Information about Sellers’ Past Behavior
in the Market for Lemons∗
Kyungmin Kim†
May 2012
Abstract
In dynamic markets under adverse selection, uninformed players’ (buyers’) inferences on the quality of goods rely on the information they have about informed
players’ (sellers’) past behavior. This paper examines the roles of different pieces of
information about sellers’ past behavior. Agents match randomly and bilaterally, and
buyers make take-it-or-leave-it offers to sellers. It is shown that if market frictions are
small (low discounting or fast matching), the observability of time-on-the-market improves efficiency, while that of number-of-previous-matches deteriorates it. If market
frictions are not small, the latter may improve efficiency. The results suggest that market efficiency is not monotone in the amount of information available to buyers but
depends crucially on what information is available under what market conditions.
The results have implications for policy makers and market designers who can control information about participants’ trading histories and for a general modeling issue
in the literature on dynamic adverse selection.
JEL Classification Numbers: C78, D82, D83.
Keywords : Adverse selection; decentralized market; information flow; time-onthe-market; number-of-previous-matches; bargaining with interdependent values.
∗ I am grateful to Yeon-Koo Che, Pierre-André Chiappori, In-Koo Cho, Jay Pil Choi, Jan Eeckhout, Srihari
Govindan, Ayca Kaya, Stephan Lauermann, Ben Lester, Jin Li, Marco Ottaviani, Santanu Roy, Wing Suen,
Charles Zheng, and Tao Zhu for helpful comments. I also thank seminar audiences at Columbia, CUHK,
HKU, HKUST, Midwest Theory conference, North America Winter Meeting of the Econometric Society,
Southern Methodist, UIUC, Western Ontario, and Yonsei.
† University of Iowa. Contact: [email protected]
1
1 Introduction
Consider an investor (he, buyer) who is interested in acquiring a company of an entrepreneur (she, seller).1 He understands that the entrepreneur has superior information
about her own company, and thus there is an adverse selection problem. A low price
may be rejected, while a high price runs the risk of overpaying for a lemon. To mitigate
this problem, the investor can rely on the information he has about the entrepreneur’s
past behavior. If the entrepreneur rejected a good price in the past, it would indicate that
her company is even more worthwhile. However, access to the information about the entrepreneur’s past behavior may be limited. Regulation or market practice may not allow
it, or relevant records simply may not exist. There are several possibilities. The investor
may not get any information or may find out how long the company has been on sale. Or,
he may know how many other investors have approached the entrepreneur or even what
prices they have offered to her. The investor’s inference will depend crucially on what
information he has. What makes the problem non-trivial is the entrepreneur’s strategic
incentive. She has an incentive to reject an acceptable price today if, by doing so, she
can extract an even better offer tomorrow. The investor must take into account that the
entrepreneur has behaved strategically in the past and will respond strategically to his
offer.
This paper examines the roles of different pieces of information about informed players’ (sellers’) past behavior in the market for lemons. In particular, it investigates the
relationship between market performance and the amount of information available to
uninformed players (buyers): Does allowing buyers to get access to more information
about sellers’ past behavior improve efficiency? Intuitively, there are two opposing arguments. On the one hand, as in reputation games, information about informed players’
past behavior provides information about their intrinsic types, and thus more information would enable uninformed players to better tailor their actions. On the other hand,
as in signalling games, if informed players’ current behavior is better observable by future uninformed players, then informed players have a stronger incentive to signal their
types, which may cause efficiency losses.
The problem is relevant to market designers and policy makers. When setting up or
running a marketplace or an industry, they must decide how much and what information
to disclose about sellers (and perhaps also about buyers). The decision is crucial, because
the information-disclosure policy affects participants’ incentives in the market. In reality,
different marketplaces adopt different policies. For example, Craigslist does not provide
1 This
is just an example. The same story applies to any other context where adverse selection matters.
2
any information about sellers, other than that provided by sellers themselves.2 To the
contrary, the Multiple Listing Service (MLS) keeps track of the history of each posted
property. In particular, it exhibits how many days each property has been on sale. Why
do different marketplaces adopt different information policies? Which policy is optimal?
This paper sheds some light on these questions.
In addition, it has general implications for the literature on dynamic adverse selection.
The literature is growing fast, partly due to the recent financial crisis and partly due to
theoretical developments within economics.3 However, it is silent about one crucial modeling issue: information flow about informed players’ past behavior. Most studies in the
field take a particular form of information flow, typically no information flow in a fullblown market setting and full information flow in a reduced-form strategic setting, but
do not provide enough justification for their use. A natural question arises: Would (and,
if so, how) the predictions obtained under one information-flow assumption carry over
or extend to the environments with different assumptions? This paper offers a unique
perspective on this question.
The model is a dynamic decentralized version of Akerlof’s market for lemons (1970).
In each unit of time, equal measures of buyers and sellers enter the market for an indivisible good. Buyers are homogeneous, while there are two types of sellers. Some sellers
possess a unit of low quality, while the others own a unit of high quality. A high-quality
unit is more valuable to both buyers and sellers. There are always gains from trade, but
type (quality) is private information to each seller. Agents match randomly and bilaterally according to a Poisson rate. In a match, the buyer offers a price and the seller decides
whether to accept it or not. If an offer is accepted, trade takes place and the pair leave the
market. Otherwise, they stay and wait for the next trading opportunity.
The following three information regimes are considered:
• Regime 1 (no information) : Buyers do not receive any information about their partners’ past behavior.
• Regime 2 (time on the market) : Buyers observe how long their partners have stayed
on the market.
• Regime 3 (number of previous matches): Buyers observe how many other buyers
2 Craigslist
shows the posting date of each item. But the information does not convey any real information, because sellers can freely repost their ads and, in fact, they have an incentive to do so, in order to be
exposed as much as possible.
3 A non-exclusive list of recent works includes Camargo and Lester (2011), Chang (2010), Chari,
Shourideh and Zetlin-Jones (2010), Chiu and Koeppl (2011), Daley and Green (2011), Guerrieri and Shimer
(2011), Hörner and Vieille (2009a), Kurlat (2010), and Moreno and Wooders (2010).
3
have approached their partners, that is, how many offers their partners have rejected before.
A priori the regimes are not fully ordered in the amount of information available to buyers. In particular, the ranking between Regime 2 and Regime 3 is not straightforward.4
It will be shown later that number-of-previous-matches dominates time-on-the-market
in the sense that Regime 3 outcome is essentially independent of whether time-on-themarket is jointly observable or not (that is, Regime 3 outcome obtains if both time-on-themarket and number-of-previous-matches are observable). Therefore, the three regimes
are de facto fully ordered.
I fully characterize steady-state market equilibrium of each regime,5 and compare the
performance of the regimes in terms of market efficiency. I first show that if market frictions are small (agents are patient or matching is fast), then the observability of numberof-previous-matches reduces efficiency. There is a cutoff level of market frictions below
which Regime 1 weakly Pareto dominates Regime 3: Buyers weakly prefer Regime 1 to
Regime 3, while (low-type) sellers strictly prefer Regime 1 to Regime 3. The underlying
reason for this result is as follows. In any regime, high-type sellers, due to their higher
cost, stay on the market relatively longer than low-type sellers. Therefore, the proportion
of high-type sellers in the market is higher than the corresponding proportion among
new sellers. In Regime 1, all sellers look identical to buyers, that is, all cohorts of sellers
are completely mixed. Therefore, a higher proportion of high-type sellers in the market makes buyers more willing to offer a high price and, therefore, facilitates trade. In
Regime 3, the observability of number-of-previous-matches prevents mixing of different
cohorts of sellers. Therefore, a higher proportion of high-type sellers in the market does
not facilitate trade.
This result is in accordance with the result of Hörner and Vieille (2009a) and seems
to suggest that less information could always be preferable. Hörner and Vieille study a
reduced-form game in which a single seller faces a sequence of buyers and show that the
observability of previously-rejected-prices can cause bargaining impasse.6 Precisely, provided that adverse selection is severe and discounting is low, only the first buyer offers
4 On
the one hand, in dynamic environments, time-on-the-market enables informed players to credibly
signal their types and/or uninformed players to effectively screen informed players. Therefore, the role
of number-of-previous-matches might be just to provide an estimate of time-on-the-market. On the other
hand, number-of-previous-matches purely reflects the outcomes of informed players’ decisions, while timeon-the-market is compounded with search frictions.
5 The characterization of each regime is interesting itself, because it can be used to address other substantive questions and also shows how crucially market outcome depends on buyers’ information about
sellers’ past behavior.
6 They also study the case where buyers do not observe past offers (private offers). Their setup is in
discrete time without search frictions. Since time-on-the-market and number-of-previous-matches are in-
4
a price that can be accepted with a positive probability by the seller. In the two-type
case, the low-type seller trades with probability less than 1 at her first match. All subsequent buyers make only losing offers, so trade never occurs afterward. If their game is
embedded in the market setting of this paper, no agent obtains a positive expected payoff.7 Therefore, the observability of previously-rejected-prices unambiguously impedes
market efficiency.
The observability of time-on-the-market, however, enhances efficiency. I show that if
market frictions are sufficiently small, realized market surplus is strictly higher in Regime
2 than in Regime 1. To understand this, consider a new seller who quickly meets a buyer.
In Regime 3, the opportunity cost of accepting a current offer is rather large, as the seller’s
rejection would be observable by future buyers, who would update their beliefs accordingly. In Regime 2, the corresponding opportunity cost is smaller, as future buyers would
not know that the seller rejected a price. This difference yields the following consequence:
in Regime 2 a low-type seller who quickly meets a buyer trades with probability 1, while
in Regime 3 she does not. In Regime 1, the opportunity cost of accepting a current offer
is independent of a seller’s private history, and thus a low-type seller often rejects a low
price at her first match. Consequently, within a cohort, the proportion of high-type sellers
increases faster in Regime 2 than in Regime 1, which induces buyers to offer a high price
relatively quickly in Regime 2. In Regime 2, as in Regime 3, different cohorts of sellers are
not mixed, which negatively affects efficiency. When market frictions are small, it turns
out that the former positive effect more than offsets the latter negative effect, and thus
Regime 2 is more efficient than Regime 1.
This result demonstrates that efficiency is not monotone in the amount of information
available to buyers. It also shows that the results on number-of-previous-matches and
previously-rejected-prices stem from the nature of such pieces of information, and not
from any general relationship between efficiency and information.
In addition, depending on market conditions, the observability of number-of-previousmatches may contribute to efficiency. I show that if market frictions are not sufficiently
small, there are cases where Regime 3 is more efficient than the other two regimes. This
distinguishable and always observable by buyers, their private case corresponds to the reduced-form game
of the regime in which both time-on-the-market and number-of-previous-matches are observable (so, effectively the reduced-form game of Regime 3).
7 The intuition behind the payoff result is as follows. Given that high-type units never trade, buyers offer
only the reservation price of low-type sellers, and thus low-type sellers do not obtain a positive expected
payoff. Buyers’ zero expected payoff is due to the endogenous distribution of sellers in the market. A buyer
gets a positive expected payoff if and only if he meets a new seller. However, some low-type sellers and
all high-type sellers stay in the market forever, and thus the probability of a buyer meeting a new seller is
negligible.
5
reinforces the argument that efficiency is not monotone in the amount of information. It
also points out that what matters is what information is available under what market conditions.
