Competing Dynamic Matching Markets

Sanmay Das? John P. Dickerson† Zhuoshu Li? Tuomas Sandholm†

?Washington University in St. Louis{sanmay,zhuoshuli}@wustl.edu

†Carnegie Mellon University{dickerson,sandholm}@cs.cmu.edu

ABSTRACTWhile dynamic matching markets are usually modeled inisolation, assuming that every agent to be matched entersthat market, in many real-world settings there exist rivalmatching markets with overlapping pools of agents. We ex-tend a framework of dynamic matching due to Akbarpour etal. [2] to characterize outcomes in cases where two such ri-val matching markets compete with each other. One marketmatches quickly while the other builds market thickness bymatching slowly. We give an analytic bound on the loss—theexpected fraction of unmatched vertices—of this two-marketenvironment relative to one in which all agents enter eitherone market or the other, and numerically quantify its exactloss, demonstrating that rival markets increase overall losscompared to a single market that builds thickness. We thenlook at two competing kidney exchanges, where patients withend-stage renal failure swap willing but incompatible donors,and show that matching with rival barter exchanges per-forms qualitatively the same as matching with rival match-ing markets—that is, rival markets increase global loss.

Categories and Subject DescriptorsJ.4 [Social and Behavioral Sciences]: Economics

General TermsEconomics, Experimentation, Theory

KeywordsMatching markets, dynamic matching, greedy matching, tradefrequency, kidney exchange

1. INTRODUCTIONIn matching problems, a central clearinghouse pairs agents

with other agents, transactions, or contracts. Most classicalmatching problems—matching medical residents to hospi-tals, matching students to schools—are static, where agents

.

and items exist at the same time, are matched, and thenthe market disappears; however, many real-world matchingproblems are dynamic, with agents arriving and departingover time in a persistent market.

Furthermore, many dynamic matching applications in-volve multiple competing clearinghouses with overlappingsets of participants. For example, a lonely graduate stu-dent may register on two dating websites (e.g., Match.comand OkCupid), or choose to only register on one. Thus, amember of both sites can be matched to any member of ei-ther site, while single-site members can only be matched tomembers of their specific dating market. The clearinghousesthen compete on a metric like total number of matches. Itis also common for patient-donor pairs in kidney exchangeto register on multiple exchanges, an application we explorein detail later.

In this paper, we explore, in a dynamic matching setting,how rival clearinghouses a↵ect global social welfare in termsof total agents matched relative to a world in which all agentsenter exactly one market, which can optimize for how tomatch them independently.

1.1 Our contributionThis paper’s major contribution is the extension of a re-

cent framework of dynamic matching due to Akbarpour etal. [2] to two rival matching markets with overlapping pools.Specifically, we formalize a two-market model where agentsenter one market or both markets; they can then be matchedto other agents who have joined the same market or bothmarkets. The markets adhere to di↵erent matching policies,with one matching greedily and the other building marketthickness through a patient policy. We provide an analyticlower bound on the loss, or the expected fraction of verticeswho enter and leave the pool without finding a match, of thetwo-market model and show that it is higher than runninga single “patient” market. We also provide a quantitativemethod for determining the loss of the two-market model.

Our work draws motivation from kidney exchange, an in-stantiation of barter exchange where patients paired withwilling but medically incompatible donors swap those donorswith other patients. In the United States, multiple fieldedkidney exchanges exist, and patient-donor pairs are enteredsimultaneously into one or more of these markets, based ongeographical location, travel preferences, home transplantcenter preferences, or other logistical reasons. Individualkidney exchange clearinghouses have incentive to competeon number of matches performed within their specific pools;yet, fragmenting the market across multiple exchanges op-erating under di↵erent matching policies may lower global

welfare. In this paper, we provide the first experimental evi-dence on dynamic kidney exchange graphs showing that thismay indeed be the case.

1.2 Related workMost related to our work is a recent paper by Akbarpour

et al. [2], which presents a general framework for bilateraldynamic matching in a single market and analyzes the e�-cacy of a variety of matching policies over time. We builddirectly on that framework and delay a more in-depth reviewof that work until Section 2.

Dynamic matching in a single market has been exploredin many domain-specific applications. Some examples aregiven below for both one-sided and two-sided traditionalmatching markets, as well as for barter exchanges; this listis not exhaustive.One-sided markets. In these settings, only one side (theagents) has preferences over the other (the items). Wait-ing lists are used in many applications as a mechanism forallocating the items, which are scarce resources, to agents.Both agents and items arrive over time, and an agents’ pri-ority for an arriving item can be set by a variety of fac-tors. Examples of waiting list applications include publichousing assignment [19, 20, 24] and cadaveric organ alloca-tion [9, 33, 35]. In a two-period dynamic housing allocationproblem, agents can either apply for a public good (e.g., ahouse) in the first stage and receive priority in that stage, oropt out in the first stage and receive priority in the secondstage [1]. Other variants of the dynamic housing allocationproblem have also been addressed where, e.g., agents arriveand depart and, upon departure, an agent’s allocated itemis then given to an existing agent in the waiting pool [10,23].Two-sided markets. In two-sided markets, participatingagents belong to one of two disjoint sets (e.g., “firms” or“workers”), but an agent on either side will have preferencesover those on the other. In online labor marketplaces likeoDesk, employers and applicants arrive and depart over timeand are interested in finding an acceptable match [5, 18].In the dynamic school choice problem, schools exist perma-nently and indefinitely, but students arrive and depart peri-odically in a discrete time model [22]. Students matched toa school at one time period may be matched to a new schoolat a di↵erent time period. Schools and students have prefer-ence orderings over each other, based on the utility providedto one side by being allocated an element or elements of theother. Finally, generalizations of the online bipartite match-ing problem as originally introduced by Karp et al. [21] haverecently seen great real-world impact in Internet ad alloca-tion [25,26].

