1. society
2. welfare

Optimal Dynamic Matching
Mariagiovanna Baccara
SangMok Leey
Leeat Yarivzx
May 19, 2015
Abstract
We study a dynamic matching environment where individuals arrive sequentially.
There is a tradeo¤ between waiting for a thicker market, allowing for higher quality
matches, and minimizing agents’ waiting costs. The optimal mechanism cumulates a
stock of mismatched pairs up to a threshold and matches all others in an assortative
fashion instantaneously. In decentralized settings, a similar protocol ensues in equilibrium, but expected queues are ine¢ ciently long. We quantify the welfare gain from
centralization, which can be substantial, even for low waiting costs. We also evaluate
welfare improvements generated by transfer schemes and by matching individuals in
…xed time intervals.
Keywords: Dynamic Matching, Mechanism Design, Organ Donation, Market
Design.
Olin School of Business, Washington University in St. Louis. E-mail: [email protected]
Department of Economics, University of Pennsylvania. E-mail: [email protected]
z
Division of the Humanities and Social Sciences, Caltech. E-mail: [email protected]
x
We thank Larry Samuelson for very helpful comments. We gratefully acknowledge …nancial support from
the National Science Foundation (SES 0963583).
y
1
Introduction
1.1
Overview
Many matching processes are inherently dynamic, with participants arriving and matches
being created over time. According to the Organ Donation and Transplantation Statistics, a
new patient is added to the kidney transplant list every 14 minutes and about 3000 patients are
added to the kidney transplant list each month. A signi…cant fraction of these patients arrive
at the list with a living donor – in 2013, about a third of nearly 17; 000 kidney transplants
that took place in the US involved such donors. A similar picture emerges when considering
child adoption. According to the Census 2000, about 1:6 million or 2:5 percent of all children
were adopted. The precise statistics on the entry of parents and children into the process
are poorly documented. However, data from one US adoption facilitator who links adoptive
parents and children indicates a rate of about 11 new potential adoptive parents and 13 new
children entering the facilitator’s operation each month.1 Likewise, many labor markets entail
unemployed workers and job openings that become available at di¤erent periods – the US
Bureau of Labor Statistics reports approximately …ve million new job openings and slightly
fewer than …ve million newly unemployed workers each month this year.
Nonetheless, by and large, the extant matching literature has taken a static approach
to market design – participants all enter at the same time and the market’s operations are
restricted in their horizon (see the literature review below for several notable and recent
exceptions). In the current paper, we o¤er techniques for extending that approach to dynamic
settings.
All of the examples we mentioned above share two important features. First, the quality
of matches varies. In other words, agents care about whom they match with. Second, waiting
for a match is costly, be it for …nancial costs of keeping lawyers on retainer for potential
adoptive parents, children’s hardship from growing older in the care of social services, the
lack of wages and needed employees in labor markets, or medical risks for organ patients and
psychological waiting costs for donors. These two features introduce a Some crucial trade-o¤ –
on the one hand, a thick market can help generate a greater match surplus; on the other hand,
1
This adoption facilitator is one of 25 registered in its state of operation. See Baccara, Collard-Wexler,
Felli, and Yariv (2014) for details.
1
instantaneous matching cuts down on waiting costs. The goal of this paper is to characterize
the resolution of this trade-o¤ in both centralized and decentralized environments. Namely,
we identify the optimal protocol by which a social planner would match agents over time. We
also identify the conditions under which decentralized matching processes would especially
bene…t from such centralized intervention.
Speci…cally, we consider a market that evolves dynamically. There are two classes of agents,
which we refer to as squares and rounds. At each period, a square and round pair enters the
market. Squares and rounds each have two types, one type more appealing to the other side
than the other. For instance, if we think of squares and rounds as children relinquished for
adoption and potential adoptive parents, types can stand for gender of children and wealth
levels of potential adoptive parents, respectively (see Baccara, Collard-Wexler, Felli, and Yariv,
2014). Alternatively, if we think of the two classes of agents as patients and donors, types
can represent age, and both younger patients and younger donors are more appealing than
older patients or donors (see, e.g., Gjertson, 2004 and Oien et al., 2007). We assume that
preferences are super-modular so that the (market-wide) assortative matching maximizes joint
welfare. We also assume that, once agents arrive at the market, waiting before being matched
comes at a per-period cost.
We start by analyzing the optimal matching mechanism in such settings. We show that
the optimal mechanism takes a simple form. Whenever congruent pairs of agents are present
in the market (i.e., a square and round that are both of the more appealing type or the less
appealing type), they are matched instantaneously. When only mismatched agents are present
in the market (i.e., a square and a round of the more appealing and less appealing type, or vice
versa), they form a queue. When the stock of pairs in the queue exceeds a certain threshold,
mismatched pairs are matched in sequence, until the queue length falls within the threshold.
Such thresholds induce a Markov process, where states correspond to the length of queues
of mismatched pairs of agents. Any threshold yields a di¤erent steady-state distribution
over possible queue lengths, and we evaluate the expected welfare of the mechanism in the
steady-state. The optimal mechanism utilizes the threshold that maximizes welfare. When
waiting costs are vanishingly small, the welfare under the optimal mechanism approaches the
maximum feasible, that generated by no matches of mismatched pairs. Nonetheless, as time
frictions, or waiting costs increase, the welfare generated by the optimal mechanism decreases.
2
This welfare decrease raises the question of the value of dynamic clearinghouses for nontrivial waiting costs in di¤erent environments, identi…ed by type distributions and preferences.
We therefore study the performance of a simple decentralized matching process in our setting.
As before, we consider agents arriving at the market in sequence. At each period in which they
are on the market, agents declare their willingness to match with partners of either type. After
these demands have been made, the maximal number of pairs of willing agents are matched
in order of arrival.2 Those who prefer to stay in the market, or have to stay for lack of willing
partners, form a queue.3 In our environment, individuals waiting in the market impose a
negative externality on those who follow them in the queue, as they force them to wait a
longer period and potentially miss appealing matches. This negative externality, which is not
internalized by individuals, leads to excessive waiting in the decentralized setting. In fact,
the matching protocol induced by equilibrium in the decentralized matching process ends up
resembling the protocol corresponding to the optimal mechanism, but with higher thresholds
for the queues’lengths.
We evaluate the welfare wedge between a centralized and a decentralized market as a
function of the underlying primitives of the environment, namely the agents’type distribution
and the cost of waiting. As the frequency of desirable types on either side of the market
increases, the option value of waiting becomes higher. The negative externalities imposed by
agents waiting in the decentralized setting lead to the welfare di¤erential between the two
processes increasing. This suggests greater bene…ts from market intervention.
The comparative statics with respect to costs of waiting are more subtle. An increase
in the cost of waiting has two e¤ects. On the one hand, for any …xed queue length, agents
experience greater waiting costs. Since expected queues are longer in the decentralized setting,
this e¤ect elevates the welfare di¤erential between the decentralized and optimal processes.
On the other hand, both the optimal threshold as well as the equilibrium threshold in the
decentralized process decrease as a function of waiting costs. Since in the decentralized setting
higher costs reduce the incentives to wait, and therefore the negative externalities imposed
on longer queues, this indirect e¤ect works to mute the welfare di¤erence between the two
2
This process is reminiscent of a double auction, as each agent submits a “demand function” specifying
which types of agents she would be interested in matching with immediately.
3
We provide a characterization of preferences that assure that the process is individually rational for all
participants.
3
processes. We show that the combination of these e¤ects leads to a welfare wedge that is locally
increasing in costs (formally, it is piece-wise increasing), but exhibits a general decreasing
trend. Ultimately, when costs are prohibitively high, both processes lead to instantaneous
matches and identical welfare levels.
Finally, we ask in which ways one can improve upon decentralization with interventions
that are simpler than the full-‡edged optimal mechanism. Indeed, centralization may be hard
to implement for two main reasons: …rst, it requires the central planner to be able to force
matches upon individuals and, second, it requires the central planner to monitor arrivals and
to possibly create matches at every period, yielding potentially high administrative costs.
To address the …rst issue, we consider a decentralized setting in which per-period taxes are
introduced for the agents that decide to wait. Our characterization of the optimal mechanism
allows us to identify a budget-balanced tax scheme that implements the optimal levels of
welfare. Such a tax scheme can be tailored so that it does not distort agents’incentives to
enter the market to begin with. Nonetheless, even such a taxation plan may be viewed as
cumbersome from an administrative perspective as it entails the collection or distribution of
resources that depend on agents’location in the queue. To address this problem, as well as
avoid administrative costs due to market intervention at every period, we also analyze the
performance of a simple mechanism in which all matches occur at …xed time intervals. The
length of these time intervals can be chosen to balance the costs of waiting and the quality
of the resulting matches, which we characterize. We show that such a simple procedure can
improve welfare substantially relative to full decentralization, though is still inferior to the
fully optimal matching mechanism.
1.2
Related Literature
The interest surrounding dynamic matching is quite recent and the literature on this topic is
still relatively limited. Much of this literature originally stemmed from the organ donation
application. Zenios (1999) develops a queueing model to explain the di¤erences between
waiting times of di¤erent categories of patients anticipating a kidney transplant. In the
context of kidney exchange, Ünver (2010) focuses on a market in which donors and recipients
arrive stochastically, preferences are compatibility-based, and the goal of a central planner is
to minimize total discounted waiting costs. Under some conditions, he shows that the e¢ cient
4
two-way matching mechanism always carries out compatible bilateral matches as soon as they
become available. However, when multi-way matches are possible, some two-way matches
could be withheld in order to allow future multi-way matches.4
Akbarpour, Li, and Oveis Gharan (2014) is possibly the closest to our study as they also
inspect the bene…ts of di¤erent mechanisms in a dynamic two-sided matching environment.
In their setting, however, preferences are compatibility-based according to a network mapping
the set of exchange possibilities. Agents in the system (thought of as patient-donor pairs)
become “critical” at random dates, and perish immediately after if they remain unmatched.
Therefore, when waiting costs are negligible, the goal of the planner is to minimize the number
of perished agents. Market thickness is bene…cial in that it guarantees the availability of
immediate matches for agents who become critical. Left to their own devices, agents in that
setting would match quickly and useful mechanisms induce agents to wait. In contrast, in our
setting, agents in a decentralized process wait too long and useful mechanisms induce agents
to wait shorter times on the market. In addition, while the welfare benchmark in Akbarpour,
Li, and Oveis Gharan (2014) is that of an omniscient planner, our di¤erent focus allows us to
characterize the optimal mechanism, which serves as the benchmark for welfare comparisons.
In a somewhat di¤erent realm, Leshno (2014) studies a one-sided market in which objects
(say, public houses) need to be allocated to agents who wait in a queue. Welfare maximization
always requires agents to be matched to their preferred objects. However, if agents’preferences
are unknown to the planner and their preferred item is in short supply, agents may prefer a
mismatched item earlier to avoid costly waiting. Leshno (2014) shows that the welfare loss
from mismatches can be reduced substantially through a policy under which all agents who
decline a mismatched item face the same expected wait for their preferred item.5 Anderson,
Ashlagi, Gamarnik, and Kanoria (2014) study an environment in which each agent is endowed
with an item that can be exchanged with an item owned by someone else. Compatibility is
stochastic, and three classes of feasible exchanges are considered: two-way exchanges only,
two- and three-way cycles, and any kind of chain. They …nd that a policy that maximizes
4
Some recent models in inventory control have a similar ‡avor to the compatibility-based matching process
considered by Ünver (2010), see e.g. Gurvich and Ward (2014).
