Matching in Dynamic Imbalanced Markets
MMatching in Dynamic Imbalanced Markets
Itai Ashlagi Afshin Nikzad Philipp Strack ∗ June 14, 2019
Abstract
We study dynamic matching in exchange markets with easy- and hard-to-match agents. Agreedy policy, which attempts to match agents upon arrival, ignores the positive externality thatwaiting agents generate by facilitating future matchings. We prove that this trade-off betweena “thicker” market and faster matching vanishes in large markets; A greedy policy leads toshorter waiting times, and more agents matched than any other policy. We empirically confirmthese findings in data from the National Kidney Registry. Greedy matching achieves as manytransplants as commonly-used policies (1.6% more than monthly-batching), and shorter patientwaiting times (23 days faster than monthly-batching).
We study how to optimally match agents in a dynamic random exchange market. Matching agentsfaster reduces waiting times but at the same time makes the market thinner, leaving more agentswithout a compatible partner. This trade-off naturally arises for kidney exchange platforms thatseek to form exchanges between incompatible patient-donor pairs. Waiting to match may increasethe number of patients receiving a kidney, but comes at a cost: receiving a transplant earlier doesnot only improve the quality of life for the patient but also leads to substantial savings in dialysiscosts for society. In the last decade kidney exchange platforms in the United States gradually ∗ We want to thank Mohammad Akbarpour, Aaron Bodoh-Creed, Yeon-Koo Che, Yuichiro Kamada, and OlivierTercieux for useful comments and suggestions. Ashlagi: MS&E, Stanford, [email protected]. Nikzad: Eco-nomics, University of California, Berkeley, [email protected]. Strack: Economics, University of California, Berke-ley, [email protected]. Ashlagi acknowledges the research support of the National Science Foundation grant(SES-1254768). For some early work on kidney exchange in static pools and the importance of creating a thick marketplace, seeRoth et al. (2007, 2004). The savings from a transplant over dialysis is estimated by over $270,000 (Held et al., 2016) per year over thefirst five years. a r X i v : . [ ec on . T H ] J un oved from matching roughly every month to matching daily. Practitioners are concerned thatthis behavior, some of which is driven by competition between Kidney exchanges, is harmful, es-pecially for the most highly sensitized patients. In contrast, kidney exchange programs in Canada,Australia, and the Netherlands match periodically every 3 or 4 months (Ferrari et al., 2014).This article analyses the trade-off between agents’ waiting times and the percentage of matchedagents in dynamic markets. We find that, maybe surprisingly, matching greedily minimizes thewaiting time and simultaneously maximizes the chances to find a compatible partner for all agentsfor sufficiently large markets. We further quantify the inefficiency associated with other commonlyused policies like monthly matching using data from the National Kidney Registry.To analyze this question we propose a stochastic compatibility model with easy-to-match andhard-to-match agents. Easy-to-match agents can match with all other agents with a positive prob-ability p , whereas hard-to-match agents can match only with easy-to-match agents with a positiveprobability q . The main focus of our analysis is on the case where the majority of agents arehard-to-match, which is inline with kidney exchange pools. This model captures two empiricalregularities of the patient-donor data from the National Kidney Registry (NKR): First, as themarket grows large, the fraction of patient-donor pairs that are matched in a maximal matchingdoes not approach 1, which is a consequence of the imbalance between different pairs’ blood typesin kidney exchange (Saidman et al., 2006; Roth et al., 2007). Second, as the market grows large,the fraction of agents that cannot be matched in any matching goes to zero. Our parsimonioustwo-type model captures the above regularities and no single-type model can account for both ofthem (Propositions 1 and 2).We study a dynamic model based on the above two-type compatibility structure in which easy-and hard-to-match agents arrive to the market according to independent Poisson processes withrates m E and m H . Agents depart exogenously at rate d . The market-maker observes the realizedcompatibilities and decides when to match compatible agents. We evaluate a policy based on threemeasures: match rate , matching time , and waiting time . The match rate is the probability withwhich an agent is matched. The waiting time is the average difference between the time an agentarrives and the time she leaves, either matched or unmatched. The matching time measures howlong an agent has to wait on average before being matched, conditional on being matched.We start by analyzing the greedy policy , which matches every agent upon its arrival if possible. The National Kidney Registry (NKR) and the Alliance for Paired Donation (APD) search for matches on a dailybasis, whereas the United Network for Organ Sharing (UNOS) search for matches twice a week. From personal communication with the kidney exchange directors. That is, patients who have common antibodies that will attack foreign tissue. See also Agarwal et al. (2018), who study a production function of a kidney exchange platform in order to quantifythe marginal benefits of different types of pairs and altruistic donors. A patient-donor pair cannot be matched in any matching if it cannot form a (two-way) exchange with any otherpatient-donor pair due to biological compatibility.
2e first derive the distribution of the number of hard- and easy-to-match agents waiting in themarket in steady state. We find that, as the market grows large, many hard-to-match agentswill wait in the market for a compatible partner at any point in time. As a consequence, almostevery easy-to-match agent is matched with a hard-to-match agent immediately upon arrival and theprobability that an easy-to-match agent leaves the market unmatched converges to zero (Proposition3). As their match rate is close to one and their waiting time is close to zero the greedy policyis asymptotically optimal for easy-to-match agents in large markets. As hard-to-match agents areincompatible with each other and almost every easy-to-match agent is paired with a hard-to-matchagent, greedy also maximizes the match rate of hard-to-match agents. Maybe less intuitively,greedy-matching also minimizes the waiting time of hard-to-match agents compared to any otherpolicy when the market grows large (Proposition 4). We establish this result by first showing thatweakly more hard-to-match agents wait for a partner in any other policy. Then, we use a version ofLittle’s law which implies that the average number of hard-to-match agents waiting in the marketis proportional to their waiting time. Together, this establishes that greedy matching will performweakly better than any other policy in a sufficiently large market.The main challenge in the proof is analyzing the steady state distribution of a two-dimensionalrandom walk which keeps track of the number of easy- and hard-to-match agents waiting in the mar-ket. Instead of analyzing the two-dimensional process directly —which is in general intractable—we use coupling techniques to derive an upper and a lower bound on the marginal distribution ofhard-to-match agents. These bounds allow us to derive the distribution of the number of easy-to-match agents waiting in steady state.Next, we quantify the inefficiency associated with batching policies, which are commonly usedin practice. A batching policy periodically (e.g. monthly) matches as many agents as possible. Wederive a lower bound on the waiting time and an upper bound on the match rate under batchingpolicies. As the batching period gets longer the bound on match rate decreases and the bound onthe waiting time increases. Our bounds imply, that in a large market, greedy matching dominatesany batching policy, as it leads to strictly shorter waiting times and strictly higher match rates.Quantitatively, our results imply that in a large market the match rate of any batching policy, forboth easy- and hard-to-match agents, is at most the match rate under the greedy policy minushalf the length of the batching period multiplied by the departure rate. For example if agentsexogenously depart on average after a year and the batching period is one month (i.e. / of ayear) the batching policy will match at least / = 4 .
2% fewer agents.We also analyze the patient matching policy introduced by Akbarpour et al. (2017). This policyassumes that agents’ exogenous departure times are observable. It matches an agent upon departureif possible, and otherwise the agent leaves the market unmatched. We show that the patient policy3eads to the same match rate as the greedy policy when the market becomes large. In both policiesalmost all easy-to-match agents are matched almost upon arrival in a large market. We show thathard-to-match agents wait longer (in first order stochastic dominance) under the patient policycompared to the greedy policy. Quantitatively, the waiting time of hard-to-match agents underthe greedy policy equals the waiting time under the patient policy multiplied by (cid:0) − m E m H (cid:1) where m E , m H are the arrive rates of easy and hard-to-match agents. For example when / of the agentare easy-to-match (i.e. 2 m E = m H ) hard-to-match agents will wait twice as long under the patientpolicy. Finally, we test whether the large-market predictions of our model hold in data from the NationalKidney Registry (NKR). This data differs from our assumptions along two dimensions: first, becauseof blood and tissue types it does not match our stylized two-type compatibility structure. Second, itis unclear that the market is sufficiently large for our results to apply, because only a finite numberof agents arrive every year (around 360/year). Nevertheless, the data confirms the predictions ofour model (Section 4): As the market becomes large, the waiting and matching times of patient-donor pairs who are “easier” to match approach 0, but the waiting and matching time of pairswhich are “harder” to match do not (c.f. Table 1). We further find that batching policies resultin no improvement to the match rate and lead to longer waiting times relative to greedy matching(c.f. Table 1 and Figure 8). Lastly, under greedy matching, the waiting and matching times aresignificantly lower than under patient matching. At the same time, we do not find systematicdifferences between the match rates under greedy and patient matching (Figure 11 and Table 1).
The most closely related literature studies dynamic matching on networks when agents’ preferencesare based on compatibilities, motivated by kidney exchanges. This literature was initiated by ¨Unver(2010). It is useful to organize this literature into two perspectives:The papers taking the first perspective seek to minimize the waiting time of agents in the marketassuming that agents never depart exogenously. ¨Unver (2010) analyzes a kidney exchange modelwith linear waiting costs where compatibility is deterministically determined by blood type. Hefinds that for pairwise exchanges, greedy matching is optimal. Anderson et al. (2017) consider amodel where compatibilities are based on a random graph model, in which each agent is compatiblewith any other agent with some fixed probability. They find that greedy matching minimizes average The differences with Akbarpour et al. (2017) are discussed in detail in Sections 1.1 and 3.2. To capture the theoretical notions of hard-to-match and easy-to-match in our data set, we categorize patient-donorpairs based on notions of “over-demanded” and “under-demanded” notions, as defined in Section 4. Further, using thresholds to facilitate three-way matchings is beneficial, but generates relatively small improve-ments. Ashlagi et al. (2016) add asymmetrictypes to this random compatibility model where one type has a non-vanishing probability of beingmatched with any other agent. In particular, in contrast to our model, any two types can potentiallymatch. They too find that greedy matching minimizes waiting time. This strand of literaturethus establishes in various models that greedy matching minimizes average waiting time.The second perspective one could take to dynamic matching in kidney exchanges is to analyzehow many agents are matched. Akbarpour et al. (2017) consider a model with departures, inwhich each agent is compatible with any other agent with some fixed probability. They find thatthe patient policy leads to an exponentially smaller loss rate (i.e., fraction of unmatched agents)compared to the greedy policy.Each of the above perspectives studies one of two objectives: either to minimize the time untilan agent is matched or to minimize the number of agents that leave the market unmatched. Giventhe different objectives the two perspectives lead to different conclusions about the optimality ofgreedy and suggest a trade-off between matching agents quickly and matching as many agents aspossible. Our main contribution with respect to this literature is to study this trade-off and showthat it vanishes in large kidney-exchange markets with asymmetric agents.Technically, our paper is the first to analyze a model with both exogenous departures andheterogeneous agents and to also analyze the distribution of waiting and matching times, ratherthan just averages.From a modeling perspective there are two major differences between our paper and most ofthe the above literature. First, compatibilities in our model depend also on the agent’s type (i.e.donor’s blood type). Second, in contrast to Anderson et al. (2017) and Akbarpour et al. (2017)we focus on markets, where the matching probabilities do not vanish. Their models intend tocapture sparse compatibility networks and small markets. In contrast we are interested in largemarkets where the agents’ types are independent of the market size as arguably natural for kidneyexchanges. Whether a given market is approximated by either compatibility model depends highlyon the specific context, and is ultimately an empirical question. Our simulations reveal that kidneyexchange markets of even moderate size behave as predicted by our large market model.Similar type of asymmetries across agents also appear in Nikzad et al. (2017). They are con-cerned with a proposal for global kidney exchange, which incorporates international pairs to domes- Their results hold under different types of feasible matchings: pairwise, pairwise and three-way cycles, and chains. Note that types differ only by the probability to match with other agents. They also study the relationship between the balance in the market between types of agents and the desiredmatching technology. One may interpret our model as combining both blood types and randomness due to tissue-type incompatibilities.We note that Dickerson et al. (2012) develop a heuristic to approximate the full dynamic program and overcome theinfamous “curse of dimensionality.” A kidney exchange pool can be represented by a compatibility graph G . Each node in the graphrepresents an agent (a patient-donor pair), and a link between two nodes exists if and only if thetwo corresponding agents are compatible with each other (so a bilateral exchange between the nodesis feasible). We restrict attention to bilateral exchanges.A matching µ is a set of non-overlapping compatible pairs of agents. Denote by M ( G ) the setof matchings in G . For every compatibility graph G let | G | denote the number of agents in thegraph, and for every matching µ let | µ | denote the number of agents in that matching.Define the (normalized) size of the maximum matching (SMM) in a graph G to be the fractionof matched agents in a maximum matching:SMM = max µ ∈ M ( G ) | µ || G | . Define the fraction of agents without a partner (FWP) to be the fraction of agents that are notmatched in any matching (thus have no compatible agent):FWP = |{ i ∈ G : ( i, j ) / ∈ M ( G ) for all j }|| G | . See also related results in queueing models (Leshno, 2014; Bloch and Cantala, 2016). The paper restricts attention to matching only pairs of agents and not through chains. For the effect of matchingthrough chains see, e.g., Ashlagi et al. (2011) and Anderson et al. (2017). <
1. This is a natural consequence of the different bloodtypes (Roth et al., 2007). Second, when the market grows large, the fraction of pairs that have nocompatible pairs decreases. Roughly 8% of pairs are incompatible with any other pair in this data(this is the FWP). Since compatibility depends only on the characteristics of the patients anddonors, it is independent of pool size, and thus in a sufficiently large pool one would expect thatthe FWP would further decrease to zero.
100 400 700 1000 1300 1600 1900 2200 2500 2800 3100 3400Number of pairsSMM FWP SMM FWP
Figure 1: The average fraction of patient-donor pairs without a compatible partner in blue andthe (normalized) size of the maximum matching in red, for a random subset of patient-donor pairsfrom NKR, APD, and UNOS data.
Fact 1.
As the kidney exchange patient-donor pool grows large, the compatibility graph (Figure 1)is such that the size of the maximal matching (SMM) stays bounded away from and the fractionof patient-donor pairs without a compatible partner (FWP) goes to . The change in both the SMM and FWP measures captures the benefit of a larger market. Sincea matching policy in a dynamic environment trades off the benefits of a larger market with thewaiting costs incurred by the agents, having a model that accurately represents the SMM and theFWP is important to correctly describe the costs and benefits of waiting to match. In practice some patients can receive a kidney from blood-type incompatible donors due to advanced technology.For the sale of simplicity we ignore this in our simulations, but it is worth noting that the FWP drops to roughly 4%when this form of compatibility is allowed. .1 A compatibility model To capture the features of kidney exchange identified in Fact 1 we adopt a stylized and tractablemodel with random compatibilities. There are two types of agents, easy-to-match or hard-to-match ,denoted by E and H, respectively. There are more hard-to-match than easy-to-match agents. Anypair of hard-to-match and easy-to-match agents are compatible independently with probability p >
0, any pair of two easy-to-match agents are compatible independently with probability q > qp Figure 2: The random compatibility model.Proposition 1 shows that this simple model is indeed able to capture the features of real kidneyexchanges identified in Fact 1:
Proposition 1.
