environment: Time is continuous and extends from zero to inﬁnity. We consider an openended process in which a seller with an indivisible object negotiates with two asymmetricbuyers, denoted by i = 1, 2, to determine who obtains the object and at what price. All partiesare risk neutral and there is no time discounting. Each buyer has a private valuation Xi forthe object, which is drawn from some distribution Fi over the interval Wi ≡ [wi , wi ], wi ≥ 0
4
Dilme and Li (2012) consider a model with two types, high and low, where high-type buyers ﬂow into themarket at a constant rate.

4

with its corresponding density fi . We make the following assumption on the distributionfunctions.
i (x)Assumption 1 For i = 1, 2, ψi (x) ≡ x − 1−Ffi (x) , i.e., the virtual valuation, is strictlyincreasing in x and fi (x) > 0 for all x ∈ Wi .
Note that the buyers are possibly asymmetric in that their valuations might be drawn
from diﬀerent distributions. Letting xi denote the realized valuation and mi the payment tothe seller, buyer i’s payoﬀ is given by zi xi − mi where zi = 1 if buyer i obtains the object andzi = 0 otherwise. The object yields no value to the seller who simply maximizes the expected
payment she collects from the buyers.
Negotiation process: At each instance t, the seller oﬀers a pair of (possibly asymmetric)
prices p(t) ≡ (p1 (t), p2 (t)), where pi (t) denotes the price oﬀered to buyer i: strictly for
expositional purposes, we refer to each pi (t) as a price sequence and to p(t) as a price path.
Oﬀers are made in public, so that each buyer knows not only the price oﬀered to him but
also the price oﬀered to the other buyer. The negotiation process ends either when one of
the buyers accepts or when the seller decides to terminate the process without selling theobject. If both of the buyers accept at the same time, each of them obtains the object withprobability one half. We assume that the seller adjusts the price oﬀers continuously overtime, so that each price sequence pi (t) cannot make discrete jumps.
Assumption 2 The price sequence for each buyer, pi (t), is continuous and weakly decreasing
in t.
The negotiation process described above includes the Dutch auction as a special case, inwhich case p1 (t) = p2 (t) is required for all t.5 Within this environment, we ask whether andto what extent the seller can beneﬁt from this larger degree of discretion in price setting, i.e.,the discretion to price discriminate, both with and without commitment. We say that thevalue of price discrimination is positive (negative) if the seller’s expected payoﬀ when she isallowed to price discriminate increases (decreases) from the payoﬀ level which she can attainby running the Dutch auction.
5

As long as p1 (t) = p2 (t) is satisﬁed for all t, any pair of weakly descending price sequences would replicate
the Dutch-auction outcome.

5

3

Optimal Outcome with Commitment

In order to investigate the role of commitment power in this negotiation environment, we ﬁrst
consider the case, as a benchmark, where the seller can fully commit to any price path before
the negotiation begins. The allocation rule of this trading environment speciﬁes who obtains
the object for each pair of realized valuations (x1 , x2 ). In particular, the allocation rule of any
incentive compatible and individually rational mechanism can be written as x̂2 (x1 ), wherex̂2 (x1 ) indicates the cutoﬀ type of buyer 2 as a function of buyer 1’s valuation, i.e., given x1 ,buyer 2 obtains the object if his valuation exceeds x̂2 (x1 ). Given this formulation, we set up
the the seller’s problem as the one to ﬁnd the optimal pair of (cutoﬀ) buyer type sequences
rather than the one to ﬁnd the pair of price sequences.6 Since the optimal mechanism mayassign zero probability to types in the lower end, we let the domain of x̂2 (x1 ) be [x1 , w1 ],x1 ≥ w1 . Note that in the absence of time discounting, the timing of transaction is payoﬀirrelevant as in most standard auction environments and is hence not included in the allocation
rule. To be more precise, a price path {p̃1 (αt), p̃2 (αt)}, with or without commitment, resultsin the same allocation for any α > 0. Throughout the analysis, therefore, we abstract awayfrom the time dimension.
To derive the optimal path, we ﬁrst establish in Theorem 1 that, with commitment,
the seller can implement any incentive compatible and individually rational mechanism byappropriately tailoring a pair of weakly descending price sequences. The fact that the seller
can implement Myerson’s optimal outcome with full commitment power then directly follows
from this result.
Theorem 1 With commitment, the seller can implement any incentive compatible and individually rational mechanism.
The optimal allocation rule characterized by Myerson (1981) is to allocate the object
to the buyer whose virtual valuation is the greatest and also positive. A direct corollary
from Theorem 1 is that Myerson’s optimal outcome can be implemented with commitment.
Since the optimal outcome cannot be implemented by standard auctions in the presence ofasymmetric buyers, the result implies that the value of price discrimination is strictly positive
when the buyers are ex ante asymmetric.
6
Given the allocation function which pairs the two buyer types, it is relatively straightforward to derive
the associated pair of price sequences which implements this allocation. See the proof of Theorem 1 for detail.

6

Corollary 1 By committing to a speciﬁcally designed price path, the seller can implement
Myerson’s optimal outcome. With full commitment, the value of price discrimination cannot
be negative in general and is strictly positive when the buyers are ex ante asymmetric.
It is fairly straightforward to construct the optimal price path from the allocation functionx̂2 (x1 ). The seller’s task here is simply to design a price path along which the buyer withthe higher virtual valuation always accepts earlier. Without loss of generality, assume that
ψ1 (w1 ) ≥ ψ2 (w2 ). Following the notations deﬁned above, let x1 = ψ1−1 (0) if ψ1 (w1 ) < 0,and x1 = w1 otherwise, so that the virtual valuation of buyer 1 is positive if his valuation
is greater than x1 . Finally, let the function characterizing the allocation rule be x̂2 (x1 ) ≡ψ2−1 (ψ1 (x1 )). Then, buyer 1 with valuation x1 has the same virtual valuation as buyer 2
with valuation x̂2 (x1 ), i.e., ψ1 (x1 ) = ψ2 (x̂2 (x1 )), and given x̂2 (x1 ), by committing to theprice path constructed in the proof of Theorem 1, the seller can implement Myerson’s optimal
outcome.
Theorem 1 suggests that, with full commitment, the seller can accomplish various goals,
including revenue maximization, by carefully designing a pair of weakly descending pricesequences. However, as a growing body of literature on mechanism design emphasizes (see,e.g., McAfee and Vincent, 1997; McAdams and Schwarz, 2007; Skreta, 2006, 2011; Hornerand Samuelson, 2011; Chen, 2012; Vartiainen, 2013), it is often prohibitively costly for theseller to make full commitment in advance. It is hence crucial to see how much the seller cangain from her discretion to oﬀer asymmetric prices when she lacks such commitment powerat her disposal. We dedicate the next section to investigate this issue.

4

Equilibrium without Commitment

4.1

Preliminaries

We now examine the case in which the seller cannot commit to any price path, or any“mechanism,” in advance. The problem is now substantially more complicated because every
price oﬀer must be sequentially rational at any continuation game. In particular, to implement
the optimal outcome, the seller must have an incentive not to deviate from the initiallyintended optimal path after each rejection.
Before we formally deﬁne the equilibrium concept for the non-commitment case, it is
helpful to establish some equilibrium properties which allow us to narrow down the class of7strategies we need to consider. To be more precise, we show that any equilibrium of thisgame has the following properties:• Each buyer’s purchasing decision is characterized by a cutoﬀ strategy; 7• The seller does not terminate the negotiation until the object is sold.
In what follows, we prove these properties in turn.
We ﬁrst show that each buyer’s strategy attains some sort of monotonicity. The ﬁrst
property directly follows from the next lemma and corollary.
Lemma 1 Buyer i’s equilibrium strategy has the property that, given any history, belief, andcurrent price oﬀer, if buyer i with valuation xi accepts, then buyer i with valuation higher
than xi also accepts.
The lemma implies that given any history, a player’s belief about buyer i’s valuation can
be characterized by a cutoﬀ representing the supremum of buyer i’s valuation. Throughout
the analysis, therefore, we generically denote a player’s belief at any continuation game byω ≡ (w1 , w2 ) ∈ W ≡ W1 × W2 where wi is the supremum of buyer i’s valuation. This
property in particular implies that the equilibrium allocation of this non-commitment case
can also be characterized by an allocation function x̂2 (x1 ) which pairs the two buyer types.
We will hence attempt to characterize the seller’s strategy in terms of buyer types rather
than of prices, as we did for the commitment case. Given the set of equilibrium strategies,
it is relatively straightforward to construct the associated price path which implements theequilibrium allocation.
We next show that although we allow the seller to terminate the negotiation at any point,
this option is never exercised in equilibrium when the seller lacks commitment power.
Lemma 2 In equilibrium, the seller never terminates the negotiation process when w1 > w1
and w2 > w2 . Therefore, the object is sold with probability one.
7
In a dynamic bargaining game where a seller makes oﬀers to a buyer who discounts the future and has
private information about his value, the buyer also adopts a cutoﬀ strategy in equilibrium. In that setting,time functions as a screening device since diﬀerent types of buyer value time diﬀerently. In our model, insteadof time, buyers are screened by the fear that they will lose the trade to a competitor, who might accept thecurrent price.