5
Bloch and Cantala (2014) also study dynamic allocations. In their setting, a mechanism is a probability
distribution over all priority orders consistent with the waiting list. Given a priority order, whenever a new
object becomes available, agents are proposed the object in sequence and can either accept or reject. They
show the bene…ts of a strict seniority order.
5
immediate exchanges without withholding them in the interest of market thickness performs
nearly optimally.6;7
In yet another context, Budish, Cramton, and Shim (2015) consider …nancial exchanges
and argue that high-frequency trading leads to ine¢ ciencies, while frequent batch auctions,
uniform-price double auctions that occur at …xed and small time intervals, can provide e¢ ciency improvements. These results are reminiscent of our observations regarding the welfare
improvements generated by matchings that occur at the end of each …xed window of time.
There is also a recent theoretical literature that considers decentralized matching processes
that are dynamic, considering both informational and time frictions (see, e.g., Haeringer and
Wooders, 2011, Niederle and Yariv, 2009, and Pais, 2008). In that literature, the number
of agents on each side of the market is …xed at the outset and agents on one side can make
directed o¤ers to agents on the other side. The main goal is the identi…cation of market
features that guarantee that an equilibrium of the induced game generates a stable matching.
Another related strand of literature is the search and matching literature (e.g., Burdett
and Coles, 1997, Eeckhout, 1999, and the survey by Rogerson, Shimer, and Wright, 2005).
There, each period, workers and …rms randomly encounter each other, observe the resulting
match utilities, and decide jointly whether to pursue the match and leave the market or to
separate and wait for future periods. As time frictions vanish, such markets generate outcomes
that are close to a stable matching. A crucial di¤erence with our setting is the stationarity of
the market –the perceived distribution of potential partners does not change with time, and
each side of the market solves an option value problem.8
Last, there is a rather large literature that considers dynamic matching of buyers and sellers
and inspects protocols that increase e¢ ciency or allow for Walrasian equilibrium outcomes to
emerge as agents become increasingly patient (see, e.g., Satterthwaite and Shneyerov, 2007
and Taylor, 1995).
6
On the bene…ts of algorithms that create thicker pools in sparse dynamic allocation environments, see also
Ashlagi, Jaillette, and Manshadi (2013).
7
While most of this literature has been focusing on the design of algorithms to achieve socially desirable
matchings Doval (2014) introduces a notion of stability in dynamic environments, and provides conditions
under which dynamically stable allocations exist.
8
In the context of marriage markets, Kocer (2014) considers learning over time and models the choice
of temporary interactions with di¤erent potential partners as a multi-armed bandit problem. Choo (2015)
develops a new model for empirically analyzing dynamic matching in the marriage market and applies that
model to recent changes in the patterns of US marriages.
6
2
Setup
We study an in…nite-horizon dynamic matching market. There are two kinds of agents: squares
and rounds. Squares and rounds can stand for potential adoptive parents and children relinquished for adoption, workers and employers, patients and donors, etc.9
At each time t, one square and one round arrive at the market. Each square can be of
either type A or B with probability p or 1
or
with probability p or 1
p, respectively, and each round can be of type
p, respectively. These types correspond to the attributes of
participants – they can stand for the wealth of parents and race of children in the adoption
application, level of education and wages or interest of employers in labor markets, age or
tissue types in the organ donation context10 , etc.
In our model, squares seek to match with rounds and vice versa. We denote by Ux (y)
the surplus for a type-x participant from matching with a type-y participant. We assume
that A-squares are more desirable for all rounds and that -rounds are more desirable for all
squares. That is,
UA ( ) > UA ( );
UB ( ) > UB ( );
U (A) > U (B);
U (A) > U (B):
It will be convenient to denote:
UA
UB
UA ( ) + U (A);
UB ( ) + U (B);
UA
UB
UA ( ) + U (A);
UB ( ) + U (B);
as well as
U
UA + UB
UA
UB :
We will further assume that U > 0 so that the utilitarian e¢ cient matching in a static
market matches the maximal number of (A; ) and (B; ) pairs. The value of U captures the
e¢ ciency gain from the assortative matching relative to the anti-assortative matching. Notice
9
The organ donation application shares some features of our model when considering live donations from
good samaritan donors, in which case the matching process is two-sided in nature.
10
This is a simpli…ed representation that aims at capturing heterogeneity in types and the quality of di¤erent
matches it implies. For organ donation, patients and donors are often classi…ed into coarse categories based
on age. However, more than two tissue types are often considered. For example, for kidney transplantation,
the medical community currently looks at six tissue types, called major histo-compatibility complex or HLA
antigens.
7
that if we assumed fully symmetric utilities, Ux (y) = Uy (x), U > 0 would be tantamount to
assuming super-modular assortative preferences (a-la Becker, 1974) and U can be thought of
as the degree of super-modularity preferences exhibit.
We assume that each square and round su¤er a cost c > 0 for each period they spend on the
market waiting to be matched. We also assume that agents leave the market only by matching.
In Section 7.1 we provide bounds on the utility of agents from remaining unmatched that
assures this assumption is consistent with individual rationality in the processes we analyze.11
Several assumptions merit some discussion. The assumptions of super-modular preferences
and waiting costs that are identical for both sides of the market are made for presentation
simplicity. Both of these assumptions are common in the literature and, as we describe
in Section 5, lead to a conservative comparison of the optimal and decentralized matching
protocols.
The assumption that the distribution of types of rounds mirrors that of squares also simpli…es our analysis substantially as will be seen. It essentially implies that if we drew a large
population of rounds and squares, the realized distributions of types would be approximately
balanced with high probability. This may be a fairly reasonable assumption for certain applications, such as organ donation. Indeed, the distribution of tissue types of donors and
patients is arguably similar. Furthermore, the age of a donor is known to have a strong impact on the expected survival of a graft (see, e.g., Gjertson, 2004 and Oien et al., 2007) and
younger recipients have been suggested as the natural recipients of higher-quality organs (see
Stein, 2011). Our assumptions then …t a world in which the classi…cation of patients and
donors into “young” or “mature” categories is similar. Our assumption also approximately
holds for certain attributes in the online dating world (see Hitch, Hortacsu, and Ariely, 2010).
Nonetheless, it might be a rather harsh assumption for other applications. In Section 7.2 we
discuss how the techniques we introduce can be used were we to relax that assumption. We
replicate some of our analysis for general asymmetric settings in the Online Appendix.
The other strong assumption we make is that a pair of agents arrives at the market in
each period. The analysis would remain virtually identical were we to assume that pairs
arrive at random times following, say, a Poisson distribution. However, the assumption that
11
We show that individual rationality is guaranteed when all agents are acceptable and when any -round
receives a utility lower than U (B) 1 p p [U (A) U (B)] when leaving the market unmatched (analogously
for B-squares).
8
participants arrive in pairs is important for the techniques we use. This assumption assures
that the market is balanced throughout the matching process. It is a reasonable assumption
for some applications. For example, in the adoption process presumably potential adoptive
parents and children make important decisions (whom to match with, whether to leave the
process, etc.) at spaced-out intervals. Given the limited variability in the volume of entrants
on a monthly basis, the assumption of balanced arrivals provides a decent approximation of
reality.12 This assumptions also approximately holds in certain labor markets. Indeed, Hall
and Kruger (2012) suggest that a large fraction of jobs have posted wages. Naturally, these
wages may re‡ect general equilibrium wages tailored to the precise composition of the market.
Nonetheless, the fact that wages do not ‡uctuate dramatically suggests that the distributions
of agents is fairly stable. Allowing for di¤erent arrival rates on both sides of the market
introduces new considerations as matching participants in thin markets, while still entailing
low waiting costs, imposes now a loss in terms of both the quality of matches and the number
of individuals matched. The extensions to di¤erent arrival rates is therefore left for the future.
3
Optimal Dynamic Matching
3.1
The Matching Process
In this section we characterize the socially optimal dynamic mechanism. At each period, after
a new square-round pair arrives, the social planner determines which matches to implement,
if any, to maximize the aggregate social welfare.
At any time t, before a new square-round pair enters the market, a queue corresponds to
a vector (kA ; kB ; k ; k ), where each argument is the stock of squares or rounds of each type
waiting in line from the previous period.
It is immediate to see that an optimal mechanism would match an A-square and an
-
round whenever both are available, since having both in the queue only imposes waiting
costs.13 Also, when both a B-square and a
-round are available, an optimal mechanism
would match them immediately. The only reason to maintain a B-square in the queue is to
12
Our results extend directly if we assume that each period a …xed number of square-round pairs (possibly
greater than one) enter the market.
13
As waiting costs are linear, the identity of those matched has no welfare implications.
9
match her with an -round, and similarly the only reason to keep a -round in the queue is
to match him with an A-square. However, because of our super-modularity assumption, if
such matches are to be possible, they would be inferior to the (B; ) and (A; ) matches that
would then be feasible instead. Matching a B-square and a -round immediately therefore
generates at least as high ultimate match surplus and entails a lower waiting cost. Therefore,
we have the following:
Lemma 1. Any optimal mechanism requires (A; ) and (B; ) pairs to be matched as soon
as they become available.
The lemma implies that at any point in time an optimal dynamic mechanism entails queues
of only A-squares and -rounds or only B-squares and -rounds. That is, the queue can take
the form of either (k; 0; 0; k) or (0; k; k; 0); for some k
0:
The optimal dynamic mechanism is then identi…ed by the maximal stock of A-squares (and
-rounds) and the maximal stock of -rounds (and B-squares) that are kept waiting. In the
following proposition we characterize the structure of the optimal mechanism.
Proposition 1 (The Optimal Mechanism). An optimal dynamic mechanism is identi…ed by
a pair of thresholds (kA ; k ) 2 Z+ such that
1. whenever more than kA A-squares are present, kA
kA pairs of type (A; ) are matched
immediately, and
2. whenever more than k
-rounds are present, k
k pairs of type (B; ) are matched
immediately.
From the symmetry of our environment, an optimal mechanism corresponds to symmetric
thresholds: k = kA = k :14
A dynamic mechanism with symmetric thresholds (k; k) as de…ned in Proposition 1 is
depicted in Figure 1, where kA = kA
k captures the di¤erence between the length of the
queue of A-squares and the length of the queue of -rounds. We call kA the (signed) length
of the A- queue. Agents of congruent types who enter the market lead to no change in the
queue lengths, whereas mismatched pairs can change the queue length, unless the threshold
is reached.
14
In Section 7.2 we describe the extension to asymmetric environments, where the two thresholds may di¤er.