Consider a compatibility graph with m easy-to-match agents and (1 + λ ) m hard-to-match agents where λ > . Compatibilities between pairs of agents are generated as described inSection 2.1. As m grows large the SMM goes to λ and the FWP goes to zero almost surely: lim m →∞ SMM = 22 + λ , (1)lim m →∞ FWP = 0 . (2)That the size of the maximal matching cannot exceed λ is intuitive: since H agents cannotmatch with each other and there are more H agents than E agents, some H agents must remainunmatched when the pool is large. An upper bound on the fraction of agents that can be matchedequals twice the fraction of E agents λ . Furthermore, note that this fraction is achieved wheneverthere exists a matching, in which all E agents are matched with H agents. It follows from a standardresult in random graph theory that the probability that such a “perfect matching” exists approaches1 as the pool grows large. Furthermore, as the pool grows large any H agent will be compatible withsome E agent, since compatibilities between agents are drawn independently. Thus, the fraction ofagents without a partner converges to 0.The parameter λ of the model measures the degree of imbalance between hard- and easy-to-match agents. So λ = 0 corresponds to a balanced pool. Figure 1 suggests that the size of8he maximal matching in the national kidney exchange data converges to roughly 60% when thepool becomes large, implying that λ ≈ .
33 in the context of our model. This reduced-formcalibration is roughly consistent with a value of λ ≈ .
33 that one obtains for the same datawhen defining hard-to-match agents directly as those who cannot match with each other due toblood-type incompatibilities (see Section 4).Proposition 1 establishes that our two-type model can match the empirical behavior of the SMMand FWP measures. Proposition 2 establishes that no model with a single type can replicate theempirical features of real kidney exchanges observed in Fact 1, even when allowing the probabilityof compatibility between two agents to depend on the market size in arbitrary ways.
Proposition 2.
Consider a model with m homogeneous agents, in which every pair of agents arecompatible independently with probability p ( m ) > that may depend on the market size. Thefollowing two conditions cannot be satisfied simultaneously almost surely: lim m →∞ SMM < , and (3)lim m →∞ FWP = 0 . (4)The proof of Proposition 2 is constructive. It begins with assuming that every agent has acompatible partner when the pool grows large, i.e., that (4) is satisfied. It then constructs analgorithm which selects a single matching for any given compatibility graph and proves that thematching selected by this algorithm will include all agents with high probability as the pool growslarge. This implies that (3) and (4) cannot be simultaneously satisfied in any random graph modelwith homogeneous agents.Economically, this observation implies that heterogeneity of agents plays a major role in kid-ney exchanges. Our two-type model is arguably the simplest random compatibility model thatcaptures these features of the compatibility graph.
We embed the static compatibility model from Section 2.1 in a dynamic model to allows to studymatching policies in a dynamic setting. We consider an infinite-horizon dynamic model, in whichagents can match bilaterally. Easy-to-match agents arrive to the market according to a Poissonprocess with rate m , and hard-to-match agents arrive to the market according to an independent This is consistent with Roth et al. (2007) and Agarwal et al. (2018), who demonstrate that the types of patientsand donors play a crucial role for efficiency. λ ) m . We assume that the majority of agents are hard-to-match,that is λ >
0, unless explicitly stated otherwise.An agent that arrives to the market at time t becomes critical after Z units of time in themarket, where Z is distributed exponentially with mean d , independently between agents. We referto d as the exogenous departure rate , or departure rate for the sake of brevity. The latest time anagent can match is the time she becomes critical, t + Z ; immediately after this time the agent leavesthe market unmatched. Matching policies.
Denote by G t the compatibility graph induced by the agents that are presentat time t . A dynamic matching policy selects at any time t a matching µ t ∈ M ( G t ), which may beempty. Whenever a non-empty matching is selected, all matched agents leave the market.Several kidney exchange platforms in the United States typically match in a greedy manner,attempting to match a patient-donor pair as soon as it arrives to the market (see Ashlagi et al.(2018)). A tractable approximation of this behavior is a greedy matching policy. Definition 1 (Greedy) . In the greedy policy an agent is matched upon arrival with a compatibleagent if such an agent exists. If she is compatible with more than one agent, H agents are prioritizedover E agents and otherwise ties are broken randomly. Some platforms identify matches periodically (thus less frequently than a greedy matching pol-icy), allowing the pool to thicken and possibly offer more matching opportunities. For example,UNOS matches twice a week, whereas national platforms in the United Kingdom and the Nether-lands identify matches every three months (Biro et al., 2017). This behavior is approximated withthe following batching policy.
Definition 2 (Batching) . A batching policy executes a maximal match every T days. If there aremultiple maximal matches, select randomly one that maximizes the number of matched H agents. The last policy we consider is a patient matching policy, proposed by Akbarpour et al. (2017),which attempts to match an agent only once she becomes critical. In the context of kidney exchangethis means that two patient-donor pairs in the pool are matched only if the condition of one ofthese pairs is such that it cannot match at a later point in time for medical or any other reason. Definition 3 (Patient) . In the patient policy an agent that becomes critical is matched with acompatible agent if one exists. If she is compatible with more than one agent, H agents are prioritizedover E agents, and ties are broken randomly. Such a policy is practical if the times at which pairs become critical are observable.
Measures for performance.
To study the performance of a matching policy we focus on twomeasures. One is the match rate of each type Θ ∈ {
E, H } , which is the fraction of agents of type Θthat match at the steady state. The other is the expected waiting time (or simply waiting time ) anagent of type Θ spends in the market, whether eventually matched or not. For type Θ ∈ { H, E } we denote its match rate by q Θ and its expected waiting time by w Θ , where the policy will beclear from the context. Another measure we analyze is the matching time of a type Θ, which is theaverage time agents of type Θ who eventually match spend in the market.In kidney exchange the match rate corresponds to the probability of exchanging a kidney withanother patient-donor pair. Because waiting for a kidney is often spent on dialysis, which is costly(both financially and physically), these quantities have a direct impact on welfare.We are interested in optimal policies for large pools. Formally, we consider the following notionof optimality: Definition 4 (Asymptotic optimality) . A policy is asymptotically optimal if for every (cid:15) > there exists an m (cid:63) such that if the arrival rate is large m ≥ m (cid:63) , every type of agent improves itsmatch rate and expected waiting time by at most (cid:15) when changing to any other policy. This optimality notion is demanding, since it requires the policy to be optimal for every type ofagent simultaneously. It is unclear whether an asymptotically optimal policy exists, since a policythat is optimal for H agents might be suboptimal for E agents.
In this section we present the main result of the paper and discuss its implications. We discussthe logic underlying the results in Section 3.3. The main finding is a characterization of the matchrates and waiting times associated with the greedy, batching and patient matching policies.
Theorem 1.
The greedy policy is asymptotically optimal, whereas the batching and patient policiesare not asymptotically optimal.
We further compute the match rates and expected waiting times under these policies.
Proposition 3.
As the arrival rate m grows large:(i) The match rates of hard- and easy-to-match pairs under greedy approach ( q GH , q GE ) = ( λ , ,respectively, and their expected waiting times approach ( w GH , w GE ) = ( λ d λ , . ii) The batching policy, which matches every T periods, achieves match rates of at most ( q BH , q BE ) =( − e − T/d (1+ λ ) T/d , − e − T/d
T/d ) . Furthermore, the expected waiting and matching time for each type Θ ∈ { E , H } is at least w B Θ = d (1 − q Θ ) . Also, q B Θ < q G Θ , whereas q B Θ approaches q G Θ as T approaches . In addition, w B Θ > w G Θ , whereas w B Θ approaches w G Θ as T approaches .(iii) The match rates of hard- and easy-to-match pairs under the patient policy approach λ and , respectively, and their expected waiting times approach d and , respectively. Figure 3 illustrates the match rates and waiting times of H and L agents under the differentpolicies as found in Theorem 3. GreedyPatientBatching 7 daysBatching 14 daysBatching 30 days0.40 0.41 0.42 0.43 0.440100200300 match rate w a i t i ng t i m e i nda ys hard to match agents Patient / GreedyBatching 7 daysBatching 14 daysBatching 30 days0.90 0.95 1.00 1.050100200300 match rate easy to match agents
Figure 3: Illustration of Theorem 1 when there are twice as many hard-to-match agents as easy-to-match agents ( λ = 1 .
33) and expected departure time is 360 days. The blue dots represent thepredictions of our model as the arrival rate goes to infinity.In Figure 3 we calibrated the model such that it matches the data from the National KidneyRegistry, where agents depart on average after 360 days and λ ≈ .
33. As Figure 3 illustrates, thebatching policy leads agents to wait longer and get matched with a smaller probability than underthe greedy approach. The losses resulting from this are substantial. For example, under a monthlybatching policy hard-to-match agents wait on average 6 days and easy-to-match agents 15 dayslonger and get matched with 1 .
7% and 4% lower probability. Similarly, the patient policy matchesequally many agents as the greedy policy, but leads to a substantially longer expected waiting timefor hard-to-match agents (155 more days).We now provide a rough intuition for the differences among greedy, batching and patient match-ing policies. In Section 3.3 we provide a more precise intuition and a sketch of the argument forthe various parts of the results. As there are more hard- than easy-to-match agents, hard-to-matchagents will accumulate and a large number of them will be present at any time under any policy.This implies that under greedy matching, easy-to-match agents will have upon arrival, with high12robability, a compatible hard-to-match agent and are therefore matched immediately. As a con-sequence, every easy-to-match agent is matched with a hard-to-match agent, which implies thatgreedy matching asymptotically achieves the optimal match rate.Under the batching policy each agent waits at least from the time of her arrival until thenext time a matching is identified. Thus each agent waits on average at least half the length ofthe batching interval. Furthermore, each agent departs during that time with strictly positiveprobability. Thus, easy-to-match agents are worse off under the batching policy than under thegreedy policy where they get matched immediately with probability 1. As some easy-to-matchagents leave the market unmatched, hard-to-match agents are matched with a smaller probability.Consequently there are, on average, more hard-to-match agents waiting in the market. Little’slaw, which states that the arrival rate multiplied by the average waiting time equals the averagenumber of waiting agents, implies that hard-to-match agents also wait longer under any batchingpolicy than under a greedy matching policy. As both types are worse off, batching policies are notasymptotically optimal.By analyzing the dynamics of the market we show that under patient matching, so many hard-to-match agents accumulate that an easy-to-match agent will match, with high probability, with acritical hard-to-match agent almost immediately upon arrival. This implies that patient matchingasymptotically achieves the optimal match rate. As hard-to-match agents get matched only whenthey become critical, the distribution and expectation of their waiting time is the same as if they donot match at all. Hence, hard-to-match agents get matched faster under a greedy policy, implyingthat patient matching is not asymptotically optimal.Finally, observe that the smaller the imbalance in the market, the faster hard-to-match agentsmatch under the greedy policy. Under patient matching, however, the waiting time distribution ofhard-to-match agents is independent of the market imbalance. So as m H m E approaches 1, the ratiobetween the average waiting times under patient and greedy matching policies approaches infinity. Remark 1.
For completeness, we prove in Appendix G a counterpart for some parts of Theorem1 for the empirically irrelevant case λ <
0. In that appendix we establish the intuitive fact that,when the majority of agents are easy to match, the expected waiting and matching times approach0 under both greedy and patient matching policies as the market size m approaches infinity. Remark 2.
Akbarpour et al. (2017) find, in apparent contrast to our results, that patient matchingperforms better than greedy matching. The difference stems from a combination of two factors.First, in their model, there is a single type of agent. Second, the likelihood that two agents matchvanishes in the arrival rate as p ( m ) = cm for some constant c >
0. We set the likelihood of matchingto be independent of the size of the market and allow for agents to have differing abilities to match13ith other agents (in line with the empirical structure of kidney exchanges – recall Proposition 1and Proposition 2).A further difference is that Akbarpour et al. (2017) measure the ratio between loss rates ofdifferent matching policies, whereas we are interested in the match rate. Analyzing the matchrate allows us to show that greedy matching is an asymptotically optimal policy whenever theagent has risk-neutral expected utility preferences. While the ratio between loss rates is an intuitivemeasure it has no direct relation to expected utility preferences.
In this section we provide a proof sketch for the various parts of Theorem 1 and Proposition 3 aswell as additional results on the matching time distributions. It is useful to first establish an upperbound on the performance of any policy.
Proposition 4 (Upper bound on the performance of any policy) . For any (cid:15) there exists m (cid:15) suchthat for any market size m > m (cid:15) and any policy the match rate of H agents is at most λ + (cid:15) andthe expected waiting and matching time is at least λ d λ − (cid:15) . Proposition 4 is shown by considering the hypothetical situation where each E agent can matchwith any H agent and thus in a large market no E agent remains unmatched. Since H agents cannotmatch with other H agents, this provides an upper bound on the probability of an H agent beingmatched, which is at most the ratio of E to H agents, λ . Next we analyze the performance of greedy matching as the market grows large. The followingproposition includes the results in the first part of Proposition 3.
Proposition 5 (Performance of the greedy matching policy) . Consider the greedy policy as themarket grows large ( m → ∞ ). The match rate of H agents converges to λ and the waiting andmatching times converge to an exponential distribution with mean λ d λ . The match rate of E agentsconverges to and their waiting and matching times converge to . We first provide intuition for the waiting time distribution. Consider greedy matching in adeterministic setting where every E agent is compatible with every H agent, and agents arrive and In our model, as the market grows large, the ratio between the loss rates under the greedy and patient matchingpolicies approaches 1 as does the ratio between the match rates. Their model, however, predicts an exponential ratiobetween the loss rates, with the exponent being proportional to the average degree of the compatibility graph. As theaverage degree grows larger, the ratio between loss rates grows. However, the ratio between match rates approaches1. (Loss rates under both policies approach 0 and match rates approach 1.) x the steady state number of H agents present in the market. Per unit of time,(1 + λ ) m H agents arrive to the market and m of them are matched with E agents. Further, xd of the waiting agents are expected to depart unmatched per unit of time. In the steady state thenumber of departing agents equals the number of unmatched arriving agents. Thus, x solves thebalance equation xd = λ m ⇒ x = λ m d . Thus, if the matching partner for an E agent is chosen at random, each H agent has a probability of mλ m d = λ d of being chosen per unit of time. The time at which a never-departing H agent would bematched is therefore exponentially distributed with rate λ d . The time until an H agent departs themarket is exponentially distributed with rate d . Since the minimum of two exponentially distributedrandom variables is again exponentially distributed with rate equal to the sum of the rates, thewaiting time of an H agent is exponentially distributed with rate λ d + d = λλ d , and thus withmean λ d λ .The formal proof of Proposition 5 is more complex. The main idea is to show that for asufficiently large market m the steady state of the model with random compatibilities is close tothe steady state of the model where every E agent is compatible with every H agent and agentsarrive and depart deterministically. Our results can thus also be understood as a motivation forstudying deterministic models. To show this approximation, consider the number of waiting Hagents at time t , denoted by x t , and the number of waiting E agents, denoted by y t . We showthat under greedy matching ( x t , y t ) t is a two-dimensional continuous-time Markov process. Wederive the fixed-point equation, which characterizes the steady state distribution of this processand show that it admits a unique solution. We then prove that the stationary distribution musthave exponential tails, and the rate at which the density of the stationary distribution decays inthe tails shrinks at least at a rate proportional to √ m . We use this to show that the steady-statenumber of E and H agents in the market is not more than a factor of √ m away from the solutionfor the fixed-point equation. As this distance grows slow relative to the market size, the randomfluctuations of ( x, y ) are well approximated by their expectations which correspond to the dynamicsof the deterministic setting described earlier. A complication in this analysis is the two-dimensionalnature of the stochastic process ( x, y ), which requires analyzing some auxiliary problems that wedescribe in detail in the appendix.Next, we explain why the matching time of H agents follows the same distribution as their See also Ashlagi et al. (2019), who analyze a two-dimensional Markov chain. t d be the (random) departure time for this agent.Let t m be the time at which the agent would be matched with another agent, assuming that theagent never departs. The waiting time of h then is t = min { t d , t m } . The conditional distributionof the minimum of two independent exponential random variables t d , t m is independent of whichone is smaller. Thus, the matching time t m of H agents has the same distribution as t .As Proposition 5 shows, greedy matching achieves the upper bound derived for arbitrary policiesin Proposition 4 and we conclude that greedy matching is asymptotically optimal. We next quantify the performance of patient matching policy. The following proposition includesthe third part of Proposition 3.