8

The second property suggests that the seller cannot help lowering at least one of the prices
pi to the lower bound of buyer i’s valuation wi , so that the object is sold with probability one.
This draws a clear contrast to the full-commitment case where the seller can set a reserve
price at which she terminates the negotiation without selling the object. More precisely, thisproperty alone suggests that Myerson’s optimal outcome cannot be implemented withoutcommitment if w1 −1−F1 (w1 )1−F2 (w2 )f1 (w1 ) < 0 and w 2 − f2 (w2 ) < 0, in which case the object should beleft unsold with some positive probability in the optimal mechanism.
The equilibrium concept we adopt is the Markov perfect equilibrium, with the posterior
belief ω as the state variable. The two properties mentioned above, along with the notion of
the Markov perfect equilibrium, allow us to characterize the equilibrium in a clearer manner.
The strategies of the game can be deﬁned as follows.
Buyer: Each buyer’s strategy is characterized by a set of functions {Pi,ω (xi )}ω∈W , where
Pi,ω (xi ) indicates the maximum price that buyer i with valuation xi is willing to accept, given
the current belief ω. Deﬁne buyer i’s marginal strategy by Pi (wi , wj ) ≡ Pi,ω (wi ), i.e., themaximum price that the cutoﬀ type is willing to accept. Although Pi (wi , wj ) does not fully
describe buyer i’s strategy, it provides enough information to characterize the equilibriumof this game, due to our focus on continuous strategies in which the seller can adjust theprices only continuously.8 In what follows, therefore, we simply refer to Pi (xi , xj ) as buyer
i’s strategy wherever it is not confusing.
Seller: The seller’s strategy is represented by a set of functions {x2,ω (x1 )}ω∈W , wherex2,ω (x1 ) indicates the pair of buyer types that are induced to accept at the same time,
given the current belief ω.9 Let x1,ω (x2 ) ≡ x−12,ω (x2 ) = inf {x1 ∈ [w 1 , w1 ] | x2,ω (x1 ) ≥ x2 }.

4.2

The Buyer’s Problem

We begin with the buyer’s problem. Taking the seller’s strategy {x2,ω (x1 )}ω∈W as given, tosatisfy buyer 1’s incentive compatibility constraints for all types, the following equation must
Note that along the seller’s equilibrium strategy, buyer i’s strategy, i.e., Pi,ω (x) for x ∈ Wi , can be fully
derived. We will make this point when we solve the buyer’s problem in Section 4.2.
9
It is certainly possible to represent the seller’s strategy by a pair of prices to be oﬀered. An advantage ofour current approach is that the domain of the optimization problem is clearly deﬁned. On the other hand,
if we set up the seller’s problem as the one to choose a price path, the relevant domain is not clearly deﬁned
because there exists a path along which no buyer type would accept.
8

9

hold:
F2 (w2 ) [w1 − P1 (w1 , w2 )] =∫ w1F2 (x2,ω (x)) dx + F2 (x2,ω (w1 )) [w1 − P1 (w1 , x2,ω (w1 ))] . (1)w1Equation (1) comes from the revenue equivalence principle.10 The left-hand side is the expected payoﬀ of buyer 1 with valuation w1 . On the right-hand side,F2 (x2,ω (w1 )) (w1 − P1 (w1 , x2,ω (w1 )))indicates the expected payoﬀ of buyer 1 with the lowest valuation, and∫ w1

w1 F2 (x2,ω (x)) dx isthe additional information rent received by buyer 1 with valuation w1 . If the expected payoﬀ
of buyer 1 with the lowest valuation w1 is zero, (1) is reduced to∫ w1
w F2 (x2,ω (x)) dx.
P1 (w1 , w2 ) = w1 − 1
F2 (w2 )Similarly, P2 (w2 , w1 ) must satisfy∫ w2
F1 (x1,ω (x)) dx + F1 (x1,ω (w2 )) [w2 − P2 (w2 , x1,ω (w2 ))] . (2)F1 (w1 ) [w2 − P2 (w2 , w1 )] =w2

We assume that both P1 and P2 are continuous and weakly increasing in their respective
arguments, so that the seller can implement {x2,ω (x1 )}ω∈W by oﬀering a pair of descending
price sequences.
While we only use the marginal strategies to characterize the equilibrium, it is conceptually straightforward to construct each buyer’s fully speciﬁed strategy, i.e., Pi,ω (xi ) for all xi .
Given the current belief ω, and P1 (x1 , x2,ω (x1 )) and P2 (x2,ω (x1 ) , x1 ) satisfying equations(1) and (2) for all x1 ∈ [w1 , w1 ], we can prove that the following strategy is optimal forbuyer i: buyer i with valuation xi ≥ wi accepts the current price Pi (wi , wj ), and buyer iwith value xi < wi waits and accepts at price Pi (xi , xj,ω (xi )).11 This implies that given thecurrent belief ω, the maximum price that buyer i with valuation xi ≥ wi is willing to accept
is Pi (wi , wj ), and the maximum price that buyer i with valuation xi < wi is willing to accept
10

In our model, given an incentive compatible mechanism, the revenue equivalence principle implies that∫ xi
qi (xi ) xi − mi (xi ) = qi (wi ) wi − mi (wi ) +qi (t) dt,
wiwhere mi (xi ) is the expected payment of buyer i with valuation xi , qi (xi ) is the probability that the buyer
obtains the object, and therefore, qi (xi ) xi − mi (xi ) is the expected payoﬀ for the buyer.
11
To prove this, we can apply the same argument as in the proof of Theorem 1.

10

is xi −Fj (xj,ω (xi ))[xi − Pi (xi , xj,ω (xi ))].12
Fj (wj )Finally, we can show that the buyers’ strategies, derived from (1) and (2), are robust to
the seller’s deviation from the equilibrium strategy. To see this, suppose that the seller deviates and unexpectedly reaches some belief ω ′ = (w1′ , w2′ ). Even in this contingency, since thedeviation is made by a player who does not have any private information, the belief about thebuyers’ valuations cannot be arbitrary and must be consistent with the buyers’ continuation
strategies which maximize their payoﬀs given their expectations about the seller’s continuation strategy, i.e., x2,ω′ (x). Therefore, P1 (x1 , x2,ω′ (x1 )) and P2 (x2,ω′ (x1 ) , x1 ) derived from
(1) and (2) for all x1 ∈ [w1 , w1′ ] continue to be optimal and characterize the two buyers’
strategies for a continuation game starting with any given ω ′ .