We note that our work does not assume a bipartite struc-ture in the matching graph—as in [1, 5, 10, 17–20, 22–24,33–35] and much of the static matching mechanism designliterature—and involves more than a single market.Barter exchange. In barter exchange, agents can directlyswap goods with other agents in cycles of length greater thanor equal to two. One fielded example is kidney exchange [29],where patients with end-stage renal failure and willing butincompatible paired donors swap those donors with other pa-tients. Unver [34] was the first to address dynamic kidneyexchange, where patient-donor pairs arrive and depart overtime, with recent follow-up work by Ashlagi et al. [6] and An-derson et al. [3]. All three papers look at matching policiesthat aim to maximize (discounted) social welfare. Partic-

ularly relevant to real-world kidney exchanges are batchingpolicies, where a market clearing occurs at a fixed interval;some theoretical and empirical explorations of this class ofpolicy has been performed [3, 4, 6, 8]. Learning approacheshave also been used to determine more complex matchingpolicies that adhere to specific data distributions [13,14].

To our knowledge, no work in the general barter exchangeor kidney exchange literature has addressed multiple com-peting exchanges, a problem that is especially relevant inthe US now, and, as kidney exchanges move to internationalswapping, will soon become relevant worldwide.

Interacting mechanisms have been studied in a variety ofdomains like auctions [12, 27], adaptations of settings fromthe classical multi-agent systems literature [32], and in two-sided networks that typically exhibit winner-take-all dynam-ics, where only one or a few large players (e.g., credit cardcompanies, computer operating systems, HMOs) prevail dueto network e↵ects [15]. Recent work by Ostrovsky [28] gen-eralizes traditional two-sided matching to a supply chainmodel with interconnected markets represented as nodes ina path, such that an “upstream” neighbor’s supply overlapswith its “downstream” neighbor’s demand. Relatively lit-tle work focuses on markets competing based on variablescheduling or clearing policies, with notable exceptions incloud or grid computing [7] and in financial markets [11].To the best of our knowledge, no work looks at competitionbetween two markets in a general framework of dynamic bi-lateral matching, as this paper does.

2. GREEDY AND PATIENT EXCHANGESWe begin by restating some of the most important results

of Akbarpour et al. [2], which will serve as the foundation forour model of competing exchanges. Akbarpour et al. ana-lyze “greedy” and “patient” matching policies—and interpo-lations between the two—by building stochastic continuous-time bilateral matching models of exchanges running thesepolicies, then measuring the e�cacy of the policies in termsof discounted social welfare.

More specifically, an exchange is running in the continuous-time interval [0, T ], with agents arriving according to a Pois-son process with rate parameter m � 1. The exchange de-termines whether potential bilateral transactions betweenagents are either acceptable or unacceptable. The probabil-ity of an acceptable transaction existing between any pair ofdistinct agents is defined as d/m, 0 d m, and is indepen-dent of any other pair of agents in the market. Each agenta remains in the market for a sojourn s(a) drawn indepen-dently from an exponential distribution with rate parameter� = 1; the agent becomes critical immediately before hersojourn ends, and this criticality is known to the exchange.An agent leaves either upon being matched successfully bythe exchange or upon becoming critical and remaining un-matched, at which point she perishes.

At any time t � 0, the network of acceptable transactionsamong agents forms a random graph G

t

= (At

, E

t

), wherethe agents in the exchange at time t form the vertex setA

t

, and the acceptable transactions between agents formsthe edge set E

t

. We assume A0 = ;. Let A

n

t

denote theset of agents who enter the exchange at time t, such thatwith probability 1, |An

t

| 1 for any t � 0. Finally, letA = [

tT

A

n

t

.Akbarpour et al. [2] present a parameterized space of

online matching policies, with a focus specifically on two:

Patient and Greedy. (In the next section, we will present anovel model of two overlapping exchanges, one running thePatient policy and the other running the Greedy policy.) Asdescribed above, vertex arrivals are treated as a continuous-time stochastic process. These policies behave as follows.Greedy. The Greedy matching algorithm attempts to matcheach entering agent immediately by selecting one of its neigh-bors (if a neighbor exists at the time of entry) uniformly atrandom. One obvious consequence of this is that the re-maining graph of unmatched agents at any instant is alwaysempty. We refer to a market running this policy as theGreedy market or simply Greedy for the rest of the paper.Patient. The Patient matching algorithm attempts to matcheach agent only at the instant she becomes critical. As withGreedy, if a critical agent has multiple neighbors, only one isselected uniformly at random. We refer to a market runningthe Patient policy as a Patient market or simply Patientwhen appropriate.

If the random graph model is Erdos-Renyi [16] when notconsidering arrivals, departures, and matching, then the re-maining graph at any instant is also Erdos-Renyi with pa-rameter d/m; furthermore, d is the average degree of theagents. Both the Patient and Greedy policies maintain thisobservation.

The main result of Akbarpour et al. [2] is that waitingto thicken the market can be substantially more importantthan increasing the speed of transactions. Formally, thePatient exchange dramatically reduces the number of agentswho perish (and thus leave the exchange without finding amatch) compared to the Greedy exchange.

In the Akbarpour et al. [2] paper, an agent a receiveszero utility if she perishes, or u(a) = 0. If she is matched,she receives a utility of 1 discounted at rate �, or u(a) =e

��s(a). In this work, we focus on the special case of � = 0in this paper (i.e., we only consider whether or not an agentis matched), and leave the � 6= 0 case for future research.Let ALG(T ) := {a 2 A : a is matched by ALG by time T}.Then, in this model, the loss of an algorithm ALG is definedas the ratio of the expected number of perished agents to theexpected size of A, as shown in Equation 1.

L(ALG) =E[|A�ALG(T )�A

T

|]E(|A|)

=E[|A�ALG(T )�A

T

|]mT

(1)At any time t 2 [0, T ], let Z

g,t

, Z

p,t

represent the sizeof the pools under the Greedy and Patient matcing poli-cies, respectively. Then, Akbarpour et al. [2] proved thatthe Markov chain on Z·,t has a unique stationary distribu-tion under either of those policies. Furthermore, let ⇡

g

,⇡

p

:N ! R+ be the unique stationary distribution of the Markovchain on Z

g,t

, Z

p,t

, respectively, and let ⇠g

:= EZg⇠⇡g [Zg

], ⇠p

:=E

Zp⇠⇡p [Zp

] be the expected size of the pool under the sta-tionary distribution under Greedy and Patient. Then, thefollowing observations can be made.Loss of Greedy. If a Greedy exchange is run for a su�-ciently long time, then L(Greedy) ⇡ ⇠g

m

. The intuition hereis that the Greedy pool is (almost) always an empty graph.Equation (2) formalizes the loss.