The implications of the results go beyond the narrow confines of this paper. Positively,
the results illustrate the importance of making the correct information-flow assumption
or obtaining more robust results. The predictions obtained under one information-flow
assumption may not generalize to the environments with other assumptions. Therefore,
when analyzing a market and discussing relevant policy issues, it is crucial to identify
true information flow or provide the results that are robust to the potential misspecification of information flow. Normatively, the results suggest that a policy maker can
indirectly influence market performance by adjusting the disclosure policy or controlling
the information about agents’ trading histories, but they must be done with caution. Market efficiency depends not on the amount of information available to buyers, but on the
nature of each piece of information and market conditions. Therefore, if a policy maker
can control several pieces of information, then he should examine the role of each piece
of information and also understand the characteristics of the market, instead of simply
deciding how much information to reveal.
1.1 Related Literature
Various dynamic versions of Akerlof’s market for lemons have been developed.8 To my
knowledge, none of the previous studies concerns the central issue of this paper, namely
information about informed players’ past behavior. One paper that deserves mention is
Moreno and Wooders (2010), who study a discrete-time version of Regime 1. The equilibrium characterization of Regime 1 is similar to theirs, but the continuous-time setting of
this paper has two advantages. First, it allows a complete characterization under a mild
assumption on the discount factor (Assumption 2).9 The complete characterization is important, because some of my main substantive results build upon it. Second, it uncovers
the main driving force for their main welfare result (that social welfare is higher in the
decentralized market than in the competitive benchmark). In the continuous-time setting
8 To name a few seminal contributions,
Janssen and Karamychev (2002) and Janssen and Roy (2002, 2004)
examine settings, with constant inflow of agents or one-time entry, where a single price clears each spot
market. Inderst and Müller (2002) study competitive search equilibrium with constant inflow of agents.
Wolinsky (1990), Serrano and Yosha (1993, 1996), Blouin and Serrano (2001), and Blouin (2003) consider
various settings, with constant inflow of agents or one-time entry and with two-sided uncertainty or onesided uncertainty, in which agents meet bilaterally and play a simple bargaining game with only two possible transaction prices. Hendel and Lizzeri (1999, 2002) and Hendel, Lizzeri, and Siniscalchi (2005) study
dynamic durable goods markets where units are classified according to their vintages.
9 Moreno and Wooders (2010) restrict attention to the case where agents are sufficiently patient.
6
of this paper, as long as market frictions are small, social welfare in Regime 1 is exactly the
same as that of the competitive benchmark, independently of the discount rate and the
matching rate (see Proposition 1). This shows that their result is driven by their assumption that matching occurs only at the beginning of each discrete-time period. If matching
occurs at the end of each period, the difference between the decentralized market and the
competitive benchmark also disappears in the discrete-time setting.
A few papers study related issues in reduced-form game settings. In a dynamic version of Spence’s signalling model, Nöldeke and Van Damme (1990) show that, although
there are multiple sequential equilibria, there is an essentially unique sequential equilibrium outcome that satisfies the never a weak best response requirement (Kohlberg and
Mertens (1986)). As the offer interval tends to zero, the unique equilibrium outcome converges to the Riley outcome. Swinkels (1999) points out that the result depends crucially
on information flow. He shows that if offers are not observable by future uninformed
players (which are observable in Nöldeke and van Damme), then the unique equilibrium
outcome is complete pooling with no delay. Using the terminology of this paper, full histories of informed player’s past behavior are observable by future uninformed players
in Nöldeke and van Damme, while only time-on-the-market and number-of-previousmatches are observable in Swinkels. Hörner and Vieille (2009a) can be interpreted as considering the two cases (public and private offers) in the bargaining-with-interdependentvalues context. In a related two-period model, Taylor (1999) shows that the observability of previous reservation price and inspection outcome is efficiency-improving, that is,
more information about past trading outcome is preferable. The key difference is that
buyers’ herding, rather than sellers’ signaling or buyers’ screening, is the main concern in
his paper.10
Two papers study the effects of decreasing information asymmetry in the market for
lemons. Levin (2001) considers a static setting and shows that less information asymmetry may or may not improve trade. Daley and Green (2011) consider a dynamic setting
in which a single seller faces a competitive market and news (public signals about the
seller’s type) arrives over time. They show that increasing the news quality may or may
not improve efficiency. Despite apparent similarities, the question and the result of this
paper are fundamentally different from those of the two papers. They study the effects
of exogenously changing the level of information asymmetry given an information struc10 In his model, buyers have no incentive to trade with a low-type seller (no gains from trade of a low-type
unit) and winners conduct an inspection prior to exchange. These cause buyers to get more pessimistic over
time, that is, the probability that a seller owns a high-type unit is lower in the second period than in the
first period. The better observability of past outcomes improves efficiency by weakening negative buyer
herding.
7
ture,11 while this paper examines the effects of changing the information structure. As a
result, they are mainly concerned with quantitative aspects of sellers’ incentives in the
market for lemons, while this paper is more about qualitative aspects.12
Lastly, the setting of this paper can be interpreted as bargaining taking place in a market. In this regard, this paper is also related to the literature on bargaining with interdependent values. Evans (1989) and Vincent (1989) offer early results and insightful examples. Deneckere and Liang (2006) provide a general characterization for the finite type
case and explain the source and mechanics of bargaining delay due to adverse selection.
They find an equilibrium structure that is quite similar to the ones in this paper. I explain
similarities and differences in Section 7.
The remainder of the paper organizes as follows. Section 2 introduces the model.
The following three sections analyze each regime. Section 6 compares the welfare consequences of the regimes. Section 7 concludes by discussing a few relevant issues.
2 The Model
2.1 Physical Environment
The model is set in continuous time.13 In each unit of time, unit measures of buyers and
sellers enter the market for an indivisible good. Buyers are homogeneous, while there are
two types of sellers. A fraction qb of new sellers possess a unit of low quality (low type),
while the others own a unit of high quality (high type). A unit of low (high) quality costs
c L (c H ) to a seller and yields utility v L (v H ) to a buyer. A high-quality unit is more costly
to a seller (c H > c L ≥ 0) and more valuable to a buyer (v H > v L ). There are always gains
from trade (v H > c H and v L > c L ), but the quality of each unit is private information to
each seller.
Agents match randomly and bilaterally according to a Poisson rate λ > 0. In a match,
the buyer offers a price and the seller decides whether to accept it or not. If an offer is
accepted, then they receive utilities and leave the market. All agents are risk neutral. If
11 In
Daley and Green (2011), buyers observe the seller’s time-on-the-market and number-of-previousmatches.
12 Borrowing from Hörner and Vieille (2009a), “it is important to distinguish between how much information can possibly be revealed given the information structure and the information that is actually revealed
in equilibrium.” Levin (2001) and Daley and Green (2011) exogenously control the amount of information
that is revealed, while this paper (and Hörner and Vieille (2009a)) examines the relationship between the
information structure and the information that is revealed in equilibrium.
13 This is only for tractability. All the results can be established in the corresponding discrete-time setting,
but only with rather significant notational complexity.
8
a price p is accepted by a low-type (high-type) seller, then the buyer’s utility is v L − p
(v H − p) and the seller’s is p − c L ( p − c H ). Otherwise, they stay in the market and wait
for the next trading opportunity. The common discount rate is r > 0. It is convenient to
R∞
define δ ≡ λ/ (r + λ) = 0 e−rt d(1 − e−λt ). This is the effective discount factor in this
environment that accounts for search frictions as well as discounting.
I focus on the case where (1) adverse selection is severe (so high-type units do not
trade in the static Walrasian benchmark) and (2) market frictions are small (so agents
have non-trivial intertemporal considerations). Formally, I make use of the following two
assumptions.
Assumption 1 (Severe adverse selection)
qbv L + (1 − qb) v H < c H .
This inequality is a familiar one in the adverse selection literature. The left-hand side is a
buyer’s willingness-to-pay to a seller who is randomly selected from an entry population.
The right-hand side is a high-type seller’s reservation price. When the inequality holds,
no price can yield non-negative payoffs to both buyers and high-type sellers, and thus
high-type units cannot trade. For future use, let q be the value such that
qv L + (1 − q)v H = c H ⇔ q =
vH − cH
.
vH − vL
A necessary condition for a buyer to be willing to offer c H to a seller is that he believes
that the probability that the seller is the low type is less than or equal to q. Assumption 1
is equivalent to qb > q.
Assumption 2 (Small market frictions)
v L − c L < δ (c H − c L ) =
λ
(c H − c L ) .
r+λ
This assumption states that a low-type seller never accepts any price that a buyer is willing to offer to her (at most v L ) if she expects to receive an offer that a high-type seller is
willing to accept (at least c H ) at her next match. Given Assumption 1, this assumption is
satisfied when δ is rather large (r is small or λ is large).
9
2.2 Strategies and Equilibrium
In order to highlight the economic role of each piece of information, I focus on steadystate equilibrium in which agents of each type employ an identical strategy. In particular,
buyers’ offers condition only on sellers’ observable types, and sellers’ acceptance strategies condition only on their intrinsic and observable types.14
Formally, let Ξ be the information set of buyers, that is, the set of sellers’ feasible states
that are observable by buyers.15 A buyer’s strategy is a Lebesgue-measurable function
B : Ξ × R+ → [0, 1] where B(ξ, p) denotes the probability that the buyer offers p to a seller
whose observable type is ξ. A seller’s strategy is a function S : { L, H } × Ξ × R+ → [0, 1]
where S(z, ξ, p) denotes the probability that a type (z, ξ ) seller accepts a price p. Finally,
define a Lebesgue-measurable function G : { H, L} × Ξ → R+ so that G (z, ξ ) is the steadystate measure of type z sellers whose observable types are less than or equal to ξ.
A steady-state equilibrium consists of an offer strategy B(·, ·), an acceptance strategy
S(·, ·, ·), and a measure G (·, ·) that satisfy the following requirements: (i) Given G (·, ·)
and S(·, ·, ·), any p such that B(ξ, p) > 0 maximizes the expected payoff of the buyer who
is facing a type ξ seller. (ii) Given B(·, ·), S(z, ξ, p) > 0 only when p − cz is not less than
the continuation payoff of a type (z, ξ ) seller. (iii) Given constant inflow of new agents,
B(·, ·) and S(·, ·, ·), G (·, ·) is time-invariant.
2.3 Preliminary Observations
I restrict attention to the equilibrium in which each buyer offers the reservation price of
a low-type seller or that of a high-type seller. Formally, denote by VS (z, ξ ) a type (z, ξ )
seller’s expected payoff in the market. Consider a buyer who is facing a seller whose
observable type will become ξ if she rejects the buyer’s offer.16 I assume that the buyer
offers either c L + VS ( L, ξ ) or c H + VS ( H, ξ ). This restriction incurs no loss of generality.
First, it is strictly suboptimal for each buyer to offer more than c H + VS ( H, ξ ) or between
c L + VS ( L, ξ ) and c H + VS ( H, ξ ).17 Second, in all the three regimes, each seller’s future
14 In
other words, each buyer’s offer is independent of his own history. Each seller’s action depends on
her own history, but only through its effect on her observable type. For example, suppose in Regime 2 there
are two sellers who have met different numbers of buyers or been offered different prices. If they have
stayed on the market for the same length of time, then they are assumed to behave identically.
15 The set Ξ is a singleton in Regime 1. It is isomorphic to the set of non-negative real numbers R in
+
Regime 2 and to the set of non-negative integers N0 in Regime 3.