4.3

The Seller’s Problem

We now turn to the seller’s problem, which is far more complicated in the absence of commitment power because every price oﬀer must now be sequentially rational given the current
belief. The seller’s problem is deﬁned as choosing a function x2 (x1 ) which maximizes her
expected payoﬀ given the current belief ω and the buyers’ strategies Pi (xi , xj ), i = 1, 2, under the restriction that the function be weakly increasing.13 Fixing the buyers’ strategies,
the principle of optimality then ensures that if some function x2,ω (x1 ) is the optimal path
for a game starting with ω, then for any continuation game starting with (x, x2,ω (x)) wherex ∈ (w1 , w1 ), x2,ω (x1 ) for x1 ≤ x is also the optimal path, so that it gives a sequentiallyrational solution.
More precisely, given the current belief ω and each buyer’s strategy Pi (xi , xj ), the seller’s
optimal strategy x2,ω (x1 ) is obtained as the solution to the following problem:∫ w1
x2,ω (x1 ) = arg
max
P1 (x1 , x2 (x1 )) F2 (x2 (x1 )) f1 (x1 ) dx1x2 (x1 ):[w1 ,w1 ]→[w2 ,w2 ] w
1

∫ w2
+

P2 (x2 , x1 (x2 )) F1 (x1 (x2 )) f2 (x2 ) dx2

(P 1)

w2

s.t.
12

dx2
≥ 0.
dx1

Given the price p = xi −Fj (xj,ω (xi ))Fj (wj )[xi − Pi (xi , xj,ω (xi ))], buyer i with value xi feels indif-

ferent between accepting now and accepting later at price Pi (xi , xj,ω (xi )), i.e., Fj (wj ) (xi − p) =Fj (xj,ω (xi )) [xi − Pi (xi , xj,ω (xi ))].
13
The restriction that the seller’s strategy must be weakly increasing comes from the fact that the buyer
with the higher valuation accepts earlier.

11

The ﬁrst and the second integrals are the seller’s expected payoﬀs received from buyer 1
and buyer 2, respectively. The ﬁrst integrand represents the seller’s payoﬀ increment frombuyer 1 when he decreases x1 by dx1 : the object is sold to buyer 1 at price P1 (x1 , x2 (x1 ))if buyer 2’s value is below x2 (x1 ) and buyer 1’s value is x1 , which occurs with probability
F2 (x2 (x1 )) f1 (x1 ) dx1 . Similar reasoning applies to the second integrand. We derive theseller’s optimal strategy x2,ω (x1 ) from (P1) for all beliefs ω ∈ W , including those oﬀ theequilibrium path. Given the solution to this problem, we can also straightforwardly construct
the actual price path submitted by the seller (see Appendix B for more detail).

4.4

A Markov perfect equilibrium

We are now ready to obtain an equilibrium of this game. The set of strategies (P1 (x1 , x2 ),P2 (x2 , x1 ), {x2,ω (x1 )}ω∈W ) constitutes a Markov perfect equilibrium if:• Taking the seller’s strategy {x2,ω (x1 )}ω∈W as given, P1 (x1 , x2 ) and P2 (x2 , x1 ) solve (1)and (2), respectively, for all (x1 , x2 ) ∈ W .
• Taking the buyers’ strategies as given, x2,ω (x1 ) solves (P1) for all ω ∈ W ;We now establish the next result which constitutes the main contention of the paper.
Theorem 2 When the seller cannot commit to a price path in advance, there always exists
a Markov perfect equilibrium whose allocation is the same as that of the second-price auction.
The theorem states that there always exists an equilibrium in which the seller’s expected
proﬁt is driven down to the second-price auction level. The equilibrium is eﬃcient, but ismore likely to favor the buyers rather than the seller.14 It is now well known that, withasymmetric buyers, the seller’s payoﬀ in the Dutch auction (or the ﬁrst-price auction) is
greater than that in the second-price auction in a range of circumstances, including the casewhere the buyers’ valuations are distributed uniformly (Vickrey, 1961; Maskin and Riley,
2000) . A striking corollary of this result, combined with this conclusion in the literature, is
14
Vartiainen (2013) considers an auction setting in which both the seller and the buyers lack commitment
power and shows that the only implementable mechanism is the English auction because it reveals just theright amount of information to the seller. Both Vartiainen’s and our results seem to suggest that mechanisms
resulting in eﬃcient allocations are generally robust against commitment problems. Although it is only aspeculation and a lot remains to be seen at this stage, it is of some interest to explore the link between
eﬃciency and commitment in other contexts as well.

12that the seller could end up with a lower payoﬀ than in the Dutch auction in the absence ofcommitment power even though her strategy space is strictly larger.
Corollary 2 In the absence of commitment power, the value of price discrimination can be
negative.
To understand the intuition behind this result, consider a simple setting in which thevaluations of the two buyers are uniformly distributed on [0, w1 ] and [0, w2 ], respectively.
Assume w2 > w1 , so that buyer 1 (buyer 2) is the weak (strong) buyer. To implement the
optimal mechanism in this speciﬁcation, the seller must submit prices in a way to pair buyer
2
1 with valuation w1 with buyer 2 with valuation w1 +w. Suppose that the seller starts the2negotiation with the prices which pair buyer 1 with valuation w1 with buyer 2 with valuation
2. If buyer 2 rejects this oﬀer, the seller’s belief about buyer 2 (buyer 2’s highest
w′2 ≡ w1 +w
2

possible value) is then updated to w′2 , while she gains no relevant information about buyer 1
from the rejection, with her belief remaining at w1 . At the next instance, therefore, the seller
faces a diﬀerent problem in which buyer 2’s valuation is distributed on [0, w′2 ], rather thanon [0, w2 ], in which he needs to pair buyer 1 with valuation w1 with buyer 2 with valuation
w1 +w′2.
2

By repeating this process over and over, the situation eventually converges to thesymmetric case where the valuations of both players are distributed on [0, w1 ]. Knowing this,the strong buyer would not bid as aggressively as under the optimal mechanism because there
is virtually no risk of losing the trade to the weak buyer. Theorem 2 shows that this intuition
generally holds in the current trading environment.

4.5

Uniqueness of the equilibrium

In the previous subsection, we established that there always exists an equilibrium whose
allocation coincides with the second-price auction outcome when the seller cannot commit toa price path in advance. This result does not necessarily imply, though, that the seller cannot
do any better than in the second-price auction because we do not rule out other equilibria
as possible outcomes. What is especially critical from the seller’s viewpoint is whether there
exists any other Markov perfect equilibrium in which the seller can beneﬁt from the discretion
to price discriminate.
One way to address this question is to ﬁnd a set of conditions under which the equilibrium
identiﬁed in Theorem 2 is the unique equilibrium outcome of this negotiation process. Al13

though establishing the uniqueness of equilibrium is in general a far more challenging problem
to deal with, and especially so in our inherently dynamic setup with incomplete information,
we can nonetheless establish the uniqueness by imposing two mild (and purely technical)
restrictions, one for the seller and the other for the buyers, on the feasible sets of strategies,
as summarized below.
Assumption 3 (i) Given any belief ω, the seller’s strategy x2,ω (x1 ) is continuous, piecewise()
diﬀerentiable on [w1 , w1 ], and strictly increasing on the interval x1,ω , x1,ω wherex1,ω ≡ sup {x1 ∈ [w1 , w1 ] | x2,ω (x1 ) < w2 } and x1,ω ≡ inf {x1 ∈ [w1 , w1 ] | x2,ω (x1 ) > w2 } ;∫w(ii) The two functions characterizing the buyers’ strategies, C1 (w1 , w2 ) ≡ w 1 F2 (x2,ω (x)) dx
1∫w
and C2 (w1 , w2 ) ≡ w 2 F1 (x1,ω (x)) dx, are continuous, and the ﬁrst derivatives of C1 (x1 , x2 )2and C2 (x1 , x2 ) with respect to x1 and x2 exist.
These restrictions are not at all stringent. The ﬁrst restriction implies that the probability
that a buyer gets the object increases continuously with his valuation. The restriction does
not entail much loss of generality, because this is an equilibrium property presenting itself inmany mechanisms, including the ﬁrst-price and second-price auctions. The second restrictionis also a natural one to impose, as it simply states that the function x2,ω (x) does not change∫wmuch when the belief ω is slightly modiﬁed, so w 1 F2 (x2,ω (x)) dx is continuous in w1 and
1w2 . Figure 1 shows several examples of this class of paths.
[Figure 1 about here]
Without loss of generality, we focus on the equilibria in which buyer 1 with valuation w1
and buyer 2 with valuation w2 get no information rent. We can then obtain the following
result which establishes the uniqueness of the equilibrium.
Theorem 3 Suppose that the two buyers’ valuations are uniformly distributed on [w1 , w1 ]and [w2 , w2 ] respectively. Then, under Assumption 3, the eﬃcient equilibrium is the uniqueequilibrium.
As is well known, with asymmetric buyers, the seller’s expected payoﬀ in the secondprice auction is lower than in the Dutch auction when the buyers’ valuations are uniformly
14

