Optimal Provision-After-Wait in Healthcare
aa r X i v : . [ c s . G T ] D ec Optimal Provision-After-Wait in Healthcare ∗ Mark Braverman † Jing Chen ‡ Sampath Kannan § Abstract
We investigate computational and mechanism design aspects of optimal scarce resource al-location, where the primary rationing mechanism is through waiting times. Specifically weconsider the problem of allocating medical treatments to a population of patients. Each patienthas demand for exactly one unit of treatment, and can choose to be treated in one of k hospitals, H , . . . , H k . Different hospitals have different costs, which are fully paid by a third party —the“payer”— and do not accrue to the patients. The payer has a fixed budget B and can onlycover a limited number of treatments in the more expensive hospitals. Access to over-demandedhospitals is rationed through waiting times: each hospital H i will have waiting time w i . In equi-librium, each patient will choose his most preferred hospital given his intrinsic preferences andthe waiting times. The payer thus computes the waiting times and the number of treatmentsauthorized for each hospital, so that in equilibrium the budget constraint is satisfied and thesocial welfare is maximized.We show that even if the patients’ preferences are known to the payer, the task of optimizingsocial welfare in equilibrium subject to the budget constraint is NP-hard. We also show that,with constant number of hospitals, if the budget constraint can be relaxed from B to (1+ ǫ ) B foran arbitrarily small constant ǫ , then the original optimum under budget B can be approximatedvery efficiently.Next, we study the endogenous emergence of waiting time from the dynamics between hospi-tals and patients, and show that there is no need for the payer to explicitly enforce the optimalequilibrium waiting times. When the patients arrive uniformly along time and when they havegeneric types, all that the payer needs to do is to enforce the total amount of money he wouldlike to pay to each hospital. The waiting times will simply change according to the demand,and the dynamics will always converge to the desired waiting times in finite time.We then go beyond equilibrium solutions and investigate the optimization problem over amuch larger class of mechanisms containing the equilibrium ones as special cases. In the settingwith two hospitals, we show that under a natural assumption on the patients’ preference profiles,optimal welfare is in fact attained by the randomized assignment mechanism, which allocatespatients to hospitals at random subject to the budget constraint, but avoids waiting times.Finally, we discuss potential policy implications of our results, as well as follow-up directionsand open problems. Keywords: healthcare, mechanism design, budget constraint, waiting times ∗ The first author is supported by the Alfred P. Sloan Fellowship, an NSF CAREER award (CCF-1149888), NSFAward CCF-1215990, and a Turing Centenary Fellowship. The second author is supported in part by the ZurichFinancial Services and NSF grant CCF-0832797. The third author is supported by NSF Award CCF-1137084. Partof this research was done when the second author was a postdoc at the Institute for Advanced Study and the thirdauthor was visiting Princeton University. The authors would like to thank Itai Ashlagi for pointing us to importantreferences and three anonymous reviewers for their comments. † Department of Computer Science, Princeton University, [email protected]. ‡ Department of Computer Science, Stony Brook University, [email protected]. § Department of Computer and Information Science, University of Pennsylvania, [email protected].