10
Figure 1: Structure of Optimal and Decentralized Matching Processes
This process induces the following Markov chain. Let st denote the number of A-squares
(or -rounds) minus the number of
-rounds (or B-squares) – that is the value of kA – at
the beginning of time t; before the arrival of a new square-round pair. If an (A; ) or a (B; )
pair arrive in period t, the mechanism matches an (A; ) or a (B; ) pair immediately, and
the state remains the same: st = st+1 . Suppose an (A; ) pair arrives in period t. As long
as 0
st < k, the mechanism creates no matches and st+1 becomes st + 1. If st < 0, the
mechanism creates one (A; ) match and one (B; ) match, and st+1 becomes st + 1. Finally,
if st = k, the mechanism creates one (A; ) pair, and st+1 remains the same, st+1 = st = k.
Analogous transitions occur with the arrival of a (B; ) pair.
Therefore, we can describe the probabilistic transition as follows. Denote by
xt
(xtk ; xtk 1 ; :::; xt k+1 ; xt k )tr 2 f0; 1g2k+1
the timed vector capturing the state, xti = 1(st = i) –that is, xti is an indicator that takes the
value of 1 if the state is st and 0 otherwise. Then,
xt+1 = Tk xt ;
where
0
B
B
B
B
B
Tk = B
B
B
B
@
1
p(1 p)
p(1 p)
p(1 p)
1 2p(1 p)
0
p(1 p)
..
..
.
.
0
0
0
0
:::
:::
:::
...
0
0
0
0
..
.
..
.
0
:::
p(1 p)
: : : 1 2p(1 p)
p(1 p)
:::
p(1 p)
1 p(1 p)
11
1
C
C
C
C
C
C:
C
C
C
A
(1)
The above Markov chain is ergodic (i.e., irreducible, aperiodic, and positively recurrent).
Therefore, an optimal mechanism corresponds to a matching process that reaches a steady
state with a unique stationary distribution. For Tk , the steady-state distribution is uniform
so that each state s = k; k
3.2
1; : : : ; k occurs with an equal
1
2k+1
probability.
Optimal Thresholds
In order to characterize the optimal threshold, we …rst evaluate the welfare corresponding to
any arbitrary symmetric threshold.
First, we compute the average total waiting costs incurred by agents waiting in line for
one period of time. During the transition from time t at state s to time t + 1, 2jsj agents wait
in line. So the total costs of waiting incurred during this one time period is 2jsjc. Thus, a
mechanism with threshold k results in expected total costs of waiting equal to
0
1
k
X
1 @
2k(k + 1)c
2jsjA c =
:
2k + 1
2k + 1
s= k
Next we compute the average total surplus generated during one time period, tracking the
Markov process described above. A new arrived pair is of type (A; ) with probability p2 , in
which case the optimal mechanism generates a surplus equal to UA . Similarly, when a new
pair of type (B; ) arrives, which occurs with probability (1
p)2 , the optimal mechanism
generates a surplus equal to UB . Suppose an (A; ) pair arrives. If st < 0, the mechanism
creates one (A; ) and one (B; ) pair, generating a surplus equal to UA + UB . If 0
st < k,
the mechanism creates no matches (and no additional surplus), and if st = k, the mechanism
creates one (A; ) pair and generates surplus equal to UA . Analogous conclusions pertain to
the case in which a (B; ) pair arrives.
Thus, a mechanism with threshold k generates an expected total surplus equal to
p2 UA + (1
= pUA + (1
UA + UB
2p(1 p)
k (UA + UB ) +
2
2k + 1
p(1 p)U
2k + 1
p)2 UB +
p)UB
and the resulting expected total welfare for one period of time, accounting for waiting costs,
is then:
12
p(1 p)U
2k(k + 1)c
(2)
2k + 1
2k + 1
When the cost of waiting c is high, waiting for potentially higher-quality matches does
pUA + (1
p)UB
not justify any waiting and the mechanism matches agents instantaneously as they arrive.
For su¢ ciently low costs, however, the optimal mechanism exhibits non-trivial waiting. The
optimal threshold k opt maximizes the welfare as given in (2).
The following proposition summarizes our discussion and provides the full characterization
of the optimal mechanism.
Proposition 2 (Optimal Thresholds). The threshold
$r
%
p(1
p)U
k opt =
2c
identi…es an optimal dynamic mechanism. In this optimal mechanism, all available (A; ) and
(B; ) pairs, and any number of (A; ) or (B; ) pairs exceeding k opt , are matched immediately.
Furthermore, the optimal mechanism is generically unique.15
Notice that when waiting costs are prohibitively high, namely when c >
maximal queue length is k
opt
p(1 p)U
,
2
the
= 0 and all matches are instantaneous.
In our characterization of the optimal mechanism, we do not worry about individual rationality constraints. This is tantamount to assuming that the outside option each agent faces, or
any agent’s utility from remaining unmatched, is su¢ ciently low such that the expected value
of waiting as prescribed by the mechanism is greater than leaving the market unmatched. In
Section 7.1 we return to this point and provide an explicit bound on the value of leaving the
market unmatched that would guarantee individual rationality.
3.3
Welfare
We now turn to the expected per-period welfare in a stationary state. Were we to consider no
costs of waiting, the optimal mechanism would naturally entail long waits to get the maximal
possible match surplus asymptotically by matching A-squares with -rounds and B-squares
15
When
q
p(1 p)U
2c
2 Z+ ; there are two optimal thresholds:
13
q
p(1 p)U
2c
and
q
p(1 p)U
2c
1:
with -rounds. We denote the resulting welfare by:
S1
pUA + (1
p)UB :
The optimal threshold identi…ed in Proposition 2 allows us to characterize the welfare
achieved by the optimal mechanism through equation (2) and to get the following corollary.16
Corollary 1 (Optimal Welfare). The welfare under the optimal mechanism is given by W opt (c) =
S1
(c); where
p(1
p)U for all c
(c) is continuous, increasing, and concave in c; lim (c) = 0; and
c!0
(c) =
p(1 p)U
:
2
As waiting costs approach 0; the welfare induced by the optimal mechanism approaches
S1 : For costs large enough, the optimal mechanism matches square-round pairs instantaneously as they arrive and the resulting welfare is S1
p(1
p)U: For intermediate costs,
the optimal mechanism generates welfare that is naturally in between these two values.17;18
The observation that welfare under the optimal mechanism decreases as c increases is rather
intuitive. Indeed, suppose c1 > c2 : Were we to implement the optimal mechanism with waiting
cost c1 when the waiting cost is c2 ; the distribution of matches would remain identical, while
waiting costs would go down, thereby leading to greater welfare overall. This implies that the
welfare under the optimal mechanism with waiting cost c2 ; which would be at least weakly
higher, is greater than that corresponding to waiting cost c1 : The amount by which welfare
decreases when waiting costs increase depends on the number of agents expected to wait in
line in the steady state. The higher the waiting costs, the lower the number of agents waiting
in line on average. Therefore, the impact of an increase in costs by a …xed increment is greater
at smaller costs, which leads to the concavity of
16
(c).
In the Appendix, we provide the analytical formula for (c) in terms of the fundamental parameters of
our setting.
17
Notice that the value of S1 is e¤ectively the analogue of the value generated by an “omnsicient”planner
in our setting, which is used as one benchmark in Akbarpour, Li, and Oveis Gharan (2014). The corollary
suggests that the omniscient planner’s value is a valid feasible benchmark when waiting costs vanish.
18
In fact, simple algebraic manipulations imply that:
p
p
S1
2p(1 p)U c c W opt (c) S1
2p(1 p)U c + c:
14
4
Decentralized Dynamic Matching
4.1
The Decentralized Process
Many dynamic matching processes are in essence decentralized: child adoption in the US and
abroad, job searches in certain industries, etc. It is therefore important to understand the implications of decentralized dynamics, particularly when considering centralized interventions.
In our decentralized matching process, we assume individuals join the market in sequence
and decide when to match with a potential partner immediately and when to stay in the
market and wait for a potentially superior match.
Formally, we assume that at each period t, there are three stages. First, a random square
and round enter the market with random attributes as before: with probability p the square
is an A-square and with probability p the round is an
-round. Second, each square and
round declare their demand –whether a square will match only with an -round, only with
a -round, or any round and whether a round will match only with an A-square, only with
a B-square, or with any square. In the third stage, given participants’demands, individuals
are matched in order of arrival. That is, the process follows a …rst-in-…rst-out protocol.19 Any
remaining participants proceed to period t + 1 (at an additional cost of c).20 As before, the
market e¤ectively entails four possible queues of rounds and squares of all types and we denote
by kx the length of the queue corresponding to an x type of agent, where x = A; B; ; .
For simplicity, we will assume from now on a symmetric setting. That is, we assume:
UA ( )
UA ( ) = U (A)
U (B) and UB ( )
UB ( ) = U (A)
U (B):
Last, we assume that the environment is regular in that
p (UA ( )
UA ( )) 6= kc and 2p(1
p) (U (A)
U (B)) 6= kc
for all natural numbers k 2 N: Regularity assures that neither squares nor rounds are ever
19
The assumption that matches are made in order of seniority in the market simpli…es our analysis substantially. There is some anecdotal evidence that order of arrivals in several markets a¤ects the sequence of
matching. For instance, in international adoption many countries follow a …rst-in-…rst-out criterion to match
abandoned children to adoptive parents. For example, see the protocol adopted by the China Center of Children’s Welfare and Adoption (CCCWA) here: http://www.aacadoption.com/programs/china-program.html.
20
The outcomes of the process we describe would be identical had we allowed participants to report no
demand at a period and ruled out weakly dominated actions.
15
indi¤erent between waiting in queue and matching immediately with an available partner.21
In what follows, we will focus on trembling-hand equilibria of the process. Roughly speaking, we do so to rule out situations in which a square and a round prefer to match with one
another immediately but both choose to reject one another since such mutual rejection is
robust to a unilateral deviation.22
4.2
Equilibrium Characterization
Given our assumptions on utilities, squares and rounds are always willing to match with rounds and A-squares, respectively, as soon as possible. Therefore, an A-square can always
match with a round immediately (indeed, an equal number of squares and rounds arrive
at the market and an equal number of them exit by matching, so there is always a round
available). If only -rounds are available, depending on the status of the queue, an A-square
may decide to wait in line for the chance of a superior match with an -round later on or to
match immediately with a -round. Consequently, a -round may wait in the queue either
by choice, hoping to match with a willing A-square at later periods, or by necessity, when
all squares in the queue are unavailable as they refuse to match with
-rounds. A similar
description holds for -rounds and B-squares.
4.2.1
Agents’Decision to Wait in Line
At the beginning of a period, the queue cannot entail both A-squares and -rounds, as that
implies an ideal match was not formed and waiting costs have been incurred at no bene…t to
some participants in the previous period. As before, we denote the (signed) length of the Aqueue at the beginning of a period by
kA
kA
k :
We …rst consider the decisions of an A-square (analogous analysis holds for an -round).
When an A-square arrives at the market and an -round is available (one that had either been
waiting in the queue or that has just arrived at the market as well), an A-square is matched
21
The assumption of regularity is not crucial and similar analysis follows without regularity for any arbitrary
tie-breaking rule. However, the presentation is far simpler for regular environments.