L(Greedy) =1

mT

E

Z

T

0

Z

g,t

dt

�

=1

mT

Z

T

0

E [Zg,t

] dt (2)

Loss of Patient. If a Patient exchange is run for a su�-ciently long time, at any point in time it is an Erdos-Renyi

random graph. So once an agent becomes critical, she hasno acceptable transaction with probability (1� d/m)Zp,t�1.

Thus, L(Patient) ⇡ ⇠p(1�d/m)⇠p�1

m

. Equation (3) formalizesthe loss of a Patient market.

L(Patient) =1

mT

E

Z

T

0

Z

p,t

(1� d/m)Zp,t�1dt

�

=1

mT

Z

T

0

Eh

Z

p,t

(1� d/m)Zp,t�1i

dt

(3)

3. OVERLAPPING EXCHANGESThe key result of Akbarpour et al. [2] is that a greedy dy-

namic matching market leads to significantly lower global so-cial welfare than a patient matching market with full knowl-edge of criticality. The central question of this paper iswhat happens in a situation where a greedy exchange anda patient exchange exist simultaneously and compete witheach other to match some shared portion of the population.Agents in this overlapping subset of the population join bothexchanges simultaneously and accept the first match o↵erfrom either of the constituent exchanges.

Drawing on Section 2, we model this in a similar stochas-tic, continuous-time framework as follows. Agents arrive atthe Competing market (a model for the whole system, incor-porating both the Greedy and Patient exchanges) at somerate m according to a Poisson process. For each agent, theprobability of entering both the Greedy exchange and thePatient exchange is �, the probability of entering the Greedyexchange alone is (1��)↵, and the probability of entering thePatient exchange alone is (1� �)(1�↵), where �,↵ 2 [0, 1].The probability that a bilateral transaction between eachpair of agents is acceptable remains d/m, conditioned on bothagents being mutually “visible” to an exchange. The agents’rates of perishing, received utility for being (un)matched,and other settings are otherwise the same as in Section 2.

We analyze the Competing market as three separate evolv-ing pools:

Greedyc

is the pool consisting of agents who enter theGreedy exchange only (with probability ↵(1� �)).

Patientc

is the pool consisting of agents who enter thePatient exchange only (with probability (1�↵)(1��)).

Bothc

is the pool consisting of agents who enter both ex-changes (with probability �).

We use Z

g,t

, Z

p,t

and Z

b,t

to denote the size of Greedyc

,Patient

c

and Bothc

, respectively, at any time t. Similarto an exchange running a single Greedy or Patient match-ing policy, the Markov chain on Z·,t also has a unique sta-tionary distribution. Let ⇡· : N ! R+ be the unique sta-tionary distribution of the Markov chain on Z·,t, and let⇠· := E

Z·⇠⇡·[Z·] be the expected size of the pool under the

stationary distribution. Using this, we will define the loss ofGreedy

c

, L(Greedyc

), the loss of Patientc

, L(Patientc

), andthe loss of Both

c

, L(Bothc

).First, note that the graph formed by the agents in Greedy

c

is empty, so the loss—as in Equation (2)—can be approxi-

mated by L(Greedyc

) ⇡ ⇠g

m

.Next, we consider the agents in Both

c

. If an edge existsbetween an agent in Both

c

and an existing agent in Greedyc

or another agent in Bothc

, she will be matched immedi-ately by the Greedy exchange (and thus does not contribute

to the loss). Similar to the Greedyc

case, at any point intime t, the Both

c

pool is an empty graph; thus, any un-matched agents who become critical in Both

c

will only bematched to agents in Patient

c

. Thus, these leftover agentsin Both

c

have no acceptable transactions with probability

(1 � d/m)Zp,t . Since each agent becomes critical with rate1, letting Competing market run for a su�ciently long time

results in L(Bothc

) ⇡ ⇠b(1�d/m)⇠p

m

, where ⇠

b

, ⇠

p

are the pre-viously defined expected sizes of Both

c

and Patientc

.Finally, we consider the Patient

c

pool. At any time t,the agents who remain in Patient

c

potentially have accept-able transactions with only the agents in Both

c

and theagents in Patient

c

. Hence, in Z

p,t

, once an agent is criti-cal, she has no acceptable transactions with probability (1�d/m)Zp,t+Zb,t�1. Similarly, each agent becomes critical withrate 1; thus, if we allow the Competing market a su�ciently

long execution window, L(Patientc

) ⇡ ⇠p(1�d/m)⇠p+⇠b�1

m

.Because the three pools of agents—Greedy

c

, Patientc

, andBoth

c

—are disjoint (although they may be connected viapossible transactions in the ways listed above), we can definethe total loss of the Competing market as follows.

L(Competing) ⇡ ⇠

g

+ ⇠

p

(1� d/m)⇠p+⇠b�1 + ⇠

b

(1� d/m)⇠p

m

.

(4)A more precise version of Equation (4) follow as Equa-

tion (5); we will make use of this form in Section 5.

L(Competing) =1

mT

E

Z

T

0

Z

p,t

(1� d/m)Zp,t+Zb,t�1

+ Z

b,t

(1� d/m)Zp,t + Z

g,t

dt

�

=1

mT

Z

T

0

E⇥

Z

p,t

(1� d/m)Zp,t+Zb,t�1

+ Z

b,t

(1� d/m)Zp,t + Z

g,t

⇤

dt

(5)

Unfortunately, we do not have a closed form expressionfor the stationary distribution or the expected size of thepool under the stationary distribution. We note that eachof ⇠

g

, ⇠

p

, and ⇠

b

can be approximated well using Monte Carlosimulations—thus, Equation (4) can be solved numerically.We do this in Section 5.1 for two parameterizations of therival market setting.