Introduction
In this paper we study computational and mechanism design issues in the context of optimalhealthcare provision. Specifically, we consider the setting where waiting times, and not payments,are used to allocate scarce care resources among patients. Waiting times in healthcare provision isan important topic of public debate worldwide. For example, it has a central role in the ongoingdebate surrounding the Patient Protection and Affordable Care Act (“Obamacare”) in the UnitedStates. In a large number of countries with public health coverage financing, including Australia,Canada, Spain, and the United Kingdom, procedures such as elective surgery are rationed bywaiting [26, 11]. While in the public perception waiting times are often associated with poorresource management, in the economics literature it is well-understood that queues of consumerswill form whenever a good is priced below the good’s perceived value, as long as supply is scarce[4, 21, 16] – independently of the ultimate distribution mechanism. In particular, waiting timesin this context are dictated by economic incentive constraints and not by stochastic fluctuationsas in classical queuing theory. Therefore, whenever “correct” monetary pricing is impossible orundesirable, waiting times should be incorporated explicitly into the allocation models.We focus on providing a single non-urgent healthcare service (such as a particular surgery)to a population of patients, and define the
Provision-after-Wait problem for this scenario.In our model, a population of patients arrives in each time unit (say, 1 month), seeking for thedesired service at some hospital. There are k hospitals providing the service under different costs.The patients have different preferences about the hospitals, and the composition of the patientpopulation in each time unit is the same. Each patient needs to be served exactly once. Theservice is fully financed by a third party —a “payer”, e.g., the government or an insurer. Thereforethe patients’ choices of hospitals are not affected by the (monetary) costs. But the payer, takento be the government in the rest of this paper for concreteness, has a fixed budget B that he iswilling to spend on providing the service to the entire patient population in each time unit, and itis unaffordable to let every patient go to his favorite hospital (otherwise the provision problem isalready solved at the very beginning). Without loss of generality, we assume that the governmenthas enough budget to treat all patients in the cheapest hospital. This can always be achieved byadding a dummy hospital which has cost 0 and is the least preferred by all patients, representingthe option of not getting any service.The government rations the patients’ demand subject to his budget by setting for each hospital H i a waiting time w i , measured using the same time unit. Every patient going to H i has to wait for w i before he can be served. There is no co-pays, and thus the waiting time is the only cost directlyincurred by the patients. We assume that waiting times are known to the patients before theymake decisions. Each patient P j has value v ij for hospital H i , representing his utility for beingtreated in H i right away. Similar to [10], we assume that the patients have quasi-linear utilitieswith respect to waiting time, that is, patient P j ’s utility for being treated at H i with waiting time w i is u ij , v ij − w i . The primary reason for this choice is that it is the most natural way toensure that patients are treated equally by welfare-optimizing mechanisms. Since, as mechanismdesigners, we do not have full access to the u ij ’s of individual patients but can observe waitingtimes, our welfare-loss due to waiting will just be the sum of all the waiting times in the system . Adding co-pays to the model would be interesting follow-up work, but the space of possible models is far vasterwith co-pays. Issues in introducing co-pays include dealing with different people having different time/money trade-offs, and defining the patients’ utility properly (with the usual ethical question: do people with higher utility formoney have lower utility for health, a.k.a. “should poor people count for less”?). In this paper we avoid theseproblems, since time is fair to everybody and our patients utility is measured in waiting-time equivalents. For example, the patients can observe the length of the lines before deciding which one to join, or they can beinformed explicitly when trying to make an appointment. We can relax this assumption to allow utility functions of the form u ij = v ij − U ( w i ), where U ( w ) is a function social welfare of anequilibrium is defined to be the total utility of the patients in each time unit. The government’sgoal when solving the Provision-after-Wait problem is to find the optimal equilibrium waitingtimes and assignments of patients to hospitals that maximize social welfare, subject to the budgetconstraint.Our model is formally defined in Section 2. Below we would like to emphasize three mainfeatures of it.
Two non-interchangeable “currencies”.
Firstly, as money is still involved, the setting leadsto two non-interchangeable “currencies” of money and waiting time. This complicates the designproblem, both conceptually and computationally. As we shall see from the first part of our mainresults, even if money and waiting time are kept separate and only the latter affects the demand,the fact that they cannot be “traded” for each other (thus reducing the setting to one currency)makes the problem much more difficult.
Indirect control of waiting times.
Secondly, although waiting time is modeled as a parameterwhose optimal value is decided by the government, there is no need for the government to enforceit explicitly. Instead, as we shall show in the second part of our main results, the government cansimply decide the amount of money it is willing to pay to each hospital in each time unit, and thedesired waiting times at different hospitals will emerge endogenously among the hospitals and thepatients. Indeed, the role of waiting time in our model is similar to that of price in markets. In amarket, it is the price that ultimately drives consumers to different purchases, but the producersdo not get to dictate it. They can only control the price indirectly by adjusting their supply levels,and the “correct” price will emerge endogenously from the market. This analogy makes it morereasonable to adopt our model in reality: it is more natural for the government to control theamount of money it pays and tell a hospital “I’ll only pay you $5,000 each month for this service”,than for it to control waiting times and tell a hospital “you have to make each patient using thisservice wait for 3 months”.