distributed. The theorem thus implies that without commitment, the seller is necessarily
made worse oﬀ in this negotiation environment when she is endowed with the discretion
to price discriminate at will. Conversely speaking, this is where auctions can be especiallyvaluable, as they instantaneously provide credible commitment not to price discriminate,
which renders the buyers bid more aggressively and consequently raises the expected proﬁt.
Corollary 3 Under Assumption 3, the value of price discrimination is strictly negative if
the two buyers’ valuations are uniformly distributed.
We use a simple example to illustrate what the equilibrium price path looks like andhow diﬀerent types of buyer accept along the path. Suppose that the two buyers’ valuations
are uniformly distributed on [0, w1 ] and [0, w2 ], respectively. In this case, under Assumption
3, it is straightforward to show that the following set of strategies constitutes the uniqueequilibrium:{{
w2
x1 , for 0 ≤ x1 ≤ min {w1 , w2 }wi − 2wij , if wj ≥ wi.
x2,ω (x1 ) =and Pi (wi , wj ) =wj
w2, for min {w1 , w2 } < x1 ≤ w1
, if wj < wi2

Figure 2 shows the equilibrium price path (p1 , p2 ) submitted by the seller and the corresponding cutoﬀ path (x1 , x2 ) when w2 > w1 . Note that x2 (x1 ) = x1 for xi ∈ [0, w1 ], so that theequilibrium allocation is eﬃcient.
[Figure 2 about here]

4.6

Why do sellers prefer auctions?

We would like to conclude the analysis by revisiting our motivating question concerning thepervasive use of auctions, i.e., why sellers prefer auctions to more ﬂexible forms of negotiation.
The most distinctive feature of standard auctions arguably lies in their simplicity, with thetrading process characterized by a minimal set of rules which oﬀers both costs and beneﬁts.
An apparent beneﬁt of auctions is that they are easy to implement, and therefore entail low
implementation costs. There is also a drawback, however, because the simplicity of auctions
necessarily restricts the seller’s freedom to pursue strategies to maximize proﬁt. To justify theuse of auctions, therefore, the “beneﬁt of simplicity” must be traded oﬀ against the potential
loss of proﬁt which stems from those restrictions.
The current paper provides a framework to evaluate the potential cost of one such restriction that the seller must oﬀer one price for all buyers and therefore cannot price discriminate15

among them.15 As we have seen, this restriction can be quite costly if the seller is endowedwith the ability to fully commit to a price path in advance. In a more realistic environment
where the seller cannot make any credible commitment, however, the restriction is not costly
after all and can even be beneﬁcial under many plausible circumstances. In particular, under
the conditions obtained in section 4.5, the value of price discrimination must be negative,
meaning that there is no loss on the seller’s part to give up the discretion to price discriminate
even when the beneﬁt of simplicity is negligibly small. We argue that these ﬁndings could
explain the deep-rooted popularity of auctions even in environments where potential buyers
are apparently heterogeneous.

5

Conclusion

This paper studies the environment in which a seller with an indivisible object negotiateswith two asymmetric buyers to determine who obtains the object and at what price. The
seller repeatedly submits price oﬀers to the two buyers until one of them accepts. Unlike theDutch auction, the two prices oﬀered to the two buyers can be diﬀerent. We show that if
the seller can commit to a price path in advance, the payoﬀ realized in Myerson’s optimalmechanism is achievable. However, if commitment is not possible, there instead exists anequilibrium in which the seller’s expected proﬁt is driven down to the second-price auctionlevel. The result suggests that having the discretion to price discriminate is not beneﬁcial
after all, and even harmful in many cases, which could explain the pervasive use of auctionsin practice.
As a ﬁnal note, one important extension of our analysis is to explore whether there is any
Markov perfect equilibrium which implements Myerson’s optimal outcome. Although we do
not know of any counterexample at this point, it is a far more challenging task to formally
prove this. Still, one way to attack this problem is to characterize conditions under which theequilibrium is unique; along with Theorem 2, this necessarily implies that the equilibrium isalways eﬃcient. Theorem 3 thus provides a preliminary answer to this issue, showing thatasymmetric uniform distributions of the buyers’ valuations yield a suﬃcient condition.
15
Note that our methodology and analysis can also be generalized to the n-buyer case. With commitment, we can construct a set of functions {p2 (p1 ) , p2 (p1 ) , · · · , pn (p1 )} specifying the relationship among
the prices to implement Myerson’s optimal outcome. Without commitment, we ﬁnd a set of functions{P1 (x1 , · · · , xn ) , P2 (x1 , · · · , xn ) , · · · , Pn (x1 , · · · , xn )} and {x2,ω (x1 ) , x3,ω (x1 ) , · · · , xn,ω (x1 )}ω to constitute a Markov perfect equilibrium. We believe that our conclusion still holds in an n-buyer case and will
leave such exploration to future research.

16

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Appendix A: the proofs
Proof of Theorem 1: Given an incentive compatible and individually rational mechanism
characterized by x̂2 (x1 ) : [x1 , w1 ] → [w2 , w2 ], let∫ x1
x F2 (x̂2 (x)) dx
for x1 ∈ [x1 , w1 ] ,b1 (x1 ) ≡ x1 − 1
F2 (x̂2 (x1 ))∫ x2
x F1 (x̂1 (x)) dx
for x2 ∈ [x2 , w2 ] ,b2 (x2 ) ≡ x2 − 2
F1 (x̂1 (x2 ))where x2 ≡ x̂2 (x1 ), x̂1 (x) ≡ x̂−12 (x). If Fj (x̂j (xi )) = 0, let∫ xi)(
xi Fj (x̂j (x)) dx
bi (xi ) = xi = lim xi −.
xi ↓xiFj (x̂j (xi ))Note that both b1 (x) and b2 (x) are strictly increasing. Suppose that the seller commits to a
price path on which the price to buyer 1, p1 , and the price to buyer 2, p2 , decrease continuously))( (in a relationship whereby p2 (p1 ) = b2 x̂2 b−1
for p1 ∈ [x1 , b1 (w1 )]. Then if buyer 2
1 (p1 )
with valuation x2 > x̂2 (x1 ) accepts at b2 (x2 ), we show that accepting at b1 (x1 ) is optimal forbuyer 1 with valuation x1 . To see this, note that if the current price for buyer 1 is p1 > b1 (x1 ),the payoﬀ for buyer 1 from accepting now is( )x1 − p1 = x1 − b1 x′1 , where x′1 = b−1
1 (p1 ) > x1 ,
2 (x1 ))
whereas the expected payoﬀ from accepting later at b1 (x1 ) is FF2 (x̂(x1 − b1 (x1 )). Since′
2 (x̂2 (x1 ))∫ x′
1( ( ′ )) (( ′ ))() ( ( ′ ))′F2 (x̂2 (x)) dx
F2 x̂2 x1
x1 − b1 x1= x1 − x1 F2 x̂2 x1 +

∫ x1
<

x1

F2 (x̂2 (x)) dx = F2 (x̂2 (x1 )) (x1 − b1 (x1 )) ,x1

19

buyer 1 would be better oﬀ by waiting and accepting later. If the current price is b1 (x1 ), on
the other hand, the payoﬀ from accepting now is x1 − b1 (x1 ), whereas the expected payoﬀ
F2 (x̂2 (x′′ ))from accepting later at b1 (x′′1 ) is F2 (x̂2 (x11 )) (x1 − b1 (x1 )). Since∫ x′′
1( ( ′′ )) (( ′′ ))() ( ( ′′ ))′′F2 x̂2 x1
x1 − b1 x1= x1 − x1 F2 x̂2 x1 +F2 (x̂2 (x)) dx
∫ x1
<

x1

F2 (x̂2 (x)) dx = F2 (x̂2 (x1 )) (x1 − b1 (x1 )) ,x1

it is strictly better for buyer 1 to accept now at b1 (x1 ). The same argument applies tobuyer 2. This shows that buyer i with valuation xi accepting at price bi (xi ) constitutes anequilibrium.
Given that buyer i with valuation xi accepts bi (xi ), if the two price oﬀers, p1 and p2 , sat( ())−1isfy the relationship p2 (p1 ) = b2 x̂2 b−1
1 (p1 ) , then the type accepting p1 , b1 (p1 ), and the
( −1
)
−1
type accepting p2 , b−1
2 (p2 ), follow the relationship b2 (p2 ) = x̂2 b1 (p1 ) . Therefore, buyer
1 with valuation x1 accepts earlier than buyer 2 with valuation x2 if and only if x2 < x̂2 (x1 ).