4. A BOUND ON TOTAL LOSSWhile we do not have a closed form for the exact ex-

pected loss of the Competing market as described by Equa-tion (4), we can provide bounds on the overall loss. In thissection, we give one such bound for the global loss underthe constraint that Greedy

c

is more likely to receive agentsthan the overlapping Both

c

exchange. Formally, this occurswhen � 0.5 and ↵ � �

1��

. We also impose some looserequirements on the arrival rate of vertices to the exchangeand the probability of an acceptable transaction existing be-tween two agents; intuitively, the exchange cannot be “toosmall” or “too sparse,” which we formalize below. Underthese assumptions, we use the bound to prove Theorem 1,which states that a single Patient market outperforms theCompeting market.

Theorem 1. Assume � 0.5, m > 10d, and ↵(1� �) �max

n

�,

12e

�d/2(1 + 3d)o

. Then, as m ! 1 and T ! 1,

almost surely

L(Competing) > L(Patient).

Proof. We prove the theorem by giving a lower boundon L(Greedy

c

), the loss of only the greedy portion of theCompeting market. In our model, the fraction of agents en-tering only the Greedy

c

side of the market is ↵(1 � �); fornotational simplicity, we use x := ↵(1��) in this proof. Sim-ilarly, the fraction of agents entering Both

c

is �; again, fornotational simplicity, we use y := � throughout this proof.

As before, let Zg,t

be the size of Greedyc

at any t 2 [0, T ],and ⌧ the expected size of the Greedy

c

pool. Similarly, letZ

b,t

be the size of Bothc

at any t 2 [0, T ], and ⌘ the expectedsize of the Both

c

pool. That is,

⌧ := Et⇠unif[0,T ]

h

Z

g,t

i

and ⌘ := Et⇠unif[0,T ]

h

Z

b,t

i

.

By assumption, ↵(1 � �) � �; that is, the arrival rateof Greedy

c

is greater than or equal to the arrival rate ofBoth

c

. In this case, ⌧ � ⌘; the Greedy matching policyremoves verties from both Both

c

and Greedyc

, while thePatient matching policy removes vertices from only Both

c

,which means the matching rate for Both

c

is greater than thematching rate for Greedy

c

.From Akbarpour et al. [2], we know the expected rate of

perishing of the individual Greedy exchange is equal to thepool size because the Greedy matching policy does not reactto the criticality of an agent at any time t in its pool andeach critical agent will perish with probability 1. Therefore,we can draw directly on Equation (2) to write

L(Greedyc

) =1

xmT

E[Z

T

t=0

dt Z

g,t

] =⌧

xm

. (6)

We know x and m, so lower bounding ⌧ will result in ananalytic lower bound on L(Greedy

c

). Following the ideasof Akbarpour et al. [2], we do this by lower bounding theprobability that an agent a does not ever have an acceptabletransaction for the duration of her sojourn s(a). Becausethese agents cannot be matched by any matching policy,this directly gives a lower bound on L(Greedy

c

). Towardthis end, fix an agent a 2 A who enters Greedy

c

at timet0 2 unif[0, T ] and draws a sojourn s(a) = t. Let f

sa(t) bethe probability density function at t of s(a). Then we canwrite the probability that a will never have a neighbor (i.e.,possible match) as

P [N(a) = ;] =Z 1

t=0

f

sa(t)Eh

(1� d/m)Zg,t0+Zb,t0

i

Eh

(1� d/m)|AG

nt0,t0+t+AB

nt0,t0+t|

i

dt,

where AG

n

t0,t0+t

(resp. AB

n

t0,t0+t

) denotes the set of agentswho enter Greedy

c

(resp. Bothc

) in time interval [t0, t0 + t].The first expectation captures the probability that agent a

has no matching at the moment of entry and the secondexpectation considers the probability that no new agentsthat can match with a arrive during her sojourn.

Using Jensen’s inequality, we have

P [N(a) = ;] �Z 1

t=0

e

�t(1� d/m)E[Zg,t0+Zb,t0]

(1� d/m)E[|AG

nt0,t+t0

+AB

nt0,t+t0

|]dt

=

Z 1

t=0

e

�t(1� d/m)⌧+⌘(1� d/m)(x+y)mt

dt.

From the assumptions in the theorem statement, d

m

<

110 ,

so 1�d/m � e

�d/m�d

2/m

2. Also, as described earlier, ⌧ � ⌘

(when � 0.5 and ↵ � �

1��

, as assumed). Therefore,

L(Greedyc

) � P [N(a) = ;]

� e

�(⌧+⌘)(d/m+d

2/m

2)

⇥Z 1

t=0

e

�t�(x+y)td�(x+y)td2/mdt

� 1� (⌧ + ⌘)(1 + d/m)d/m1 + (x+ y)d+ (x+ y)d2/m

� 1� 2⌧(1 + d/m)d/m1 + (x+ y)d+ (x+ y)d2/m

,

(7)

where the third inequality is obtained from the fact thate

�z � 1� z when z � 0, here z = (⌧ + ⌘)(d/m+ d

2/m

2).Combining Equation (6) and Equation (7),

L(Greedyc

) =⌧

xm

� 1� 2⌧(1 + d/m)d/m1 + (x+ y)d+ (x+ y)d2/m

,

which gives us a lower bound for ⌧ ,

⌧ � xm

1 + (3x+ y)d+ (3x+ y)d2/m.

Thus, as m ! 1, we get,

L(Greedyc

) � 11 + (3x+ y)d+ (3x+ y)d2/m

� 11 + 3d

.

We are interested in bounding the total loss of the Com-peting market, which is L(Competing) = xL(Greedy

c

) +(1� ↵)(1� �)L(Patient

c

) + yL(Bothc

). By definition, bothL(Patient

c

) � 0 and L(Bothc

) � 0, and by Equation (4),L(Greedy

c

) � 11+3d . Thus,

L(Competing) � x

1 + 3d.

Akbarpour et al. [2] showed that running an individualPatient market results in exponentially small loss L(Patient) <12e

�d/2. Thus, as T,m ! 1, we can get,

L(Competing) > L(Patient). (8)

We note that the result of Theorem 1 holds for only asection of the possible parameterizations of a Competingmarket—specifically, when � 0.5 and ↵ � �

1��

. In thenext section, we will give numerical results showing thatthis result—that the loss of the Competing market is greaterthan the loss of an individual Patient exchange—appears tohold for a vastly larger space of values of � and ↵. Indeed,experimentally, we will see that the loss of the Competingmarket is sometimes greater than the loss of an individualGreedy exchange, which itself is substantially greater thanthe loss of an individual Patient exchange.