16 Rejecting an offer does not affect the seller’s type in Regimes 1 and 2, but increases the seller’s type by
1 in Regime 3.
17 c > c implies c + V ( H, ξ ) ≥ c + V ( L, ξ ) for any ξ ∈ Ξ in all the regimes.
H
L
H
L
S
S
10
type does not depend on the current price.18 Therefore, whenever a buyer makes a losing offer (offer that will be rejected for sure), the offer and the corresponding acceptance
probability can be set to be equal to VS ( L, ξ ) and 0, respectively.
The following result, which is a straightforward generalization of the Diamond paradox, greatly simplifies the subsequent analysis.
Lemma 1 In equilibrium, buyers never offer strictly more than c H . Therefore, high-type sellers’
expected payoffs are always equal to 0 (VS ( H, ξ ) = 0 for any ξ ∈ Ξ in any regime), and c H is
always accepted by any seller.
From now on, for notational simplicity, I denote by VS (ξ ) a type ( L, ξ ) seller’s expected
payoff and by VS a low-type seller’s expected payoff in the market. In addition, I denote
by VB a buyer’s expected payoff.
3 Regime 1: No Information
In this section, the set Ξ is a singleton. Denote by 0 its only element. Since buyers cannot
differentiate different cohorts of sellers, by definition, buyers cannot screen sellers, and
sellers cannot signal their types.
Under severe adverse selection, one may think that no information flow (and the consequent impossibility of screening and signaling) would cause no trade of high-type units.
Such intuition does not apply to the current dynamic setting. Suppose only low-type units
trade. Due to constant inflow of agents, the proportion of high-type sellers would keep
increasing and eventually the market would be populated mostly by high-type sellers. A
buyer would then be willing to offer a high price, which is a contradiction.19 On the other
hand, it cannot be that buyers always offer c H and any match turns into trade. If so, the
proportion of low-type sellers in the market would be equal to qb, which implies, together
with Assumption 1, that buyers’ expected payoff would be strictly negative.
The discussion above suggests that in equilibrium, buyers randomize between p∗ ≡
c L + VS and c H , which are the reservation prices of low-type and high-type sellers, respectively. Intuitively, an equilibrium is sustained as follows. High-type sellers accept only
c H , while low-type sellers accept both c H and p∗ . Since high-type sellers stay relatively
longer than low-type sellers, the proportion of high-type sellers in the market is greater
18 This
claim does not hold if buyers observe previously rejected prices. In fact, the dependence of future
offers (and, consequently, sellers’ continuation payoffs) on the current price is the key to the bargaining
impasse result of Hörner and Vieille (2009a).
19 Technically, trade of high-type units is a necessary condition for the existence of a steady-state equilibrium.
11
than 1 − qb. This provides an incentive for buyers to offer c H and trade also with high-type
sellers. In equilibrium, buyers offer p∗ and low-type sellers accept p∗ with just enough
probabilities so that, with the resulting proportion of high-type sellers in the market, buyers are indifferent between p∗ and c H .
Formally, the steady-state equilibrium requirements are translated as follows. For notational simplicity, I suppress sellers’ unique observable status and simply let B( p) ≡
B(0, p), S(z, p) ≡ S(z, 0, p), and G (z) ≡ G (z, 0). In addition, let q∗ ≡ G ( L)/(G ( L) +
G ( H )) be the proportion of low-type sellers in the market.
Buyer optimality:
A buyer must be indifferent between offering p∗ and c H . Therefore,
q∗ S( L, p∗ ) (v L − p∗ ) + (1 − q∗ S( L, p∗ )) VB = q∗ v L + (1 − q∗ ) v H − c H .
(1)
The left-hand side is a buyer’s expected payoff by offering p∗ . The offer is accepted only
by a low-type seller, whose probability is q∗ , and conditional on that, with probability
S( L, p∗ ). If the offer is not accepted, which occurs with probability 1 − q∗ S( L, p∗ ), the
buyer returns to the market and receives market utility VB . The right-hand is his expected
payoff by offering c H , which is accepted for sure by both types of sellers. To pin down VB ,
it suffices to consider a buyer who would offer c H at the next match. Therefore,
VB = δ (q∗ v L + (1 − q∗ ) v H − c H ) .
Seller optimality:
Since p∗ is the reservation price of low-type sellers,
p∗ − c L = VS = δ ( B( p∗ ) p∗ + (1 − B( p∗ )) c H − c L ) .
(2)
The left-hand side is a low-type seller’s payoff by accepting p∗ , while the right-hand side
is her expected payoff by rejecting it. If she rejects, then she needs to wait for a new buyer
(thus δ), who would offer p∗ with probability B( p∗ ) and c H with the complementary
probability.
Steady-state condition:
A high-type seller leaves the market if and only if she meets
a buyer and receives the offer c H and, therefore, her market exit rate is λB(c H ). Since
a low-type seller accepts p∗ with probability S( L, p∗ ), her exit rate is λ( B( p∗ )S( L, p∗ ) +
B(c H )). In steady state, as usual, the inflow of agents of each type must be identical to the
12
corresponding outflow. Therefore,
qb = G ( L)λ( B( p∗ )S( L, p∗ ) + B(c H )),
and
1 − qb = G ( H )λB(c H ).
Using q∗ = G ( L)/(G ( L) + G ( H )), the two steady-state conditions can be reduced to
qb
q∗ B( p∗ )S( L, p∗ ) + B(c H )
=
.
1 − qb 1 − q∗
B (c H )
(3)
Intuitively, the left-hand side is the ratio of low-type sellers to high-type sellers among
new sellers, while the right-hand side is the corresponding ratio among leaving sellers.
The proportion of low-type sellers in the market is invariant only when the two ratios are
identical.
A steady-state equilibrium in Regime 1 can be described by a tuple (p∗ , B( p∗ ), S( L, p∗ ),
q∗ ) that satisfy (1), (2), and (3).20 The following proposition characterizes the equilibrium.
Proposition 1 If
δ≥
qb (v L − c L )
,
qb (v L − c L ) + (1 − qb) (v H − c H )
(4)
then p∗ = v L , VB = 0, and VS = v L − c L . Otherwise, p∗ < v L , VB > 0, and VS < v L − c L . The
equilibrium is unique as long as (4) holds or δ ≤ qb.21
Proof. There are 4 equilibrium variables (p∗ , B( p∗ ), S( L, p∗ ), q∗ ), but only 3 equilibrium
conditions ((1),(2), and (3)). This problem can be resolved by considering an additional
obvious constraint that, since p∗ is accepted only by low-type sellers, p∗ ≤ v L . The two
cases in the proposition correspond to the case where this condition is binding (p∗ = v L )
and the case where it is not (p∗ < v L ). In the former case, it is straightforward that
20 The
full description of the equilibrium is as follows:
• B( p) = 0 if p 6= p∗ , c H (buyers offer only p∗ or c H ), and B(c H ) = 1 − B( p∗ ).
• S( L, p) = 0 if p < p∗ and S( L, p) = 1 if p > p∗ (p∗ is low-type sellers’ reservation price).
• S( H, p) = 0 if p < c H and S( H, p) = 1 if p ≥ c H (c H is high-type sellers’ reservation price).
21 The
uniqueness is not guaranteed only when
qb < δ <
qb (v L − c L )
,
qb (v L − c L ) + (1 − qb) (v H − c H )
which can happen only when v H − c H < v L − c L .
13
VB = 0 and VS = v L − c L . The other equilibrium variables can also be easily found.
In the latter case, if p∗ is not accepted by low-type sellers with probability 1 (that is, if
S( L, p∗ ) < 1), then a buyer has an incentive to deviate and offer a slightly higher price,
which is a contradiction. Therefore, S( L, p∗ ) = 1, and all the other variables solve the
three equations. See the Appendix for a formal proof.
Intuitively, if δ is large, then low-type sellers are willing to accept p∗ only when it is
sufficiently high. However, buyers know that p∗ would be accepted only by low-type
sellers and, therefore, p∗ ≤ v L . In equilibrium, the inequality binds, and all other results
follow from there. If δ is relatively small, sellers are eager to trade quickly and buyers can
exploit such incentive. As a result, low-type sellers obtain less than v L − c L , while buyers
receive a strictly positive expected payoff.
4 Regime 2: Time on the Market
In this section, buyers observe how long their matched sellers have stayed on the market.
The set Ξ is isomorphic to the set of non-negative real numbers R+ . A typical element
t ∈ Ξ represents the length of time a seller has stayed on the market.
Since, unlike in Regime 1, a direct characterization of market equilibrium is very involved, I divide the analysis into the following two stages, as is often done in the search
literature. First, I consider the problem between a single seller and a sequence of buyers who sequentially contact the seller. At this stage, I assume that buyers’ continuation
payoff (outside option) VB is exogenously given and characterize buyers’ optimal offer
strategies B(·, ·) and the seller’s optimal acceptance strategy S( L, ·). Second, I embed the
outcome into the market and characterize market equilibrium. In particular, I deduce the
steady-state measure G (·) and endogenize buyers’ market expected payoff VB .
4.1 Reduced-form Game
The first-stage problem is formally equivalent to the following game. There is a seller
whose type is L with probability qb and H with probability 1 − qb. Starting from time 0,
buyers arrive at Poisson rate λ. Each buyer offers a price, conditioning only on the length
of time the seller has stayed in the game (equivalently, the time the buyer arrives). If a
price is accepted, then the seller and the buyer receive utilities (p − cz and vz − p, respectively, where z is the seller’s type), and the game ends. Otherwise, the buyer receives his
outside option VB , and the seller waits for a new buyer. I refer to the buyer who arrives
14
at time t as time t buyer and the seller who has stayed in the game for t length of time
as time t seller. I assume that VB ∈ [0, min { v L − c L , v H − c H }). This is necessary for the
existence of a steady-state market equilibrium.
Using the same notation as in the original problem, denote by B(t, p) the probability
that a buyer who arrives at t offers p and by S(z, t, p) the probability that a type z seller
accepts p at time t. In addition, denote by q(t) the probability that the seller is the low
type at time t.22 Obviously, q(0) = qb.
As in other dynamic games with asymmetric information, time-on-the-market serves
as a screening device. The reservation price of the low type, due to her lower cost, is
strictly smaller than that of the high type. Therefore, the low type finishes the game
relatively faster than the high type. Buyers offer a low price initially and a high price if
the seller has stayed long. Of course, in equilibrium, the seller must wait long enough to
receive a high price. Otherwise, the low type would mimic the high type.
A class of simple equilibria. I start by constructing a class of equilibria that implement
the aforementioned dynamic structure in the simplest way. Let t be the value that satisfies
VB =
qbe−λt
1 − qb
v
+
vH − cH .
L
qbe−λt + (1 − qb)
qbe−λt + (1 − qb)
Such t is well-defined, because VB ≥ 0 > qbv L + (1 − qb)v H − c H . In addition, let t be the
value such that
v L − c L − VB = e−r(t−t)
Z ∞
t
λ
e − r ( t − t) ( c H − c L ) d 1 − e − λ ( t − t ) = e − r ( t − t )
( c H − c L ).
r+λ
Assumption 2 guarantees that t > t. Lastly, for any t̃ ∈ [t, t], let α(t̃ ) be the value such
that
v L − c L − VB = e
−r (t̃− t)
Z ∞
t̃
λα(t̃ )
( c H − c L ).
e−r(t−t̃) (c H − c L )d 1 − e−λα(t̃)(t−t̃) = e−r(t̃−t)
r + λα(t̃ )
r (v − c −V )
It is straightforward to show that α(t̃ ) is strictly increasing in t̃ from λ(c L −vL +VB ) to 1.