22
Such behavior cannot be ruled out by simply eliminating weakly dominated strategies without further
assumptions on the process.
16
immediately to an -round, the identity of whom is prescribed by the order of arrival. If the
arriving A-square is the …rst to arrive in line, that square is matched to an -round. If there
are A-squares already in queue, this implies that our A-square arrived with an -round and
the round will be matched with the …rst A-square in the queue. In that case, the A-square
has a choice of whether to wait in line or match with a -round. However, notice that this
square’s decision is equivalent to the last A-square who had arrived and decided to wait. In
that case, the new A-square waits and queues remain as they were.
Suppose now that an -round is not available when an A-square enters the market. This
implies that there is at least one -round available (that the square can match with). The
square decides to wait in the queue only based on the number of A-squares in the queue. An
immediate match with a -round delivers UA ( ), whereas waiting in line till matching with
an -round delivers UA ( ) at an uncertain cost of waiting.
It is worth noting that once an A-square decides to wait in the queue (rather than match
immediately with a -round), she will wait until matching with an -round (rather than leave
the queue by matching with a -round at a later point). Indeed, as matches form on a …rstin-…rst-out basis, her position in the queue moves up over time, and the expected time until
matching with an
-round becomes shorter. That is, if it is optimal for her to wait in the
queue upon entry, it is optimal for her to wait at any later period. The expected waiting time
till a match with an -round is therefore solely determined by the number of other A-squares
who precede the square in the queue.
The following lemma identi…es bounds on the size of the kA queue:
k dec
Lemma 2. In all periods,
k dec
kA
k dec where
kc
< UA ( )
p
kc
= max k 2 Z+ j
< U (A)
p
max k 2 Z+ j
UA ( )
U (B) :
Intuitively, the time till an -round enters the market is distributed geometrically (with
parameter p). The expected time till an
-round arrives at the market is therefore p1 : An
A-square who is k-th in line in the queue will be matched when k
is expected to occur within
k
p
-rounds arrive, which
periods. The expected waiting costs are therefore
generate an increase in match utility of UA ( )
17
kc
;
p
which
UA ( ) (relative to matching with a -round
immediately). An A-square will wait as long as the expected bene…t of waiting exceeds its
costs, which is the comparison underlying the maximal size of the queue as described in the
lemma. Our regularity assumption further guarantees that an A-square or an
-round are
never indi¤erent between waiting in line and matching immediately.
The lemma implies that A-squares’and -rounds’behavior is essentially captured by the
maximal queue length k dec . Whenever there are fewer than k dec A-squares in queue, a new
A-square will wait in the market; whenever there are k dec or more A-squares in the queue, that
square will prefer to match with a -round immediately. An analogous description holds for
-rounds and our symmetry assumptions assure that the maximal queue length is identical
for A-squares and -rounds.
We now turn to the decisions of B-square and -rounds. A -round (similarly, a B-square)
may decide to wait voluntarily by hoping to match with an A-square who will become available
when the line for A-squares exceeds k dec . In principle, there are two forces at work. The …rst is
similar to that experienced by the A-squares waiting in line: the longer the queue of -rounds
waiting ahead in line, the longer the new -round has to wait. The second force, however,
suggests that the longer is the queue, the closer are A-squares to reach the threshold k dec
and be matched with -rounds. The next lemma illustrates that the sum of these two forces
assures that B-squares and -rounds do not coexist in the market.
Lemma 3. There can never be both a B-square and a -round waiting in the market.
To understand the intuition of Lemma 3, consider the …rst -round arriving at the market.
In this case, there cannot be other A-squares waiting in the market since any such squares
would have arrived with
-rounds, in contradiction to our
-round being the …rst in line.
Suppose that the -round arrives with a B-square. To match with an A-square, she would
have to wait for the arrival of at least k dec +1 A-squares, since the …rst k dec A-squares will wait
in line if not matched with -round. This implies that the cost of waiting for the -round is
dec
(kdec +1)c
at least
, which is strictly greater than the cost k p c faced by the last A-square willing
p
to wait in the market for her desirable match. From our super-modularity assumption, the
bene…t from waiting is
U (A)
U (B) < U (A)
U (B) = UA ( )
18
UA ( ):
Therefore, since an A-square is not willing to wait for a
-round when facing a queue of
k dec + 1; nor would the -round. Similar calculations follow for longer queues of -rounds (see
Appendix). Given our focus on trembling-hand perfect equilibria, it follows that we can never
have both -rounds and B-squares stay in the market at the same time.
Lemmas 2 and 3 together imply that, in equilibrium, the decentralized process follows a
protocol similar to that implemented by the optimal mechanism and depicted in Figure 1,
though the thresholds governing them may be di¤erent. Indeed, whenever an A-square and
an -round, or a B-square and a -round, enter the market together, a match is generated
immediately (of either an (A; ) pair or a (B; ) pair, respectively) and the lengths of the
queues do not change in that period. The length of the queues can increase or decrease only
upon the arrival of an (A; ) pair or a (B; ) pair. As long as the queue (of either A-squares
or
-rounds) is strictly shorter than its maximum of k dec , the arrival of a new mismatched
pair can either increase or decrease the length of the queue by precisely one.
4.2.2
Steady State of Decentralized Matching
As for the optimal mechanism, the length of the A-
queue kA follows a Markov process.
Therefore, the transition matrix follows the description in Section 3.1. Denote by xt
(xtkdec ; xtkdec 1 ; :::; xt kdec +1 ; xt kdec )tr 2 f0; 1g2k
dec +1
the state of the market in period t –that is,
xti is an indicator that takes the value of 1 if the queue length is precisely i and 0 otherwise.
Then,
xt+1 = Tkdec xt ;
where Tkdec is that described in (1). Since this Markov chain is ergodic for any threshold, it is
characterized by a unique stationary distribution, which is uniform across the 2k dec + 1 states.
The following proposition provides the characterization of the equilibrium steady state of
the decentralized process.
Proposition 3 (Decentralized Steady State). There is a unique trembling-hand equilibrium
under the decentralized matching process. This equilibrium is associated with a unique steady
state distribution over queue lengths, such that the length of the A- queue kA = kA
uniformly distributed over f k
dec
; k
dec
+ 1; : : : ; k
either
19
dec
k is
g, and in any period, the queues contain
1. kA A-squares and -rounds, and no -rounds and B-squares, or
2. jkA j -rounds and B-squares, and no A-squares and -rounds.
The threshold k dec is determined by the decisions of A-squares and -rounds to wait, as
speci…ed in Lemma 2. The crucial di¤erence between the decentralized and optimal mechanism
is the threshold placed on the maximal stock of A-squares or -rounds in waiting. Notice that
a decision to wait in the market imposes a negative externality on succeeding agents, as
it potentially a¤ects their waiting time, and possibly the quality of their matches.23 In the
decentralized process, agents do not internalize this externality and therefore wait “too much”.
Recall that we denoted by k opt the threshold characterizing the optimal mechanism (speci…ed
in Proposition 2). We then have the following corollary:
Corollary 2 (Threshold Comparison). Maximal waiting queues are longer in the decentralized
process than they are under the optimal mechanism. That is, k opt
k dec :
As mentioned, in Section 7.1, we handle individual rationality constraints explicitly. Namely,
we provide a bound on the value of leaving the market unmatched that assures our decentralized process (and, from the corollary above, also the optimal mechanism) is individually
rational.
4.3
Welfare
Since the protocols are similar except for the queues’ thresholds, the expected per-period
welfare in the steady state characterized in Proposition 3 can be found using an analogous
derivation to that carried out for the optimal mechanism. This derivation leads to an expression mirroring equation (2), accounting for the decentralized process’threshold k dec : That is,
the expected per-period welfare is given by:
W dec (c) = S1
p(1 p)U
2k dec + 1
23
2k dec (k dec + 1)c
;
2k dec + 1
From a welfare perspective, the externality on the quality of the match is of less importance. As long as
a social planner views identical agents as interchangeable, an immediate mismatch or a later mismatch have
similar welfare consequences.
20
where, from Lemma 2,
k dec =
p (UA ( )
UA ( ))
c
=
p (U (A)
U (B))
c
:
(3)
To summarize, we have the following corollary:
Corollary 3 (Decentralized Welfare). The welfare under the decentralized process is given by
W dec (c) = S1
c
p (UA ( )
(c); where lim (c) = p (UA ( )
c!0
UA ( )) ; and
(c) = p(1
p)U for all
UA ( )) :
Recall Corollary 1, which characterized the welfare under the optimal mechanism. By
de…nition, the welfare generated under the optimal mechanism is higher than that generated
by the decentralized process, so that
(c): In contrast with the optimal mechanism
(c)
that generates welfare that is decreasing in waiting costs, this is not necessarily the case
under the decentralized process. Furthermore, while the welfare under the optimal mechanism
approaches S1 as waiting costs diminish, this is not the case under the decentralized process.
As waiting costs become very small, there is a race between two forces. For any given threshold,
the overall waiting costs decline. However, in equilibrium, decentralized thresholds increase,
leading to greater expected wait times. As it turns out, the balance between these two forces
generates signi…cant welfare losses, given by p (UA ( )
UA ( )), even for vanishingly small
costs. The next section provides a detailed comparison of the two procedures in terms of
welfare.
5
Welfare Comparisons
By construction, the optimal mechanism generates welfare that is at least as high as that
generated by the decentralized process. In this section we inspect how the welfare wedge
responds to the underlying parameters of the environment, suggesting the settings in which
centralized intervention might be particularly useful.
The following proposition captures the e¤ect on the welfare wedge W opt (c)
W dec (c) of
both waiting costs c, the frequency p of A-squares or -rounds, and the utility bene…t for an
-round from matching with an A-square rather than a B-square (equivalently, the utility
bene…t for an A-square from matching with an -round rather than a -round).
21
Proposition 4 (Welfare Wedge –Comparative Statics).
1
1. For any interval [c; c); where c > 0; there is a partition f[ci ; ci+1 )gM
i=1 ; where c =
c1 < c2 < ::: < cM = c; such that W opt (c)
W dec (c) is continuous and increasing over
(ci ; ci+1 ) and
W opt (ci )
for all i = 1; :::; M
W dec (ci ) > W opt (ci+1 )
W dec (ci+1 )
1:
2. As c becomes vanishingly small, the welfare gap W opt (c)
that is increasing in p 2 [0; 1) and in U (A)
W dec (c) converges to a value
U (B):
To see the intuition for the comparative statics corresponding to waiting costs, notice that
an increase in costs has two e¤ects on the welfare gap. Since the equilibrium threshold under
the decentralized process is greater than the optimal threshold (Corollary 2), an increase in
waiting costs has a direct e¤ect of magnifying the welfare gap. Nonetheless, there is also
an indirect e¤ect of an increase in waiting costs that arises from the potential changes in the
induced thresholds. Consider a slight increase in waiting costs such that the optimal threshold
does not change, but the decentralized threshold decreases. In some sense, the decentralized
process is then “closer” to the optimal process –both the matching surplus and the waiting
costs are closer and the welfare gap decreases. In fact, as costs become prohibitively high,
both processes lead to instantaneous matches and identical welfare levels. As we show in
the proof of Proposition 4, the indirect e¤ect overwhelms the direct e¤ect at precisely such
transition points and acts to shrink the welfare gap. The construction of the partition is
done as follows. Each atom [ci ; ci+1 ) of the partition corresponds to constant thresholds under
the decentralized process. Over these intervals, the direct e¤ect described above dominates
and the welfare gap is increasing. Each of the endpoints fci gi corresponds to a decrease of
the decentralized threshold by one. That is, the decentralized threshold is not constant over
[ci ; ci+2 ) for all i: Therefore, when comparing two such endpoints, the indirect e¤ect kicks in
and the decreasing trend of the welfare gap emerges. Figure 2 depicts the resulting pattern
the welfare gap exhibits for UA ( ) = U (A) = 3; UA ( ) = U (B) = UB ( ) = U (A) = 1;
UB ( ) = U (B) = 0; and p = 0:3: As suggested by the proposition, W opt (c)
piece-wise increasing in c. Nevertheless, overall, the gap has a decreasing trend.