Proposition 6 (Performance of the patient matching) . Consider the patient policy when the poolgrows large ( m → ∞ ). The match rate of H agents converges to λ and the waiting and matchingtime converge to an exponential distribution with mean d . The match rate of E agents converges to and their waiting and matching times converge to . To get a rough intuition for this result, again consider the hypothetical case where all H andE agents can match and agents arrive deterministically. There exists a steady state under thepatient policy such that there are no E agents in the market and the number of H agents in themarket is approximately (1 + λ ) m d . The reason this is a steady state is as follows. H agents getcritical and attempt to match with E agents at a rate of (1+ λ ) m dd = (1 + λ ) m , whereas E agentsenter the market only at rate m . This means that E agents are matched immediately and thesteady-state number of E agents in the market remains zero. Since no E agent becomes critical,there are no H agents who are matched with a critical E agent. Thus, H agents depart at the rate (1+ λ ) m dd = (1 + λ ) m and arrive at the rate (1 + λ ) m , which implies that this is a steady state. Astandard argument implies that the steady state is unique. Since H agents are the ones that initiatematches, their average waiting time equals the average time d until they depart exogenously. Bythe same argument given for greedy matching it follows that the waiting and matching times havethe same distribution.The main technical difficulty in proving Proposition 6 is the same as in the proof Proposition5; we need to approximate the stochastic process describing the number of waiting E and H agents That is, for any z > P (cid:2) t m < z (cid:12)(cid:12) t m = t (cid:3) = P [ t < z ] . This holds because P (cid:2) t m < z (cid:12)(cid:12) t m = t (cid:3) = P [min { t < z, t m = t } ] P [ t m = t ] = P [ t < z ] · P [ t m = t ] P [ t m = t ] = P [ t < z ] . The second equality follows from the fact that the events t < z and t m = t are independent, which holds because t d , t m are independent exponential random variables. Figures 4 and 5 graphically compare the bounds provided by Proposition 3 on the match rate andwaiting times of H agents under the batching policy to the match rates and waiting times of Hagents under the greedy and patient matching policies.Figure 4: Upper bound on the match rateof H agents under the batching policy when λ = 1. Figure 5: Lower bound on the waiting timeof H agents under the batching policy when λ = 1.The bounds given by Proposition 3 for H agents are derived by analyzing a simpler stochasticprocess in which (i) easy-to-match nodes are not compatible, and (ii) the probability of compatibilitybetween an easy-to-match and a hard-to-match node is 1. A straightforward coupling exercise showsthat the match rate of H agents is larger in the simplified process than in the original process andtheir waiting time smaller. This allows us to analyze the simplified process instead of the originalprocess. The bounds that we derive this way are in fact tight (up to vanishing factors) for theoriginal process as well. Figures 4 and 5 illustrate the convergence of these bounds to those forthe greedy policy (given in Proposition 5) as T approaches 0.The bounds given in Proposition 3 for E agents are calculated simply based on the fact thatan arriving E agent should wait until the next matching period and may not get matched if shebecomes critical before that. The bounds for E agents also are tight for the original process andapproach to those of the greedy policy as T approaches 0. This can be proved formally using proof techniques similar to the ones we used to analyze the greedy matchingpolicy. .4 On convergence rates and the effect of market imbalance In the previous section we have shown that greedy matching is optimal when the market is large.Here, using simulations, we (i) explore the convergence of match rates and waiting times undergreedy matching as the market grows large for a fixed λ , and (ii) demonstrate that greedy matchingis also optimal in small imbalanced markets with a large share of hard-to-match agents.Figure 6 plots the match rates for both types of agents and the matching and waiting timesfor a fixed λ = 1 while varying the arrival rate m . Observe that the measures converge quicklyto their limit value as m increases (note that the theory predicts that in the limit one half of thehard-to-match agents are matched). arrivals per day, λ=1H match time H wait time E match time E wait time H matched E matched
Figure 6: Sensitivity analysis over the arrival rate m for the greedy matching policy. λ is set to1 and agents depart (exogenously), on average, after 360 days. The arrival rate in the NKR datais roughly 1. The left y-axis represents times in days and the right y-axis represents the fractionmatched.Figure 7 illustrates the optimality of greedy matching in small imbalanced markets. It plots thesame measures as we will see in Figure 6, but this time we fix a small arrival rate m = 0 .
25 andvary the imbalance parameter λ . We observe that the average number of matches for both typesof agents quickly converges to their upper bound as the imbalance parameter λ increases.To gain some intuition about the effect of imbalance on the optimality of greedy matching, it isuseful to consider the loss ratio of a policy, defined to be 1 minus the ratio of the expected numberof agents matched under that policy per unit of time to the expected number of agents matchedby the omniscient policy per unit of time. The omniscient policy has access to the whole sample path of the stochastic process, involving arrivals, departures,and compatibilities, and therefore makes the maximum number of matches. λ, m=0.25 H match time H wait time E match timeE wait time H matched E matched
Figure 7: Sensitivity analysis over the imbalance λ for the greedy matching policy. m is fixed to0 .
25 per day and average days in the pool is 360.We argue that the loss ratio under greedy matching can be bounded by O ( e − pλm/ ), whichapproaches zero exponentially fast in the imbalance parameter, λ . The key observation is thatthe steady-state distribution of the number of H agents in the market first-order stochasticallydominates the Poisson distribution with parameter λm . (This can be verified by a straightforwardcoupling argument.) Standard tail bounds of the Poisson distribution imply that, upon the arrival ofan E agent, the probability that the number of H agents in the market is at least λm/ O ( e − λm ).Conditioned on having at least λm/ − p ) − λm/ ≤ e − pλm/ . A union bound thenimplies that the probability that an E agent is not matched with an H agent, and therefore the lossratio, is bounded by e − pλm/ + e − λm = O ( e − pλm/ ). While we have formally shown that greedy matching is optimal in a large or imbalanced market,whether real markets are sufficiently large or imbalanced for this prediction to hold is ultimatelyan empirical question. To address this question we complement our theoretical findings with sim-ulations using comparability data from a real kidney exchange platform (the National KidneyRegistry, or NKR). These simulations indicate that greedy matching dominates the batching andpatient policies even in relatively small markets. We further provide small market simulations basedon the stylized model (and not real data) in Appendix H, which also indicate that greedy matchingoutperforms other commonly used policies.The data from the NKR includes 1364 de-identified patient-donor pairs from July 2007 toDecember 2014 (the focus of the paper is on bilateral matching and we therefore ignore altruistic19onors in the data). The data includes patients’ and donors’ blood types and antigens as well theantibodies for each patient, which allowed us to verify (virtual) compatibility between each donorand each patient. On average, approximately one patient-donor pair arrives per day to the NKR,and the average exogenous departure time of a pair is estimated to be 360 days. We note thatour simulation results are very similar when merging the APD, UNOS, and NKR data.In our simulations, arrivals of patient-donor pairs to the pool are generated by a Poisson processwith a fixed arrival rate. Arrival rates are varied from 0 .
01 to 4 pairs per day, capturing market sizesfrom one-tenth to four times the size of the NKR. Varying the rate of arrival allows us to observethe effect of thickening the market exogenously (see also Agarwal et al. (2018)). Each arrivingpair is sampled uniformly at random (with replacement) from the NKR data. Pairs depart fromthe pool according to an independent exponential random variable unless matched prior to thatpoint. The mean of the exponential random variable is set to 360 (days), based on the empiricalestimate. The compatibility of two pairs is determined from real blood types, antigen and antibodycompatibility. We simulate greedy, patient, and batching matching policies until the arrival of the 200,000thpair to the pool and report statistics for waiting time, matching time, and match rate by takingaverages over all or a predefined subset of pairs that belong to some collection of types. A batchingpolicy matches every T days the maximum number of pairs in the market, while prioritizing hard-to-match pairs. We experimented with batching frequencies T = 7 , , and 60 days. Figure 8 reports the fraction of matched pairs (left) and average waiting times (right) undergreedy, patient, and batching matching policies for different arrival rates. Patient matching resultsin the highest fraction matched, and the greedy and batching policies with T = 7 result in a slightlylower match rate (large batch sizes lead to lower match rates). Moreover, the average waiting timeunder greedy matching is the smallest among all policies. Table 1 reports the fraction of matchedpairs, average waiting time, and average matching time across all pairs in the simulation. We alsonote that the difference between the average matching times under batching and greedy matchingis between 10 and 40 days (depending on the arrival rate and the batch size).An interesting empirical observation is that waiting for more pairs to arrive does not increasethe match rate (which can be seen by comparing greedy and batching policies as shown in Figure8); however, increasing the arrival rate increases the match rate (see also Ashlagi et al. (2018)). Hazard rates vary slightly across pair types, but for the sake of simplicity we aggregate all pairs and used a simplehazard rate model to estimate departures rate. For more detailed estimates see Agarwal et al. (2018). In practice some patients can receive a kidney from a donor with an incompatible blood type, but for simplicityof exposition we ignore this possibility here. Similar findings hold without this assumption after adjusting theclassification of hard- and easy-to-match pairs in the data. We also examined weighted optimization using various weights and found no any qualitative differences. This isconsistent with Ashlagi et al. (2019), which found that prioritization is negligible when there are more hard-to-match Greedy Patient 7 30 60 (a) Fraction matched (b) Average waiting time
Figure 8:
Fraction of pairs matched (left) and average waiting time in days (right) under greedy, patient, andbatching matching policies in simulations using NKR data. The x-axis represents the arrival rate measured by thenumber of pairs arriving on average per day.arrival rate match rate matching time waiting timeGreedy Patient Batch30 Greedy Patient Batch30 Greedy Patient Batch300.01 0.108 0.108 0.10 150.07 301.50 176.27 320.82 329.90 321.630.05 0.225 0.231 0.216 130.45 253.41 164.17 277.84 318.33 281.020.1 0.283 0.286 0.27 119.79 233.50 160.62 258.10 306.44 261.980.5 0.391 0.392 0.373 98.91 204.53 131.68 218.39 285.20 224.131 0.431 0.439 0.415 92.12 198.36 115.17 204.07 279.23 209.312 0.458 0.468 0.443 85.63 192.00 100.33 194.59 272.53 197.124 0.485 0.489 0.463 77.13 183.51 89.28 183.59 268.62 188.17
Table 1:
Fraction of pairs matched, average matching time, and average waiting times in days over all pairs insimulations using NKR data.
We also compute the average matching time and waiting time for different types of pairs.Figure 9 reports these statistics for over-demanded pairs (left) and under-demanded pairs (right).Under-demanded pairs are ABO incompatible with each other and consist of the following types:O-X patient-donor pairs where X (cid:54) =O; A-AB; and B-AB. Over-demanded pairs are bloodtype-compatible with each other and consist of the following types: X-O patient-donor pairs whereX (cid:54) =O; AB-A; and AB-B. Over-demanded and under-demanded pairs are roughly equivalent toeasy-to-match and hard-to-match agents in our model; over-demanded pairs can potentially matchwith each other as well as other with under-demanded pairs whereas under-demanded pairs canmatch only with over-demanded pairs. We note that some patients are much more sensitized(that is, they have a variety of antibodies that will attack foreign tissue) than others even within agents in the markets. An X-Y patient-donor pair contains a patient with bloodtype X and a donor with bloodtype Y. (a) Over-demanded pairs (b) Under-demanded pairs
Figure 9:
Average matching times (MT) and waiting times (WT) in days under greedy (G) and patient (P) matchingpolicies in simulations using NKR data. The x-axis represents the arrival rate, which is the mean number of pairsarriving per day.
Observe in Figure 9 that the matching and waiting times of over-demanded pairs steadilydecrease as the market becomes thicker, whereas the average matching and waiting times of under-demanded pairs changes little. This finding is in line with the predictions from Proposition 3.Despite the heterogeneity in the data, the theoretical predictions (of the stylized two-type model)are aligned with the experiments when we categorize pairs as either over-demanded or under-demanded. Moreover, the patterns hold even though patients belonging to over-demanded pairsare, on average, more sensitized than those in under-demanded pairs. Figure 10 plots the fractionmatched and average waiting time of over-demanded pairs in which the patient also has a PanelReactive Antibody of at most 95 (that is, at least a 5% chance of being tissue-type compatiblewith a random donor). So even though many of them are quite sensitized, almost all of them getmatched and their waiting time is very low as the market grows large.Next, we run greedy and patient matching under the base case scenario (with an arrival rateof 1 pair per period) until 700,000 pairs arrive, which again are drawn uniformly at random withreplacement from the NKR data. For each pair, we compute the average waiting time over thecopies of this pair sampled in the simulation as well as the fraction of the copies that are matched(i.e., the empirical probability of getting matched). This gives, for each of the 1364 pairs in theoriginal data set, an average waiting time and an empirical probability of being matched under boththe greedy and patient matching policies. The results appear given in Figure 11. Figure 11a shows In fact, more than 40% of patients in over-demanded pairs have less than a 5% chance of being tissue-typecompatible with a random donor. Furthermore, about 10% of over-demanded pairs are not compatible with anyother pair within this data set, which is why the average matching and waiting times do not drop all the way to zero. Figure 10:
Fraction of pairs matched (FM) and average waiting time (WT) for over-demanded pairs whose patientshave PRA at most 95. The x-axis represents the arrival rate measured by the number of pairs arriving on average perday. The left y-axis represents the average waiting time in days matched and the right y-axis represents the averagefraction matched. that for each pair in the data set, the expected waiting time under greedy matching is shorter thanthe expected waiting time under patient matching (as all of the blue dots are above the 45 ◦ line).This observation suggests that the waiting time distribution under greedy matching stochasticallydominates the waiting time distribution under patient matching. Figure 11b reports, for anarrival rate of 1, the match rates under greedy and patient matching policies for each pair in thedata. Observe that for most pairs the empirical probabilities of matching under the greedy andpatient policies are “close” to each other.
450 0 50 100 150 200 250 300 350 400 p a t i e n t greedy (a) waiting times
300 350 400 450 p a t i e n t greedy (b) chance of matching Figure 11:
Each dot represents one of the 1364 pairs in the data. The left figure scatters the average waiting timefor each pair (averaged over its copies). The right figure scatters the empirical probability of being matched for eachpair under the greedy and patient policies. The horizontal and vertical axes correspond to the greedy and patientpolicies, respectively. A detailed analysis of the simulation results confirms that this is indeed the case. We omit the details.
This paper studies matching policies in a random dynamic market, in which some agents areeasier to match than others. We show theoretically as well as empirically that when the market issufficiently large, the greedy matching policy is optimal for all types of agents. This finding hasdirect practical implications for kidney exchanges that may not employ greedy matching policiesout of concern that greedy matching may harm highly sensitized patients. Our numerical simulations further provide evidence that matching frequently does not harm thenumber of transplants even for practical market sizes. While we only simulated pairwise matchingsand ignored frictions that occur in practice, the simulations of Ashlagi et al. (2018) accountfor such frictions and show, using simulations, that among batching policies, the policy with theshortest batching time window (essentially, the greedy policy) is optimal among a class of batchingpolicies.This paper has some limitations in the context of kidney exchange. First, we focus on matchingonly pairs of agents. Some kidney exchange programs, however, match pairs using chains initiatedby altruistic donors or cycles with three pairs. We note that numerous programs have very lowenrollment of nondirected (or altruistic) donors that initiate chains and, in some countries likeFrance, Poland, and Portugal, chains are not even feasible since altruistic donation is illegal (Biroet al., 2017). Finally we discuss a few technical limitations. First, matches are chosen uniformly at random. If,however, longer-waiting agents were to be assigned a higher priority, the waiting time and matchingtime would not be distributed exponentially (while their average values remains the same). But asthe market grows large, the match rates converge to optimal under both policies, and we conjecturethat the waiting and matching times under greedy matching would stochastically dominate theircounterparts under patient and batching policies. Another limitation is that we focus on the numberof matches and not the quality of matches. Studying matching policies under heterogeneous matchvalues remains an intriguing challenge. See, e.g., Ferrari et al. (2014). For instance, matches do not always translate into transplants due to refusals or blood test (crossmatch) failures. A simple extension of our model to have directed links will in fact predict that chains are not beneficial in largemarkets, which contrasts with the finding by Anderson et al. (2017). The difference stems from the fact that theirmodel assumes vanishing probabilities. Moreover, this prediction is likely to hold in such an extended model with a fixed market size (but is hard to analyze). eferences Nikhil Agarwal, Itai Ashlagi, Eduardo Azevedo, Clayton R Featherstone, and Omer Karaduman.Market failure in kidney exchange. Mimeo, 2018.Mohammad Akbarpour, Shengwu. Li, and Shayan Oveis Gharan. Thickness and information indynamic matching markets. Mimeo, 2017.Ross Anderson, Itai Ashlagi, David Gamarnik, and Yash Kanoria. Efficient dynamic barter ex-change.