H
L
B
For each t̃ ∈ [t, t], consider the following strategy profile (see Figure 1 for the structures
of the two extreme cases):
22 When
the reduced-form game is embedded into the market, q(t) corresponds to the proportion of lowtype sellers among those whose observable type is t.
15
Price
Price
cH
p(t)
vL
VB
p(t)
cL
t
t
0
Low type
t
No trade
t̃ = t
0
Both types
t̃ = t
Low type
Both types
Figure 1: The left panel shows the structure of the equilibrium in which all buyers offer
c H for sure after some time (t̃ = t in the example), while the right panel shows that of the
equilibrium in which all buyers play an identical strategy after t (t̃ = t in the example).
• Let


R∞
λα(t̃)
c L + t e−r(s−t) (c H − c L )d 1 − e−λα(t̃)(s−t) = r+λα(t̃) (c H − c L ),
if t ≥ t̃,
p (t) ≡
R ∞ −r (s − t )
 cL +
(c H − c L )d 1 − e−λα(t̃)(s−t̃) = e−r(t̃−t) r+λαλα(t̃()t̃) (c H − c L ), if t < t̃.
t̃ e
If t < t̃, then the buyer offers p(t) with probability 1. If t ≥ t̃, then the buyer offers
c H with probability α(t̃) and p(t̃ ) with the complementary probability.
• The low-type seller accepts any price strictly greater than p(t) and rejects any price
strictly less than p(t). She accepts p(t) with probability 1 if t < t̃ and rejects it with
probability 1 if t ≥ t̃.
• The high-type seller accepts a price if and only if it is greater than or equal to c H .
The strategy profile is an equilibrium for the following reasons:
• p(t) is the low-type seller’s reservation price. Each buyer offers the reservation price
of the low type or that of the high type. Therefore, for the purpose of calculating
the low type’s reservation price, she can be assumed to accept only c H . Since all the
buyers after t̃ offer c H with probability α(t̃), the low-type seller’s reservation price at
λα(t̃)
t ≥ t̃ is p(t) = c L + r+λα(t̃) (c H − c L ). Since no buyer offers c H before t̃, the low-type
λα(t̃ )
seller’s reservation price at t < t̃ is p(t) = c L + e−r(t̃−t) r+λα(t̃) (c H − c L ).
• Buyers’ beliefs evolve as follows:
q(t) = max
qbe−λt
qbe−λt
.
,
qbe−λt + 1 − qb qbe−λt + 1 − qb
16
The low-type seller finishes the game whenever a buyer arrives before t and, therefore, the probability that the seller is the low type decreases according to the matching rate. The decrease stops once it reaches t. After t, either no type trades (between
t and t̃) or both types trade (after t̃). Therefore, q(t) is constant after t.
• Given buyers’ beliefs, it is straightforward that buyers’ strategies are also optimal.
If t < t, then the buyer strictly prefers offering p(t) to c H , because
q(t)(v L − p(t)) + (1 − q(t))VB > VB > q(t)v L + (1 − q(t))v H − c H .
If t ≥ t, then the buyer is indifferent between offering c H and p(t), because q(t )v L +
(1 − q(t ))v H − c H = VB , while p(t) is not accepted for sure (in which case the buyer
simply gets his outside option VB ).
Characterization. The equilibrium strategy profiles above share several notable properties. In all equilibria, the buyers before t offer p(t) = c L + e−r(t−t) (v L − c L − VB ), which
is accepted by the low-type seller with probability 1. Buyers’ beliefs about the seller’s
intrinsic type evolve exactly in the same way. The buyers after t behave differently, but
are indifferent between offering c H and taking their outside option. The following proposition shows that the properties also hold in any other equilibrium.
Proposition 2 Given VB , let t be the value such that
VB =
1 − qb
qbe−λt
v L + −λt
vH − cH .
−
λt
qbe
qbe
+ (1 − qb)
+ (1 − qb)
A strategy profile ( B(·, ·), S(·, ·, ·)) is a sequential equilibrium of the reduced-form game if and
only if the following properties hold:
• Let
p (t) ≡ c L +
Z ∞
t
Rs
e−r(s−t) (c H − c L )d 1 − e− t λB(x,c H )dx .
p(t) − c L is the expected payoff of the low-type seller at time t when she accepts only c H .
• Buyers:
B(t, p(t)) =
(
1,
1 − B(t, c H ),
if t < t,
otherwise, and
B(·, c H ) satisfies
p(t ) = v L − VB , and
17
(5)
p(t) ≥ v L − VB , ∀t ≥ t.
(6)
In words, buyers offer p(t) with probability 1 until t and randomize between p(t) and c H
after t.
• Low type:
S( L, t, p) =
(
1, if p > p(t) or p = p(t) and t < t,
0, otherwise.
That is, the low type accepts a price above p(t) for sure and rejects a price below p(t) for
sure. She accepts p(t) if and only if t < t.
• High type:
S( H, t, p) =
(
1, if p ≥ c H ,
0, otherwise.
That is, the high type accepts a price if and only if it is greater than or equal to c H .
Proof. See the Appendix.
One subtle equilibrium requirement is (6). It states that the low-type seller’s reservation price p(t) should not fall below v L − VB . The condition is necessary, because if
p(t) < v L − VB for some t > t, then the buyer would strictly prefer offering p(t) to c H
(because q(t )(v L − p(t)) + (1 − q(t ))VB > VB = q(t )v L + (1 − q(t ))v H − c H ). In all the
equilibria in the example above, p(t) is strictly increasing between t and t̃ and constant
after t̃. Therefore, the condition is never violated. However, depending on buyers’ offer
strategies, it is possible that p(·) decreases over some regions.
The two extreme equilibria depicted in Figure 1 are of particular interest in this regard.
In the case where t̃ = t, p(t) is constant after t. Therefore, (6) binds for any t ≥ t. This
means that buyers cannot offer c H more quickly than in the equilibrium: If buyers offer
c H with greater probabilities than α(t ) near t, then (6) would be violated. Now consider
the case in which t̃ = t (or α(t) = 1). In this case, p(t) increases fastest. Therefore, the
equilibrium satisfies (6) most effectively. This means that buyers cannot offer c H more
slowly than in this equilibrium. These properties are not particularly notable in the analysis of the reduced-form game, but turn out to be crucial in the characterization of the set
of market equilibria.
Although there is a continuum of equilibria, all of them are payoff-equivalent. This
is because the evolution of buyers’ beliefs is common across all equilibria. It uniquely
determines the low-type seller’s reservation prices before t. Since the expected payoff of
the buyer before t depends only on his belief about the seller’s type and the low-type
18
seller’s reservation price, it is also common across all equilibria. Finally, buyers’ beliefs
evolve so that once they reach a critical level q(t ), they are indifferent between offering c H
and taking their outside option. Therefore, although they behave differently in different
equilibria, their expected payoffs are identical.
Corollary 1 In any equilibrium of the reduced-form game, the low-type seller’s expected payoff is
p(0) − c L , and time t buyer’s expected payoff is q(t)(v L − p(t)) + (1 − q(t))VB (> VB ) if t < t
and VB otherwise, where
p(t) − c L = e−r(t−t) (v L − c L − VB ), ∀t ∈ [0, t],
and
q(t) = max
qbe−λt
qbe−λt
,
qbe−λt + (1 − qb) qbe−λt + (1 − qb)
.
Despite the fact that all equilibria are payoff equivalent, the choice of a reduced-form
game equilibrium still matters for market equilibrium. As will be explained shortly, it is
because different reduced-form equilibria induce different distributions of sellers in the
market. Consequently, agents’ payoffs in market equilibrium turn out to differ, depending on which reduced-form game equilibrium is adopted in the market setting.
4.2 Market Equilibrium
Now I characterize market equilibrium by embedding a reduced-form game outcome into
the stationary market environment. I first deduce the measure G (·, ·) and then endogenize VB .
Steady-state distribution. Fix an equilibrium of the reduced-form game. A low-type
seller stays (does not trade) until t with probability e−λt if t < t and with probability
e−λt e
−
Rt
t
λB ( x,c H )dx
if t ≥ t. A high-type seller stays until t with probability 1 if t < t and
−
Rt
λB ( x,c )dx
H
if t ≥ t. This means that considering a cohort of sellers
with probability e t
(who enter the market at the same time), the fraction of low-type sellers who stay until
t is equal to e−λt if t < t and e−λt e
−
Rt
t
λB ( x,c H )dx
if t ≥ t.
Rt
− t λB ( x,c H )dx
The corresponding fractions of
high-type sellers are equal to 1 if t < t and e
if t ≥ t.
Now consider the market in which all cohorts of sellers are present simultaneously.
Using the results in the previous paragraph, in symmetric steady-state equilibrium, the
19
measure of low-type sellers who have stayed shorter than t is equal to
Similarly,
 R
 qb t e−λs ds,
if t < t,
Rs
0R
R
G ( L, t) =
t − λs
t
−
λB
(
x,c
)
dx
H
 qb 0 e ds + e−λt t e t
ds , otherwise.

R
 (1 − qb) t 1ds,
if t < t,
0R
R t − R s λB(x,c H )dx G ( H, t) =
t
 (1 − qb) 0 1ds + t e t
ds , otherwise.
The function G (·, ·) depends on the equilibrium of the reduced-form game. A vast
multiplicity of reduced-form game equilibria implies that G (·, ·) takes various forms after
t.23 For example, if the equilibrium depicted in the left panel of Figure 1 (the one in which
t̃ = t) is employed, then
 −λt
 qb 1−e + e−λt (t − t) ,
λ
G ( L, t) =
 qb 1−e−λt + e−λt t − t + 1−e−λ(t−t)
,
λ
λ
and
G ( H, t) =
(
(1 − qb)t,
(1 − qb) t +
1− e−λ( t−t)
λ
if t ≤ t < t.
if t ≥ t,
if t < t,
,
if t ≥ t.
If the one in the right panel of Figure 1 (the one in which t̃ = t) is adopted, then whenever
t ≥ t,
!
− λα(t)(t− t)
1 − e−λt
− λt 1 − e
G ( L, t) = qb
,
+e
λ
λα(t )
and
where
1 − e−λα(t)(t−t)
G ( H, t) = (1 − qb) t +
λα(t )
α (t ) =
!
,
r (v L − c L − VB )
.
λ(c H − v L + VB )
As will be shown shortly, it turns out that only the total measure of sellers in the
23 In
any equilibrium, if t < t, then
G ( L, t) = qb
1 − e−λt
and G ( H, t) = (1 − qb)t.
λ
20
market is crucial for the characterization of market equilibrium. Let G (z) be the total
measure of type z sellers in the market, that is, G (z) = limt→∞ G (z, t). The following
result shows that G (z) is bounded by those for the two extreme equilibria depicted in
Figure 1.
Lemma 2 Let
1 − e−λt
1
1
− λt
Gt ( L) ≡ qb
, Gt ( H ) ≡ (1 − qb) t +
,
+e
t−t+
λ
λ
λ
and
1 − e−λt
1
Gt ( L) ≡ qb
+ e−λt
λ
λα(t )
1
, Gt ( H ) ≡ (1 − qb) t +
λα(t )
.