22
W dec (c) is
Figure 2: Welfare Gap between Optimal and Decentralized Matching as a Function of Costs
To glean some intuition on the comparative statics the welfare gap displays with respect
to p, consider two type distributions governed by p1 and p2 such that p1 < p2 = mp1 ; m > 1.
Suppose further that costs are so low that the match surplus in the decentralized process is
very close to the optimum, S1 : The individual incentives to wait are higher under p2 than
under p1 : In fact, in the decentralized setting, the distribution of steady state queue length
is the uniform distribution where, from (3), under p2 ; roughly 1
1=m of the probability
mass is allocated to queue lengths larger than those realized under p1 . For each of these large
steady state queue lengths, we have more pairs of agents waiting, i.e., increased per-period
waiting costs. The optimal mechanism internalizes the negative externalities, so the e¤ect
of the increased waiting costs is weaker. On the other hand, the bene…t of this increase in
queue length is a lower chance of producing mismatches. However, for su¢ ciently low c;
the match surplus under p1 is already close to its optimum of S1 and this e¤ect is weak;
in particular, the di¤erence in terms of match surplus that the optimal and decentralized
processes generate is similar under p1 and p2 . Therefore, for su¢ ciently low c; the dominant
e¤ect is the one produced by the di¤erential in expected waiting costs, which generates our
comparative statics. Notice that if p = 0 or p = 1; both the optimal mechanism and the
23
decentralized processes generate the same welfare.24
Going back to our assumption of super-modular preferences, note that the construction of
the optimal mechanism would remain essentially identical were preferences sub-modular (with
an appropriate relabeling of market participants). However, in the decentralized setting,
sub-modular preferences would lead to a negative welfare e¤ect compounding the negative
externalities present in our setup. Namely, individual incentives would be misaligned with
market-wide ones. In that respect, our comparison of optimal and decentralized processes
assuming super-modular preferences is a rather conservative one.
Similarly, considering waiting costs that di¤er across the two sides of the market, which may
be reasonable for many applications, would lead to a greater welfare wedge as well. Intuitively,
suppose that squares experience a waiting cost of c1 and rounds experience a waiting cost of
c2 ; where c1 > c2 ; with an average cost of c = (c1 + c2 )=2: The optimal mechanism with
asymmetric costs would coincide with that corresponding to identical costs of c since per-pair
costs are the same in both cases: The decentralized process would also lead to an equilibrium
similar to that we described, but thresholds would no longer be symmetric. Indeed, A-squares
would be willing to wait when the queue of A-squares is no longer than kSdec and
-rounds
would be willing to wait when the queue of -rounds is no longer than kRdec ; where
kSdec =
p (UA ( ) UA ( ))
c1
and kRdec =
p (U (A) U (B))
:
c2
From convexity, it follows that the threshold k dec corresponding to identical costs of c satis…es
k dec
(kSdec + kRdec )=2: Therefore, the excessive waiting decentralized processes exhibit would
be even more pronounced when costs are asymmetric across market sides.25
6
Transfers and Fixed Matching Windows
So far, we have shown that intervention in dynamic matching markets can have a substantial
impact on welfare, at least when centralization is done using the optimal dynamic mechanism.
However, the full-‡edged optimal mechanism may be hard to implement. It requires that the
formation of matches, even those of individuals who would prefer to wait in the market,
24
25
In particular, there is a discontinuity of the welfare gap around p = 1:
We provide a full characterization of this case in the Online Appendix.
24
be within the purview of the centralized planner. It also requires the central planner to
monitor the market continuously to determine when matches should be formed, which may
be administratively costly. In this section we show that improvements to decentralization can
be achieved by mechanisms that relax each of these two requirements. We start by showing
that a tax scheme imposed on those who wait in the market can yield the optimal waiting
patterns, if tailored appropriately. We then analyze a simple centralized matching protocol
that, while not optimal, can provide substantial welfare improvements over the decentralized
matching process, and does not require the clearinghouse to monitor the market continuously.
Namely, we consider matching processes in which the centralized clearinghouse matches all
available agents every …xed number of periods.
6.1
Optimal Taxation
Certainly, a …xed per-period tax on waiting
0 can be set so that the resulting de-facto
cost of waiting, c + , is such that the decentralized process generates the optimal threshold
(with costs c).26 Namely,
can be set so that
p (UA ( ) UA ( ))
=
c+
r
p(1
p)U
2c
:
(4)
In fact, if collected taxes in period t are given to new entrants in period t + 1; they would
have no e¤ects on either overall welfare or individual incentives. Nonetheless, there is a risk
that such a policy would introduce a strong incentive for agents to enter the market only in
order to gather the previous generation’s taxes. As it turns out, there is a tax scheme that
is budget balanced such that the expected tax (or subsidy) for an entering agent who is not
privy to the pattern of queues in place is nil. Under such a scheme, no agent is tempted to
enter the market only for the sake of reaping the bene…ts of taxes.
Indeed, consider a linear tax scheme –an agent who is k-th in line pays
k to the matching
institution, regardless of whether they are a square or a round and regardless of their type.
To achieve budget balance, the collected taxes are equally redistributed back to the existing
agents in the market in each period.
^ the resulting net tax (that is added to the …xed cost c) for
When the length of queue is k,
26
This is always possible from Corollary 2.
25
an agent who is k-th in line is
k
2
Pk^
k=1
2k^
k
=
k
k^ + 1
2
!
:
^
In particular, the net added tax on waiting for the last person in the queue is (k2 1) ; which
^ Thus, using the de…nition of from equation (4), we
is increasing in the queue’s length k.
want the tax levied on the last agents in the queue of length k opt to satisfy
(k opt
2
1)
=
()
=
2
k opt
1
:
Such a taxation policy might still be di¢ cult to administer in terms of the …nancial activity
it would entail. Without taxation, the optimal mechanism may be viewed as too complex for
certain applications as it requires matchings that depend on the state of the market. Indeed,
optimality dictates matchings that occur at points in time that are not perfectly predictable
at the outset of the process and the clearinghouse needs to monitor the queues of individuals
continuously. This complexity could translate to excessive administrative burdens. In the next
subsection we o¤er the analysis of a simple mechanism that, while not optimal, can generate
substantial welfare improvements relative to the decentralized process.
6.2
Matching with Fixed Windows
We now consider the class of mechanisms that are identi…ed by a …xed-window size – every
…xed number of periods, the e¢ cient matching for the participants arriving at the market in
that time window is formed and the market is cleared. Larger windows then allow for thicker
markets and potentially more e¢ cient matchings. However, larger windows also correspond to
longer waiting times for all participants. We identify the optimal …xed window and illustrate
that it can generate substantially greater welfare than that produced by the decentralized
process. However, the welfare it generates is still notably lower than that produced by the
optimal mechanism.
The arrival of squares and rounds is the same as that analyzed so far. When …xed matching
windows are used, a window size n governs the process. Namely, every n periods, the most
e¢ cient matching pertaining to the n squares and n rounds who arrived within the window
is implemented.
26
Suppose that kA and k are the numbers of A-squares and -rounds who arrived during
a window of n time periods, respectively. Given our assumptions on match utilities, e¢ cient
matchings correspond to a unique distribution of pair-types (number of (A; ) pairs, (A; )
pairs, etc.) and generate the maximal number of (A; ) and (B; ) pairs. The total matching
surplus generated by a matching as such (the generated welfare not accounting for waiting
costs) is
k UA + (kA k )UA + (n kA )UB if kA k ;
kA UA + (k
kA )UB + (n k )UB otherwise.
Consider now the expected waiting costs when the window size is n: The …rst square and
S(kA ; k )
round to arrive wait for n
1 periods, the second square and round to arrive wait for n
2
periods, etc. Thus, the total waiting cost is
2 c ((n
1) + (n
2) +
+ 0) = cn(n
1):
Therefore, the expected per-period welfare for each square-round pair generated by a
window size n is
Wn
1 X
n 0 k ;k
A
c(n
n
n
kA
n kA
pA (1
k
pA )n
kA k
p (1
p )n
k
S(kA ; k )
1):
Notice that for any window size n, the matching surplus S(kA ; k ) is at most nUA ; while
expected waiting costs are cn(n
nUA
cn(n
1): Denote by nmax the largest window size n such that
1): Every window of size n > nmax would then generate a lower welfare than
that generated by a window of size 1; corresponding to instantaneously matching individuals.
In particular, an optimal window size exists within the …nite set f1; 2; :::; nmax g:
6.2.1
Optimal Window Size
A characterization of the precise optimal window size is di¢ cult to achieve. In what follows,
we describe bounds on the optimal window size. In turn, these bounds land themselves to
bounds on the maximal expected welfare generated by using …xed windows, which we can
then compare to the welfare generated by the optimal mechanism as well as the decentralized
process.
Consider the ex-ante marginal bene…t produced by increasing the window size from n to
n + 1 that is incurred by the …rst n square-round pairs who arrive at the market. Suppose that
27
an e¢ cient matching corresponding to the …rst n pairs generates at least some mismatches
(i.e., (A; ) or (B; ) pairs). The (n + 1)-th pair could be bene…cial for the …rst n pairs by
correcting a mismatch. If there is no mismatch among the …rst n pairs, expanding the window
size only leads to additional waiting costs for the …rst n pairs.
Denote the probability that any e¢ cient matching with the …rst n squares and rounds has
a mismatch (i.e., there is an unequal number of A-squares and -rounds) by
k j
P r (jkA
1; n) :
Since we assumed identical distribution of types of squares and rounds, an e¢ cient matching from n pairs has mismatches of type (A; ) or (B; ) with equal probability. A mismatch of
type (A; ) is corrected by a new (n + 1)-th pair of type (B; ), which occurs with probability
p(1
p), and the total bene…t for the pair of originally mismatched square and round is
UA ( ) + U (B)
UA ( )
U (A):
A similar derivation follows for the bene…t of “correcting” a mismatched pair of type (B; ):
Conditional on any e¢ cient matching entailing a mismatch, the expected bene…t to the …rst
n square-round pairs is then
p(1
p)
2
+
p(1
p)
2
(UA ( ) + U (B)
UA ( )
U (A))
(U (A) + UB ( )
U (B)
UB ( )) ;
which is equal to
p(1
p) U
:
2
Thus, the ex-ante marginal welfare from expanding the window size from n to n + 1 for each
of the n …rst square-round pairs is:
+ Wn
p(1
p) U
P r (kA 6= k ; n)
2n
2c:
The optimal window size no is then determined by
+ Wno
0
+ Wno 1 :
That is, it is (weakly) bene…cial to increase the window size from no
bene…cial to increase the window size from no to no + 1.