5. EXPERIMENTAL VALIDATIONIn this section, we provide experimental validation of the

theoretical results presented in Sections 3 and 4. Section 5.1quantifies the loss due to competing markets as described byEquation (4), while Section 5.2 expands the model to kidneyexchange and draws from realistic data to quantify the lossof competing kidney exchange clearinghouses.

5.1 Dynamic matchingIn Section 3, we gave a method for computing the expected

loss due to competing markets as Equation (4); however,we were unable to derive closed forms for the expected sizeof the competing, patient, and greedy pools (⇠

b

, ⇠

p

, and⇠

g

, respectively) under the stationary distribution. Thesequantities can be estimated using Monte Carlo simulationfor di↵erent entrance rates m. We do that now.

Figures 1 and 2 simulate agents entering the Greedyc

,Both

c

, and Patientc

markets according to a Poisson processwith rate parameter m = 1000 and remaining for a sojourndrawn from an exponential distribution with rate parameter� = 1. An agent chooses to enter Both

c

with probability �,only Greedy

c

with probability ↵(1 � �), and only Patientc

with probability (1� ↵)(1� �), as in the theory above. Wevary ↵ 2 {0, 0.1, . . . , 1} and � 2 {0, 0.1, . . . , 1}, and plot theglobal loss realized for each of these parameter settings.

0 0.2 0.4 0.6 0.8 1

�

0.00

0.01

0.02

0.03

0.04

0.05

0.06

Aver

age

LossMatching, m=1000, �=1

PatientGreedy� = 0.0

� = 0.1

� = 0.3

� = 0.5

� = 0.7

� = 0.9

Figure 1: Average loss (y-axis) as the overlap be-tween markets � increases (x-axis), with entrancerate parameter m = 1000 and d = 20, for di↵erent val-ues of ↵. The loss of individual Patient and Greedymarkets are shown as thick black and thick dashedbars, respectively.

Immediately obvious is that running a single Patient mar-ket results in dramatically less loss than competing markets,for all di↵erent values of ↵ and �. Furthermore, we see thatthe loss of a single Greedy market is also dramatically higherthan the loss of a single Patient market, as predicted by Ak-barpour et al. [2]. Indeed, from Equation (3) we would ex-pect the single Patient market to have essentially zero loss,so these experiments show that adding in a rival Greedy

c

market increases loss. In fact, as the left side of Figure 1and the right side of Figure 2 show, it is the case that if

0 0.2 0.4 0.6 0.8 1

�

0.00

0.01

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0.05

0.06Av

erag

eLo

ssMatching, m=1000, �=1

PatientGreedy� = 0.0

� = 0.1

� = 0.3

� = 0.5

� = 0.7

� = 0.9

Figure 2: Average loss (y-axis) as the probability↵ of entering Patient

c

or Greedyc

changes (x-axis),with entrance rate parameter m = 1000 and d = 20,for di↵erent values of the market overlap �. The lossof individual Patient and Greedy markets are shownas thick black and thick dashed bars, respectively.

the markets do not overlap substantially (i.e., � is low) andagents are more likely to enter the greedy side of the market(i.e., ↵ is near 1), then the loss of the competing market isworse than running a single Greedy market! This is due inpart to the decrease in market thickness on the Patient

c

sideof the market—a behavior we will see exacerbated below andin the kidney exchange experiments of Section 5.2.

Figure 3 decreases the rate parameter of the entrancePoisson process to m = 100, while holding the probabil-ity of an acceptable transaction between two agents at thatof Figures 1 and 2 (so d = 2, leading to 2/100 = 2%).With fewer participants in the market overall, all the qual-itative results of the m = 1000 markets above are ampli-fied. The individual Greedy market’s loss is now 5.9% worsethan the individual Patient market (as opposed to 3.3% inthe m = 1000 case); both individual markets’ losses aresubstantially higher as well. Similarly, the parameter set-tings for which the competing market scenario has higherloss than either individual market are much broader thanthe m = 1000 case, which is a product of market thinness.

5.2 Dynamic kidney exchangeIn this section, we expand our matching model to one of

barter exchange, where agents endowed with items partici-pate in directed, cyclic swaps of size greater than or equalto two. One recently-fielded barter application is kidney ex-change, where patients with kidney failure swap their willingbut incompatible organ donors with other patients. We fo-cus on that application here. Dynamic barter exchange gen-eralizes the matching model presented above, so we wouldnot expect the earlier theoretical results to adhere exactly.Interestingly, as we show in Sections 5.2.1 and 5.2.2, thequalitative ranking of matching policy loss (with a patientmarket outperforming a greedy market, both of which out-perform two rival markets) remains.This section’s experiments draw from two kidney exchange

0 0.2 0.4 0.6 0.8 1

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Loss

Matching, m=100, �=1PatientGreedy� = 0.0

� = 0.1

� = 0.3

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� = 0.9

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Loss

Matching, m=100, �=1PatientGreedy� = 0.0

� = 0.1

� = 0.3

� = 0.5

� = 0.7

� = 0.9

Figure 3: Average loss as the probability ↵ of enter-ing Patient

c

or Greedyc

(top) or the overlap betweenthe two markets � (bottom) changes, with entrancerate parameter m = 100 and d = 2. The loss of in-dividual Patient and Greedy markets are shown asthick black and thick dashed bars, respectively.

compatibility graph distributions. One distribution, whichwe call Saidman, was designed to mimic the characteris-tics of a nationwide exchange in the United States in steadystate [31]. Yet, kidney exchange is still a nascent conceptin the US, so fielded exchange pools do not adhere to thismodel. With this in mind, we also include results per-formed on a dynamic pool generator that mimics the UnitedNetwork for Organ Sharing (UNOS) nationwide exchange,drawing data from the first 193 match runs of that exchange.We label the distribution derived from this as UNOS.