The total measure of type z sellers associated with any equilibrium of the reduced-form game cannot
be smaller than Gt (z) and cannot be larger than Gt (z), that is, G (z) ∈ [ Gt (z), Gt (z)] for both
z ∈ { L, H }.
Proof. See the Appendix.
To understand this result, suppose B(t, c H ) marginally increases for some t > t. This
increases the low-type seller’s expected payoff at t. Since the equilibrium requires that her
reservation price at t be always equal to v L − VB (see (5) in Proposition 2), this means that
B(s, c H ) must decrease at some s 6= t. Suppose s > t. The increase of B(t, c H ) decreases
G (z) (because the seller finishes the game faster now), while the decrease of B(s, c H ) increases G (z). The net effect on G (z) is, however, positive. This is because sellers discount
future payoffs, while the discount rate plays no role for G (z). If the seller’s discount
rate is zero, then B(t, c H ) and B(s, c H ) are perfect substitutes. Therefore, the decrease of
B(s, c H ) necessary to keep the seller indifferent between the changes of B(·, c H ) is exactly
the same as that to keep G (z) constant. When the discount rate is positive, the former
is larger than the latter. Therefore, the changes of B(·, c H ) always increases G (z). Since
B(·, c H ) increases fastest in the equilibrium with t̃ = t and slowest in the equilibrium with
t̃ = t, G (z) ∈ [ Gt (z), Gt (z)].
Endogenizing VB .
Now I close the model by endogenizing VB . Fix an equilibrium in
the reduced-form game, and let G (t; VB ) ≡ G ( L, t; VB ) + G ( H, t; VB ) be the measure of the
sellers who have stayed shorter than t. In addition, let G (VB ) be the measure of all sellers
in the market. I include VB as an argument of all these functions and other equilibrium
variables (such as t(VB ) and p(t; VB )) in order to make explicit their dependence on VB .
21
In market equilibrium, buyers’ outside option VB in the reduced-form game must coincide with their market utility. Therefore, VB must satisfy
VB = δ
Z
t(VB )
0
dG (t; VB )
(q(t; VB )(v L − p(t; VB )) + (1 − q(t; VB ))VB )
+
G (VB )
dG (t; VB )
.
VB
G (VB )
t(VB )
Z ∞
If a buyer meets a seller who has stayed shorter than t, trade occurs, at price p(t; VB ),
if and only if the seller is the low type, whose probability is q(t; VB ). In all other cases,
the buyer obtains VB , whether trade occurs or not. Rearranging the terms with VB =
v L − p(t; VB ),
Z t(VB )
δ
qb
e−λt ( p(t; VB ) − p(t; VB ))dt.
VB =
1 − δ G (VB ) 0
Define a correspondence Φ : [0, min {v L − c L , v H − c H }) ⇒ [0, min {v L − c L , v H − c H })
so that
Z t(VB )
δ
qb
− λt
Φ(VB ) =
e ( p(t; VB ) − p(t; VB ))dt : G (VB ) ∈ [ Gt (VB ), Gt (VB )] .
1 − δ G (VB ) 0
Then it is a market equilibrium if and only if VB is a fixed point of the correspondence Φ.
Proposition 3 There exists a continuum of market equilibria in Regime 2. Buyers’ expected
payoff is maximized (minimized), while low-type sellers’ expected payoff is minimized (maximized)
when the reduced-form game equilibrium in which t̃ = t (t̃ = t) is embedded into the market
environment.
Proof. t(VB ) always satisfies
qbe−λt(VB ) v L +(1− qb)v H
qbe−λt(VB ) +(1− qb)
≥ c H . Therefore, t(VB ) is bounded away
from 0 even if VB tends to zero. In addition, Gt (VB ) is finite for any VB . Therefore, the
minimum value of Φ(0) > 0. On the other hand, if VB is sufficiently close to min{v L −
c L , v H − c H }, then the maximum value of Φ(VB ) < VB , because no buyer can obtain more
than min{v L − c L , v H − c H } from any seller, but matching takes time. The existence of a
continuum of equilibria follows from the fact that Φ(VB ) is an interval (convex) for any
VB 24 and Φ(VB ) changes smoothly in VB . The results regarding the equilibria that maximize and minimize agents’ payoffs follow from the fact that the maximum (minimum) of
Φ(VB ) is achieved in the equilibrium in which t̃ = t (t̃ = t).
24 It
is straightforward that the class of simple equilibria I constructed in the reduced-form game spans
the interval.
22
5 Regime 3: Number of Previous Matches
In this section, the set Ξ is isomorphic to the set of non-negative integers, N0 . A typical
element n ∈ Ξ represents the number of matches a seller has experienced in the market.
As in Regime 2, I first analyze the reduced-form game between a seller and a sequence
of buyers and then characterize market equilibrium.
5.1 Reduced-form Game
The reduced-form game is almost identical to that of Regime 2. The only difference is that
now buyers can condition only on the number of previous buyers, that is, on how many
offers the seller has rejected until then. Using the same notation as in the original market
setting, denote by B(n, p) the probability that a buyer offers p to a seller who has rejected
n offers in the past. Also, denote by S(z, n, p) the probability that a type z seller accepts
p in her (n + 1)-th match. As in Regime 2, I first present a class of simple equilibria and
then provide a complete characterization of the set of all equilibria.
A class of simple equilibria. Let α be the value such that
∞
v L − c L − VB =
δα
∑ δk (1 − α)k−1 α(c H − c L ) = 1 − δ + δα (c H − c L ).
k=1
α uniquely exists, because Assumption 2 ensures v L − c L − VB ≤ v L − c L < δ(c H − c L ).
For any α ∈ [α, 1], let n(α) and β(α) be the values such that n(α) is an integer, β(α) < α,
and
!
∞
v L − c L − VB = δn(α)−1
= δ
n (α)−1
β(α) + (1 − β(α))
∑ δ k (1 − α ) k − 1 α
k=1
δα
β(α) + (1 − β(α))
1 − δ + δα
(c H − c L )
( c H − c L ).
The pair of n(α) and β(α) uniquely exist: As α increases, β(α) continuously decreases.
When β(α) reaches 0, n(α) increases by 1, β(α) jumps to α, and this process continues.
Fix α ∈ [α, 1], and consider the following strategy profile (see Figure 2 for the structures of the two extreme cases):
23
Price
Price
cH
cH
p(n)
vL
vL
cL
cL
p(n)
n
n
0
1
2
3
4
0
1
2
3
4
Figure 2: The left panel shows the structure of the equilibrium in which all buyers offer c H
for sure once the seller experiences a certain number of matches (α = 1 in the example),
while the right panel shows that of the equilibrium in which all buyers offer c H with
probability α after the first match.
• Let
p (n) ≡
(
δα
if n ≥ n(α) − 1,
c L + 1−δ+δα (c H − c L ),
δα
n
(
α
)−
n
−
1
β(α) + (1 − β(α)) 1−δ+δα (c H − c L ), if n < n(α) − 1.
cL + δ
If n < n(α) − 1, then the buyer offers p(n) with probability 1. If n = n(α) − 1,
then the buyer offers p(n) with probability β(n) and c H with the complementary
probability. If n > n(α) − 1, then the buyer offers p(n) with probability α(n) and c H
with the complementary probability.
• The low-type seller accepts any price strictly above p(n) and rejects any price below
1− qb − c H −VB
p(n). She accepts p(0) with probability 1 − qb vc HH −
v L +VB , while rejects p(n) for any
n > 0.
• The high-type seller accepts a price if and only if it is greater than or equal to c H .
The strategy profile is an equilibrium for the following reasons:
• p(n) is the low-type seller’s reservation price when her observable type is n. For
the same reason as in Regime 2, her reservation price can be calculated by assuming
that she would accept only c H . p(n) is constructed exactly that way.
• Buyers’ beliefs about the seller’s intrinsic type are q(0) = qb and
q(n) =
1− qb v H − c H −VB
qb c H − v L +VB
1− qb v H − c H −VB
qb qb c H −v L +VB + (1 − qb)
qb
24
=
v H − c H − VB
, ∀n > 0.
vH − vL
q(n) is constant after the seller’s first match, because the seller accepts only c H thereafter.
• The first buyer obtains VB if he follows the strategy. If he offers c H , then his expected
payoff is negative, because qbv L + (1 − qb)v H < c H . Therefore, he has no incentive to
deviate. All the other buyers are indifferent between offering c H and p(n): Observe
that q(n) is such that q(n)v L + (1 − q(n))v H − c H = VB .
Characterization. As in Regime 2, various properties of the simple equilibria generalize
to any other equilibrium.
Proposition 4 A strategy profile ( B(·, ·), S(·, ·, ·)) is an equilibrium of the reduced-form game if
and only if the following properties hold:
• Let
∞
p (n) ≡ c L +
∑
k= n +1
!
−1
δk−n Πkl =
n +1 (1 − B(l, c H )) B(k, c H ) (c H − c L ).
• Buyers:
B(n, p(n)) =
(
1,
1 − B(n, c H ),
if n = 0,
otherwise, and
B(·, c H ) satisfies
p(0) = v L − VB , and
(7)
p(n) ≥ v L − VB , ∀n ≥ 1.
(8)
In words, the first buyer offers p(0) with probability 1, while all the other buyers mix between p(n) and c H .
• Low type:



 1,
S( L, n, p) =
1−


 0,
if p > p(n),
1− qb v H − c H −VB
qb c H − v L +VB ,
if n = 0 and p = p(n),
otherwise.
That is, the low type accepts a price above p(n) for sure and rejects a price below p(n) for
sure. She accepts p(n) with a positive probability only at her first match (n = 0).
25
• High type:
S( H, n, p) =
(
1, if p ≥ c H ,
0, otherwise.
That is, the high type accepts a price if and only if it is greater than or equal to c H .
Again, as in Regime 2, all the equilibria are payoff equivalent.
Corollary 2 In any equilibrium of the reduced-form game, the low-type seller’s expected payoff is
δ(c L − v L − VB ), and the expected payoff of any buyer is VB .
Proof. The low-type seller is indifferent between accepting p(0) = v L − VB at her first
match. Therefore, her market expected payoff is δ(v L − c L − VB ) (note that she needs to
wait until the first buyer arrives). If p(0) is accepted, then the first buyer’s payoff is VB .
Since he obtains VB even if the offer is not accepted, his expected payoff is VB . It is optimal
for any other buyer to make a losing offer. Therefore, their expected payoff is also VB .
One difference from Regime 2 is that even the first buyer does not obtain more than
VB . To see why, suppose the first buyer can obtain more than VB . Since c H is obviously
suboptimal for the first buyer (whose belief about the seller’s intrinsic type is qb), this
means that the low-type seller’s reservation price must be strictly less than v L − VB . This
implies that p(0) must be accepted with probability 1: Otherwise, the buyer has an incentive to offer a slightly higher price, because it would be accepted with probability 1 then.
This in turn implies that the second buyer would believe that the seller is the high type
with probability 1 and, therefore, offer c H . But then, by Assumption 2, the low-type seller
would strictly prefer rejecting p(0), which is a contradiction. It will be shown shortly
that this difference has significant implications for market equilibrium. In particular, unlike in Regime 2, the choice of the reduced-form game equilibrium does not affect agents’
payoffs in market equilibrium.