28
(5)
1 to no , and it is not
While it is di¢ cult to accomplish a closed-form solution for the optimal window size, the
following proposition utilizes inequality (5) to establish bounds on no .27
Proposition 5 (Optimal Window Size).
1. An upper bound for the optimal window size is given by
no
p(1
p)U
4c
:
2. For any " > 0, there exists c such that if c < c , a lower bound for the optimal window
size is given by
no
p(1 p)U
:
(4 + ")c
Recall that U captures the extent to which preferences exhibit super-modularity, the welfare advantage of an assortative matching relative to an anti-assortative one. Intuitively, the
bounds on the optimal window size increase with U and decrease with the cost of waiting. To
understand the dependence on p(1
p); let m be the number of pairs in the market of type
(A; ) or (B; ): The designer’s aim is to achieve a small number of such mismatches. Clearly,
when m is odd, the probability of at least one mismatch is one. Suppose m is even, m = 2l:
1 2l
2
2l
l
The conditional probability of having a mismatch is then 1
which is increasing in
l: If we focus on large n; the probability of an even or an odd m are approximately equal.
Furthermore, when p(1
p) increases, there is a greater probability of (A; ) or (B; ) pairs
and the expected value of m is higher. Fixing all else, a greater p(1
p) would then intuitively
lead to a lower match surplus. Thus, the optimal window size grows with p(1
p):
Notice that Proposition 5 implies that the optimal window size is smaller than the threshold
induced by the decentralized process, at least for su¢ ciently low costs. That is, no < k dec
for low waiting costs. In particular, the bounds we identify in Section 7.1 on the value of
leaving the market unmatched that guarantee individual rationality under the decentralized
process, also assure individual rationality under the …xed-window mechanism. It also follows
27
The di¢ culty stems from the fact that inequality (5) depends crucially on
P r (kA 6= k ; n) = 1
n
X
l=0
n l
p (1
l
2
p)n
l
;
and state-of-the-art combinatorics has little to say about this function’s behavior with changes in n:
29
that the choice of the optimal window size allows for some alleviation of the excess waiting
the decentralized process exhibits. Indeed, as we show in the next subsection, the welfare
generated by the optimal …xed window dominates that produced by the decentralized process.
6.2.2
Welfare Bounds for Fixed Windows
If the waiting cost c is small, so that the optimal window size is large, we expect to have an
approximate fraction p of A-squares and -rounds, in which case per-pair surplus is close to
S1 : The bounds on the optimal size of the window allow us to provide bounds on how far the
match surplus is from S1 and how costly the wait is. The following proposition illustrates
the resulting bounds on the welfare generated by the optimal window size.
Proposition 6 (Fixed-Window Welfare).
1. A lower bound on the optimal welfare is given by
r
2c
W f ix (c) S1
(UA
UB )
U
p(1
p)U
2
+ 2c:
2. For any " > 0; there exists c such that if c < c , an upper bound on the optimal welfare
is given by
W f ix (c)
S1 +
r
(2 + ")c
(UA
U
p(1 p)U
+ 2c:
2+"
UB )
Notice that as the cost of waiting becomes very small, some of the terms in the above
bounds vanish. Furthermore, the bounds converge to one another and we get:
limW f ix (c) = S1
c!0
Intuitively, p(1
p(1
p)U
2
:
p) re‡ects the probability of a mismatched pair entering the market.
As that probability grows, welfare decreases. The value of U re‡ects the value of e¢ ciently
matching individuals. Mismatches are more costly when U is higher, and so U a¤ects the
expected welfare negatively as well.
Proposition 6 allows us to compare the performance of the optimal …xed-window mechanism with the performance of the optimal mechanism and the decentralized process.
30
Corollaries 1 and 3 illustrate the limit values of W opt (c) and W dec (c) as costs become very
small. These limits combined with Proposition 6 lead to the following corollary.
Corollary 4 (Relative Performance of Fixed Window Mechanisms).
W opt (c)
c!0 W opt (c)
lim
W f ix (c)
(1 p)U
=
:
dec
W (c)
2 (UA ( ) UA ( ))
Notice that the right-hand side of the equality in Corollary 4 is lower than 1. Therefore, the
corollary suggests that the optimal …xed-window mechanism potentially provides a substantial
improvement over the decentralized process. Whenever 1
p; U > 0; the right hand side of
the equality in Corollary 4 is strictly positive, which implies that the welfare generated by
the optimal …xed-window mechanism is still signi…cantly lower than that generated by the
optimal mechanism. As p decreases, or U increases, the welfare generated by the …xed-window
mechanism provides a greater improvement over the decentralized process.
7
Conclusions and Extensions
We considered a dynamic matching setting and identi…ed the optimal matching mechanism
in an environment as such. The optimal mechanism always matches immediately congruent
pairs and holds on to a stock of mismatched pairs up to a certain threshold. When matching
follows a decentralized process, a similar matching protocol ensues in equilibrium, but the
induced thresholds for waiting in the market are larger as individuals do not internalize the
negative externalities they impose on those who follow. This di¤erence generates a potentially
signi…cant welfare wedge between decentralized processes and centralized clearinghouses, even
when waiting costs are vanishingly small. Our results provide guidance as to the features of
the economy that could make centralized intervention more appealing.
We also o¤er several simple interventions to decentralized markets – transfer schemes
that can induce optimal outcomes, and …xed-window mechanisms, which are arguably far
less complex than the full-‡edged optimal mechanism and can provide substantial welfare
improvements relative to decentralization.
In what follows, we discuss two channels by which our results could be expanded. First,
we discuss individual rationality constraints and illustrate the class of utilities under which
31
all of the processes discussed in the paper are individually rational. Second, we describe how
our techniques can be used for more general asymmetric environments.
7.1
Individual Rationality
Throughout the paper we assumed that agents leave the market only after being matched.
Agents do not leave the market unmatched, regardless of their expected utility. Certainly,
if remaining unmatched generates zero utility, our assumption violates individual rationality
as some agents (in particular, B-squares or -rounds) sometimes stay in the market simply
since no other available agent will match with them. These agents could consequently earn
a negative expected utility even when Ux (y) > 0 for all x; y. Assume that any agent prefers
to match with any other agent immediately over remaining unmatched. We now provide a
bound on the value of remaining unmatched, or the value of an outside option agents have,
that assures all the matching protocols we discuss are individually rational.
Notice that since, from Corollary 2, the decentralized threshold is higher than the optimal
threshold, it su¢ ces to …nd such a bound that guarantees that the decentralized process is
individually rational.
In the decentralized process, A-squares or -rounds always have the possibility of matching
with -rounds or B-squares instantaneously when they decide to wait in the market. Therefore, when deciding to wait they expect an even greater utility and individual rationality holds
for them. Consider now -rounds (analogously, B-squares). A -round who enters as k-th in
line can always declare all squares as acceptable in each period. Notice that, by construction,
k
k dec : The time between arrivals of B-squares is distributed geometrically with probability
1
p: Therefore, with such a strategy, the expected time for the -round to match with a
B-square is at most k=(1
p); yielding a match utility of U (B): The wait time till match-
ing with a B-square could be even shorter if -rounds who precede the -round in question
are willing to match only with A-squares. Furthermore, the -round could end up matching
with an A-square before k B-squares arrive at the market. It follows that such a strategy
guarantees an expected utility of at least
U (B)
kc
1
p
U (B)
k dec c
1 p
U (B)
32
p
1
p
[U (A)
U (B)]
U min :
If -rounds follow a di¤erent strategy in equilibrium, it must be that their utility is at least as
high. Therefore, as long as the value of remaining unmatched is lower than U min , individual
rationality holds under both the optimal and the decentralized processes (analogous calculations follow for B-squares and, under full symmetry of utilities, the bound corresponding to
them is also U min ). In addition, since the optimal …xed-window size is smaller than the decentralized equilibrium threshold, the construction above assures that this bound on remaining
unmatched assures individual rationality for the …xed-window protocol as well.
7.2
Introducing Asymmetries
Our symmetry assumptions, while potentially relevant for several applications, may appear too
strong for others. As it turns out, relaxing symmetry complicates somewhat the analysis, but
similar techniques to those introduced in the paper can be utilized and most of our qualitative
results carry through. In this section we focus on asymmetries in the distribution of types
among squares and rounds. In the Online Appendix, we provide a more general analysis of the
asymmetric environment, allowing for both asymmetries in payo¤s and in type distributions.
Suppose, without loss of generality, that the probability of a square being an A-square is
pA and that the probability of a round being an -round is p ; where pA > p : Let us …rst
look at the optimal mechanism. Absent waiting costs, the optimal mechanism would entail
very long waits in order to achieve the maximal feasible welfare of:
S1 = p UA + (pA
p )UA + (1
pA )UB :
Now, consider the following mechanism for waiting costs c > 0: For each A-square that arrives
at the market, the mechanism relabels the square as a B-square with probability 1
p =pA ;
while all other squares and rounds maintain their original labels. The resulting market (with
the altered labels) is symmetric – there are p squares labeled as A-squares and p rounds
labeled as -rounds. Suppose further that the mechanism under consideration then mimics the
optimal mechanism for this symmetric distribution as described in Section 3. From Corollary
1, it follows that as waiting costs diminish, the welfare from this mechanism generates welfare
arbitrarily close to the maximal feasible for the symmetric distribution, i.e., a fraction close to
one of the squares labeled as A-squares are matched with -rounds. This, in turn, implies that
33
the mechanism achieves welfare approaching S1 in the asymmetric market as waiting costs
become very low. In particular, even in the asymmetric market, for low costs, the optimal
mechanism generates welfare close to that maximally feasible, in analogy with Corollary 1.
For intermediate costs, the structure of the mechanism is similar to that identi…ed in
Section 1. However, the thresholds corresponding to A-squares and -rounds may be di¤erent.
Intuitively, consider the case as above in which pA > p : In such a setting, the expected time
between any two -rounds arriving at the market is smaller than the expected time between
any two A-squares arriving at the market and so the willingness to queue by A-squares would
be lower than that of -rounds.
Formally, for any threshold pair (kA ; k ); there is a transition process analogous to that
described in Section 3.1 through the transition matrix in (1) as follows. Denote by xt
(xtkA ; xtkA 1 ; :::; xt k
1(st = i) –that is,
; xt k )tr 2 f0; 1gkA +k
+1
xti is
+1
the timed vector capturing the state, xti =
an indicator that takes the value of 1 if the state is st (i.e., the queue
length kA is st ) and 0 otherwise. Then,
xt+1 = TkA ;k xt ;
where
TkA ;k
0
B
B
B
B
B
=B
B
B
B
@
1
(1 pA )p
(1 pA )p
0
..