Proof of Lemma 1: We show that the buyers adopt cutoﬀ strategies in equilibrium, that is,a buyer with a higher valuation always accepts earlier. To see this, consider a continuation
game starting with price pair (p′1 , p′2 ). Suppose that the belief is such that buyer 1’s andp′p′buyer 2’s values are in sets S1 1 and S2 1 , respectively.16 The buyers expect that the seller
will oﬀer two continuously decreasing price sequences in a relationship whereby p2 (p1 ) for
p1 ≤ p′1 , and S1p1 and S2p1 are the updated beliefs regarding the sets of possible values ofthe buyers, conditional on that no one has accepted any price before the price pair reachesp′(p1 , p2 (p1 )) (note that Sip1 ⊂ Si 1 ).
Now suppose that with price pair (p′1 , p′2 ), buyer i with value xi rejects the oﬀer. Thisimplies that buyer i with value xi obtains a weakly higher payoﬀ if he accepts later with someprice pair (p′′1 , p′′2 ), where p′′2 = p2 (p′′1 ). That is,()
p′′
Pr xj ∈ Sj 1 ()() xi − p′′i ,xi − p′i ≤′pPr xj ∈ Sj 1
where

(
)
p′′
Pr xj ∈Sj 1()
p′
Pr xj ∈Sj 1(3)≤ 1 is the probability that the object is not taken by buyer j that isyet conditional on the hypothesis that buyer i waits until the price pair reaches (p′′1 , p′′2 ). If
16

p′

p′

The belief assigns positive support to all the elements in S1 1 and S2 1 .

20

(
)
p′′
Pr xj ∈Sj 1() < 1, for x′ < xi ,ip′Pr xj ∈Sj 1()
p′′
Pr xj ∈ Sj 1 ()() x′i − p′′i ,x′i − p′i <p′1
Pr xj ∈ Sj
so buyer i with value x′i < xi will also reject the oﬀer p′i .

If

(
)
p′′
Pr xj ∈Sj 1()
p′
Pr xj ∈Sj 1= 1, forthe case where p′′i < p′i , buyer i with value x′i will certainly reject the oﬀer p′i ; forthe case where p′′i = p′i , without loss of generality, we assume that buyer i with value x′i
also rejects the oﬀer p′i . Therefore, buyer i with a higher value accepts earlier in equilibrium.

Proof of Lemma 2: Given that w1 > w1 and w2 > w2 , if the seller terminates thenegotiation, she obtains a payoﬀ of 0; if he continues the game, there must exist a price pathwith which the seller’s expected payoﬀ is greater than 0, unless buyer 1 with valuation in[w1 , w1 ] and buyer 2 with valuation in [w2 , w2 ] only accept at price 0. If that is the case,given that buyer 2 accepts at price 0, buyer 1 must expect that buyer 2’s price will belowered to 0 later than buyer 1’s price, so buyer 1 with a valuation close to w1 is willing to
wait and accept at 0. Likewise, buyer 2 must also expect that buyer 1’s price will be lowered
to 0 later than buyer 2’s price. This is a contradiction. Therefore, the situation where thetwo buyers only accept at price 0 cannot arise in equilibrium, and there always exist pricepaths which generate a positive payoﬀ for any continuation game.

Proof of Theorem 2: We prove that
w2 for w1 ≤ x1 < max {w1 w2 } ,x2,ω (x1 ) = x1 for max {w1 , w2 } ≤ x1 ≤ min {w1 , w2 } ,
w2 for min {w1 , w2 } < x1 ≤ w1
∫ x1
P1 (x1 , x2 ) = x1 −w1 F2 (x2,ω (x)) dx

F2 (x2 ),∫ x2
P2 (x2 , x1 ) = x2 −w2 F1 (x1,ω (x)) dx

F1 (x1 )21,

are a set of functions satisfying the equilibrium conditions.17 Note that x2,ω (·) represents aforty-ﬁve degree line for x1 ∈ [max {w1 , w2 } , min {w1 , w2 }], which means that buyers withthe same valuation in [max {w1 , w2 } , min {w1 , w2 }] will accept at the same time, and thus
implies that the buyer with the higher valuation will accept earlier so that the allocation iseﬃcient and identical to that of the second-price auction.
Given {x2,ω (·)}ω , P1 (x1 , x2 ) and P2 (x2 , x1 ) are derived from (1) and (2), so we only
need to show that given P1 (x1 , x2 ) and P2 (x2 , x1 ), {x2,ω (·)}ω is the seller’s best response.
Without loss of generality, assume that w1 ≥ w2 . First notice that given P1 (x1 , x2 ) andP2 (x2 , x1 ), the price accepted by buyer 2 with valuation x ∈ [w2 , w1 ) is lower than w1 , theprice accepted by buyer 1 with valuation w1 ; in addition, being paired with buyer 2 withvaluation x ∈ [w2 , w1 ) will not raise the prices accepted by buyer 1.18 Therefore, given any
ω, a path x2 (x1 ) going below w1 (i.e., x2 (x) < w1 for some x ∈ [w1 , w1 ]) is dominated andcannot occur in equilibrium.
Next we show that, given P1 (x1 , x2 ) and P2 (x2 , x1 ), all the paths that stay above w1 yield
the same expected payoﬀ to the seller. Let (w1 , w2 ) be the initial belief. First consider thecase where w2 > w1 (the other case can be proved in a similar manner). Given P1 (x1 , x2 ) andP2 (x1 , x2 ), if the seller chooses a function x2 (x1 ) such that x2 (x1 ) ≥ x1 for all x1 ∈ [w1 , w1 ],17

If w2 < w1 , x1,ω (x2 ) = w1 for x2 < w1 . If w2 > w1 , x1,ω (x2 ) = w1 for x2 > w1 . If xj = wj , let∫ xi
w Fj (xj,ω (x))dx.
Pi (xi , xj ) = xi − limxj ↓wj i F xj( j)18Given any x1 , P1 (x1 , x) ≤ P1 (x1 , y) for y ≥ w1 .

22

i.e., x2 (x1 ) is above the forty-ﬁve degree line, then the expected payoﬀ is
[
]
∫
∫
w1

w1

∫ w1 [+

x1 F1 (x1 ) −w1+ w1 F1 (w1 ) −=F2 (x) dx f1 (x1 ) dx1
w1

[

∫ w1

x1

x1 F2 (x2 (x1 )) −

]

∫ x1

∫ w1

F1 (x) dx f2 (x2 (x1 ))
w1]

dx2 (x1 )dx1
dx1

F1 (x) dx [F2 (w2 ) − F2 (x2 (w1 ))]w1{x1 [F2 (x2 (x1 )) − F2 (x1 )] + w1 F2 (w1 )} dF1 (x1 ) +w1∫ w1 ∫ x1
xdF2 (x) dF1 (x1 )w1∫ x2 (w1 ) ∫ x−1 (x2 )

w1

2

+

xdF1 (x) dF2 (x2 )w∫ w1 1+

w1

xdF1 (x) [F2 (w2 ) − F2 (x2 (w1 ))]w∫ w1 1∫ x1=∫ w1
xdF2 (x) dF1 (x1 ) +

w1

w1

∫ w1

+

{x1 [F2 (x2 (x1 )) − F2 (x1 )] + w1 F2 (w1 )} dF1 (x1 )w1

x1 [F2 (x2 (w1 )) − F2 (x2 (x1 ))] dF1 (x1 )w∫ w1 1x1 [F2 (w2 ) − F2 (x2 (w1 ))] dF1 (x1 )w1∫ w1 ∫ x1
∫ w1