Formally, we represent a kidney exchange pool with n

patient-donor pairs as a directed compatibility graph G =(V,E), such that a directed edge exists from patient-donorpair v

i

2 V to patient-donor pair v

j

2 V if the donor at v

i

can give a kidney to the patient at vj

. Edges exist or do notexist due to the medical characteristics (blood type, tissuetype, relation, and many others) of the patient and potential

donor, as well as a variety of logistical constraints. Ourgenerators take care of these details; for more informationon how edge existence checking is done in the Saidman andUNOS distributions, see Saidman et al. [31] or Dickersonand Sandholm [14], respectively. Importantly, under eitherdistribution, there is no longer a costant probability“d/m”ofan acceptable transaction existing between any two agents.

Vertices arrive via a Poisson process with rate parameterm = 100 and depart according to an exponential clock withrate parameter � = 1 as before, and choose to enter eitherexchange or both with the previously-defined probabilities �and ↵. However, a “match” now only occurs when a vertexforms either a 2-cycle or 3-cycle with one or two other ver-tices, respectively.1 Section 5.2.1 performs experiments on2-cycles alone, which adheres more closely to the theoreticalsetting above (2-cycles can be viewed as a single undirectededge between two vertices), while Section 5.2.2 expands thisto both 2- and 3-cycles.

Code to replicate the experiments in this section is avail-able at github.com/JohnDickerson/KidneyExchange. Thiscodebase includes our experimental framework, dynamic ex-change simulator, and graph generators but, due to privacyconcerns, does not include the real match runs from theUNOS kidney exchange.

5.2.1 Kidney exchange with 2-cycles only

We now present results for dynamic matching under com-peting Patient

c

and Greedyc

kidney exchanges, both of whichuse only 2-cycles. Figure 4 and Figure 5 show losses in-curred in our parameterized market when run on Saidman-generated and UNOS-generated pools, respectively.

0 0.2 0.4 0.6 0.8 1

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Loss

Saidman, 2-cycles, m=100, �=1PatientGreedy� = 0.0

� = 0.1

� = 0.3

� = 0.5

� = 0.7

� = 0.9

Figure 4: Average loss under various values of � and↵ for the Saidman distribution with 2-cycles only.

1In fielded kidney exchange, cycles longer than some shortcap L (e.g., L = 3 at the UNOS exchange and many oth-ers) are typically infeasible to perform due to logistical con-straints, and thus are not allowed. We adhere to thatconstraint here. Fielded exchanges also realize gains fromchains, where a donor without a paired patient enters thepool and triggers a directed path of transplants through thecompatibility pool. We do not include chains in this work.

0 0.2 0.4 0.6 0.8 1

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Loss

UNOS, 2-cycles, m=100, �=1PatientGreedy� = 0.0

� = 0.1

� = 0.3

� = 0.5

� = 0.7

� = 0.9

Figure 5: Average loss under various values of � and↵ for the UNOS distribution with 2-cycles only.

While the barter exchange environment under either theSaidman or UNOS distributions clearly breaks the struc-tural properties of the stationary distribution of the underly-ing Markov process used in our theoretical results, the quali-tative results of these experiments align with the traditionaldynamic matching results of Section 5.1. The overall loss re-alized by UNOS is substantially higher than that realized bySaidman because, in general, UNOS-generated graphs aremore sparse than those from the Saidman family. Similarly,in either distribution there exist “highly-sensitized” vertextypes that are extremely unlikely to find a match with an-other randomly selected vertex, and thus almost certainlycreate loss. Indeed, both Figure 4 and 5 exhibit higher lossthan the similarly-parameterized Figure 3 of Section 5.1.

5.2.2 Kidney exchange with both 2- and 3-cycles

We now extend our experiments to allow for “matches”that include both 2- and 3-cycles. Unlike Section 5.1 or 5.2.1,where a matched edge was chosen uniformly at random fromthe set of all acceptable transactions between a distinguishedvertex and its neighbors, in these results we may wish todistinguish a potential match from others (for example, bychoosing a 3-cycle before a 2-cycle, as the former results ina larger myopic decrease in the market’s loss). Thus, givena set of possible 2- and 3-cycle matches, we consider twomatching policies: Uniform selects a cycle at random fromthe set of possible matches, regardless of cycle cardinality,while Uniform3 selects a 3-cycle randomly (if one exists),otherwise a random 2-cycle.

Figures 6 and 7 show results for the Saidman and UNOSdistributions, respectively, under the Uniform match se-lection policy. Intuitively, one might expect the loss of amatching policy run in the 2- and 3-cycle case to be lessthan the same policy run in the 2-cycle case alone, as theset of possible matches weakly increases in the former case.We see this behavior when comparing the Saidman resultsof Figure 6 to the earlier 2-cycle-only Saidman results ofFigure 4, witnessing a drop in global loss of around 4% forany parameter setting. We see a similar decrease in losswhen comparing the new UNOS results of Figure 7 to those

in the 2-cycle case shown in Figure 5.

0 0.2 0.4 0.6 0.8 1

�

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Aver

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Loss

Saidman, 2- and 3-cycles, m=100, �=1PatientGreedy� = 0.0

� = 0.1

� = 0.3

� = 0.5

� = 0.7

� = 0.9

Figure 6: Average loss under various values of � and↵ for the Saidman distribution with both 2- and 3-cycles, under the Uniform matching policy.

0 0.2 0.4 0.6 0.8 1

�

0.79

0.80

0.81

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Aver

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Loss

UNOS, 2- and 3-cycles, m=100, �=1PatientGreedy� = 0.0

� = 0.1

� = 0.3

� = 0.5

� = 0.7

� = 0.9

Figure 7: Average loss under various values of �

and ↵ for the UNOS distribution with both 2- and3-cycles, under the Uniform matching policy.

We now consider the Uniform3 matching policy, whichwould likely be closer to how a fielded exchange would act.Figures 8 and 9 show results for the Saidman and UNOSfamilies of compatibility graphs, respectively. The loss ofthe individual Patient market does not change in either dis-tribution, which is likely a byproduct of the thicker marketsinduced by its match cadence. Curiously, the loss of the in-dividual Greedy market drops dramatically—to around thePatient loss in the UNOS case, and below Patient in theSaidman case. This large drop in Greedy loss is likely duein part to Greedy now “poaching” larger 3-cycles from theleftover market from which the Patient policy draws. Theother qualitative results of earlier sections are repeated, with

rival markets hurting global loss relative to either individualmarket for nearly all settings of � and ↵.