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Proof of Proposition 1.
Note that as there are more E agents than H agents and H agents cannotmatch to themselves λ is an upper bound on the fraction of agents which can be matched forany m . Note, that the size of the maximal matching (SMM) equals λ if the bipartite graph with m easy-to-match agents and m hard-to-match agents on the other side admits a perfect matching.Becasue the probability that such a perfect matching exists converges to one as m → ∞ (see forexample Theorem 5.1 page 77 in Frieze and Karo´nski (2015)), it follows that SMM → λ .The probability that a hard-to-match agent has no partner is given by (1 − p ) m . Becausethe compatibilities between hard-to-match and easy-to-match agents are drawn independently theprobability that all hard-to-match agents have at least one partner is given by(1 − (1 − p ) m ) m (1+ λ ) . This probability converges to one as m → ∞ . The same argument shows that the probability thatall easy-to-match agents have at least one partner converges to one. Proof of Proposition 2.
The proof is by contradiction; suppose such p ( m ) exists. The chance thatan agent has no other compatible agents is (1 − p ( m )) m . If p ( m ) = O (1 /m ), then we have(1 − p ( m )) m ≤ e − mp ( m ) = e − O (1) , which implies that (4) cannot be satisfied. Therefore, suppose that p ( m ) = ω ( m ) m , where lim m →∞ ω ( m ) = ∞ . Next, we use this property to show that (3) cannot be satisfied.The proof is constructive. We propose a simple algorithm that chooses a matching µ with size | µ | such that lim m →∞ | µ | m = 1. Our algorithm is a greedy algorithm, defined as follows. It ordersagents of the graph from 1 to m , and visits the agents one by one. When visiting agent i , if thereare no agents left that are compatible with agent i , then the algorithm passes agent i and movesto agent i + 1. Otherwise, the algorithm chooses one of the neighbors of agent i arbitrarily, namelyagent j , and adds the pair ( i, j ) to the matching. The algorithm then visits the next available agentin the ordering. This process continues until the algorithm visits all agents.We claim that the algorithm produces a matching µ which satisfies lim m →∞ | µ | m = 1. Let φ ( m )be a function that grows faster than mw ( m ) but slower than m . Then, during the algorithm, so longas there are φ ( m ) agents left in the graph, the chance that a visited node has no compatible agents27s (1 − w ( m ) /m ) φ ( m ) ≤ e − w ( m ) φ ( m ) m = o (1) . That said, so long as there are φ ( m ) agents left in the graph, the agent visited by the algorithm willbe matched with a probability at least q ( m ) where lim m →∞ q ( m ) = 0. By linearity of exception,the expected number of agents that are left unmatched by the end of the algorithm is then at most φ ( m ) + ( m − φ ( m )) · q ( m ). Noting thatlim m →∞ φ ( m ) + ( m − φ ( m )) · q ( m ) m = 0completes the proof. Proof of Proposition 4.
Consider a random process which is similar to the original random process(described by M ), with the following differences:1. Matches are made greedily upon arrival of nodes, and only between an easy-to-match and ahard-to-match node.2. Easy-to-match nodes do not leave the pool before getting matched.3. The probability of compatibility of an easy-to-match and a hard-to-match node is 1.We represent the difference between the number of hard-to-match and easy-to-match nodes inthis random process with a one-dimensional MC M l , which is defined as follows. The state spaceof M l is V ( M l ) = Z . The MC is in state x when the number of hard-to-match nodes minus thenumber of easy-to-match nodes is x . The transition rates from state x to its left and right neighborsare respectively defined by l x = m + max { x, } , r x = (1 + λ ) m. It is straightforward to verify that M l is ergodic and therefore has a unique stationary distributionthat we denote by π . For notational simplicity, let x = E π [max { x, } ] and x ∗ = λm .The expected waiting time under any policy τ is at least xm (1+ λ ) and the expected match rateis at most 1 − xm (1+ λ ) . The proof of this statement is by a straightforward coupling of the samplepaths of M l and the stochastic process corresponding to τ . We skip the tedious details.The proposition is proved in two steps. The first step shows that x ≤ x ∗ + o ( x ∗ ), and the secondstep shows that x ≥ x ∗ − o ( x ∗ ). This would prove the claim.28 roof of Step i. We start by writing the balance equations, according to which π i +1 π i = r i +1 l i . (5)Suppose i = x ∗ + y , for some y >
0. Then, π i +1 π i = (1 + λ ) mx ∗ + y + m = (1 + λ ) mx ∗ + y + m = 1 − yx ∗ + y + m . Let n = x ∗ + m . When y > n / , then the above equation implies that π i +1 π i ≤ − n / = 1 − n / ,π i π x ∗ ≤ − n / . The above equations imply that for any i with i − x ∗ > n / we have π i π x ∗ ≤ (cid:16) − n − / (cid:17) i − x ∗ − n / ≤ e − n − / · ( i − x ∗ − n / ) = e − i − x ∗√ n (6)Now, we can establish that x ≤ x ∗ + O ( √ n ). This is done by applying (6) as follows¯ x = E x ∼ π [max { x, } ] ≤ x ∗ + ∞ (cid:88) i =0 π x ∗ + i √ n · ( i + 1) √ n ≤ x ∗ + ∞ (cid:88) i =0 e − x ∗ + i √ n − x ∗√ n · ( i + 1) √ n (7)= x ∗ + √ n ∞ (cid:88) i =0 e − i · ( i + 1) = x ∗ + O ( √ n ) = x ∗ + o ( x ∗ ) , (8)where (7) holds by (6), and (8) holds because x ∗ > n δ for a δ > / Proof of Step ii.
In this step we prove that ¯ x ≥ x ∗ − o ( x ∗ ). The proof is similar to the previousstep. Now, suppose i = x ∗ − y , for some y >
0. Equation (5) then implies π i π i +1 = x ∗ − y + λn = n − yn = 1 − yn . (9)29he above bound, followed by a calculation similar to how we derived (6) implies π i π x ∗ ≤ e − x ∗− i √ n (10)for all i < x ∗ .Also, observe that π i π i +1 = 11 + λ (11)holds for all i <
0. Define π < = (cid:80) i< π i . Equations (10) and (11) imply that π < ≤ e − x ∗√ n = o (1) . (12)Next, we use (10) and (12) to provide a lower bound for ¯ x :¯ x = E x ∼ π [max { x, } ] ≥ (1 − π < ) · x ∗ − x ∗ / √ n (cid:88) i =0 π x ∗ − i √ n · ( i + 1) √ n ≥ (1 − π < ) · x ∗ − √ n x ∗ / √ n (cid:88) i =0 π x ∗ · e − i · ( i + 1) (13)= (1 − π < ) · x ∗ − O ( √ n ) = x ∗ − o ( x ∗ ) (14)where (13) holds by (10) and (14) holds by (12). B Proof of Theorem 1
Theorem 1 is a direct consequence of Propositions 3 and 4. Proposition 4 was proved in Section A.To prove the theorem, we prove the rest of the propositions. Before starting the proofs, we statesome preliminaries in Section C. The greedy, patient, and batching policies are then analyzed inSections D, E, and F, respectively.
Proof of Proposition 3.
The first part of this proposition is a direct consequence of Proposition 5.The second part of the proposition is proved in Section F. The third part of the proposition is adirect consequence of Proposition 6.
Proof of Proposition 5.
The claim on the match rate under greedy matching is proved in Lemma 5.The claim on waiting time is proved in Lemma 6 and Lemma 7. These lemmas are stated and30roved in Section D.
Proof of Proposition 6.
The claim on the match rate under patient matching is proved in Lemma 9.The claim on the waiting time is proved in Lemma 10 and Lemma 11. These lemmas are statedand proved in Section E.
C Preliminaries
We use the terms
E pool and
H pool to denote the pools containing E agents and H agents, re-spectively. The terms
E arrival and
H arrival respectively refer to the arrival of an E agent andthe arrival of an H agent. The term departure clock refers to the exponential random variable thatdetermines the exogenous time that an agent becomes critical if she does not get matched priorto that time. When the departure clock ticks , the agent leaves the pool if she has not left witha match already. We use the term departure to refer to the event of an agent leaving the pool,either because of being matched to another agent or because her departure clock ticks. Similarly,the terms
E departure and
H departure refer to the departure of an E agent and the departure ofan H agent.Each of the policies, greedy or patient, defines a continuous-time stochastic process whose statespace is { ( x, y ) : 0 ≤ x + y ≤ C, x, y ∈ Z } , where ( x, y ) denotes a state with x hard-to-match and y easy-to-match agents. These stochasticprocesses could be modeled as a Markov chain in the natural way. (The Markov chains would havethe same state space as the above) We call the corresponding Markov chains the greedy Markovchain , and the patient Markov chain , respectively.Under either of these policies, an agent might have to search for a compatible match. This hap-pens in greedy matching upon the arrival of agents, and in patient matching upon their departure.In either of the policies, we suppose that an agent searches for a compatible match in the followingway: she orders all the agents in the H pool in a random order, and gets matched to the firstagent compatible with her, in that order. If no compatible agent is found, then the agent ordersall the agents in the E pool in a random order and gets matched to the first compatible agent inthat ordering. It also helps the analysis to define the notion of offer . When a searching agent a checks her compatibility with another agent b , if a and b are compatible, then we say that a makes That is, she draws a permutation over all the H agents, uniformly at random.
31n offer to b . By the definition of our policies, offers are always accepted. In our analysis, however,we sometimes consider scenarios in which an agent does not accept any offers. We always assumeall offers are accepted unless we explicitly mention otherwise.Suppose f, g : R + → R + . We use the notation g = o ( f ) when lim n →∞ g ( n ) f ( n ) = 0 and g = O ( f )when lim n →∞ g ( n ) f ( n ) < ∞ .We say an event E holds with high probability (whp) when lim m →∞ P (cid:2) ¯ E (cid:3) = 0. For notationalsimplicity, we also write this as P (cid:2) ¯ E (cid:3) = o (1). We say an event E holds with low probability (wlp)when ¯ E holds whp. We say an event E holds with very high probability (wvhp) when P (cid:2) ¯ E (cid:3) ≤ e − m α ,for some constant α >
0. We say an event E holds with very low probability (wvlp) when ¯ E holdswvhp. C.1 Markov chains
In case of existence, we denote the unique stationary distribution of a Markov chain N by π ( N ).Sometimes we slightly abuse the notation and use π to denote the stationary distribution of aMarkov chain that is clearly known in the context. Proposition 7.
Anderson et al. (2017) Suppose that X t is positive recurrent and that there exists α, β, γ > , a set B ⊂ X , and functions U : X → R + and f : X → R + such that for x ∈ X \ B , E x [ U ( X ) − U ( X )] ≤ − γf ( x ) , (15) and for x ∈ B , f ( x ) ≤ α, (16) E x [ U ( X ) − U ( X )] ≤ β. (17) Then E [ f ( X ∞ )] ≤ α + βγ . (18) Embedded Markov Chain
To apply Proposition 7, we need to work with a discrete-time MC. The Markov chains that we havereferred to, however, so far have been continuous-time Markov chains. Rather than working withthe continuous time Markov chain N , we typically work with a well-known discrete-time Markovchain that is called the embedded Markov chain for N , which we denote by (cid:98) N . For completeness,we state the definition of (cid:98) N below. Let N be a continuous-time Markov chain with transition rate32 i,j from state i to state j , for any i (cid:54) = j . Let N be the transition rate matrix for N , i.e., N i,j = n i,j for i (cid:54) = j , and the entries on the diagonal of N are set so each row in N sums to 0. Definition 5.
The embedded Markov chain of N , denoted by (cid:98) N , is a discrete-time Markov chainwith V ( (cid:98) N ) = V ( N ) . The transition probability from state i to state j in (cid:98) N is denoted by (cid:98) n i,j andis defined by (cid:98) n i,j = n i,j (cid:80) k (cid:54) = i n i,k if i (cid:54) = j if i = j. Given that a finite-state Markov chain is ergodic, it is straightforward to show that its em-bedded Markov chain has a unique stationary distribution that we denote by (cid:98) π Wikipedia (2015).Intuitively, (cid:98) N is a discrete-time Markov chain that “behaves” similar to N . We formalize thenotion of similarity that we use in our analysis below. Definition 6.
Let N = (cid:104)N m (cid:105) m ≥ and N (cid:48) = (cid:104)N (cid:48) m (cid:105) m ≥ represent two infinite sequences of Markovchains where V ( N m ) = V ( N (cid:48) m ) and | V ( N m ) | < | V ( N m +1 ) | for all m ≥ . Suppose π m , π (cid:48) m respec-tively denote the unique stationary distributions for N m , N (cid:48) m . We say N approximates N (cid:48) , anddenote it by N ∼∼∼ N (cid:48) , if there exist constants m > and θ, ¯ θ > such that for all m > m and all x ∈ V ( N m ) we have θ · π (cid:48) m ( x ) ≤ π m ( x ) ≤ ¯ θ · π (cid:48) m ( x ) . The next lemma shows that, under certain conditions, the expected values of a function are not“too far” from each other, where the expectations are taken over the stationary distribution of aMarkov chain and the stationary distribution of its embedded Markov chain. We have not tried tostate the most general version of this lemma, by any means; the following version comes in handyin the analysis.
Lemma 1.
Let N = (cid:104)N m (cid:105) m ≥ and N (cid:48) = (cid:104)N (cid:48) m (cid:105) m ≥ be two infinite sequences of Markov chains withstate spaces defined on Z such that N ∼∼∼ N (cid:48) . Also, let f : Z → R + . Then, if E x ∼ π ( N m ) [ f ( x )] ≤ o ( m ) , we would also have E x ∼ π ( N (cid:48) m ) [ f ( x )] ≤ o ( m ) . Proof.
Let π, π (cid:48) respectively denote π ( N m ) , π ( N (cid:48) m ), and θ, ¯ θ be the coefficients defined in Definition Recall that o ( m ) denotes a term that grows with a rate slower than m .
33. Then, o ( m ) ≥ E x ∼ π [ f ( x )]= (cid:88) x π ( x ) f ( x ) ≥ (cid:88) x θπ (cid:48) ( x ) f ( x ) = θ · E x ∼ π (cid:48) [ f ( x )] , which implies E x ∼ π (cid:48) [ f ( x )] ≤ o ( m ). D Analysis of Greedy Matching
In this section, we first analyze the stochastic process corresponding to the greedy policy. The maintechnical results established by this analysis are stated in Subsection D.3. Using these results wewill be able to prove the results on the match rate and waiting time for greedy matching (mentionedin Subsection 3.2). We establish the result on the match rate in Subsection D.7 and the waitingtime result in Subsection D.8.