.
0
0
0
pA (1 p )
pA p +(1 pA )(1
(1 pA )p
:::
p ) :::
:::
0
0
0
0
0
0
..
.
..
.
..
.
..
0
0
0
:::
pA (1 p )
: : : pA p +(1 pA )(1
:::
(1 pA )p
.
0
p )
pA (1 p )
1 pA (1 p )
The above Markov chain is ergodic and, as in the symmetric case, an optimal mechanism
corresponds to a matching process that reaches a steady state with a unique stationary distribution over states. In the Online Appendix, we derive the stationary distribution of any
such process and illustrate how to determine the optimal threshold. We also show how, as
waiting costs become small, a one-threshold mechanism approximates the optimal mechanism.
Namely, for small costs a policy maker could consider a mechanism in which -rounds wait
in the market regardless of the length of the queue of -rounds ahead of them and a cap kA
is (optimally) placed on the number of A-squares who might be waiting in line.
34
1
C
C
C
C
C
C;
C
C
C
A
The analysis of the decentralized process is also analogous to the one described in the body
of the paper and we can associate the matching process with the Markov process governed by
TkA ;k as above, though with thresholds that are determined in equilibrium. As waiting costs
diminish, the distance between the resulting welfare and the maximal feasible S1 is bounded
below by p (UA ( ) UA ( )); which mirrors the result of Corollary 3 for the symmetric setting.
In particular, the wedge between the welfare generated by the optimal mechanism and the
decentralized process can be substantial even when waiting costs are minuscule.
8
Appendix
8.1
Proofs Regarding the Optimal Mechanism
Proof of Proposition 1: By Lemma 1, under the optimal mechanism, at any point in time,
either only A-squares and -rounds, or only B-squares and -rounds are waiting in the market.
Consider a queue that contains only A-squares and -rounds, and consider a scenario in
which an (A; ) pair enters the market (if any other pair enters the market, matches should be
immediately implemented). The expected welfare at this time period is uniquely determined
by the number of (A; ) pairs created and the number remaining agents waiting in line until
the next period. From the stationarity of the process, and since waiting costs are linear, if it
is optimal to retain k (A; ) pairs in the market at some period, it is also optimal to retain
any k 0
k such pairs at any other periods. Let kA be the largest number of (A; ) pairs
that might be allowed to wait in line. Such a number clearly exists as the optimal mechanism
cannot allow an inde…nite number of (A; ) pairs to wait in line for one more period. The
value of kA corresponds to the threshold that the optimal mechanism uses: i.e., whenever the
queue of A-squares (or -rounds) gets longer than kA , the mechanism creates (A; ) matches
such that exactly kA number of A-squares (or -rounds) wait for one more period. Similar
arguments follow for queues composed of (B; ) pairs.
Proof of Proposition 2: De…ne the marginal increase of welfare when increasing the
threshold by one as:
M W (k)
W (k + 1)
2 p(1 p)U 2c(k + 1)2
W (k) =
:
(2k + 1)(2k + 3)
35
For a non-trivial (i.e., non-zero) optimal threshold, it is necessary that M W (0) > 0, or
equivalently c <
p(1 p)U
:
2
Suppose c is small enough so that this is the case. Ignoring the integer constraint on k,
d2 W (k)
1
=
2
dk
(2k + 1)3
2c
p(1 p)
< 0 as 2p(1
4U
p)U > c:
Thus, accounting for the optimal threshold being an integer, there exists either a unique
optimal threshold or two adjacent optimal thresholds.
An optimal threshold k opt satis…es either
M W (k opt
M W (k opt
1) > 0 and M W (k opt ) 0; or
1) 0 and M W (k opt ) < 0:
We rewrite the two conditions as
q
p(1 p)U
q 2c
1
p(1 p)U
2c
k
opt
1 < k opt
and the Proposition’s claim follows.
<
q
q
p(1 p)U
;
2c
or
p(1 p)U
;
2c
Proof of Corollary 1: Using the optimal thresholds from Proposition 2, we get that for
c
p(1 p)U
;
2
v(c)
w(c)
We can then de…ne
p(1 p)U
=
2k opt + 1
2
p(1
q
p)U
p(1 p)U
2c
; and
+1
(2k opt + 1)c
c
2k opt (k opt + 1)
c
=
=
opt
opt
2
2k + 1
2(2k + 1)
2
3
$r
%
!
6
7
p(1 p)U
1
1
7 c:
= 6
+
q
4
5
2c
2
p(1 p)U
4
+
2
2c
(c) = v(c) + w(c)
to get the representation of W opt (c) in the corollary.
36
p(1 p)U
for which
2
0
opt
28
Take any c <
jc c0 j < , k opt (c ) = k
(c).
k opt 2
= Z+ . There exists " > 0 such that for every c0 with
Thus, W opt is di¤erentiable at c. Moreover, for any c <
p(1 p)U
2
such that k opt 2 Z+ , W opt (c) is semi-di¤erentiable. Hence, W opt (c) is continuous.
At any di¤erentiable point c (around which k opt (c) is constant)
dW opt
@W (k opt (c); c)
=
=
dc
@c
2k opt (k opt + 1)
< 0:
2k opt + 1
Furthermore, convexity of W opt (c) follows from the fact that at any semi-di¤erentiable but
not di¤erentiable point c,
8.2
d W opt
d+ W opt
<
:
dc
dc
Proofs Regarding Decentralized Processes
Proof of Lemma 2: The time till an -round appears in the market is distributed geometrically, and so the expected time for arrival of any -round is given by p1 : Consider an A-square
who enters the market as the k-th in line when only -rounds are available. The A-square
will wait in the queue if and only if
kc
< UA ( )
p
UA ( ):
From regularity, the A-square is never indi¤erent between waiting in queue and matching
immediately, and the lemma’s conclusion follows. Analogous analysis holds for -rounds.
Proof of Lemma 3:
Consider the …rst moment the market contains at least one -round and at least one Bsquare and suppose the queue of -rounds is of length k: We show that at least one -round
is willing to match with a B-square immediately. Toward a contradiction, suppose this is
not the case. Since A-squares and -rounds cannot be waiting in the market simultaneously,
and not all squares are A-squares (as there is at least one B-square available), it follows that
kA < k: Notice that if a -round is not willing to match with a B-square at a certain period,
she will not be willing to match with a B-square in future periods as well. Thus, for the
28
We slightly abuse our notation and make the dependence of k opt on the cost c explicit here.
37
last -round in queue to match with an A-square, all preceding k
to match with A-squares …rst. From Lemma 2, at least k dec
of A-squares would therefore be necessary for the last
A-square. Hence, the expected waiting cost for that
1 -rounds would need
k dec + 1 arrivals
kA + k
-round in queue to match with an
-round is at least
(kdec +1)c
:
p
However,
by super-modularity, the marginal bene…t for the -round from matching with an A-square
instead of a B-square, U (A)
U (B); is lower than U (A)
U (B). Since an -round is not
willing to wait to match with an A-square when faced with a queue longer than k dec , the last
-round in queue would prefer not to wait either, in contradiction.
Similar arguments follow for the B-squares in the market. Therefore, in any tremblinghand equilibrium, there cannot be both B-squares and -rounds waiting in the market.
Proof of Proposition 3: First, the (signed) length of the A-
queue, denoted by kA
constitutes an ergodic Markov chain. Following previous arguments, the unique steady state
distribution of kA is the uniform distribution over f k dec ; k dec + 1; : : : ; k dec g.
At any time t, suppose that kA > 0. Clearly, the queue has no
-rounds. As equal
numbers of squares and rounds enter and exit the market, it must be that kA + kB = k :
Lemma 3 guarantees that kB = 0; therefore k = kA . Similarly for kA
Proof of Corollary 2: Whenever c >
agents immediately and k opt = 0 and k opt
p(1 p)U
;
2
dec
k
0:
the optimal mechanism matches arriving
:
Suppose, then, that c < p(1 2p)U : We then have
r
p(1 p)U
pU
p(1 p)U
<
2c
2c
2c
p(UA ( )
UA ( ))
c
:
and the result follows from the de…nition of k opt and k dec :
8.3
Proof Regarding Welfare Comparisons
Proof of Proposition 4:
1. As in the proof of Corollary 1, W opt (c) W dec (c) is di¤erentiable at any c <
p(1 p)U
2
such
that k opt ; k dec 2
= Z+ . In a small neighborhood around any such c, the thresholds corresponding
to both the optimal and decentralized thresholds are constant in c. Therefore,
d(W opt (c) W dec (c))
=
dc
2k opt (k opt + 1) 2k dec (k dec + 1)
+
2k opt + 1
2k dec + 1
38
0;
where the inequality follows from Corollary 2, which implies that k dec
k opt . Furthermore,
the proof of Corollary 1 implies that W opt (c) is continuous in c and so W opt (c)
W dec (c) is
increasing in any point c for which k dec 2
= Z+ :
Denote by:
p (UA ( )
k dec (c)
UA ( ))
and k opt (c)
c
$r
p(1
p)U
2c
%
;
essentially explicitly expressing the dependence of both the decentralized and optimal thresholds on the waiting costs, respectively.
Let fdk g1
k=1 denote the decreasing sequence of costs such that k =
p(UA ( ) UA ( ))
:
dk
That is,
cost dk corresponds to a threshold of k in the decentralized setting.