+
=

{x1 [F2 (w2 ) − F2 (x1 )] + w1 F2 (w1 )} dF1 (x1 ) ,xdF2 (x) dF1 (x1 ) +
w1

w1

w1

which is independent of x2 (x1 ).
The ﬁrst equation comes from integration by
∫ x1
∫ x1
parts:The second equation
w1 Fi (x) dx = Fi (x1 ) x1 − Fi (w 1 ) w 1 − w1 xdFi (x).
∫ x2 (w1 ) ∫ x−1 (x2 )2
xdF1 (x) dF2 (x2 ) =comes from changing the order of integration:
w1
w1∫ w1 ∫ x2 (w1 )wx2 (x) xdF2 (x2 ) dF1 (x). Therefore, the seller obtains the same expected payoﬀ for all
1

x2 (x1 ) such that x2 (x1 ) ≥ x1 . Similarly, we can prove that the seller receives the same
expected payoﬀ for all x2 (x1 ) such that x2 (x1 ) ≤ x1 , and can further show that the seller
obtains the same expected payoﬀ for any path x2 (x1 ).19 Therefore, P1 (x1 , x2 ), P2 (x2 , x1 ),and {x2,ω (x1 )}ω constitute a Markov perfect equilibrium in which the buyer with thehigher valuation obtains the object, and a buyer whose valuation is smaller than or equal tomax {w1 , w2 } receives zero payoﬀ.
19

Note that although the seller receives the same payoﬀ for all x2 (x1 ), this does not mean that all of thosefunctions can be sustained in equilibrium. If the seller switches to another function x̃2 (x1 ), P1 (·) and P2 (·),
which characterize the buyers’ strategies, will change accordingly. Then x̃2 (x1 ) might no longer be optimalfor the seller.

23

Proof of Theorem 3: We prove the uniqueness of the equilibrium under the two restrictions in Assumption 3. The ﬁrst restriction is on the seller’s side and states that, given
ω, the path x2,ω (x1 ) is continuous, piecewise diﬀerentiable on [w1 , w1 ], and strictly in()
creasing on the interval x1,ω , x1,ω where x1,ω ≡ sup {x1 ∈ [w1 , w1 ] | x2,ω (x1 ) < w2 } andx1,ω ≡ inf {x1 ∈ [w1 , w1 ] | x2,ω (x1 ) > w2 }.20 Under this restriction, program (P1) can be
rewritten as∫ x1x2,ω (x1 ) = arg max

x2 (x1 ) w
1

P1 (x1 , x2 (x1 )) F2 (x2 (x1 )) f1 (x1 )+ P2 (x2 (x1 ) , x1 ) F1 (x1 ) f2 (x2 (x1 )) u (x1 ) dx1+ 1(x1 <w1 ) P1 (x1 , x2 (x1 )) F2 (x2 (x1 )) [F1 (w1 ) − F1 (x1 )]+ 1(x2 (x1 )<w2 ) P2 (x2 (x1 ) , x1 ) F1 (x1 ) [F2 (w2 ) − F2 (x2 (x1 ))]
s.t.

(P 2)

either x1 = w1 and x2 (w1 ) ≤ w2 , or x1 ≤ w1 and x2 (x1 ) = w2 ,x2 (w1 ) ≥ w2 ,dx2 (x1 )= u (x1 ) ,dx1
dx2 (x1 )< ∞ for x1 ∈ (x1 , x1 ) ,0<dx1

where 1(·) is the indicator function and x1 ≡ sup {x1 ∈ [w1 , w1 ] | x2 (x1 ) < w2 }. When x1 =w1 and x2 (w1 ) < w2 , it means that buyer 2 with valuation between x2 (w1 ) and w2 obtains
the object with the same probability, so in equilibrium all those types accept the same priceP2 (x2 (x1 ) , x1 ); this event occurs with probability F1 (x1 ) [F2 (w2 ) − F2 (x2 (x1 ))] and yields
the payoﬀ shown in the forth term of the objective function. Similarly, when x1 < w1 andx2 (x1 ) = w2 , buyer 1 with valuation between x1 and w1 accepts price P1 (x1 , x2 (x1 )); thisoccurs with probability F2 (x2 (x1 )) [F1 (w1 ) − F1 (x1 )] and yields the payoﬀ shown in thethird term.
We also need to verify that P1 (x1 , x2 ) and P2 (x2 , x1 ) satisfy (1) and (2), respectively.
Since the expected payoﬀs of the lowest types of the two buyers are zero, by (1) and (2),Ci (w1 , w2 ),Fj (wj )(4)

Fj (xj,ω (x)) dx.

(5)

Pi (wi , wj ) = wi −where
Ci (w1 , w2 ) ≡

∫ wi
wi

20

This class of paths satisﬁes the equilibrium property described in Lemma 2.

24

Here, we impose an additional restriction on the buyers’ side that C1 (x1 , x2 ) and C2 (x1 , x2 )are continuous, and the ﬁrst derivatives of C1 (x1 , x2 ) and C2 (x1 , x2 ) with respect to x1 andx2 exist.21
To prove the uniqueness of the equilibrium, we ﬁrst use Pontryagin’s maximum principle
to derive necessary conditions for the seller’s equilibrium strategy. Along with (4), we showthat given the strategy spaces described in Assumption 3, there is only one set of strategies
and beliefs satisfying all the conditions and the requirements.
Consider the seller’s optimal control problem in (P2). Deﬁne the initial value function as{[x1 F2 (x2 (x1 )) − C1 (x1 , x2 (x1 ))] [F1 (w1 ) − F1 (x1 )]if x1 < w1 , x2 (x1 ) = w2 ,I (x1 , x2 (x1 )) =[x2 (x1 ) F1 (x1 ) − C2 (x1 , x2 (x1 ))] [F2 (w2 ) − F2 (x2 (x1 ))] if x1 = w1 , x2 (x1 ) < w2 ,and the Hamiltonian function H as
H (x1 , x2 , u, λ) = G (x1 , x2 , u) + λg (x1 , x2 , u) ,where
G (x1 , x2 , u) = [x1 F2 (x2 ) − C1 (x1 , x2 )] f1 (x1 ) + [x2 F1 (x1 ) − C2 (x1 , x2 )] f2 (x2 ) u,g (x1 , x2 , u) = u.
The following theorem is a restatement of Pontryagin’s maximum principle:
Theorem 4 If a control u (·) with a corresponding state trajectory x2 (·) is optimal, there
exists an absolutely continuous function λ : [w1 , w1 ] 7→ R such that the maximum condition
H (x1 , x2 (x1 ) , u (x1 ) , λ (x1 )) = max {H (x1 , x2 (x1 ) , u, λ (x1 )) | 0 ≤ u ≤ ∞} ,the adjoint equation
λ′ (x1 ) = −∂H (x1 , x2 (x1 ) , u (x1 ) , λ (x1 ))
,
∂x2

21

The optimal control theory only requires that the integrand of the objective function in program (P2) becontinuous and that the ﬁrst derivative of the integrand with respect to x2 exist. However, in our problem,
instead of solving for x2,ω (x1 ), we can change the formulation to solve for x1,ω (x2 ). Therefore, we requirethat the ﬁrst derivatives with respect to both x1 and x2 exist.

25

and the transversality conditionsλ (w1 ) ≥ 0;
x2 (w1 ) ≥ w2 ;(x2 (w1 ) − w2 ) λ (w1 ) = 0;
∂I
;
∂x2
∂I
= 0.
< w1 and x2 (x1 ) = w2 , H (x1 ) +∂x1

if x1 = w1 and x2 (w1 ) < w2 , λ (w1 ) =

(6)

if x1

(7)

are satisﬁed.
For later convenience, let c1 (x1 , x2 )=(w2 − w2 ) C1 (x1 , x2 ) and c2 (x1 , x2 )=(w1 − w1 ) C2 (x1 , x2 ). Given this result, we now prove the theorem through a series of lemmas, from lemma 3 to 10.
Lemma 3 c1 (x1 , x2 ) = (x1 − w1 ) (x2 − w2 ) − c2 (x1 , x2 ) .
Proof: By (5),∫ x1,ω
C1 (w1 , w2 ) =F2 (x2,ω (x)) dx + F2 (x2,ω (x1,ω )) [w1 − x1,ω ] ,

w1

and

∫ x2,ω (x1,ω )C2 (w1 , w2 ) =F1 (x1,ω (x)) dx + F1 (x1,ω ) [w2 − x2,ω (x1,ω )] .

w2

Multiplying both sides of the two equations by (w2 − w2 ) and (w1 − w1 ) respectively, we
obtain∫ x1,ω
c1 (w1 , w2 ) =(x2,ω (x) − w2 ) dx + (x2,ω (x1,ω ) − w2 ) [w1 − x1,ω ] ,

(8)

w1

∫ x2,ω (x1,ω )c2 (w1 , w2 ) =(x1,ω (x) − w1 ) dx + (x1,ω − w1 ) [w2 − x2,ω (x1,ω )] .