D.1 A Simplifying Assumption
Before we start the analysis, we make an additional assumption for the sake of technical simplicity.As we will explain, our theorems quantitatively remain the same with or without this assumption.
The Finite Capacity Assumption.
We suppose that the exchange program has capacity C = κ · m , where κ > κ · m , then that agent will not be added to the system.For example, by letting C = 100 m , we restrict the maximum possible number of agents at eachmoment in the system to be bounded by the expected number of pairs that arrive in the next 100time units. For practical purposes, this is a very reasonable assumption: through the analysis,we can see that the chance of having 100 m agents in the system is a low-probability event; moreprecisely, the stationary probability of this event is bounded by 2 − m (96 − λ ) ). This fact also givesan intuitive explanation of why our quantitative results do not rely on this assumption. When thisassumption is dismissed, the proof would follow similarly while bearing some additional notation.To avoid tedious notation, we present the analysis of greedy matching under this assumption. Since 1 to 5 years is a reasonable estimate for each time unit in our model, the capacity in this case would beabout the number of arriving agents in the next 100 to 500 years. .2 Modeling the dynamics We use a two-dimensional Markov chain, which we denote by M , to model the dynamics of thepool size. First we set up some notation before proceeding to the description. For brevity, we usethe abbreviation MC for Markov chain . For MC M , let V ( M ) denote the state space of M . Werepresent each state by a pair ( x, y ) where x, y respectively denote the number of H agents and thenumber of E agents. In other words, we have V ( M ) = { ( x, y ) : 0 ≤ x + y ≤ C, x, y ∈ Z } , where we recall that the term C is the capacity parameter defined in the Finite Capacity Assump-tion.The definition of M would be completed by defining the transition rates. A transition can onlyhappen from a state ( x, y ) to its (at most) four neighbors , which are { ( x (cid:48) , y (cid:48) ) ∈ V ( M ) : | x − x (cid:48) | + | y − y (cid:48) | = 1 } . See Figure 12 for a visual description of the neighbors. To simplify the definition of transition ratesfrom a node to its neighbors, we define the following notations: Let M x = (1 − p ) x and N y = (1 − q ) y .For each node ( x, y ), we denote the transition rates from this node to its four neighbors on the top,right, bottom, and left of it by u x,y , r x,y , d x,y , l x,y . These rates are defined as follows: • If x + y < C , u x,y = mM x N y is the transition rate from the node ( x, y ) to node ( x, y + 1),otherwise u x,y = 0. • If x + y < C , r x,y = m (1 + λ ) N y is the transition rate from the node ( x, y ) to node ( x + 1 , y ),otherwise r x,y = 0. • If y >
0, then d x,y = mM x (1 − N y ) + m (1 + λ )(1 − N y ) + y is the transition rate from thenode ( x, y ) to node ( x, y − d i,j = 0. • If x >
0, then l x,y = m (1 − M x ) + x is the transition rate from the node ( x, y ) to node( x − , y ), otherwise l x,y = 0.Given the above transition rates for M , it is straightforward to verify that it models our stochasticprocess. Proposition 8. M is ergodic and has a unique stationary distribution, π . , yx − , y x + 1 , yx, y + 1 x, y − m (1 − M x ) + x m (1 + λ ) N y mM x N y mM x (1 − N y ) + m (1 + λ )(1 − N y ) + y Figure 12: An illustration of the transitions from node ( x, y ) Markov chain.36 roof.
It is enough to show that M is ergodic, i.e., that it is irreducible and positive recurrent.Irreducibility is trivial: there is only one communication class. To prove that M is positive recur-rent, it is enough to note that it has a finite state space; this implies that M is ergodic and has aunique stationary distribution.We remark that the existence of the stationary distribution does not rely on the Finite CapacityAssumption (nor do the statements of our theorems, as mentioned earlier). Definition 7.
Let π denote the stationary distribution of M and π ( i, j ) denote the probabilityassigned to state ( i, j ) in π . We define π ( i, j ) = 0 when i + j > C . Let π ( i ) = (cid:80) ∞ j =0 π ( i, j ) and π j = (cid:80) ∞ i =0 π ( i, j ) . D.3 The main analytical results for greedy matching
The next two theorems are the core technical results for the analysis of greedy matching. They willbe used for proving our results on the match rate and the waiting time distribution under greedymatching (Subsections D.7 and D.8).
Theorem 2. E π [ x ] = λm + O ( √ m ) and E π [ y ] = O (1) . Theorem 3.
There exists constants θ X , θ Y > such that for any x, y ≥ , we have π ( x ) ≤ e − θ X · | x − x ∗|√ m ,π ( y ) ≤ e − θ Y · y , where x ∗ = λm . Computing E π [ y ] and showing that y is concentrated around its mean is the relatively easierpart of Theorems 2 and 3; this is done in Subsection D.6. The main idea to prove the otherparts (i.e., computing E π [ x ] and showing that x is concentrated around its mean) is definingtwo other Markov processes that are coupled with M , namely M l , M u . These two processes aredefined such that the number of unmatched hard-to-match agents in M stochastically dominatesthe number of unmatched hard-to-match agents in M l and is stochastically dominated by thenumber of unmatched hard-to-match agents in M u .In Sections D.5 and D.4, we will show that E π l [ x ] = λm + O ( √ m ) , (19) E π u [ x ] = λm + O ( √ m ) , (20)37here π l , π u respectively denote the (unique) stationary distributions of M l , M u . This impliesthat E π [ x ] = λm + O ( √ m ). To prove the concentration result, we will show that for the randomprocesses M l , M u , there exist constants θ l , θ u > π l ( x ) ≤ e − θ l · | x − x ∗ |√ m , (21) π u ( x ) ≤ e − θ u · | x − x ∗ |√ m . (22)This is done in Sections D.5 and D.4. Then, we use the fact that the PMF of x in M stochasticallydominates the PMF of x in M l and is stochastically dominated by the PMF of x in M u . This fact,together with (21) and (22), implies that there exists a constant θ X > π ( x ) ≤ e − θ X · | x − x ∗|√ m . We prove (20) and (22) in Subsection D.4, and (19) and (21) in Subsection D.5. We introducedthe required notions and tools for the analysis in Appendix C.
D.4 Definition of M u Consider a random process which is similar to the original random process (described by M ), withthe following differences:1. Matches are made only between hard-to-match nodes and easy-to-match nodes.2. Matches are made greedily only upon arrival of easy-to-match nodes.3. Easy-to-match nodes do not stay in the pool: they leave right after their arrival, if they arenot matched with a hard-to-match node.We represent the number of hard-to-match nodes in this random process with a one-dimensionalMC M u , which is defined as follows. The state space of M u is defined as V ( M u ) = { , , , . . . } .The MC is in state x when the number of hard-to-match agents in the pool is x . The transitionrates from state x to its left and right neighbors are respectively defined by l x = m (1 − N x ) + x and r x = m (1 + λ ). It is straightforward to verify that M u is ergodic and therefore has a uniquestationary distribution that we denote by π u . Lemma 2. E π u [ | x − x ∗ | ] = O ( √ m ) . roof. The proof is by Proposition 7. First, we define the functions f, U to be: f ( x ) = | x − x ∗ | ,U ( x ) = ( x − x ∗ ) . Also, we define B = { x : | x − x ∗ | < √ m } . We apply Proposition 7 to the embedded Markov chaincorresponding to M u , which we denote by (cid:100) M u . This will imply that E (cid:99) π u [ | x − x ∗ | ] = O ( √ m ) . If this is proved, the lemma is then concluded by Lemma 1; note that Lemma 1 is applicable since,by Claim 1, we have M u ∼∼∼ (cid:100) M u .Next, we bound E (cid:99) π u [∆ U ( x )], by considering two cases: x > x ∗ and x < x ∗ . For notationalsimplicity, we will drop the index (cid:98) π u from the expectation in the rest of this proof. Also, let S x = l x + r x . When x > x ∗ , we have E [∆ U ( x )] = ( m (1 + λ ) − m (1 − M x ) − x ) · x − x ∗ ) S x + 2 /S x ≤ − Ω(( x − x ∗ ) / √ m ) , where the last inequality holds, since x − x ∗ > √ m .Now, suppose x < x ∗ . Then E [∆ U ( x )] = ( m (1 − M x ) + x − m (1 + λ )) · x ∗ − x ) S x + 2 /S x ≤ − Ω(( x − x ∗ ) / √ m ) . Therefore, we can set γ = Ω(1 / √ m ). Also, it is straightforward to verify that α = O ( √ m ) and β = O (1), by the definition of B . This implies that E (cid:99) π u [ f ( x )] = O ( √ m ), which proves the promisedclaim. Claim 1.
Let (cid:100) M u denote the embedded Markov chain of M u . Then, M u ∼∼∼ (cid:100) M u .Proof. Since M u is ergodic, then (cid:100) M u has a unique stationary distribution. Let π u , (cid:98) π u denote thestationary distributions of M u and (cid:100) M u . Also, suppose M denotes the transition rate matrix of M u . Then, π u can be written in terms of (cid:98) π u as follows Wikipedia (2015): π u = − (cid:98) π u (diag( M )) − (cid:107) (cid:98) π u (diag( M )) − (cid:107) . x , we can write π u ( x ) = (cid:98) π u ( x ) · − /M x,x (cid:107) (cid:98) π u (diag( M )) − (cid:107) . Given this equation, the lemma would be proved if there exist constants θ, ¯ θ > θ < − /M x,x (cid:107) (cid:98) π u (diag( M )) − (cid:107) < ¯ θ. (23)To see why (23) holds, first note that for any x , we have − M x,x = Θ( m ), by the Finite CapacityAssumption. From this, we can imply that (cid:13)(cid:13) (cid:98) π u (diag( M )) − (cid:13)(cid:13) = Θ( m ), since the left-hand-side is aconvex combination of the diagonal entries. These two facts together imply (23), which concludesthe lemma. D.5 Definition of M l Consider a random process which is similar to the original random process (described by M ), withthe following differences:1. Matches are made greedily upon arrival of nodes, and only between an easy-to-match and ahard-to-match node.2. Easy-to-match nodes do not leave the pool before getting matched.3. The probability of compatibility of an easy-to-match and a hard-to-match node is 1.We represent the difference between the number of hard-to-match and easy-to-match nodes inthis random process with a one-dimensional MC M l , which is defined as follows. The state spaceof M l is V ( M l ) = Z . The MC is in state x when the number of hard-to-match nodes minus thenumber of easy-to-match nodes is x . The transition rates from state x to its left and right neighborsare respectively defined by l x = m + max { x, } , r x = (1 + λ ) m. It is straightforward to verify that M l is ergodic and therefore has a unique stationary distributionthat we denote by π u . Lemma 3. E π l [ | x − x ∗ | ] = O ( √ m ) . roof. The proof is by Proposition 7. First, we define the functions f, U to be: f ( x ) = | x − x ∗ | ,U ( x ) = ( x − x ∗ ) . Also, we define B = { x : | x − x ∗ | < √ m } . We apply Proposition 7 to the embedded Markov chaincorresponding to M l , which we denote by (cid:100) M l . This would imply that E (cid:98) π l [ | x − x ∗ | ] = O ( √ m ) . If this is proved, the lemma is then concluded by Lemma 1; note that Lemma 1 is applicable since,by Claim 2, we have M u ∼∼∼ (cid:100) M u .Next, we bound E (cid:98) π l [∆ U ( x )], by considering two cases: x > x ∗ , and x < x ∗ . For notationalsimplicity, we will drop the index (cid:98) π l from the expectation in the rest of this proof. Also, let S x = l x + r x . When x > x ∗ , we have E [∆ U ( x )] = ( m (1 + λ ) − m − x ) · x − x ∗ ) S x + 2 /S x ≤ − Ω(( x − x ∗ ) / √ m ) , where the last inequality holds since x − x ∗ > √ m .Next, we consider the second case, x < x ∗ . See that E [∆ U ( x )] = ( m + max { x, } − m (1 + λ )) · x ∗ − x ) S x + 2 /S x ≤ − Ω(( x ∗ − x ) / √ m ) . Therefore, we can set γ = Ω(1 / √ m ). Also, it is straightforward to verify that α = O ( √ m ) and β = O (1), by the definition of B . This implies that E (cid:98) π l [ f ( x )] = O ( √ m ), which proves the claim. Claim 2.
Let (cid:100) M l denote the embedded Markov chain of M l . Then, M l ∼∼∼ (cid:100) M l .Proof. The proof is identical to the proof of Claim 1.
D.6 Analysis of the Markov chain on the vertical axis
Lemma 4.
Let π denote the stationary distribution of M . Then, E π [ y ] = O (1) . Moreover, y isconcentrated around its mean, in the following sense: there exists θ Y > such that for any y ≥ ,we have π ( y ) ≤ e − θ Y · y . roof. It is enough to show that there exist constants y > θ > y ≥ y wehave π ( y ) /π ( y + 1) > θ .Let Up ( y ) = { u x,y : x ≥ } , Down ( y ) = { d x,y : x ≥ } . Also, let y be the smallest positive integer for which N y < /
3. Now, see thatmin { d : d ∈ Down ( y + 1) } max { u : u ∈ Up ( y ) } ≥ m (1 + λ )(2 / m/ > . Therefore, by the balance equations, we can set θ = 2. D.7 Match rate under greedy matching
Lemma 5.
Under greedy matching, the match rate of hard-to-match agents is λ − O ( λ ) √ m ) and the match rate of easy-to-match agents is − o (1) .Proof. First, we define some notation. Let m E = m and m H = (1 + λ ) m . We use E Gt , H Gt respectively to denote the number of E, H agents under the greedy policy at time t .For any type Θ ∈ { E , H } , let G (Θ) denote the match rate of Θ under greedy matching. Thedeath rate (the rate of agents who depart the market unmatched) would then be 1 − G (Θ). Astraightforward calculation shows that 1 − G (Θ) = Θ G m Θ . This is a consequence of the fact that the departure rate for all agents is equal to 1. An applicationof linearity of expectation and the ergodic theorem imply the above equality. We then can useTheorem 2 to write 1 − G ( E ) = E G m = O (1 /m )and 1 − G ( H ) = H G (1 + λ ) m = λm + O ( √ m )(1 + λ ) m = λ λ + O (1 / √ m ) . This proves the claim for greedy matching.
D.8 Distribution of waiting time under greedy matching
We show that the waiting time for matching matched hard-to-match agents has an exponentialdistribution with rate 1 + 1 /λ . This result is derived from the assumption that the ties betweenhard-to-match agents are broken randomly. As a consequence, we will be able to compute the42xpected waiting time of hard-to-match agents, conditioned on being matched or unconditionally.(Recall that the former is called the “matching time” and the latter the “waiting time”.) Lemma 6.
Under the greedy policy, as m approaches infinity, the waiting time and matching timeof an E agent converge in distribution to the degenerate distribution at .Proof. First, we prove the result for waiting time. Fix an E agent, e , and let w e denote the waitingtime for e . For any fixed constant t >
0, we will show that lim m →∞ P [ t > w e ] = 0. This will provethe claim. First, see that E [ w e ] = 1 m · E ( x,y ) ∼ π [ x ] = 1 m · O (1)where the second equality holds because of Theorem 2. Therefore, by Markov inequality, P [ w e > t ] < O ( t/m )holds, which means lim m →∞ P [ t > w e ] = 0.Now, we prove the result on the matching time (i.e., the waiting time of matched agents). Let M e denote the event in which e leaves the pool with a match. Our goal is to show that, for anyfixed constant t >
0, lim m →∞ P [ t > w e | M e ] = 0 . See that P [ w e = 0 | M e ] = P [ w e = 0] P [ M e ] = 1 − o (1) , where the second equality holds because of Theorem 3. Therefore, P [ w e > | M e ] = o (1) , which implies that P [ w e > t | M e ] = o (1) . This proves the claim.