For an arbitrary m; we will show that
W opt (dm+1 )
W dec (dm+1 ) > W opt (dm )
W dec (dm );
W opt (dm+1 )
W opt (dm ) > W dec (dm+1 )
W dec (dm ):
or equivalently that
(6a)
We …rst focus on W opt (c): When k opt (c) > 0; k opt (c) 2 Z+ ; both k opt (c) and k opt (c)
1
generate the same welfare when waiting costs are c under the optimal mechanism (so, in
particular, what follows holds even if dm is a transition point corresponding to the thresholds
under the optimal mechanism). Note that W opt (c) is piece-wise linear and continuous. It
follows that:
W
q
Let k 0
opt
(dm+1 )
W
opt
Z
(dm )
dm
dm+1
p(1 p)U
2dm
2k opt (c)(k opt (c) + 1)
dc:
2k opt (c) + 1
. For any c 2 [dm+1 ; dm ], k opt (c)
2k opt (c)(k opt (c) + 1)
1
=
opt
2
2k (c) + 1
(2k opt (c) + 1)
opt
Z
k 0 and
1
opt
2k (c) + 1
2k 0 (k 0 + 1)
2k 0 + 1
Thus,
W
(dm+1 )
W
opt
(dm )
dm
2k 0 (k 0 + 1)
2k 0 (k 0 + 1)
dc
=
(dm
2k 0 + 1
2k 0 + 1
dm+1
0 0
2k (k + 1)
1
p(UA ( ) UA ( ))
0
2k + 1
m
0 0
2k (k + 1) p(UA ( ) UA ( ))
=
:
2k 0 + 1
m(m + 1)
=
39
dm+1 )
1
m+1
Next, we consider W dec (c). Denote by
W (k; c)
S1
p(1 p)U
2k + 1
2k(k + 1)c
:
2k + 1
Note that
W dec (dm+1 )
W dec (dm ) = W (m + 1; dm+1 )
= W (m + 1; dm+1 )
W (m; dm )
W (m + 1; dm ) + W (m + 1; dm )
W (m; dm ):
We use the following two results:
2(m + 1)(m + 2)
(dm dm+1 )
2m + 3
2(m + 1)(m + 2) p(UA ( ) UA ( ))
=
2m + 3
m(m + 1)
2(m + 2)p(UA ( ) UA ( ))
=
;
m(2m + 3)
W (m + 1; dm+1 )
W (m + 1; dm ) =
and
W (m + 1; dm )
2m(m + 1)dm p(1 p)U
2(m + 1)(m + 2)dm
p(1 p)U
+
2m + 1
2m + 1
2m + 3
2m + 3
2
2p(1 p)U 4(m + 1) dm
=
(2m + 1)(2m + 3)
1
4(m + 1)2 p(UA ( ) UA ( ))
=
2p(1 p)U
:
(2m + 1)(2m + 3)
m
W (m; dm ) =
Then,
W dec (dm+1 )
W dec (dm )
=
2(m+2)p(UA ( ) UA ( ))
m(2m+3)
+
2p(1 p)U
(2m+1)(2m+3)
4(m+1)2 p(UA ( ) UA ( ))
:
m(2m+1)(2m+3)
We want to prove that (6a) holds. It su¢ ces to show that
2k 0 (k 0 + 1) 2m + 3
2m
(1 p)U
> 2(m + 2) +
0
2k + 1 m + 1
2m + 1 UA ( ) UA ( )
2m
(1 p)U
2m
=
+
:
2m + 1 2m + 1 UA ( ) UA ( )
To prove the above inequality, we consider the following two cases:
40
4(m + 1)2
2m + 1
(7)
Case 1: k 0
2. Note that
U
UA ( )
<
UA ( )
(UA ( )
UA ( )) + (U (A)
UA ( ) UA ( )
U (B))
= 2:
(8)
Since the left hand side of (7) is increasing in k 0 ; for (7) to hold, it su¢ ces that
2m
4m
6m
12 2m + 3
>
+
=
;
5 m+1
2m + 1 2m + 1
2m + 1
which holds for all m:
Case 2: k 0 = 1. One su¢ cient condition for (7) using (8) is
4(2m + 3)
6m
>
;
3(m + 1)
2m + 1
which holds for m = 1; 2; or 3.
Since k 0 = 1;
(1 p)U m
p(1 p)U
=
< 4:
2dm
2(UA ( ) UA ( ))
Thus, another su¢ cient condition for (7) in this case is
4(2m + 3)
2m + 16
>
;
3(m + 1)
2m + 1
which holds for every m
4.
To construct the partition in the proposition, let m = max fi j di
cg and m = minfi j
di < cg: Now de…ne c1 = c; cM = c: If m = m; set M = 2 and the partition has only one atom.
Otherwise, if m < m; set ci = dm
i+1
for i = 2; :::; m
m + 1 and M = m
m + 2:
2. Notice that
lim W opt (c)
c!0
W dec (c) = lim (c)
c!0
lim (c) = p (UA ( )
c!0
In particular, as c becomes vanishingly small, W opt (c)
is increasing in both p and UA ( )
UA ( ), as needed.
41
UA ( )) :
W dec (c) converges to a value that
8.4
Proofs Regarding Fixed Matching Windows
Proof of Proposition 5:
1. Notice that
+ Wn
=
p(1
As the upper bound of
p) U
P r(kA 6= k ; n)
2n
+ Wn
p(1
2c
p)U
2n
2c:
decreases in n, let x be the unique solution of
p(1
p)U
2x
2c = 0:
It follows that
no
2. We …nd an upper bound of
x=
+ Wn
p(1
p)U
4c
:
that will provide us with a lower bound on the
optimal window size.
Fix a window size n, and de…ne a multinomial random variable Xt for each time t =
1; 2; : : : ; n such that
8
< 1
if (A; ) arrives,
1 if (B; ) arrives,
Xt =
:
0
otherwise.
Thus, Xt takes the values of 1; 1; or 0 with probabilities p(1
respectively. Notice that
kA
k =
n
X
p), (1
p)p, or 1
2p(1
p),
Xt :
t=1
Let
(:) be the cumulative distribution function of the standard normal distribution. We
use the Berry-Esseen Theorem on the speed of convergence in the Central Limit Theorem (see
Feller, 1972 and Tyurin, 2010).
Theorem 1. (Berry-Esseen) Let Y1 ; Y2 ; : : : ; Yn be i.i.d random variables with mean 0 and vari^ n the cumulative distribution function of pnZn = .
ance 2 , Zn = 1 (Y1 + Y2 + ::: + Yn ) ; and G
n
Then
^ n (y)
sup G
(y)
y
E[jY1 j3 ]
p :
2 3 n
Now, notice that
Var[X1 ] = E[jX1 j3 ] = 2p(1
42
p):
Let F^n be the empirical cumulative distribution function of pkA
k
:
2np(1 p)
By the Berry-Esseen
Theorem,
p
P r(kA = k ; n) = F^n 1= 2np(1 p)
p
(1= 2np(1 p))
1
p
2
np(1
Thus,
p(1
+ Wn
p(1
=
p)U
2n
1
p)U
2n
1
F^n (0)
1
(0) + p
2np(1
1
+p
:
p)
2np(1 p)
1
1
1
p +p
p
2
2
np(1
!
p
2 +1
p
2c:
2
np(1 p)
p)
p)
!
2c
(9)
We now show that there exists c1 > 0 such that for all c < c1 ; the optimal …xed size no > 2
by illustrating that
+ W1
+ W1
=
> 0 and
p(1
p)U
2
+ W2
> 0. Indeed,
P r(kA 6= k ; 1)
2c = p2 (1
P r(kA 6= k ; 2)
2c
p)2 U
2c > 0;
and
+ W2
=
=
p(1
p)U
2
p(1
p)U
4
p(1
p)U
4
=
whenever c < c1
1
p2 (1
p2 + (1
p2
(1
p)2 U
2
p)2 )
(1
p3 (1
p)2
p)3 U
2
2
2p2 (1
p3 (1
2c
p)2
p)3 U
2
3p2 (1
2c
2c
p)2 U
8
3p2 (1 p)2 U
:
16
We now consider n
3: From (9),
h(r)
p(1
+ Wn
p)U r2
2
h(r); where r = n 1=2 and
!
p
( 2 + 1)r
p
1
2c:
2
p(1 p)
We use the following observations, assuming that c < c2
43
4 p2 (1 p)2 U
p
27( 2 +1)2
< c1 :
2c > 0
1.
+ W3
p
h 1= 3 =
p(1 p)U
6
p
1
2
2 +1
3 p(1 p)
p
p
2c: In order to show that h(1= 3) > 0;
we show that
4 p2 (1 p)2 U
p(1 p)U
p
c<
<
12
27( 2 + 1)2
1
Indeed, the right inequality holds whenever
1
p
<
2
12
27( 2 + 1)
!
p
2 +1
p
:
2 3 p(1 p)
p
2 +1
p
;
24 3
which holds if and only if
p
2 +1
12
p
+ p
= 0:4437422::: < 1:
27( 2 + 1)2
2 3
0
2. h (r) = 0 at r = 0 and
3. h(0) < 0 and h
p
p(1 p)
p
.
3( 2 +1)
4
p
p(1 p)
p
3( 2 +1)
4
=
8 p2 (1 p)2 U
p
27( 2 +1)2
2c > 0.
4. From points 2 and 3 it follows that there exists a unique r > 0 such that h(r ) = 0 and
h0 (r ) > 0.
5. For any " > 0; there exists c3 such that for any c < c3 we have h
order to show this, note that
s
!
(4 + ") c
"
h
= 2+
c 1
p(1 p)U
2
q
(4+")c
p(1 p)U
!
p
p
( 2 + 1) (4 + ") c
p
2 U p(1 p)
> 0: In
2c
and the claim holds for
c3 =
1
4
4+"
2
!2
p
2 U p(1 p)
p
p
:
( 2 + 1) (4 + ")
6. From points 4 and 5, it follows that for c < min fc2 ; c3 g, r <
q
no
q
(4+")c
:
p(1 p)U
To summarize, for any c < c
minfc2 ; c3 g; for every window size n such that p1n
p
p
(4+")c
o
,
we
have
h(1=
n)
>
0.
On
the
other
hand,
0
W
h(1=
no ). Therefore,
+
n
p(1 p)U
p(1 p)U
.
(4+")c
44
Proof of Proposition 6: Recall that we denote by Sn the ex-ante per-pair surplus when
the window size is n. We …nd bounds on Sn starting from the following inequalities:
Sn
Sn
kA k
; g
n n
kA k
E maxf ; g
n n
E minf
kA k
; g
n n
kA k
E maxf ; g
n n
UA + 1
E minf
UA + 1
and
UB
UB :
Notice that
E min
kA k
;
n n
=
E [jkA + k ]
p
= p
= p
= p
1
2
s
k j]
E [jkA
2n
E (kA k )2
n2
1
2
=p
s
(E [jkA k j])2
n2
(Jensen’s inequality)
r
1 E [kA2 ] + E [k 2 ]
2s
n2
2E [kA k ]
1
p(1 p)
2 p2 +
2
n
r
p(1 p)
:
2n
2p2
(since E[kA2 ] = V (kA ) + (E[kA ])2 )
Similarly, we obtain
E max
kA k
;
n n
Therefore, for every window size n,
r
p(1 p)
S1
(UA
UB )
2n
p+
Sn
r
S1 +
p(1 p)
:
2n
r
p(1 p)
(UA
2n
UB ) :
The per-pair welfare from a window size n is determined by:
Wn
Sn
2c(n
1):
The optimal ex-ante per-pair welfare from a static matching with a …xed-window size is
de…ned as
Wo
Sno
2c(no
We now turn to the parts of the proposition.
45
1):
p(1 p)U
:
4c
1. From part 1 of Proposition 5, no
Wn
Sn
2c(n
1)
S1
r
Therefore,
p(1 p)
(UA
2n
UB )
2c(n
1):
We can verify that the lower bound decreases in n by di¤erentiating the right hand side of
the last inequality (ignoring integer constraints). Thus,
r
2c
f ix
W (c) = Wno S1
(UA
UB ) + 2c
U
p(1
p)U c
2
:
2. From the above inequalities,
W
f ix
(c)
S1 +
r
p(1 p)
(UA
2no
UB )
2c(no
1):
From part 2 of Proposition 5, for any " > 0; there exists c such that when c < c
p2 (1
4
p)2 U
p
;
27( 2 +1)2
no
p(1 p)U
:
(4+2")c
W
f ix
Therefore,
(c)
S1 +
r
2c
(UA
U
UB )
p(1 p)U
+ 2c:
2+"
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