0 0.2 0.4 0.6 0.8 1

�

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Loss

Saidman, 2- and 3-cycles, m=100, �=1PatientGreedy� = 0.0

� = 0.1

� = 0.3

� = 0.5

� = 0.7

� = 0.9

Figure 8: Average loss under various values of � and↵ for the Saidman distribution with both 2- and 3-cycles, under the Uniform3 matching policy.

0 0.2 0.4 0.6 0.8 1

�

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Loss

UNOS, 2- and 3-cycles, m=100, �=1PatientGreedy� = 0.0

� = 0.1

� = 0.3

� = 0.5

� = 0.7

� = 0.9

Figure 9: Average loss under various values of �

and ↵ for the UNOS distribution with both 2- and3-cycles, under the Uniform3 matching policy.

6. CONCLUSION & FUTURE RESEARCHOur main goal is to study the impact of competition be-

tween exchanges in a dynamic matching setting. In thispaper, we extended the recent dynamic matching model ofAkbarpour et al. [2] to two rival matching markets withoverlapping pools. Specifically, we formalized a two-marketmodel where agents enter one market or both markets; theycan then potentially be matched to other agents who havejoined the same market or both markets. The markets,called Greedy and Patient, adhere to di↵erent matching poli-cies. We provided an analytic lower bound on the loss of the

two-market model and showed that it is higher than runninga single Patient market. We also provided a quantitativemethod for determining the loss of the two-market model.We supported these theoretical results with extensive simu-lation. We also looked at competing kidney exchanges, andprovided (to our knowledge) the first experimental quan-tification of the loss in global welfare in a setting with twoclearinghouses using realistic kidney exchange data drawnfrom a generator due to Saidman et al. [31] and anotherbased on the United Network for Organ Sharing (UNOS)program.

We see competing dynamic matching markets as fertileground for future research, with a trove of both theoreticaland practical questions to answer. First, the model of Ak-barpour et al. [2] discounts the utility of a match by the timethe matching agent has already waited in the pool; this iswell motivated in a variety of settings, including kidney ex-change. Our results in this paper assume a discount factorof zero, so it would be valuable to consider the impact ondiscounted loss for non-zero cases. Second, in our model thechoice of market to enter is exogenously determined for eachagent. In reality, agents with di↵erent levels of knowledge,wealth, etc. may make strategic decisions on which marketsto enter. Thus, one could approach this dynamic matchingproblem from a game-theoretic point of view. Similarly, tak-ing network e↵ects (where more popular exchanges have aneasier time attracting agents, lower operating costs, higherprobabilities of two agents forming an acceptable transac-tion, and other advantages) into account would make thesemodels more applicable to many real-world settings. Finally,we only looked at two overlapping markets; generalizing thisto any number of overlapping markets would also be of in-terest.

In terms of barter exchange and, specifically, kidney ex-change, the question of how clearinghouses interact is atimely one. In the United States and, eventually, elsewhere,multi-center and single-center exchange clearinghouses arealready competing, each drawing from some (often overlap-ping) subset of the full set of patient-donor pairs available.Indeed, the dynamic barter exchange problem in a singlemarket is still not fully understood (barring very promis-ing recent work due to Anderson et al. [3]). We saw inSection 5.2.2 that including 3-cycles in the matching pro-cess results in lower loss, even when two markets overlap,compared to including only 2-cycles (a result that has beenshown repeatedly in the static [30] and dynamic [3] singleclearinghouse setting), so extending the theoretical under-pinnings of our framework to a more general setting wouldbe of great value. Finally, it is curious that the Uniform3policy had such a large e↵ect on the loss of the individualPatient and Greedy exchanges compared to the Uniformpolicy; further exploration of di↵erent matching policies (in-cluding those that use a strong prior to consider possible fu-ture states of the pool when matching now) would be helpfulin making policy recommendations to fielded exchanges.Acknowledgments. This material was funded by NSFgrant IIS-1320620, NSF CAREER Award IIS-1414452, andby a National Defense Science & Engineering Graduate (ND-SEG) Fellowship. This work used the Extreme Science andEngineering Discovery Environment (XSEDE), which is sup-ported by NSF grant OCI-1053575; specifically, it used theBlacklight system at the Pittsburgh Supercomputing Center(PSC).

7. REFERENCES[1] A. Abdulkadiroglu and S. Loertscher. Dynamic house

allocations, 2007. Working paper.[2] M. Akbarpour, S. Li, and S. O. Gharan. Dynamic

matching market design. In Proceedings of the ACMConference on Economics and Computation (EC),page 355, 2014.

[3] R. Anderson, I. Ashlagi, D. Gamarnik, andY. Kanoria. A dynamic model of barter exchange. InAnnual ACM-SIAM Symposium on DiscreteAlgorithms (SODA), 2015.

[4] E. Anshelevich, M. Chhabra, S. Das, and M. Gerrior.On the social welfare of mechanisms for repeatedbatch matching. In AAAI Conference on ArtificialIntelligence (AAAI), pages 60–66, 2013.

[5] N. Arnosti, R. Johari, and Y. Kanoria. Managingcongestion in decentralized matching markets. InProceedings of the ACM Conference on Economicsand Computation (EC), page 451, 2014.

[6] I. Ashlagi, P. Jaillet, and V. H. Manshadi. Kidneyexchange in dynamic sparse heterogenous pools. InProceedings of the ACM Conference on ElectronicCommerce (EC), pages 25–26, 2013.

[7] I. Ashlagi, M. Tennenholtz, and A. Zohar. Competingschedulers. In AAAI Conference on ArtificialIntelligence (AAAI), 2010.

[8] P. Awasthi and T. Sandholm. Online stochasticoptimization in the large: Application to kidneyexchange. In Proceedings of the 21st InternationalJoint Conference on Artificial Intelligence (IJCAI),pages 405–411, 2009.