Welfare-burning effect of waiting times.
Finally, unlike monetary transfers, nobody benefitsfrom one’s waiting time, and thus waiting times represent a net loss in welfare. That is why in ourmodel the social welfare is defined as the total utility of the patients —that is, total value minustotal waiting time—, differently from auctions where social welfare is the total value of the buyers.The welfare-burning phenomenon is common in the study of resource allocation with waiting times,and is similar to the money-burning mechanisms [14], subject to the important caveat that timeburnt is not interchangeable with money.Given the general welfare-burning effect of waiting times, it is very natural to ask whether theycan be avoided or reduced via a different allocation mechanism altogether. If monetary paymentsare not allowed, and patients are free to choose their hospitals, then the (deterministic) equilibriumsolution of the
Provision-after-Wait problem is the only one possible. What if the governmenthas sufficient control over the patients that it can tell them where to receive their treatment, orotherwise restrict their options ? The simplest such mechanism would be a randomized assignmentof patients to available slots, with the probabilities decided by the budget constraint. In suchassignment, we benefit from zero waiting time. On the downside, we incur an efficiency loss:patients may not end up in the hospitals they prefer. How does this randomized assignment (common to all patients) that maps waiting time w to utility loss caused by waiting w time units. Possible “soft” mechanisms for doing this are discussed below. and diverse preferences on where to betreated, then the free-choice equilibrium mechanism is better, since efficiency gains due to betterallocation offset the inefficiency caused by waiting. At the other extreme, if all patients have similarpreferences, then no efficiencies are to be gained from patients’ choice, and randomized assignmentmechanisms are superior. We further investigate this question in the case of two hospitals, in thethird part of our main results.
Finding optimal equilibrium waiting times and assignments
We first study the computational issues in our model, assuming that the government is fully in-formed about the hospitals’ costs and the patients’ valuations. The following theorem shows thatthe
Provision-after-Wait problem is hard to solve in general.
Theorem 1.
Finding optimal equilibrium waiting times and assignments is NP-hard.
The hardness result motivates one to ask whether one can efficiently approximate the welfareof the optimal solution. Interestingly, we show that if we relax the budget constraint to (1 + ǫ ) B with an arbitrarily small constant ǫ , we can achieve at least as much welfare as the best B -budgetequilibrium solution, using an algorithm whose running time depends on (log m ) k , where m is thenumber of patients in one time unit and k is, as already mentioned, the number of hospitals. Theorem 2. (rephrased)
There is an algorithm that runs in time O (cid:0) (log ǫ m ) k · m (cid:1) and outputsan equilibrium solution such that, the total cost is at most (1 + ǫ ) B and the social welfare is at leastas high as that of the optimal equilibrium solution with budget B . These results are formally presented in Sections 3 and 4. It remains an interesting open problemwhether there is a welfare approximation algorithm that does not exceed the budget. Also, it isunknown whether there is an approximation algorithm that is polynomial in k . Letting waiting times emerge endogenously
Next we show how the desired waiting times and the corresponding optimal social welfare canemerge endogenously as the patients arrive and choose their favorite hospitals in dynamics. Say thegovernment has decided how to spend its budget for the desired service, by using our approximationalgorithm above or by using other methods. The way of spending the budget can be enforced bysetting the quota for each hospital, namely, how many patients the government is willing to pay inone time unit (of course, the total quota must be at least the number of patients).It is natural to assume that the hospitals want to keep waiting times as low as possible, andat time 0 all hospitals have waiting time 0. When the patients arrive along time, they choosewhich hospital to go according to their own valuations and the current waiting times. If a hospitalgets over-demanded, namely, the number of patients going there exceeds the quota paid by thegovernment, then a line has to form and this hospital’s waiting time increases accordingly. If thewaiting time becomes too high due to previous demand, patients arriving later may choose not to gothere and the hospital may become under-demanded, causing its waiting time to decrease. As theremay be many waiting time vectors of the hospitals that correspond to equilibrium assignment giventhe quotas, it is not immediately clear which one the dynamics will converge to (if it converges),and how much social welfare the government can generate from the dynamics.Assuming the patients’ valuations are in a generic position as properly defined in Section 5, ourfollowing theorem characterizes the structure of the optimal equilibrium given any quotas of the3ospitals.