Lemma 7. As m approaches infinity, the waiting time and matching time of hard-to-match agentsconverge in distribution to Exp (1 + λ ) . We sketch the proof below. The formal proof is presented after the proof sketch.
Proof sketch.
Here we give a proof sketch. We define a new process, namely P , in which there areno easy-to-match agents. Rather, an exponential clock is attached to each hard-to-match agent43hich ticks at rate 1 /λ . We suppose the agent is matched if the clock ticks before the agent departs.It is not hard to show that the matching time of agents in P converges to the matching time ofhard-to-match agents in the original process in distribution, as m approaches infinity.Now, we compute the distribution for the waiting time of an agent in P . Consider an agent p and suppose it has entered the pool at time t . Note that p is matched iff it is matched beforeher departure clock ticks. Let t , t be random variables such that t ∼ exp(1 /λ ) , t ∼ exp(1);these random variables are interpreted as follows. The agent departs at time t + t if she has notreceived any matches by then (i.e., her clock has not ticked). The time t + t is the first time whenthe agent’s offer clock ticks, i.e., the first time when the agent receives an offer. So, the agent ismatched if and only if t < t . Alternatively, we can say the agent is matched iff t = t min where t min = min { t , t } .First, see that t min represents the waiting time of the agent. Therefore, the waiting time of theagent has distribution Exp (1 + 1 /λ ).Next, we compute the distribution of the matching time. Fix a constant z >
0. The probabilitythat an agent is matched before time z conditioned on being matched is P (cid:2) t < z (cid:12)(cid:12) t = t min (cid:3) = P [ t min < z (cid:86) t = t min ] P [ t = t min ]= P [ t min < z ] · P [ t = t min ] P [ t = t min ] = P [ t min < z ] . Since the above equality holds for any z , then the waiting time for an agent conditioned on beingmatched has the same distribution as the distribution of t min , which is Exp (1 + 1 /λ ). Proof of Lemma 7.
First, we show that the waiting time of an H agent converges in distributionto
Exp (1 + 1 /λ ). To this end, fix an H agent, h , upon her arrival at time t . Let Q denote thestochastic process under greedy matching starting from t and ending when h leaves the system.We couple another process with Q , namely Q (cid:48) . We define this coupling below. Roughly speaking, Q (cid:48) is the same as the greedy matching process, with the exception that the arrival of h is “ignored”in the sense that h does not interfere with the evolution of Q (cid:48) . • Q (cid:48) runs from time t to t + log m . Furthermore, the departure clock of h is set to tick attime t + log m in Q (cid:48) . • If h finds a compatible match upon her arrival (at time t ) in Q , then we stop both Q , Q (cid:48) .Otherwise, we let Q evolve according to the greedy process. By definition, Q (cid:48) has a samplepath identical to Q , until one of the following disjoint events happens:Event (i) h receives an offer in Q before time t + log m . In this case, we stop Q . In Q (cid:48) ,44 rejects the received offer as well as all offers she will receive in the future. AnyE agent who gets rejected by h will make an offer to the next compatible agentin his (random) list. Q (cid:48) will continue evolving according to the greedy process,with the exception of agent h , who does not interfere with the process.Event (ii) The departure clock of h ticks in Q before time t + log m and h departs withoutbeing matched. In this case, we stop Q but continue to run Q (cid:48) . Q (cid:48) will continueevolving according to the greedy process, with the exception of agent h , who doesnot interfere with the process (i.e., agent h rejects all the offers she receives, inthe sense clarified above). • We stop Q (cid:48) when it reaches time t + log m .For notational simplicity, we suppose t = 0 without loss of generality. Let E h ( t ) denote theevent in which h leaves the pool in Q before time t (either matched or unmatched). Also, let E (cid:48) h ( t )denote the event in which h receives at least one offer in Q (cid:48) before time t . For any constant t > m →∞ P (cid:104) E h ( t ) (cid:105) = e − t · lim m →∞ P (cid:104) E (cid:48) h ( t ) (cid:105) . (24)Therefore, if we show that lim m →∞ P (cid:104) E (cid:48) h ( t ) (cid:105) = e − t/λ , we can use (25) to imply thatlim m →∞ P (cid:104) E h ( t ) (cid:105) = e − t · (1+1 /λ ) , which proves the claim on the distribution of waiting time. To complete the proof, the followingclaim must be proved. Claim 3.
For any constant t > , lim m →∞ P (cid:104) E (cid:48) h ( t ) (cid:105) = e − t/λ . (25) Proof.
Our proof approach is as follows. First, we observe that the process Q (cid:48) can be run fromtime t = 0 to log m as follows: sample a state ( x, y ) ∼ π . Then, let the stochastic system startfrom ( x, y ) and evolve for log m units of time, under the greedy policy. By the PASTA property, the sample paths generated by this process are identical to the sample paths of Q (cid:48) (note that thearrival of h is “ignored” in Q (cid:48) , in the sense that h does not affect the evolution of Q (cid:48) ). PASTA, or Poisson Arrivals See Time Averages, is a well-known property in the queueing literature; e.g., seeHarchol-Balter (2013).
45y the above argument, we sample the state ( x, y ) ∼ π at time t = 0, and let the process rununtil time log m . We also consider an “imaginary agent” h , which exists in the H pool, but rejectsall of the proposals that are made to her. Our goal is to show that the probability that h receivesno proposals in the period [0 , t ], which we know by P (cid:104) E (cid:48) h ( t ) (cid:105) , approaches e − t/λ as m approachesinfinity.Let H t denote the history of the process until time t . (Note that the history does not includethe offers made to h , this agent does not change the evolution of the process.) Let the randomvariable n t denote the number of E agents who arrive to the pool in the time interval [0 , t ]. Also,let a , . . . , a n t denote the arrival times of these agents. Define x , . . . , x n t to be the number of Hagents in the pool at times a , . . . , a n t , respectively. Now, see that P (cid:104) E (cid:48) h ( t ) (cid:12)(cid:12) H t (cid:105) ≥ Π n t i =1 (1 − x i ) , (26) P (cid:104) E (cid:48) h ( t ) (cid:12)(cid:12) H t (cid:105) ≤ Π n t i =1 (1 − − (1 − p ) x i x i ) , (27)where (26) and (27) hold because the chance that a match happens at time a i is at most 1 and atleast 1 − (1 − p ) x i .In the rest of the proof, we will use probabilistic bounds to show that in almost all histories H t , n t is close to tm and x ∗ is close to λm , where x ∗ = min { x , . . . , x n t } . This will let us simplify (26)and (27). The simplified forms will expose that in almost all histories H t , P (cid:104) E (cid:48) h ( t ) (cid:105) approaches e − t/λ as m approaches infinity.Define event G as G : λm − √ m log m ≤ x ∗ ≤ λm + √ m log m. We will show that ¯ G happens wlp. To do this, we use the fact that the size of the H pool afterthe arrival of any H agent is close to λm (in the sense of Theorem 3). Formally, we use the PASTAproperty together with Theorem 3 and write a union bound over all arrivals of E agents in theinterval [0 , t ]. This implies that ¯ G happens wlp. More precisely, this implies that P (cid:2) ¯ G (cid:3) ≤ e − O (log m ) . (28)Define event G as G : tm − √ tm log m ≤ n t ≤ tm + √ tm log m.
46y Chebyshev’s inequality, P (cid:2) ¯ G (cid:3) ≤ log − m. (29)Note that the above inequality holds because Poisson distribution has equal mean and variance; inthis case, the random variable n t has mean and variance tm .Now, recall (26) and (27), and that our goal is to show that in almost all histories H t , n t isclose to tm and x ∗ is close to λm . Inequalities (28) and (29) show just this. More formally, theyimply that in almost all histories H t but a fraction O (1 / log m ) of them, the events G , G hold.Therefore, we can use (26) and (27) to write P (cid:104) E (cid:48) h ( t ) (cid:105) ≥ (1 − O (1 / log m )) · e − tmλm , P (cid:104) E (cid:48) h ( t ) (cid:105) ≤ e − tmλm · (1 − o (1)) . The above equations imply that lim m →∞ P (cid:104) E (cid:48) h ( t ) (cid:105) = e − t/λ . The proof is complete.To complete the proof of the lemma, it remains to show that the waiting time of matched Hagents converges in distribution to
Exp (1 + λ ). We follow the same idea used in the proof sketch.Again, we fix an H agent, h , who arrives at time t = 0. Define the (coupled) processes Q , Q (cid:48) asbefore. Let t be a random variable that denotes the first time at which h receives an offer in Q (cid:48) .If h does not receive an offer in Q (cid:48) , let t = log m . Also, let t be an (independent) exponentialrandom variable with rate 1. This variable denotes the time at which the departure clock of h ticksin Q .Define the random variable t min = min { t , t } . For any constant z >
0, observe that P (cid:2) t < z (cid:12)(cid:12) h receives an offer in Q (cid:3) = P (cid:2) t < z (cid:12)(cid:12) h receives an offer in Q before time log m (cid:3) + P (cid:2) t < z (cid:12)(cid:12) h receives an offer in Q after time log m (cid:3) . m →∞ P (cid:2) t < z (cid:12)(cid:12) h receives an offer in Q (cid:3) = lim m →∞ P (cid:2) t < z (cid:12)(cid:12) h receives an offer in Q before time log m (cid:3) = lim m →∞ P (cid:2) t < z (cid:12)(cid:12) t = t min (cid:3) . (30)Next, observe thatlim m →∞ P (cid:2) t < z (cid:12)(cid:12) t = t min (cid:3) = lim m →∞ P [ t min < z (cid:86) t = t min ] P [ t = t min ] . Now, recall from the first part of the proof that t converges in distribution to Exp (1 /λ ). (This isessentially due to Claim 3). Now, note that since t converges in distribution to Exp (1 /λ ), then t min converges in distribution to Exp (1 + 1 /λ ). Therefore, since t , t are independent random variablesand t min = min { t , t } , we can simplify the above equation further as follows.lim m →∞ P (cid:2) t < z (cid:12)(cid:12) t = t min (cid:3) = lim m →∞ P [ t min < z (cid:86) t = t min ] P [ t = t min ]= lim m →∞ P [ t min < z ] · P [ t = t min ] P [ t = t min ] = lim m →∞ P [ t min < z ] = e − z (1+1 /λ ) . The above equation together with (30) implies thatlim m →∞ P (cid:2) t < z (cid:12)(cid:12) h receives an offer in Q (cid:3) = e − z (1+1 /λ ) . (31)This finishes the proof. Corollary 1 (of Lemma 7) . The conditional expected waiting time for a hard-to-match agentconditioned on not being matched is λ − λ .Proof. Let w be a random variable denoting the waiting time of an agent p and let M be the eventin which agent p is matched. Also, let ¯ M be the complement of M . E [ w ] = P [ M ] · E (cid:2) w (cid:12)(cid:12) M (cid:3) + P (cid:2) ¯ M (cid:3) · E (cid:2) w (cid:12)(cid:12) ¯ M (cid:3) . (32)Note that the left-hand-side is equal to λ . Also, see that E (cid:2) w (cid:12)(cid:12) M (cid:3) = λ λ , P [ M ] = λ . Therefore, plugging the above equation into(32) implies that E (cid:2) w (cid:12)(cid:12) ¯ M (cid:3) = 1 + λ −
11 + λ .
E Analysis of Patient Matching
After introducing some notation, we analyze the stochastic process corresponding to the patientpolicy. First, we present our core technical result, Theorem 4, and prove it in Subsection E.1. Afterthat, we prove our results about the match rate and the distribution of waiting time under patientmatching in Subsections E.2 and E.3.For the analysis in this section, we use a two-dimensional MC, M , to model the dynamics. Let V ( M ) denote the state space of M . We represent each state by a pair ( x, y ) where x, y respectivelydenote the number of H-agents and E-agents. In other words, we have V ( M ) = { ( x, y ) : x, y ≥ , x, y ∈ Z } . The definition of M would be completed by defining the transition rates. Similar to the MC forthe greedy policy, a transition can only happen from a state ( x, y ) to its (at most) four neighbors,which are { ( x (cid:48) , y (cid:48) ) ∈ V ( M ) : | x − x (cid:48) | + | y − y (cid:48) | = 1 } . We do not define these transition rates explicitly here, since they are defined implicitly by thepolicy. Define ( x ∗ , y ∗ ) = E ( x,y ) ∈ π [( x, y )].The following technical result forms the basis of our analysis and is proved in Subsection E.1. Theorem 4.
There exists a constant σ Y > such that for any y ≥ , we have π ( y ) ≤ e − σ Y · y . The above result is essentially a concentration result for the E pool. The following concentrationresult holds for the H pool, which will not be used in our analysis. We state it for the sake ofcompleteness.
Theorem.
There exists a constant σ X > such that for any x ≥ , we have π ( x ) ≤ e − σ X · | x − x ∗|√ m . urthermore, | x ∗ − (1 + λ ) m | = o ( m ) . Below we state a weaker version of the above theorem, which will be used in the analysis. Theproof is presented in Subsection E.1.
Lemma 8.
There exists a constant γ > such that for any d ≥ , we have π ( λm/ − d ) ≤ e − γ d . E.1 Proofs
First, we prove Theorem 4.
Proof of Theorem 4.