[9] D. Bertsimas, V. F. Farias, and N. Trichakis. Fairness,e�ciency, and flexibility in organ allocation for kidneytransplantation. Operations Research, 61(1):73–87,2013.

[10] F. Bloch and D. Cantala. Markovian assignment rules.Social Choice and Welfare, 40(1):1–25, 2013.

[11] E. B. Budish, P. Cramton, and J. J. Shim. Thehigh-frequency trading arms race: Frequent batchauctions as a market design response, 2015. Workingpaper.

[12] R. Burguet and J. Sakovics. Imperfect competition inauction designs. International Economic Review,40(1):231–247, 1999.

[13] J. P. Dickerson, A. D. Procaccia, and T. Sandholm.Dynamic matching via weighted myopia withapplication to kidney exchange. In AAAI Conferenceon Artificial Intelligence (AAAI), pages 1340–1346,2012.

[14] J. P. Dickerson and T. Sandholm. FutureMatch:Combining human value judgments and machinelearning to match in dynamic environments. In AAAIConference on Artificial Intelligence (AAAI), 2015.

[15] T. Eisenmann, G. Parker, and M. W. Van Alstyne.Strategies for two-sided markets. Harvard BusinessReview, 84(10):92, 2006.

[16] P. Erdos and A. Renyi. On the evolution of randomgraphs. Publications of the Mathematical Institute ofthe Hungarian Academy of Sciences, 5:17–61, 1960.

[17] H. Hosseini, K. Larson, and R. Cohen. Matching withdynamic ordinal preferences. In AAAI Conference onArtificial Intelligence (AAAI), 2015.

[18] S. V. Kadam and M. H. Kotowski. Multi-periodmatching, 2014. Working paper.

[19] E. H. Kaplan. Managing the demand for publichousing. PhD thesis, Massachusetts Institute ofTechnology, 1984.

[20] E. H. Kaplan. Tenant assignment models. OperationsResearch, 34(6):832–843, 1986.

[21] R. M. Karp, U. V. Vazirani, and V. V. Vazirani. Anoptimal algorithm for on-line bipartite matching. InProceedings of the Annual Symposium on Theory ofComputing (STOC), pages 352–358, 1990.

[22] J. Kennes, D. Monte, and N. Tumennasan. The daycare assignment: A dynamic matching problem.American Economic Journal: Microeconomics,6(4):362–406, 2014.

[23] M. Kurino. Essays on dynamic matching markets.PhD thesis, University of Pittsburgh, 2009.

[24] J. D. Leshno. Dynamic matching in overloadedwaiting lists, 2015. Working paper.

[25] A. Mehta. Online matching and ad allocation.Theoretical Computer Science, 8(4):265–368, 2012.

[26] A. Mehta, A. Saberi, U. Vazirani, and V. Vazirani.Adwords and generalized online matching. Journal ofthe ACM, 54(5), 2007.

[27] D. Monderer and M. Tennenholtz. K-price auctions:Revenue inequalities, utility equivalence, andcompetition in auction design. Economic Theory,24(2):255–270, 2004.

[28] M. Ostrovsky. Stability in supply chain networks.American Economic Review, pages 897–923, 2008.

[29] A. Roth, T. Sonmez, and U. Unver. Kidney exchange.Quarterly Journal of Economics, 119(2):457–488, 2004.

[30] A. Roth, T. Sonmez, and U. Unver. E�cient kidneyexchange: Coincidence of wants in a market withcompatibility-based preferences. American EconomicReview, 97:828–851, 2007.

[31] S. L. Saidman, A. Roth, T. Sonmez, U. Unver, andF. Delmonico. Increasing the opportunity of livekidney donation by matching for two and three wayexchanges. Transplantation, 81(5):773–782, 2006.

[32] Y. Shoham and K. Leyton-Brown. Multiagent systems:Algorithmic, game-theoretic, and logical foundations.Cambridge University Press, 2008.

[33] X. Su and S. A. Zenios. Patient choice in kidneyallocation: A sequential stochastic assignment model.Operations Research, 53:443–455, 2005.

[34] U. Unver. Dynamic kidney exchange. Review ofEconomic Studies, 77(1):372–414, 2010.

[35] S. A. Zenios. Optimal control of a paired-kidneyexchange program. Management Science,48(3):328–342, 2002.

APPENDIXA. ADDITIONAL EXPERIMENTSIn this section, we provide additional results supporting

the dynamic kidney exchange experiments of Section 5.2.Figure 10 corresponds to the 2-cycle-only experiments ofFigures 4 and 5 in the body of the paper; instead of vary-ing the market overlap parameter � on the x-axis, they varythe probability ↵ of entering either the Greedy

c

or Patientc

market, while holding � constant for a variety of values. Sim-ilarly, Figure 11 corresponds to the 2- and 3-cycle Uniformmatching policy experiments of Figures 6 and 7. Finally,Figure 12 corresponds to the Uniform3 matching policy re-sults shown in Figures 8 and 9.

0 0.2 0.4 0.6 0.8 1

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Saidman, 2-cycles, m=100, �=1PatientGreedy� = 0.0

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UNOS, 2-cycles, m=100, �=1PatientGreedy� = 0.0

� = 0.1

� = 0.3

� = 0.5

� = 0.7

� = 0.9

Figure 10: 2-cycles-only experiments, paired withFigure 4 (left) and Figure 5 (right).

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Saidman, 2- and 3-cycles, m=100, �=1PatientGreedy� = 0.0

� = 0.1

� = 0.3

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� = 0.9

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UNOS, 2- and 3-cycles, m=100, �=1PatientGreedy� = 0.0

� = 0.1

� = 0.3

� = 0.5

� = 0.7

� = 0.9

Figure 11: 2- and 3-cycle Uniform experiments,paired with Figure 6 (left) and Figure 7 (right).

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Saidman, 2- and 3-cycles, m=100, �=1PatientGreedy� = 0.0

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UNOS, 2- and 3-cycles, m=100, �=1PatientGreedy� = 0.0

� = 0.1

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� = 0.9

Figure 12: 2- and 3-cycle Uniform3 experiments,paired with Figure 8 (left) and Figure 9 (right).