5.2 Market Equilibrium
Now I characterize market equilibrium by embedding a reduced-form game equilibrium
into the market. As in Regime 2, the steady-state measures of sellers’ types associated
with each reduced-form game equilibrium can be deduced and then buyers’ market utility VB can be endogenized. I skip the first step, because its procedure is similar to that of
Regime 2, while the following result implies that agents’ payoffs can be determined, independently of the distribution of sellers’ types (which depends on the choice of a reducedform equilibrium).
26
Lemma 3 In any market equilibrium, buyers’ expected payoff is 0.
Proof. As shown above, a buyer’s expected payoff is VB , no matter which seller he meets.
Yet, matching takes time. Therefore, VB = δVB , which implies that VB = 0.
Given this result and Proposition 4, the following result is straightforward.
Proposition 5 There exists a continuum of market equilibria in Regime 3, but all of them are
payoff equivalent. In any equilibrium, buyers’ expected payoff is 0, and low-type sellers’ expected
payoff is δ(v L − c L ).
I conclude this section with two remarks on the information structure. First, in the
current two-type case, less information suffices to produce essentially the same outcome.
The only necessary information is whether sellers have been matched before or not. If
only that information is available, the equilibrium described in the right panel of Figure
2 (the one in which α = α) is the unique equilibrium. Second, the results do not change
even if time-on-the-market is also observable. In particular, the result that no buyer obtains more than VB in the reduced-form game, which is crucial for the characterization
of agents’ payoffs in market equilibrium, is independent of the observability of time-onthe-market. The additional information may be used as a public randomization device to
enlarge the set of equilibria, but does not affect agents’ payoffs.
6 Welfare Comparison
This section compares the welfare consequences of the regimes.
6.1 Regime 1 vs. Regime 3: the role of number-of-previous-matches
Suppose market frictions are relatively small (r is small or λ is large). In particular,
qb(v − c )
for simplicity, assume that δ ≥ qb(v −c )+(L1−qbL)(v −c ) . It then immediately follows from
L
L
H
H
Propositions 1 and 5 that Regime 1 weakly Pareto dominates Regime 3: Buyers are indifferent between the two regimes, while low-type sellers are strictly better off in Regime 1
(v L − c L ) than in Regime 3 (δ (v L − c L )).25
Why is Regime 1 more efficient than Regime 3? In Regime 1, different cohorts of
sellers are completely mixed. Such mixing helps relax the incentive constraint for buyers
to offer a high price, because high-type sellers, due to their higher cost, stay on the market
25 There is another cutoff, say δ, such that if δ falls between δ and
sellers and buyers strictly prefer Regime 1 to Regime 3.
27
qb( v L −c L )
, then both low-type
qb( v L −c L )+(1−qb)( v H −c H )
relatively longer than low-type sellers and, therefore, the proportion of high-type sellers
is larger in the market than among new sellers. In Regime 3, buyers’ access to numberof-previous-matches does not allow such mixing. Consequently, a higher proportion of
high-type sellers in the market does not facilitate trade. In particular, buyers never offer
c H to new sellers in Regime 3, while they often do in Regime 1. One can easily check that
this is exactly the reason why low-type sellers’ expected payoff is higher in Regime 1 than
in Regime 3.
6.2 Regime 1 vs. Regime 2: the role of time-on-the-market
When market frictions are relatively small, Regime 1 and Regime 2 are not Pareto ranked.
Buyers always obtain a positive expected payoff in Regime 2 and, therefore, strictly prefer
Regime 2 to Regime 1. To the contrary, low-type sellers strictly prefer Regime 1 to Regime
2. Their expected payoff in Regime 2 is p (0) − c L , which is strictly smaller than v L − c L .
The two regimes can still be compared in terms of total market surplus, as the model
is of transferable utility. In addition, the environment is stationary and I have focused
on steady-state equilibrium. Therefore, attention can be further restricted to total market
surplus of a cohort, that is, qbVS + VB .26
The following proposition shows that (at least) when market frictions are sufficiently
small, Regime 2 outperforms Regime 1.27
Proposition 6 When market frictions are sufficiently small, total market surplus of a cohort is
greater in any equilibrium of Regime 2 than in the equilibrium of Regime 1.
Proof. When market frictions are small, VS = v L − c L and VB = 0 in Regime 1, and thus
qbVS + VB = qb (v L − c L ). In the Appendix, I prove that if λ is sufficiently large or r is
sufficiently small, then qbVS + VB > qb (v L − c L ) in Regime 2. The proof proceeds in two
steps. First, I show that VS and VB approach v L − c L and 0, respectively, as λ tends to
infinity or r tends to 0. I then show that qbVS + VB is decreasing at the limit.
Why does the observability of time-on-the-market improve efficiency, while that of
number-of-previous-matches deteriorates it? As with number-of-previous-matches, the
observability of time-on-the-market prevents mixing of different cohorts of sellers, which
26 Recall
that high-type sellers’ expected payoff is always zero and the measures of low-type sellers and
buyers in each cohort are qb and 1, respectively.
27 Sufficiently small market frictions are only a sufficient condition. Numerical simulations show that
Regime 2 outperforms Regime 1 even when market frictions are not so small. The difficulty of characterizing agents’ expected payoffs in Regime 2 does not allow me to perform a more comprehensive welfare
comparison.
28
is negative for efficiency. However, there is an offsetting effect. To see this, consider a new
seller who quickly meets a buyer. In Regime 3, the seller has an incentive to reject a low
price, because doing so will convince future buyers that she is the high type with a high
probability. In Regime 2, the same incentive is present but weaker than in Regime 3. The
length of time a seller has to endure to receive a high price is independent of whether, and
how many times, the seller has rejected offers. Therefore, the opportunity cost of accepting a current offer is smaller in Regime 2 than in Regime 3 (recall that a low-type seller
accepts a low price with probability 1 if she matches before t in Regime 2, while she does
with probability less than 1 at her first match in Regime 3). Consequently, the proportion
of high-type sellers within a cohort increases fast in Regime 2, which induces buyers to
offer c H relatively quickly. In Regime 1, sellers cannot signal their types by rejecting offers or waiting for a certain length of time. Still, they have an incentive to wait for a high
price. This incentive is constant over time in Regime 1, while it is increasing in Regime 2
(and Regime 3). For sellers who are quickly matched, this incentive is stronger in Regime
1 than in Regime 2. As a result, low-type sellers often reject their reservation price in
qb(v − c )
Regime 1 (recall that B( p∗ ) < 1 whenever δ > qb(v −c )+(L1−qbL)(v −c ) ). This directly slows
L
L
H
H
down trade of low-type units and indirectly discourage buyers to offer c H , by increasing
the proportion of low-type sellers in the market. When market frictions are small, this
offsetting effect turns out to dominate, and thus Regime 2 outperforms Regime 1.
6.3 The role of market frictions
The following proposition shows that depending on market conditions, the observability
of number-of-previous-matches may improve market efficiency.
Proposition 7 For a fixed δ, if v H − c H is sufficiently small, then realized market surplus of a
cohort is close to zero in Regimes 1 and 2, while it is equal to qbδ (v L − c L ) in Regime 3.
Proof. Recall that in Regime 3, VS = δ (v L − c L ) and VB = 0 as long as Assumptions 1
and 2 are satisfied. It suffices to show that in Regimes 1 and 2, both VS and VB converge
to zero as v H tends to c H . The result for Regime 1 is immediate from the characterization
in Section 3. In Regime 2, VB < v H − c H , and thus obviously VB converges to 0 as v H
tends to c H . For VS , observe that when v H is close to c H , q is close to 0. Since q (t) ≤ q
(otherwise, buyers would never offer c H ), t must be sufficiently large. This implies that
p (0) is close to c L .
The intuition for this result is as follows. In any regime, buyers offer c H to some
sellers. Their benefit of offering c H depends on v H − c H , while their opportunity cost
29
is independent of v H − c H . If buyers obtain a positive expected payoff, they are less
willing to offer c H as v H − c H gets smaller. This is exactly what happens in Regimes 1 and
2.28 In the limit, buyers offer c H with probability 0. Then, as in Diamond (1971), buyers
offer c L with probability 1 and low-type sellers do not obtain a positive expected payoff.
Buyers’ expected payoff would be also zero. A buyer would get v L − c L if he meets a lowtype seller, but the probability of a buyer meeting a low-type seller would be zero, as the
market would be populated mostly by high-type sellers. In Regime 3, the effectiveness
of number-of-previous-matches as a signalling device leaves no surplus to buyers in any
circumstance and, therefore, makes their incentive to offer c H independent of v H − c H .
This prevents all the surplus from disappearing even as v H − c H approaches zero.
Rather informally, number-of-previous-matches serves as sellers’ signalling device
(while time-on-the-market as buyers’ screening device). Using it for signalling purpose is
socially wasteful in itself, as in other signalling games. However, it also allows informed
players (sellers) to secure a certain payoff. When market frictions are small, the former
(negative) effect dominates and thus the observability of number-of-previous-matches
reduces efficiency. When market frictions are not small, the latter effect could be significant. In particular, when all agents’ expected payoffs may be driven down close to zero,
the observability of number-of-previous-matches can contribute to market efficiency by
preserving informed players’ information rents.
7 Discussion
Liquidity and Information
The presence of low-type units and information asymmetries between sellers and buyers
slow down trade of high-type units. Several recent studies utilize this phenomenon to
model the (il)liquidity of high-type units (assets) and study relevant policy issues.29 Here,
I illustrate that the liquidity of high-type units depends crucially on information flow.
The liquidity of high-type units can be measured as the discounted average time to
sell a high-type unit: E[e−rτ ] where τ is a random (stopping) time that a seller is offered
c H . This seems to be appropriate in the current dynamic environment, because it reflects
not only the length of time necessary to sell a high-type unit, but also discounting.
Notice that VS = E[e−rτ ](c H − c L ), that is, a low-type seller’s expected payoff is equal
to the measure of liquidity times c H − c L . This is because a low-type seller’s incentive
qb(v −c )
28 Buyers
obtain a positive expected payoff in Regime 1 whenever δ < qb(v −c )+(L1−qbL)(v −c )
L
L
H
H
29 See, e.g., Camargo and Lester (2011), Chang (2010), and Guerrieri and Shimer (2011).
30
compatibility is binding whenever she is offered a lower price than c H : Each buyer offers
either the reservation price of a low-type seller or c H . Therefore, a low-type seller can still
obtain VS by simply waiting for c H .
It then follows that Regime 1 is most liquid for high-type units if market frictions are
qb(v − c )
relatively small, but may not be the case otherwise. Precisely, if δ ≥ qb(v −c )+(L1−qbL)(v −c ) ,
L
L
H
H
then low-type sellers’ expected payoff and, therefore, the liquidity of high-type units are
highest in Regime 1 (v L − c L > p(0) − c L , δ(v L − c L )). However, as shown in Proposition
7, for a fixed δ, if v H − c H is sufficiently small, then low-type sellers’ expected payoff
and, therefore, the liquidity of high-type units are higher in Regime 3 than in Regime 1.
The underlying reasons for these results are essentially identical to those for the welfare
results in Section 6.
More than Two Types
Equilibrium characterization can generalize beyond the tractable two-type case, with significant technical difficulties but without any conceptual difficulties. In Regime 1, buyers
would play a mixed bidding strategy over the set of sellers’ reservation prices. The bidding strategy and sellers’ acceptance strategies would be jointly determined so that the
distribution of seller types in the market is invariant and buyers are indifferent over the
prices. In Regimes 2 and 3, the equilibrium structures would extend similarly to Deneckere and Liang (2006). Welfare comparison, however, would be significantly more
involved. Closed-form solutions, which greatly simplify the welfare comparison in the
previous section, would not be available in any regime, unless some strong restrictions
are imposed on the parameter values.