We use a coupling technique to simplify the stochastic process. We use P to denote the stochastic process governing the patient matching algorithm. Define the stochasticprocess P (cid:48)(cid:48) to be the same as P , with the following differences in departures. In P (cid:48)(cid:48)
1. E agents never leave the pool unless they have been matched, and2. H agents do not stay in the pool: an H agent leaves the pool immediately with probability e − m , without searching for a match. With probability 1 − e − m she searches for a compatibleagent (upon arrival). If a match is found, both agents leave the pool. Otherwise, the H agentleaves the pool.We use the random variables x (cid:48)(cid:48) , y (cid:48)(cid:48) to denote the number of H and E agents in P (cid:48)(cid:48) . Recall thatwe use the MC M to model P , and that π denotes the steady-state distribution of M . We usea similar MC, M (cid:48) , to model P (cid:48)(cid:48) and use π (cid:48)(cid:48) to denote its steady-state distribution. We use π Y todenote the marginal distribution over y in π . Similarly π (cid:48)(cid:48) Y denotes the marginal distribution over y (cid:48)(cid:48) in π (cid:48)(cid:48) .The proof has two steps. In Step 1, we show that π (cid:48)(cid:48) Y stochastically dominates π Y . In Step 2,we will show that there exists a constant σ (cid:48)(cid:48) Y > y (cid:48)(cid:48) ≥ π (cid:48)(cid:48) ( y (cid:48)(cid:48) ) ≤ e − σ (cid:48)(cid:48) Y · y (cid:48)(cid:48) . This directly proves the lemma for σ Y = σ (cid:48)(cid:48) Y , because of the stochastic dominance relation provedin Step 1.Before proceeding to Step 1, we define a useful notation. Let x t , y t denote the number of H andE agents in P at time t . Similarly, we use x (cid:48)(cid:48) t , y (cid:48)(cid:48) t to denote the number of H and E agents in P (cid:48)(cid:48) attime t . 50 tep 1 First, we define a mediator process, P (cid:48) , as follows. P (cid:48) is the same as P , with the followingdifferences. In P (cid:48)
1. E agents never leave the pool unless they have been matched, and2. Upon arrival, H agents draw the (exponential) random variable corresponding to their waitingtime. If this variable is larger than m , they leave the pool immediately without searching fora match. Otherwise, the waiting time of the agent is set to be m , i.e., the agent stays in thepool for m units of time and searches for a match upon departure.We use the random variables x (cid:48) , y (cid:48) to denote the number of H and E agents in P (cid:48) . Also, we use π (cid:48) to denote the steady-state distribution corresponding to the process P (cid:48) , and π (cid:48) Y to denote themarginal distribution induced by π over y (cid:48) .To complete Step 1, we will show that (i) π (cid:48)(cid:48) Y stochastically dominates π (cid:48) Y , and (ii) π (cid:48) Y = π Y .Part (ii) is straightforward: P (cid:48) is just the same as P except that the H agents who are allowed toenter the pool in P (cid:48) (those with waiting time shorter than m ) will enter with a constant delay of m . Since the delay is constant, the arrival process remains a Poisson process (with the same rate).It remains to show that part (i) holds, i.e., π (cid:48)(cid:48) Y stochastically dominates π (cid:48) Y .The proof proceeds by defining a coupling of the processes P (cid:48) , P (cid:48)(cid:48) . The joint process, denotedby Q = ( P (cid:48) , P (cid:48)(cid:48) ), will have two components corresponding to P (cid:48) , P (cid:48)(cid:48) . This process Q , in additionto being a valid coupling, will satisfy the following property: y (cid:48) t ≤ y (cid:48)(cid:48) t + m for all t ≥
0, in all samplepaths of Q . If Q satisfies this condition, then π (cid:48)(cid:48) Y must stochastically dominate π (cid:48) Y , and we are donewith Step 1. So, all that remains is defining Q so that it satisfies the above-mentioned condition.We define Q as follows. It starts with empty pools in both processes P , (cid:48) P (cid:48)(cid:48) , i.e., x (cid:48) t = y (cid:48) t = x (cid:48)(cid:48) t = y (cid:48)(cid:48) t = 0. Both processes will have identical sequences for the arrival of agents, but different departureand matching processes. To define the matching process, we need some additional notation. Let a (cid:48) ≤ a (cid:48) ≤ . . . be the sequence of departure times in P (cid:48) , i.e., a (cid:48) i is the time that the i th arrivalhappens in P (cid:48) . Also, let a (cid:48)(cid:48) ≤ a (cid:48)(cid:48) ≤ . . . be the sequence of departure times in P (cid:48)(cid:48) . We make a finalnotational convention: in case of departure of an H agent at time t , we use y (cid:48) t to denote the numberof E agents just before that departure, i.e., y (cid:48) t = lim t ∗ → t − y (cid:48) t ∗ . The same definition holds for y (cid:48)(cid:48) t . First, note that a (cid:48) i ≤ a (cid:48)(cid:48) i ≤ a (cid:48) i + m, ∀ i ≥ . (33) Note that m is a constant with respect to t . Proving y t ≤ y (cid:48)(cid:48) t obviously implies the desired stochastic dominancerelation. Proving y (cid:48) t ≤ y (cid:48)(cid:48) t + m is just as good, since shifting the sample paths y (cid:48)(cid:48) , y (cid:48)(cid:48) , . . . in time does not change thestationary distribution of y (cid:48)(cid:48) . We use this convention since it allows us to conveniently distinguish between the departure process and thematching process. P (cid:48) , P (cid:48)(cid:48) have the same arrival process; however, the departure of an agent in P (cid:48)(cid:48) can be delayed by up to m units of time, compared to her departure time in P (cid:48) .The rest of the proof in Step 1 is straightforward. We couple P (cid:48) and P (cid:48)(cid:48) in such a way that thefollowing property is satisfied: Upon the i th departure in P (cid:48)(cid:48) , a compatible match is found iff (i) acompatible match is found upon the i th departure in P (cid:48) , or (ii) y (cid:48)(cid:48) a (cid:48)(cid:48) i > y (cid:48) a (cid:48) i . We label this propertyProperty ζ . If our coupling satisfies Property ζ , then we are done with Step 1: this fact, the factthat P (cid:48) , P (cid:48)(cid:48) have the same arrival process, and (33) together would imply that y (cid:48) t ≤ y (cid:48)(cid:48) t + m holds forall t ≥ Q can be defined in a way that Property ζ is satisfied. Thisis done inductively. The induction basis is i = 1. See that a (cid:48) ≤ a (cid:48)(cid:48) , and that y (cid:48) a (cid:48) ≤ y (cid:48)(cid:48) a (cid:48)(cid:48) . We considertwo cases: either y (cid:48) a (cid:48) = y (cid:48)(cid:48) a (cid:48)(cid:48) or y (cid:48) a (cid:48) < y (cid:48)(cid:48) a (cid:48)(cid:48) . If y (cid:48) a (cid:48) = y (cid:48)(cid:48) a (cid:48)(cid:48) , then we use the same compatibility graph forthe departing agent in both processes P (cid:48) , P (cid:48)(cid:48) , i.e., the departing agent would be compatible withanother H agent in P (cid:48) , namely agent z , iff the departing agent is compatible with z in P (cid:48)(cid:48) . It isthen clear that Property ζ would be satisfied in this case.So, suppose that the case y (cid:48) a (cid:48) < y (cid:48)(cid:48) a (cid:48)(cid:48) holds. In this case, let z , . . . , z k (cid:48) denote the H agents in P (cid:48) at time a (cid:48) , where k (cid:48) = y (cid:48) a (cid:48) . Similarly, let z , . . . , z k (cid:48)(cid:48) denote the H agents in P (cid:48)(cid:48) at time a (cid:48)(cid:48) , where k (cid:48)(cid:48) = y (cid:48)(cid:48) a (cid:48)(cid:48) . According to this notation, the first k (cid:48) agents in P (cid:48)(cid:48) are the same agents as the k (cid:48) agentsin P (cid:48) . Indeed, our coupling treats these agents identically upon the i th departure in P (cid:48) , P (cid:48)(cid:48) . Thisguarantees that if the departing agent is matched to an agent z ∈ { z , . . . , z k (cid:48) } in P (cid:48) , then, thedeparting agent will also be matched to z in P (cid:48)(cid:48) . Therefore, Property ζ holds in this case as well.To complete the induction, suppose that Property ζ holds for all departures before the i thdeparture. In the induction step, we will show that Property ζ will be satisfied after the i thdeparture. First, recall that a (cid:48) i ≤ a (cid:48)(cid:48) i , by (33). This fact, together with the induction hypothesis,implies that y (cid:48) a (cid:48) i ≤ y (cid:48)(cid:48) a (cid:48)(cid:48) i . We consider two cases: either y (cid:48) a (cid:48) i = y (cid:48)(cid:48) a (cid:48)(cid:48) i or y (cid:48) a (cid:48) i < y (cid:48)(cid:48) a (cid:48)(cid:48) i . We define the couplingfor each case separately. This definition is identical to the definition of our coupling in the inductionbasis. This completes the induction. Therefore, Property ζ holds, and Step 1 is complete. Step 2
As we mentioned in the beginning of the proof, in this step we will show that there existsa constant σ (cid:48)(cid:48) Y > y (cid:48)(cid:48) ≥ π (cid:48)(cid:48) ( y (cid:48)(cid:48) ) ≤ e − σ (cid:48)(cid:48) Y · y (cid:48)(cid:48) . (34)Process P (cid:48)(cid:48) is an easily tractable process. The random variable y (cid:48)(cid:48) evolves according to twoPoisson processes (Poisson clocks). The first clock is the arrival clock, which ticks with rate m ;upon each tick, the value of y (cid:48)(cid:48) increases by 1. The second clock is the departure clock, whichticks with rate (1 + λ ) m (1 − e − m ); upon each tick, the value of y (cid:48)(cid:48) decreases by 1 with probability52 − M y (cid:48)(cid:48) .Let y (cid:48)(cid:48) > λ ) M y (cid:48)(cid:48) > λ/
2. Therefore, for any y (cid:48)(cid:48) > y (cid:48)(cid:48) , thebalance equations imply that π (cid:48)(cid:48) ( y (cid:48)(cid:48) + 1) π (cid:48)(cid:48) ( y (cid:48)(cid:48) ) ≤
11 + λ/ . The above equation, together with the fact that y is a constant, implies that there exists a constant σ (cid:48)(cid:48) Y such that (34) holds. This finishes Step 2 and completes the proof. Proof of Lemma 8.
Note that for any x ≤ λm/
2, by the balance equations we have π ( x + 1) π ( x ) ≤ x + m (1 + λ ) m ≤ λ/
21 + λ .
Letting γ = λ/ λ proves the claim. E.2 Match rate under patient matching
Lemma 9.
Under patient matching, the match rate of hard-to-match agents is λ − O ( λ ) √ m ) and the match rate of easy-to-match agents is − o (1) .Proof. Let m E = m and m H = (1 + λ ) m . We use E Pt , H Pt respectively to denote the number of E,H agents under the patient policy at time t . Let E P , H P denote the average (steady-state) numberof E, H agents under the patient policy, respectively. Also, let the steady-state number of matchesbetween E and H agents be denoted by a random variable M eh .We will show that E [ M eh ] = m − O ( √ m ). This would prove the claim on the match rates of Eand H agents. To this end, we define an event F , as follows. Consider an E agent upon her arrival,namely agent a . Let F be the event in which a is matched with another E agent. Suppose t denotesthe time when a ’s departure clock is set to tick (which is determined by a draw from Exp (1), rightafter her arrival). Let G be the event in which a leaves the pool at time t , while being matched toanother E agent. See that P [ F ] ≤ P [ G ] . (35)This holds because any match between two E agents is formed upon the departure of one of them.(In the above inequality, the probabilities are unconditional steady-state probabilities, i.e., theprobabilities are not conditional on the state of the pool that a enters to, the waiting time of a , orcompatibility of a to the other agents.) 53ext, we will show that P [ G ] = o (1). This holds essentially by Lemma 8. Roughly speaking,this lemma says that the steady-state H pool is large, and therefore, a is compatible with an Hagent wvhp. This would imply that F holds wvlp. More precisely, Lemma 8 is applicable sincethe marginal distribution of the size of the H pool just before an E departure (conditional on anE departure) is equal to the unconditional marginal steady-state distribution of the size of the Hpool. (This is a consequence of the PASTA property.) Therefore, Lemma 8 is applicable andwould imply that, wvhp, the size of the H pool is at least λm/ a . Therefore,wvhp, a is compatible with at least one of the H agents in the pool. This means G happens wvlp.This fact and (35) together imply that there exists a constant η > P [ F ] ≤ e − ηm . The above equation and linearity of expectation together imply that E [ M eh ] ≥ m − e − ηm , whichmeans E [ M eh ] = m − O ( √ m ). (Note that E [ M eh ] ≤ m .) E.3 Distribution of waiting time
Lemma 10.
Under the patient policy, as m converges to infinity, the waiting time and matchingtime of an E agent converge in distribution to the degenerate distribution at .Proof. Fix an E agent, e , and let w e denote the waiting time of e . For any fixed constant t >
0, wewill show that lim m →∞ P [ t > w e ] = 0. This will prove the claim. First, see that E [ w e ] = 1 m · E ( x,y ) ∼ π [ x ] = 1 m · O (1)where the second equality is by Theorem 4. Therefore, by Markov inequality, P [ w e > t ] < O ( t/m )holds, which means lim m →∞ P [ t > w e ] = 0. Lemma 11.
Under the patient policy, as m converges to infinity, the waiting time and matchingtime of an H agent converge in distribution to an exponential random variable with rate .Proof. Fix an H agent, namely h , upon arrival. Let E h denote the event in which agent h receivesno offers before his departure clock ticks. More precisely, E h denotes the event in which no E agent Recall that an E departure denotes the event of the departure of an E agent, whether because the agent ismatched or because the agent’s departure clock has ticked. h is compatible with him. To prove the lemma, it is enough toshow that lim m →∞ P [ E h ] → . (36) Claim 4. If (36) holds, then the waiting time and matching time of an H agent converge indistribution to an exponential random variable with rate .Proof. Let w denote the waiting time of h and let d denote the time that his departure clock is setto tick (which will be set upon the arrival of h ). Note that for any t we have P [ d < t ] ≤ P [ w < t ] ≤ P [ d < t ] + P [( d > t ) ∧ E h ] ≤ P [ d < t ] + P [ E h ] . The above inequality and (36) together imply thatlim m →∞ P [ w < t ] = P [ d < t ] , which proves the claim.To prove that (36) holds, we slightly modify the stochastic process after the arrival of h . Let Q denote the original process, which starts upon the arrival of h , namely at time t , and ends when h leaves the system. The modified process, namely Q (cid:48) , is similar to Q , except for the differencesclarified below. Roughly speaking, Q (cid:48) is the same as the patient matching process with the exceptionthat the arrival of h is “ignored” in the sense that h does not interfere with the evolution of Q (cid:48) . • Q (cid:48) runs from time t to t + log m . Furthermore, the departure clock of h is set to tick attime t + log m in Q (cid:48) . • By definition, Q (cid:48) has a sample path identical to Q , until one of the following disjoint eventshappens:Event (i) h receives an offer in Q before time t + log m . In this case, we stop Q . In Q (cid:48) , h rejects the received offer as well as all of the offers that she will receive inthe future. Any E agent who gets rejected by h will make an offer to the next We are implicitly assuming that E agents, upon departure, search for a compatible match by first visiting Hagents one by one, and match with the first compatible agent. This is of course without loss of generality, by theprinciple of deferred decisions. Q (cid:48) will continue evolving according to thepatient matching process, with the exception of agent h , who does not interferewith the process.Event (ii) The departure clock of h ticks in Q before time t + log m , and no offers are madeto h prior her departure. In this case, we stop Q but continue to run Q (cid:48) . Q (cid:48) willcontinue evolving according to the patient matching process, with the exceptionof agent h , who does not interfere with the process (i.e., agent h rejects all theoffers she receives, in the sense clarified above). • We stop Q (cid:48) when it reaches the time t + log m .Let E (cid:48) h denote the event in which h receives at least one offer in Q (cid:48) before time t + log m . Astraightforward coupling argument shows that P [ E h ] ≤ P (cid:2) E (cid:48) h (cid:3) + e − log m . Therefore, to prove that (36) holds, it is enough to show thatlim m →∞ P (cid:2) E (cid:48) h (cid:3) → . (37)Our approach for proving (37) is as follows. First, we observe that, when the arrival of h isignored, the process Q (cid:48) can be run from time t to t + log m as follows: sample a state ( x, y ) ∼ π .Then, let the stochastic system start from ( x, y ) and evolve for log m units of time, under thepatient policy. By the PASTA property, the sample paths generated by this process are identicalto the sample paths of Q (cid:48) (when the arrival of h is ignored).By the above argument, we sample the state ( x, y ) ∼ π at time t and let the process run untiltime t + log m . We also consider an “imaginary agent” h , which exists in the H pool, but rejectsall of the proposals that are made to her. Our goal is showing that the probability that h receivesat least one proposal over the period [ t , t + log m ], which we denoted by P [ E (cid:48) h ], approaches 0 as m approaches infinity.Without loss of generality, let t = 0 and t = log m for notational simplicity. Let H t denote thehistory of the process until time t . (Note that the history does not include the offers made to h ,since this agent does not change the evolution of the process.) Let the random variable n t denotethe number of E agents whose departure clock ticks in the interval [0 , t ]. Also, let d , . . . , d n t denotethe times that the departure clocks of these agents tick. Define x , . . . , x n t to be the number of Hagents in the pool at times d , . . . , d n t , respectively.56irst, see that P (cid:2) ¯ E h (cid:12)(cid:12) H t (cid:3) ≥ Π n t i =1 (1 − /x i ) . (38)In the rest of the proof, we will use probabilistic bounds to show that in almost all histories H t , n t is small and x ∗ is large, where x ∗ = min { x , . . . , x n t } . These two facts, together with (38), willfinish the proof.Define event G as x ∗ > λm/ . The goal is to show that ¯ G happens wvlp. For this, we will show that the size of the E pool afterthe arrival of any E agent is small, wvhp. Formally, we can use the PASTA property together withLemma 8 and write a union bound over all arrivals of E agents in the interval [0 , t ]. This impliesthat P (cid:2) ¯ G (cid:3) = O ( mte − γ m ) , (39)where γ = λγ / G as n t < t log m . By Theorem 4, E [ n t ] = O ( t ). Therefore, by Markov inequality, P (cid:2) ¯ G (cid:3) = O (1 / log m ) . (40)Now, recall (38), and that our goal is to show that in almost all histories H t , n t is small and x ∗ is large. (39) and (40) show just this. More formally, they imply that in almost all histories H t but a fraction O (1 / log m ) of them, events G , G hold. Therefore, using (38), we can write P (cid:2) ¯ E h (cid:3) ≥ (1 − O (1 / log m )) · (cid:18) − λm/ . (cid:19) t log m , (41) ≥ (1 − O (1 / log m )) · (cid:18) − t log mλm/ . (cid:19) = 1 − o (1) , (42)which implies that P [ E h ] = o (1). This completes the proof. F Proof of Proposition 3, Part (ii)