(9)

w2

Note that either x1,ω = w1 or x2,ω (x1,ω ) = w2 . Therefore, c1 (w1 , w2 ) + c2 (w1 , w2 ) =(w1 − w1 ) (w2 − w2 ).

Lemma 4 ci (w1 , x2 ) = 0 and ci (x1 , w2 ) = 0.

26

Proof: By Lemma 3, c1 (x1 , w2 ) + c2 (x1 , w2 ) = 0. Since ci (·) ≥ 0, c1 (x1 , w2 ) = 0 andc2 (x1 , w2 ) = 0. Similarly, c1 (w1 , x2 ) = 0 and c2 (w1 , x2 ) = 0.

Lemma 5 Let x2,ω (x1 ) be the equilibrium path of the continuation game starting with belief
ω. Then(x1 − w1 ) w2 + c1 (x1 , x2,ω (x1 )) −

∫ x1
w1

x−

∂c1 (x, x2,ω (x)) ∂c1 (x, x2,ω (x)) dx2,ω (x)+dx = 0,
∂x2
∂x2
dx1
(10)

for all x1 ∈ [w1 , x1,ω ].
Proof: By the maximum condition,u ∈ (0, ∞) if∂H= [x2 F1 (x1 ) − C2 (x1 , x2 )] f2 (x2 ) + λ = 0.
∂u

(11)

By the adjoint equation,
{[
]
[
]
}
∂C1
∂C2
′
λ (x1 ) = − x1 f2 (x2,ω (x1 )) −f1 (x1 ) + F1 (x1 ) −f2 (x2,ω (x1 )) u ,
∂x2
∂x2
]
[
]
∫ x1 [∂C1∂C2
so λ (x1 ) = λ (w1 ) − w xf2 (x2,ω (x)) − ∂x2 f1 (x) + F1 (x) − ∂x2 f2 (x2,ω (x)) u (x) dx.
1

Plugging into equation (11),−[x2,ω (x1 ) F1 (x1 ) − C2 (x1 , x2,ω (x1 ))] f2 (x2,ω (x1 )) + λ (w1 )][]∫ x1 [∂C1∂C2
xf2 (x2,ω (x)) −f1 (x) + F1 (x) −f2 (x2,ω (x)) u (x) dx = 0,
∂x2
∂x2
w1

which implies that λ (w1 ) = 0.22

Multiplying both sides by (w1 − w1 ) (w2 − w2 ) =1
f1 (x1 )f2 (x2,ω (x1 )) ,] []∫ x1 [∂c1∂c2[(x1 − w1 ) x2,ω (x1 ) − c2 (x1 , x2,ω (x1 ))] −x−+ (x − w1 ) −u (x) dx = 0.
∂x2
∂x2
w1
∫x
∂c1 dx2,ω
∂c1+ ∂xdx1 = 0 for
By Lemma 3, we obtain (x1 − w1 ) w2 + c1 (x1 , x2,ω (x1 )) − w 1 x1 − ∂x
2
2 dx1
1

all x1 ∈ [w1 , x1,ω ].

Lemma 6 Let x2,ω (x1 ) be the equilibrium path of the continuation game starting with belief
ω. Then,∂c1 (x1 , x2,ω (x1 )) ∂c1 (x1 , x2,ω (x1 ))+= x1 − w2 ,∂x1
∂x2
for all x1 ∈ [w1 , x1,ω ].
22

When x1 = w1 , the equation holds only if λ (w1 ) = 0.

27

(12)

Proof: Since equation (10) holds for all x1 ∈ [w1 , x1,ω ], by taking derivatives with respect tox1 on both sides, we obtain
∂c1
∂c1
∂c1
∂c1
+
u = (x1 − w2 ) −+u.
∂x1 ∂x2∂x2 ∂x2
Therefore,∂c1 (x1 ,x2,ω (x1 ))∂c (x ,x (x1 ))+ 1 1∂x2,ω
= x1 − w2 .
∂x1
2

Let A = {(x1 , x2,ω (x1 )) | x1 ∈ [w1 , x1,ω ] , w1 ≤ w1 ≤ w1 , w2 ≤ w2 ≤ w2 } , and A is theclosure of A. A contains all the points on the equilibrium paths of all the continuation
games. Note that given ω = (x, x) where x ∈ [max {w1 , w2 } , min {w1 , w2 }],{w2 , for x1 ∈ [w1 , max {w1 , w2 }) ,x2,ω (x1 ) =x1 , for max {w1 , w2 } ≤ x1 ≤ min {w1 , w2 } ,so (x, x) ∈ A for x ∈ [max {w1 , w2 } , min {w1 , w2 }].
Lemma 7 For all (x1 , x2 ) ∈ A, c1 (x1 , x2 ) is of the form

(x1 −w2 )2+ ϕ (x2 − x1 ).
2

Proof: If (x1 , x2 ) is on the equilibrium path of a continuation game so that it is in A, thenthe diﬀerential equation (12) has to hold. The general solution to the diﬀerential equation isc1 (x1 , x2 ) =(x1 −w2 )2+ ϕ (x2 − x1 ).
2

{
}
{
}
Let d = max x2 − x1 | (x1 , x2 ) ∈ A and d = min x2 − x1 | (x1 , x2 ) ∈ A .
Lemma 8 c1 (x1 , x2 ) =(x1 −w2 )2(max{w1 −w2 ,0})2−2
2

Proof: For (x1 , x2 ) ∈ A, c1 (x1 , x2 ) =that x2 ≥ x1 .

for x2 ≥ x1 .

(x1 −w2 )22+ ϕ (x2 − x1 ) by Lemma 7.

We ﬁrst show that for (x1 , x2 ) ∈ A, ϕ (x2 − x1 ) = −Suppose(max{w1 −w2 ,0})2,2(max{w −w ,0})21
2
and then show that for (x1 , x2 ) ∈/ A, ϕ (x2 − x1 ) = −as well. Let
2{}
U L (x2 ) = min x1 | (x1 , x2 ) ∈ A for x2 ∈ [w2 , w2 ]. Let x = U L (max {w1 , w2 }), y =inf {x2 ∈ [w2 , w2 ] | U L (x2 ) > w1 },23 and y = sup {x2 ∈ [w2 , w2 ] | U L (x2 ) < w1 }. U L (x2 )[ ]is strictly increasing on y, y ,24 but might not be continuous as shown in Figure 3. For()
x2 ∈ max {w1 , w2 } , y , (w1 , x2 ) ∈ A and c1 (w1 , x2 ) = 0; for x1 ∈ (x, max {w1 , w2 }),23

Note that either x = w1 or y = w2 by Assumption 3 (i).
If U L (·) is not strictly increasing, then there exist a and b such that a > b but U L (a) ≤ U L (b). However,this implies that for some path x2,ω (·) passing through (a, U L (a)), x2,ω (·) is not strictly increasing, andcannot be the solution to program (P2).
2428

(x −w )2(x1 , w2 ) ∈ A and c1 (x1 , w2 ) = 0; given c1 (x1 , x2 ) = 1 2 2 + ϕ (x2 − x1 ) for (x1 , x2 ) ∈ A,[]()
(max{w1 −w2 ,0})2
it is implied that ϕ (d) = −for d ∈ 0, y − x . For any d ∈ y − x, d , there
2()
exists a ∈ y, w2 such that a − U L (a) = d. Since U L is increasing, for any a2 ∈ (a, w2 ],(U L (a) , a2 ) ∈/ A. Thus, given belief w1 = U L (a) and w2 = a2 , x2,ω (U L (a)) = a. By∂c1(8), c1 (U L (a) , a2 ) = c1 (U L (a) , a) for all a2 ∈ (a, w2 ], so ∂x(U L (a) , a) = 0. Since
2(U L (a) , a) ∈ A, we know c1 (x1 , x2 ) is of the form