Comparison with Deneckere and Liang (2006)
The equilibrium structures of Regimes 2 and 3 resemble that of Deneckere and Liang
(2006) (DL, hereafter). In particular, DL’s limit outcome (as the offer interval tends to
zero) exhibits the same qualitative properties as in the equilibria described in the left
panels of Figures 1 and 2: trade occurs either at the beginning of the game or only after
some real-time delay.
There are two important differences. First, it is only the limit outcomes that have the
same qualitative properties. Away from the limit, trade occurs with a positive probability in every period in DL, while there is always an interval of time or a set of matches
during which trade does not occur in the equilibria of Regimes 2 and 3. Second, in the
31
comparable limit case (λ is arbitrarily large in Regimes 2 and 3), the equilibrium delay is half as much as that of DL. To be more precise, let τ be the value that satisfies
(v L − c L ) = e−rτ (c H − c L ). The equilibrium delay is equal to 2τ in DL, while it is equal to
τ in Regimes 2 and 3.
Further Questions
The results in the paper raise several further questions. A question particularly germane
to this paper is what information flow would be most efficient. When market frictions are
sufficiently small, is it possible to improve upon Regime 2? Also, what information flow
is optimal for buyers?30 More generally, one may ask what is the constrained-efficient
benchmark in dynamic markets under adverse selection with constant inflow of agents
(with or without search frictions). Different from static settings, with constant inflow of
agents, as shown in Regime 1, subsidization across different cohorts of sellers is possible.
Would the social planner exploit such possibility or completely separate different cohorts
of sellers? Would the constrained-efficient outcome be stationary, cyclical (for example,
high-type units trade every n periods), or non-stationary?
One potentially interesting extension, which is also suggested by Hörner and Vieille
(2009a), is to allow buyers to conduct inspections before or after bargaining and with or
without cost. With inspections, buyers’ beliefs can evolve in any direction. When a unit
remains on the market for a long time, it might be because the previous offers have been
rejected by the seller (signalling effect), or because the previous buyers have observed
bad signals about the unit (herding effect). The former inference shifts buyers’ beliefs
upward, while the latter does the opposite. It might be interesting to examine a setting in
which both effects are present and study which effect dominates and what consequences
it would have on the relationship between efficiency and information about past trading
outcomes.
Appendix: Omitted Proofs
Proof of Proposition 1:
(1) Existence when δ ≥
qb(v L − c L )
.
qb(v L − c L )+(1− qb)(v H − c H )
30 Low-type sellers’ expected payoff is bounded by v − c in any circumstance, and thus Regime 1 is the
L
L
one (of possibly many) that is optimal for sellers, as long as market frictions are relatively small.
32
Suppose p∗ = v L . Then from the equilibrium conditions,
vH − cH
,
vH − vL
δ (c H − c L ) − (v L − c L )
,
B( p∗ ) =
δ (c H − v L )
1 − B( p∗ ) qb(1 − q∗ ) − (1 − qb)q∗
.
S( L, p∗ ) =
B( p∗ )
(1 − qb)q∗
q∗ = q =
Under Assumption 2, B( p∗ ) is always well-defined. S( L, p∗ ) is well-defined if and only if
qb(1 − q∗ ) − (1 − qb)q∗
B( p ) ≥
.
qb(1 − q∗ )
∗
Applying the solutions for q∗ and B( p∗ ) and arranging terms,
δ≥
(2) Uniqueness when δ ≥
qb(v L − c L )
.
qb(v L − c L ) + (1 − qb)(v H − c H )
qb(v L − c L )
.
qb(v L − c L )+(1− qb)(v H − c H )
It suffices to show that there cannot exist an equilibrium with p∗ < v L . Suppose
p∗ < v L . Then, q∗ < q and S( L, p∗ ) = 1. From sellers’ indifference condition,
(1 − δ)( p∗ − c L )
(1 − δ)(v L − c L )
> 1−
.
∗
δ (c H − p )
δ (c H − v L )
B( p∗ ) = 1 −
On the other hand, from the steady-state condition and the fact that q∗ < q,
B( p∗ ) = 1 −
(1 − qb)q∗
(1 − qb)q
<
1
−
.
qb(1 − q∗ )
qb(1 − q)
There exists B( p∗ ) that satisfies both inequalities if and only if
Arranging terms,
(1 − δ)(v L − c L )
(1 − qb)q
<
.
δ (c H − v L )
qb(1 − q)
δ<
qb(v L − c L )
.
qb(v L − c L ) + (1 − qb)(v H − c H )
qb(v − c )
(3) Existence when δ < qb(v −c )+(L1−qbL)(v −c ) .
L
L
H
H
In this case, S( L, p∗ ) = 1. Then, from the steady-state condition and sellers’ indiffer-
33
ence condition,
B( p∗ ) =
qb − q∗
δ(1 − B( p∗ ))
∗
,
and
p
=
c
+
( c H − c L ).
L
qb(1 − q∗ )
1 − δB( p∗ )
Applying these expressions to buyers’ indifference condition,
q∗
(1 − δ + δq∗ )
δ(1 − qb)q∗
vL − cL −
(c − c L )
(1 − δ)qb(1 − q∗ ) + δ(1 − qb)q∗ H
The right-hand side strictly decreases from v H − c H to 0 as
q∗
= q ∗ v L + (1 − q ∗ ) v H − c H .
(9)
increases from 0 to q. The
left-hand side is 0 if q∗ = 0. If q∗ = q, then the second term on the left-hand side is
δ(1 − qb)(v H − c H )
(c − c L )
(1 − δ)qb(c H − v L ) + δ(1 − qb)(v H − c H ) H
(1 − δ)qb(c H − v L )(v L − c L ) − δ(1 − qb)(v H − c H )(c H − v L )
(c H − c L )
=
(1 − δ)qb(c H − v L ) + δ(1 − qb)(v H − c H )
(1 − δ)qb(v L − c L ) − δ(1 − qb)(v H − c H )
(c − c L )(c H − v L ).
=
(1 − δ)qb(c H − v L ) + δ(1 − qb)(v H − c H ) H
vL − cL −
This expression is strictly positive if and only if δ <
qb(v L − c L )
.
qb(v L − c L )+(1− qb)(v H − c H )
Therefore, there
exists q∗ that solves (9). Once q∗ is found, the other two variables are straightforward.
qb(v − c )
(4) Uniqueness when δ < qb(v −c )+(L1−qbL)(v −c ) and δ ≤ qb.
L
L
H
H
I show that if δ ≤ qb then the left-hand side in (9) is strictly concave. Since the righthand side is linear in q∗ , the result immediately follows. Let
f (q) =
and
g (q) = v L − c L −
q
> 0,
1 − δ + δq
δ(1 − qb)q
(c − c L ) > 0.
(1 − δ)qb(1 − q) + δ(1 − qb)q H
34
Then, for q ∈ (0, q),
1−δ
> 0,
(1 − δ + δq)2
2δ(1 − δ)
f 00 (q) = −
< 0,
(1 − δ + δq)2
(1 − δ)δb
q (1 − qb)
(c H − c L ) < 0,
g 0 (q) = −
((1 − δ)qb(1 − q) + δ(1 − qb)q)2
2(1 − δ)δb
q (1 − qb)(−qb + δ)
g00 (q) =
(c H − c L ) ≤ 0.
((1 − δ)qb(1 − q) + δ(1 − qb)q)2
f 0 (q) =
The second derivative of the left-hand side in (9) is f 00 (q) g(q) + 2 f 0 (q) g0 (q) + f (q) g00 (q)
and, therefore, is strictly negative. Q.E.D.
Proof of Proposition 2:
Since the sufficiency part is essentially identical to the arguments made for the class of
simple equilibria, I prove only the necessity part here.
(1) Fix any (sequential) equilibrium of the reduced-form game. Let t∗ be the first moment at which buyers become indifferent between offering c H and taking their outside
option VB . Formally, t∗ is the infimum value such that
q(t∗ )v L + (1 − q(t∗ ))v H − c H = VB .
(2) t∗ is finite, and t∗ ≥ t.
Suppose not. Then, buyers would never offer c H . Then, by the same reasoning as in
the Diamond paradox, all buyers would offer c L , and the low type would accept it for
qbe−λt
sure. This means that q(t) = qbe−λt +1−qb and, therefore, t∗ = t, which is a contradiction.
Also, since q(t) ≥
qbe−λt
qbe−λt +1− qb
for any t in any equilibrium, t∗ ≥ t.
(3) After t∗ , trade occurs only at c H and, therefore, buyers’ beliefs q(t) do not change
after t∗ , that is, q(t) = q(t∗ ) for any t ≥ t∗ .
Suppose not. Let t1 (≥ t∗ ) and t2 be the first and last moments at which trade occurs
at a different price from c H with a positive probability.31 Since buyers after t1 can ensure strictly more than VB by offering c H (q(t)v L + (1 − q(t))v H − c H > q(t∗ )v L + (1 −
31 t is obviously finite, while t is finite for the following reason: Suppose not. Since no buyer makes a
2
1
losing offer after t1 , if t2 is not finite, then q(t) converges to 0 as t tends to infinity. But, then for sufficiently
large t, q(t)v L + (1 − q(t))v H − c H > q(t)(v L − p(t)) + (1 − q(t))VB = q(t)(v L − c L ) + (1 − q(t))VB , that is,
buyers strictly prefer offering c H to p(t) and, therefore, would offer only c H .
35
q(t∗ ))v H − c H = VB ), they would never make a losing offer. This implies that all buyers
after t2 would offer c H for sure. This in turn implies that the low-type seller would accept
only a price close to δ(c H − c L ) right before t2 . Obviously, buyers would never be willing
to offer such a high price and, therefore, trade cannot occur at a lower price than c H right
before t2 . This contradicts to the fact that buyers never make a losing offer after t2 .
(4) It then follows that
p (t) − c L =
Z ∞
t
Rs
e−r(s−t) (c H − c L )d 1 − e− t λB(x,c H )dx ≥ v L − c L − VB ,
for any t ≥ t∗ and that
p(t∗ ) − c L = v L − c L − VB .
If the first condition is violated, then the low type’s reservation price would fall short
of v L − VB , in which case trade would occur at a different price from c H after t∗ . If p(t∗ ) −
c L > v L − c L − VB , then there cannot be any trade right before t∗ , because if so the buyer
would obtain strictly less than VB . But, then q(·) does not change right before t∗ , which
contradicts the definition of t∗ .
(5) t∗ = t.
This stems from the fact that the low-type seller’s reservation price strictly increases
before t∗ . At any point in time before t∗ , buyers offer the reservation price of the low-type
seller. Therefore, to calculate the low-type seller’s continuation payoff, it can be assumed
that the low type simply rejects all the offers until t∗ . This implies that the low type’s
reservation price at t < t∗ is equal to
p (t) = c L + e −r (t
∗ −t)
( p (t∗ ) − c L ) = c L + e −r (t
∗ −t)
(v L − c L − VB ) < v L − VB .
The low type is indifferent between accepting and rejecting p(t). However, in equilibrium, it must be accepted with probability 1, because otherwise a buyer would have an
incentive to slightly increase his offer. Therefore, t∗ = t. Q.E.D.