We break the proof into four parts: Proposition 9 and Proposition 10 prove the bounds on thematch rate and waiting time of H agents, and Proposition 11 and Proposition 12 prove the boundson the match rate and waiting time of E agents. These propositions are stated and proved after57he following preliminary analysis.Suppose that time is indexed by real numbers and starts from 0. The batching policy makesmatches every T units of time. Define round i to be the time window between time ( i − T + 1right after the matches are made (if any) and time iT right before the matches are made.To give an upper bound on the match rate of the batching policy, we can analyze a simplerprocess instead, which we call process B and define as follows. B is similar to the original batchingpolicy, with the following differences:1. Easy-to-match nodes are not compatible.2. The probability of compatibility between an easy-to-match node and a hard-to-match nodeis 1.A straightforward coupling argument shows that the match rate of H agents in B is smaller andtheir (average) waiting time larger than in the batching policy. (This holds in each sample pathof the coupled process.) Therefore, to provide the promised bounds for H agents in the batchingpolicy, it suffices to analyze the simpler process, B . All the following definitions are thereforedefined for the process B .Let E i , H i respectively denote the number of E agents and H agents at the beginning of round i . Let X i = H i − E i . (Note that E i · H i = 0 must always hold.) It is straightforward to verify that C = (cid:104) ( E , H ) , ( E , H ) , . . . (cid:105) is an ergodic Markov chain with its state space being the set of pairsof non-negative integers. Let π denote the steady-state distribution of C .Let E (cid:48) i , H (cid:48) i respectively denote the number of E agents and H agents who were present in thepool both at the beginning and at the end of round i . Define X (cid:48) i = H (cid:48) i − E (cid:48) i . Moreover, let E (cid:48)(cid:48) i , H (cid:48)(cid:48) i respectively denote the number of E agents and H agents who arrived after the beginning of round i and were present in the pool at the end of round i . Define X (cid:48)(cid:48) i = H (cid:48)(cid:48) i − E (cid:48)(cid:48) i .Next, observe that X i +1 = X (cid:48) i + X (cid:48)(cid:48) i . (43)Taking expectation (with respect to the stationary distribution π ) from both sides of the aboveequality implies that E π [ X i +1 ] = E π (cid:2) X (cid:48) i (cid:3) + E π (cid:2) X (cid:48)(cid:48) i (cid:3) . (44)We will compute E π [ X i ] using the above equation and the following observations.58irst, observe that E π (cid:2) X (cid:48) i (cid:3) = E π (cid:2) H (cid:48) i (cid:3) − E π (cid:2) E (cid:48) i (cid:3) = E π [ X i ] · e − γT , (45)where γ = 1 /d . This equality holds simply because an agent who is present at the beginning ofround i will also be present at the end of round i with probability e − γT .Let N i denote the number of arrivals. Second, observe that E π (cid:2) X (cid:48)(cid:48) i (cid:3) = E π (cid:2) H (cid:48)(cid:48) i (cid:3) − E π (cid:2) E (cid:48)(cid:48) i (cid:3) = n = ∞ (cid:88) n =1 P (cid:2) H (cid:48)(cid:48) i = n (cid:3) · n · (cid:90) T T · e − γ ( T − t ) d t − n = ∞ (cid:88) n =1 P (cid:2) E (cid:48)(cid:48) i = n (cid:3) · n · (cid:90) T T · e − γ ( T − t ) d t = (cid:90) T T · e − γ ( T − t ) d t · (cid:0) E π (cid:2) H (cid:48)(cid:48) i (cid:3) − E π (cid:2) E (cid:48)(cid:48) i (cid:3)(cid:1) = ( 1 T − e − γT T ) · λmT /γ. (46)We then can use (44), (45), and (46) to write E π [ X i +1 ] = E π [ X i ] · e − γT + ( 1 T − e − γT T ) · λmT /γ. We can simplify the above equation further using the fact that E π [ X i +1 ] = E π [ X i ], which holdssince π is the steady-state distribution: E π [ X i ] = ( T − e − γT T ) · λmT /γ − e − γT = λm/γ. (47)Next, we use (47) to compute an upper bound on the match rate of H agents under the batchingpolicy. Equivalently, we compute a lower bound on the death rate of H agents, where the deathrate is the expected fractions of H agents per unit of time who depart the market without gettingmatched. To this end, we just need to compute the expected number of such departures (i.e., deaths ) per round.Fix a round i and let D i denote the number of deaths in round i . The expected number ofdeaths of H agents during round i is just equal to the expected number of H agents who werepresent at the beginning of round i but not at the end, plus the expected number of H agents whoarrived sometime in round i but were not present at the end of round i . That is, E π [ D i ] = E π (cid:2) max { X i , } · (1 − e − γT ) (cid:3) + (1 + λ ) mT · (1 − γT + e − γT γT ) . (48)Verify that the second summand on the right-hand-side is indeed the expected number of H agentswho arrived sometime in round i but were not present at the end of round i by the following59quality: n = ∞ (cid:88) n =1 P (cid:2) H (cid:48)(cid:48) i = n (cid:3) · n · (cid:90) T T · (1 − e − γ ( T − t ) ) d t = (1 + λ ) mT · (1 − γT + e − γT γT ) . We now can use (48) to write the inequality E π [ D i ] ≥ E π [ X i ] · (1 − e − γT ) + (1 + λ ) mT · (1 − γT + e − γT γT ) , which can be simplified using (47) to E π [ D i ] ≥ λm/γ · (1 − e − γT ) + (1 + λ ) mT · (1 − γT + e − γT γT ) . (49)The above inequality implies E π [ D i ](1 + λ ) mT ≥ λ (1 − e − γT )(1 + λ ) T γ + (1 − γT + e − γT γT )which is the promised lower bound on the death rate of H agents (i.e., the left-hand-side of theabove inequality). This implies that the match rate of the H agents is bounded from above by1 − e − γT γT − λ (1 − e − γT )(1 + λ ) γT . (50) Proposition 9.
Fix
T > . There exists a constant r < λ such that for any m, λ > , the matchrate of H agents under the batching policy is smaller than r .Proof. The proof uses the upper bound (50) on the match rate of the batching policy. Let u denotethis upper bound. We will show that u < λ holds for any m, λ >
0. Simple algebra reveals thatthis is equivalent to showing that 1 + e γT ( − γT ) > . (51)Let the left-hand-side of (51) be denoted by f ( T ). Observe that f ( T ) is strictly increasing in T since f (cid:48) ( T ) = γ T e T >
0. Since f (0) = 0 and T >
0, then f ( T ) >
0. This proves the claim.
Proposition 10.
Fix
T > . There exists a constant r > λ λ such that for any m, λ > , theexpected waiting time of H agents under the batching policy is larger than r .Proof. Let W i denote the waiting time incurred by all H agents in round i . The expected waitingtime of H agents is just equal to E π [ W i ](1+ λ ) mT . We therefore prove the claim by computing a lower60ound on E π [ W i ]. Observe that E π [ W i ] = E π (cid:2) W (cid:48) i (cid:3) + E π (cid:2) W (cid:48)(cid:48) i (cid:3) , (52)where W (cid:48) i denotes the waiting time incurred in round i by the H agents who were present at thebeginning of round i and W (cid:48)(cid:48) i denotes the waiting time incurred in round i by the H agents whowere not present at the beginning of round i .Verify that E π (cid:2) W (cid:48) i (cid:3) = E π [max { X i , } ] · (cid:18)(cid:90) T γte − γt d t + T e − γT (cid:19) ≥ λm (1 − e − γT ) /γ , (53)where the left-hand-side is the product of the expected number of H agents that are present at thebeginning of round i and the expected waiting time incurred by any such agent in round i . Also,verify that E π (cid:2) W (cid:48)(cid:48) i (cid:3) = (1 + λ ) mT · (cid:18)(cid:90) T T (cid:18)(cid:90) T − s γte − γt d t + ( T − s ) e − γ ( T − s ) (cid:19) d s (cid:19) = (1 + λ ) m ( e − γT + γT − /γ , (54)where on the right-hand-side of the first equality the first term is the expected number of H agentswho arrive sometime in round i and the second term is the expected waiting time that each suchagent incurs. The second equality follows from simple calculations.We now can use (52), (53), and (54) to write the following lower bound on the waiting time ofbatching policies: E π [ W i ](1 + λ ) mT ≥ γT (1 + λ ) + e − γT − γ ( λ + 1) T . (55)Simple algebra shows that the right-hand-side of the above inequality is larger than λ/γ λ if and onlyif 1 + e γT ( − γT ) > . Let the LHS of the above inequality be denoted by f ( T ). Observe that f ( T ) is strictly increasing in T since f (cid:48) ( T ) = γ T e T >
0. Since f (0) = 0 and T >
0, then f ( T ) >
0. This proves the claim.
Proposition 11.
Fix
T > . The match rate of E agents is at most − e − γT γT .Proof. First we compute the chance that an E agent who arrives during round i does not become61ritical before round i ends: (cid:90) T T e − γt d t = 1 − e − γT γT . This implies that, conditioned on arriving in period i , an E agent does not get matched withprobability at least 1 − − e − γT γT . This proves the claim. Proposition 12.
Fix
T > . The waiting time of E agents is at least γ − − e − γT γ T .Proof. We compute the expected waiting time that an E agent who arrives in round i incurs duringround i : (cid:90) T T · (cid:18)(cid:90) T − s γte − γt d t + ( T − s ) e − γ ( T − s ) (cid:19) d s = e − γT + γT − γ T = 1 γ − − e − γT γ T .
This proves the claim.
G Few Hard-to-Match Agents ( λ < ) For completeness we analyze the behavior of the patient and greedy matching policies when λ < Theorem 5.
Let λ ∈ [ − , .1. The match rate for all agents is − O (1 /m ) under both policies.2. Under greedy matching, the expected waiting time and matching time of H agents are O (1 /m ) ,and the expected waiting time and matching time of E agents are O (cid:16) log m / (1 − p ) m (cid:17) .3. Under patient matching, the expected waiting time and matching time of H agents are O (1 /m ) ;moreover, the distribution of waiting time and matching time of E agents converge to theexponential random variable with mean − λ . We omit the proof of this theorem. It is a simpler version of the proof of Theorem 1.
H Simulations Based on the Stylized Model
This section presents simulation results based on the stylized model. The findings illustrate that thetheoretical predictions hold even in small markets. As in the model, no two hard-to-match agents62re compatible with each other. Every two easy-to-match agents are compatible with probability q = 0 .
04, independently. Every pair of easy-to-match and hard-to-match agents is compatiblewith probability p = 0 .
1, independently. The choice of these probabilities is motivated from somecomponents of the kidney exchange pool.
42 43
Other parameters of the model are set as follows. Agents arrive according to a Poisson processwith an average of 1 per time period (day) and become critical according to an exponential distri-bution with mean 200 days. We set λ = 0 . λ . Boththe greedy and patient matching policies are simulated until 70 ,
000 agents arrive and the waitingand matching times are recorded for all but the first 5000 agents. Under the greedy policy the waiting time and matching time distributions of hard-to-matchagents fit exponential distributions with means 64.63 and 64.6, respectively, where the mean pre-dicted by Proposition 3 (albeit in a thick enough market) is λ λ · ≈ .
66. The empiricaldistribution of waiting times, the empirical distribution of matching times, and the CDF of theexponential distribution with mean 66.66 are depicted in Figure 13 (note that they almost coincide).Under the patient policy the waiting time and matching time distributions of hard-to-matchagents fit exponential distributions with means 189.09 and 190.12, respectively. Figure 13 also plotsthese empirical distributions, as well as the exponential distribution with mean 200 (here too, theCDFs almost coincide).The fraction of hard-to-match agents that are matched is approximately 0 .
67 under both poli-cies, the fraction of easy-to-match agents that are matched is approximately 1 under both policies,and in particular, almost all easy-to-match agents are matched with hard-to-match agents. Consider patient-donor pairs with pair of blood types A-O, respectively, and O-A pairs. Since the donor and thepatient of an A-O pair are ABO compatible, the patient is likely to be sensitized and thus two O-A pairs will have alow chance of being compatible with each other. Since the donor of an O-A pair is ABO incompatible, the patient isless sensitized, and therefore such a pair has a higher probability to being compatible with an A-O pair. This choice of parameters is also interesting because it shows that the convergence rates are fast even when p > q ,and even faster for larger q , e.g., when p = 0 . q = 0 . When no matches are made at all, the expected pool size is therefore about 200. Typical kidney exchangenetworks are fairly thicker; for example, recall the estimates for arrival and departure rates from our data set inSection 4 (i.e., 1 arrival per day and a departure rate of 1/360). According to these estimates, when no matches aremade, the expected pool size would be 360. We ignore the first 5000 agents to ensure that our samples are taken when the stochastic process is (almost) inthe steady state, i.e., the corresponding Markov chain is mixed. Figure 13:
Waiting time and matching time distributions under the greedy and patient policies for λ = 0 . Table 2 presents similar results for different values of λ , while fixing the total arrival rate to 200.The values of λ are derived by setting the fraction of easy-to-match agents arriving to the pool (i.e., λ ) to 0 . , .
35 and 0 .
45, which respectively correspond to λ = 2, λ = 0 . λ = 0 . ∈ { E , H } , let W T (Θ) be the average waiting time of matched agents of typeΘ,
M T (Θ) be the average matching time of agents of type Θ, and M (Θ) be the fraction of agentsof type Θ who match. Also let ˆ M T (Θ) be the estimated average matching time of type Θ that isobtained by applying the limit result of Proposition 3, i.e., ˆ
M T ( E ) = 0 and ˆ M T ( H ) = λ λ · m approaches infinity, it can be used to estimate thematching time for small m . As Table 2 shows, these estimates are reasonably close to the empiricalaverage matching time and the average waiting time. Policy λ MT(H) WT(H) ˆ MT (H) MT(E) WT(E) ˆ MT (E) M(H) M(E)Patient 0.222 181.8 182.4 200 18.9 18.9 0 0.814 0.99Greedy 0.222 36.31 36.87 36.36 0.28 0.29 0 0.82 1Patient 0.857 188.9 192.7 200 10.43 10.43 0 0.54 1Greedy 0.857 92.5 92.7 92.3 0 0 0 0.54 1Patient 2 191.3 196.3 200 7.43 7.43 0 0.33 1Greedy 2 132.6 132.8 133.3 0 0 0 0.33 1 Table 2:
Statistics for the stylized model for different values of λ . The waiting time and matching time distributions follow exponential distributions with thecorresponding averages reported in Table 2 (plots are omitted). Note that under the patient policythe average waiting time and the average matching time of hard-to-match agents are similar for allvalues of λ and are close to the predicted waiting time, 200, which is the exogenous average time,in which hard-to-match agents become critical. Under greedy matching, for each λ the averagewaiting time and average matching time for hard-match-agents are similar and are close to the64redicted values, 200 · λ λλ