(x1 −w2 )22+ ϕ (x2 − x1 ), and hence
) ′
(
∂c1
′
′
∂x2 (U L (a) , a) = ϕ (a − U L (a)) = ϕ (d) = 0. Thus, for d ∈ y − x, d , ϕ (d) = 0.
]([)(max{w1 −w2 ,0})2
Since ϕ (d) = −for d ∈ 0, y − x and ϕ′ (d) = 0 for d ∈ y − x, d ,2[ ](max{w1 −w2 ,0})2
for d ∈ 0, d .
ϕ (d) = −2
[Figure 3 about here][ ]From the above discussion, we know that, for any (x1 , x2 ) ∈ A, if x2 − x1 ∈ 0, d ,c1 (x1 , x2 ) =(x1 −w2 )2(max{w1 −w2 ,0})2−. However, by (8), this implies that x2,ω (x1 ) = x1 for2
2

x1 ∈ [max {w1 , w2 } , min {w1 , w2 }], so U L (x2 ) = x2 for x2 ∈ [max {w1 , w2 } , min {w1 , w2 }].
(x1 −w2 )22

Given U L (x2 ) = x2 , by (8), c1 (x1 , x2 ) =−(max{w1 −w2 ,0})22

for x1∈[max {w1 , w2 } , min {w1 , w2 }] and x2 ∈ [x1 , w2 ].

Lemma 9 c2 (x1 , x2 ) =Proof:(x2 −w1 )2(max{w2 −w1 ,0})2−22

for x1 ≥ x2 .

For (x1 , x2 ) ∈ A, c2 (x1 , x2 ) = (x1 − w1 ) (x2 − w2 ) −by Lemmas 3 and 7.

Suppose that x1 ≥ x2 .

(x1 −w2 )22− ϕ (x2 − x1 )We ﬁrst show that for (x1 , x2 ) ∈2(max{w1 −w2 ,0}), and then show that for (x1 , x2 ) ∈
/ A,
2
2
2
}
{
(max{w1 −w2 ,0})1)
ϕ (x2 − x1 ) = − (x2 −x−
as well. Let LR (x1 ) = min x2 | (x1 , x2 ) ∈ A
2
2
2

1)
A, ϕ (x2 − x1 ) = − (x2 −x−2

for x1 ∈ [w1 , w1 ]. Let y = LR (max {w1 , w2 }), x = inf {x1 ∈ [w1 , w1 ] | LR (x1 ) > w2 },25
and x = sup {x1 ∈ [w1 , w1 ] | LR (x1 ) < w2 }. LR (x1 ) is strictly increasing on [x, x], but
might not be continuous as shown in Figure 4. For x1 ∈ (max {w1 , w2 } , x) , (x1 , w2 ) ∈ A()
and c2 (x1 , w2 ) = 0; for x2 ∈ y, max {w1 , w2 } , (w1 , x2 ) ∈ A and c2 (w1 , x2 ) = 0; givenc2 (x1 , x2 ) = (x1 − w1 ) (x2 − w2 ) −(max{w1 −w2 ,0})2
that ϕ (d) = − 2 −2d2(x1 −w2 )2− ϕ (x2 − x1 ) for (x1 , x2 )2

∈ A, it is implied]()
for d ∈ y − x, 0 . For any d ∈ d, y − x , there ex[

ists a ∈ (x, w1 ) such that LR (a) − a = d. Since LR is increasing, for any a1 ∈ (a, w1 ],25

Note that either x = w1 or y = w2 by Assumption 3 (i).

29

(a1 , LR (a)) ∈/ A. Thus, given belief w1 = a1 and w2 = LR (a), x1,ω (LR (a)) = a. By∂c2(a, LR (a)) = 0.
(9), c2 (a1 , LR (a)) = c2 (a, LR (a)) for all a1 ∈ (a, w1 ], so ∂x
1

Since

(x −w )2
(a, LR (a)) ∈ A, we know c2 (x1 , x2 ) is of the form (x1 − w1 ) (x2 − w2 )− 1 2 2 −ϕ (x2 − x1 ),∂c2
and hence ∂x(a, LR (a)) = LR (a) − a + ϕ′ (LR (a) − a) = d + ϕ′ (d) = 0. Thus, for
1
(
)
]
[
2
(max{w1 −w2 ,0})2d ∈ d, y − x , ϕ′ (d) = −d. Since ϕ (d) = − d2 −for d ∈ y − x, 0 and2()
2(max{w1 −w2 ,0})2ϕ′ (d) = −d for d ∈ d, y − x , ϕ (d) = − d2 −for d ∈ [d, 0].
2[Figure 4 about here]
From the above discussion, we know that, for any (x1 , x2 ) ∈ A, if x2 − x1 ∈ [d, 0],c2 (x1 , x2 ) =(max{w2 −w1 ,0})2(x2 −w1 )2−. However, by (9), this implies that x1,ω (x2 ) = x2 for22x2 ∈ [max {w1 , w2 } , min {w1 , w2 }], so LR (x1 ) = x1 for x1 ∈ [max {w1 , w2 } , min {w1 , w2 }].
Given LR (x1 ) = x1 , by (9), c2 (x1 , x2 ) =(x2 −w1 )22−(max{w2 −w1 ,0})22

for x2∈[max {w1 , w2 } , min {w1 , w2 }] and x1 ∈ [x2 , w1 ].

Finally, the following lemma completes the proof.
Lemma 10 In equilibrium,[](x1 −w2 )2(max{w1 −w2 ,0})2
 1
−
2
2
C1 (x1 , x2 ) = (w2 −w2 ) 1(w2 −w2 ) [(x1 − w 1 ) (x2 − w 2 ) − (w 1 − w 1 )C2 (x1 , x2 )] 1
1 − w 1 ) (x2 − w 2 ) − (w 2 − w 2 )C1 (x1 , x2 )](w1 −w1 ) [(x[]2C2 (x1 , x2 ) =(x(max{w2 −w1 ,0})22 −w 1 )1−2
2(w1 −w )1

if x1 ≤ x2 ,if x1 ≥ x2 ,if x1 ≤ x2 ,if x1 ≥ x2 ,A = {(x1 , x2 ) | x1 = x2 , max {w1 , w2 } ≤ x1 ≤ min {w1 , w2 }} .
Proof: From Lemmas 8, 9, and the discussion in their proofs, we know that under Assumption 3, the only possible c1 is that
 (x1 −w2 )2 − (max{w1 −w2 ,0})222[
c1 (x1 , x2 ) =2
2]
(x1 − w1 ) (x2 − w2 ) − (x2 −w1 ) − (max{w2 −w1 ,0})2

2

if x1 ≤ x2 ,if x1 ≥ x2 ,which implies that the seller’s strategy isw2 if x1 ∈ [w1 , max {w1 , w2 }),x2,ω (x1 ) = x1 if x1 ∈ [max {w1 , w2 } , min {w1 , w2 }],
w2 if x1 ∈ (min {w1 , w2 } , w1 ],given the belief ω about the buyers’ values at the beginning of a continuation game.

30

Appendix B: the derivation of the price path
With solution x2,ω (x1 ), let{[]}
Gr (x2,ω ) = (x1 , x2 ) ∈ [w1 , w1 ] × [w2 , w2 ] | x2 ∈ lim x2,ω (x) , lim x2,ω (x)x→x1 −

be

the

graph

of

x2,ω (x1 )

with

lines

connecting

x→x1 +(x1 , limx→x1 − x2,ω (x))(x1 , limx→x1 + x2,ω (x)) when x2,ω (·) is discontinuous at x1 . ThenPω ≡ {(P1 (x1 , x2 ) , P2 (x2 , x1 )) | (x1 , x2 ) ∈ Gr (x2,ω )}is the price path submitted by the seller.

31

and

Figures

Figure 1: some examples which satisfy Assumption 3

Figure 2: the actual price path and the cutoﬀ type32

Figure 3

Figure 433