R ∞ − R s λB(x,c H )dx
Proof of Lemma 2: It suffices to show that t e t
ds is minimized when B(·, c H )
increases fastest (this corresponds to the case where B(t, c H ) = 0 if t < t and B(t, c H ) = 1
if t ≥ t), while maximized when B(·, c H ) increases slowest (this corresponds to the case
where B(t, c H ) is constant). The constraints on B(·, c H ) are
p (t ) − c L =
Z
∞
t
e
−
Rs
t
(r + λB ( x,c H ))dx
λB(s, c H )ds (c H − c L ) = v L − c L − VB ,
36
and
p (t) − c L =
Z
∞
e
t
−
Rs
t
(r + λB ( x,c H ))dx
λB(s, c H )ds (c H − c L ) = v L − c L − VB ≥ 0, ∀t > t.
Since the latter constraint binds for all t > t only in the equilibrium with t̃ = t (depicted
in the right panel of Figure 1), it can be ignored for local variations.
R ∞ − R s λB(x,c H )dx
ds
Consider a marginal increase of B(t, c H ). Then, the objective function t e t
increases at rate
−λ
R
while the constraint
λ e
−
Rt
Rs
∞ −
t e
t
Z ∞
e
t
−
(r + λB ( x,c H ))dx
t (r + λB (t,c H ))dx
−
Z ∞
t
e
−
Rs
t
Rs
λB ( x,c H )dx
t
ds,
λB(s, c H )ds (c H − c L ) increases at rate
(r + λB ( x,c H ))dx
λB(s, c H )ds (c H − c L ).
These marginal changes evolve over time at rates
−
−λ
and
λe
R∞
Rt
e
t
λB ( x,c H )dx
Rs
− t λB ( x,c H )dx
t
ds
= −R ∞
t
e−
Rs
t
1
λB ( x,c H )dx
ds
Rt
−λre− t (r+λB(t,c H ))dx (c H − c L )
R
t
R ∞ − R s (r+λB(x,c H ))dx
− t (r + λB (t,c H ))dx
t
λ e
− t e
λB(s, c H )ds (c H − c L )
= −
1−
R∞
t
e−
Rs
t
r
(r + λB ( x,c H ))dx
λB(s, c H )ds
,
respectively. The absolute value of the latter is strictly larger than that of the former,
because
r
> r
Z ∞
t
Z ∞
t
Z ∞
e−
e
−
Rs
t
Rs
Rs
t
λB ( x,c H )dx
ds +
(r + λB ( x,c H ))dx
Z ∞
t
ds +
e−
Rs
Z ∞
t
t
e
(r + λB ( x,c H ))dx
−
Rs
t
λB(s, c H )ds
(r + λB ( x,c H ))dx
λB(s, c H )ds
e− t (r+λB(x,c H ))dx (r + λB(s, c H ))ds
t
Z ∞ Rs
d 1 − e− t (r+λB(x,c H ))dx = 1.
=
=
t
This means that when B(t, c H ) increases, the decrease of B(t + dt, c H ) necessary to keep
37
the constraint hold is larger than that to keep theR objective function. Therefore, the higher
R ∞ − s λB(x,c H )dx
ds is. The result then immediB(t, c H ) is than B(t + dt, c H ), the higher t e t
ately follows from the fact that B(t, c H ) is highest near t in the equilibrium with t̃ = t,
while lowest in the equilibrium with t̃ = t.Q.E.D.
Proof of Proposition 4:
As in the proof of Proposition 2, I prove only the necessity here.
(1) Since all buyers offer either c H or the reservation price of the low type, the reservation price of the low type who has rejected n(≥ 0) offers is
∞
p (n) ≡ c L +
∑
k= n +1
−1
δk−n Πkl =
n +1 (1 − B(l, c H ) B(k, c H )
!
( c H − c L ).
(2) Every buyer’s expected payoff is VB .
Let n be the minimal number of previous matches such that the (n + 1)-th buyer obtains more than VB . Then, it must be that (i) p(n) < v L − VB (the buyer obtains more than
VB by offering the low type’s reservation price) or (ii) q(n)v L + (1 − q(n))v H − c H > VB
(by offering c H ).
Suppose (i) is the case. Then the low-type seller must accept the offer for sure: Otherwise, the buyer would slightly increase his offer. Given this, the next buyer ((n + 2)-th
buyer) is certain that the seller is the high type32 and, therefore, would offer c H . But then,
due to Assumption 2, the low-type seller in her (n + 1)-th match would not accept p(n),
which is a contradiction.
Suppose (ii) is the case. Due to Assumption 1, certainly n ≥ 1. Suppose trade does not
occur at the n-th match of the seller. Then the probability that the seller is the low type
does not change between n-th and n + 1-th matches (q(n − 1) = q(n)). This implies that
the n-th buyer can obtain strictly more than VB as well, which contradicts the definition
of n. Now suppose trade occurs with a positive probability at the n-th match of the seller.
If it occurs only at c H , then the same contradiction as before arises. On the other hand,
by Assumption 2, trade cannot occur at v L − VB with a positive probability, because the
(n + 1)-th buyer offers c H with probability 1 (notice that (i) is already ruled out).
(3) The low-type seller accepts p(n) only in her first match, and p(0) = v L − VB .
Suppose p(n) is accepted with a positive probability. (2) implies that p(n) = v L − VB .
If n > 0, then p(n − 1) = δp(n) < v L − VB . This cannot be the case because if so, then the
32 Note
that this is true even if the (n + 1)-th buyer randomizes between p(n) and c H .
38
n-th buyer can offer slightly above p(n − 1) and obtain more than VB , which contradicts
(2).
1− qb
c H −VB
(4) The low-type seller accepts p(0) with probability 1 − qb vc HH −
− v L +VB .
After the first match, buyers’ beliefs about the seller’s intrinsic type become
q (1 ) =
qb(1 − S( L, p(0)))
.
qb(1 − S( L, p(0))) + (1 − qb)
(3) implies that q(n) = q(1) for any n ≥ 1. Since each buyer must be indifferent between
offering c H and taking his outside option (for the reason explained in (2)),
q(1)v L + (1 − q(1))v H − c H = VB .
Solving this equation,
S( L, p(0)) = 1 −
1 − qb v H − c H − VB
.
qb c H − v L + VB
(5) p(n) ≥ v L − VB for any n ≥ 1.
If this condition is violated, then the (n + 1)-th buyer strictly prefers offering p(n) to
c H and this offer should be accepted by the low-type seller, which contradicts (2) and (3).
Q.E.D.
Proof of Proposition 6:
Step 1: Given VB ,
Z
t(VB )
δ
qb
Φ(VB ) =
e−λt ( p(t; VB ) − p(t; VB ))dt
1 − δ G (VB ) 0
Z t(VB )
λ qb
≤
e−λt ( p(t; VB ) − p(0; VB ))dt
r G (VB ) 0
qb
(1 − e−λt(VB ) )( p(t; VB ) − p(0; VB )).
=
rG (VB )
(10)
(1) VB approaches 0 as λ tends to infinity.
The proof differs depending on whether v L − c L < v H − c H or not.
(i) v L − c L < v H − c H
Suppose λ is sufficiently large. I prove that in this case t is close to 0, which immediately implies that VB is close to 0 and VS is close to v L − c L (see Figure 1). Suppose t is
39
bounded away from 0. Then q(t ) is close to zero. Buyers then strictly prefer offering c H
to making losing offers after t, because
q (t) v L + (1 − q (t)) v H − c H ' v H − c H > v L − c L ≥ v L − p (t) .
This violates the structure of the equilibrium.
(ii) v L − c L ≥ v H − c H
(ii-1) Given VB , Φ (VB ) approaches 0 as λ tends to infinity (pointwise convergence).
Fix VB < v H − c H . If λ is large, by the same reasoning as in (i), t must be sufficiently
small. Since G (VB ) is bounded away from zero, (10) implies that Φ (VB ) is close to zero.
(ii-2) For a fixed λ, Φ (VB ) approaches 0 as VB tends to v H − c H . Together with (ii1), this implies that if λ is sufficiently large, then equilibrium VB is close to 0 (uniform
convergence).
Fix λ and suppose VB is sufficiently close to v H − c H . For q (t) v L + (1 − q (t)) v H −
c H = VB , q (t) must be sufficiently small, and thus t must be sufficiently large. Since
G (VB ) > (1 − qb) t, (10) implies that Φ (VB ) is close to 0.
(2) VB approaches 0 as r tends to zero.
Since
p (t) − c L = e−r(t−t) ( p (t) − c L ) = e−r(t−t) (v L − c L − VB ) ,
(10) is equivalent to
VB ≤
qb
G (VB )
1 − e−rt(VB ) r
1 − e−λt(VB ) (v L − c L − VB ) .
(11)
(i) v L − c L < v H − c H
Suppose r is sufficiently small. Then t must be bounded from above. Otherwise, q (t)
would be close to 0, and then the same contradiction as in (1-i) arises. In (11), the term
1 − e−rt /r is bounded from above (as r tends to zero, the term approaches t). On the
other hand, for low-type sellers’ incentive compatibility (see (5) in Proposition 2), B(·, c H )
must be low overall (more concretely, in the equilibrium with t̃ = t, t must be sufficiently
large, and in the equilibrium with t̃ = t, α(t̃) must be sufficiently small). This implies that
for r sufficiently small, G (VB ) is sufficiently large, and thus VB is close to zero.
(ii) v L − c L ≥ v H − c H
The proof is essentially identical to the one in (1). First, use the argument in (2-i)
to show that given VB , Φ (VB ) approaches 0 as r tends to zero (pointwise convergence).
Then, apply (ii-2).
40
Step 2: Recall that in Regime 2, VS = p (0) − c L and VB = v L − p ( t). From the
characterization of reduced-form game equilibrium,
Therefore,
p ( t) − p (0) = 1 − e−rt ( p (t) − c L ) .
dp (t)
d ( p (t) − p (0))
dt
= re−rt ( p ( t) − c L )
+ 1 − e−rt
.
dλ
dλ
dλ
As shown above, for λ sufficiently large, t is close to 0, and thus the second-term does not
have a first-order effect. Similarly, when λ is sufficiently large,
dVB
dq (t)
= − (v H − v L )
,
dλ
dλ
and
Using all the results,
2
qbe−λt + (1 − qb) dq ( t)
dt
=−
.
dλ
dλ
λb
q (1 − qb) e−λt
d (qbVS + VB )
dV
d ( p (0))
d ( p ( t))
= qb
− qb
+ (1 − qb) B
dλ
dλ
dλ
dλ
d ( p (t) − p (0))
dVB
= −qb
+ (1 − qb)
dλ
dλ
2
−
λt
qbe
+ (1 − qb) dq (t)
dq (t)
− (1 − qb) (v H − v L )
= qbr ( p (t) − c L )
−
λt
dλ
dλ
λb
q (1 − qb) e
!
2
qbe−λt + (1 − qb)
dq ( t)
− (1 − qb) (v H − v L )
=
qbr ( p (t) − c L )
.
−
λt
dλ
λb
q (1 − qb) e
This is negative because for λ sufficiently large, the first term is negative, while the second
one is positive (as λ tends to infinity, VB approaches zero, and thus q (t) converges to q
from the left). The proof for the case where r tends to zero is essentially the same. Q.E.D.
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