Theorem 3. (rephrased)
For any quotas of the hospitals, there is a unique optimal equilibriummaximizing social welfare. It has the minimum waiting time vector among all equilibria, and anyhospital whose quota is not fully used has waiting time 0.
Accordingly, it is reasonable to hope that the optimal equilibrium is the one implemented bythe dynamics. Our following theorem shows this is indeed the case.
Theorems 4 and 5. (rephrased)
At any point of time, the waiting time of any hospital will neverexceed its waiting time in the optimal equilibrium, and thus the social welfare generated in any timeunit will be at least the optimal social welfare given the quotas. The dynamics will always convergeto the optimal equilibrium, in time proportional to the number of hospitals, the maximum socialwelfare of the patients, and the maximum quota of the hospitals.
These results are formally presented in Section 5.
When is the randomized assignment optimal?
Finally, we turn our attention to the enlarged setting where we are not limited to mechanisms thatproduce equilibrium solutions. The two “extreme” mechanisms are the equilibrium mechanismdiscussed above that gives the patients free choices, and the randomized assignment mechanismthat assigns patients at random to available slots and does not give them any choice. In addition,there is an infinite number of various lotteries in-between these extremes. In a lottery, the patientsare presented with a set of distributions over hospitals, with an expected waiting time associatedwith each distribution. Instead of free choices among all possible (distributions of) hospitals, thepatients can only choose from the available ones in the lottery, and they make choices to maximizetheir expected utilities.Intuitively, if there are no extreme variations among the patients’ preferences, the randomizedassignment should outperform other mechanisms, since it avoids the deadweight loss of waitingtimes. We give further evidence suggesting that randomized assignment may be superior in termsof social welfare, by analyzing the case when there are two hospitals.Let the hospitals be H and H with costs c and c respectively, such that c < c . We assumewithout loss of generality that patients going to hospital H faces no waiting time . Thus patientswho prefer H over H will always choose H . We can therefore exclude them from consideration,and focus on patients who prefer H over H .We assume a continuous population of such patients, indexed by the [0 ,
1] interval. Each patient x is associated with a value v ( x ), representing how much time x is willing to wait to be treatedin H instead of H . That is, v ( x ) is the difference between x ’s utility for being treated at H immediately and his utility for being treated at H immediately. We rename the patients so that v ( x ) is a non-decreasing function on [0 , v (0 .
5) represents the median timethat patients preferring H are willing to wait to be treated there. We prove the following theoremin Section 6. Theorem 6. (rephrased) If v ( x ) is concave, then no lottery can generate more social welfare thanthe randomized assignment. Here a lottery is a set of options, each consisting of a probability of being treated in H andthe corresponding waiting time there. This shows that for a broad class of preferences, the ran-domized assignment is welfare-maximizing even when waiting times are an option available to thegovernment. As a special case, this shows that randomized assignment has better welfare than the Indeed, positive waiting time at H will give patients incentives to go to the more expensive hospital H , andthus increase the total cost while burning more social welfare. In this paper we consider two separate issues. The first one is how to optimally allocate treatmentsin equilibrium, when the government faces budget constraints and waiting times are used to rationpatients’ behavior. The second one is whether it may be beneficial to do away with the (ex-post)equilibrium requirements by limiting available options of the patients.While finding the optimal equilibrium solution in the
Provision-after-Wait problem is NP-hard, our approximation result suggests that this problem might not be as difficult in practice. Inmany cases the number of treatment facilities involved is fairly small, making running time expo-nential in k feasible. Moreover, in some cases the “hospitals” are actually treatment alternativesthat vary in costs (e.g. physiotherapy is cheaper than knee replacement), in which case k may beas low as 2. For the general case where k can be big, it would be interesting to explore restrictionson the patients’ valuations that would make the exact optimization efficient, such as when thevaluations are highly correlated so that the valuation matrix ( v ij ) has low rank. There are manyquestions one can ask about the general complexity of the Provision-after-Wait problem, forexample, whether it is strongly NP-hard, whether it has an FPTAS, whether it is fixed-parametertractable in the number of hospitals, etc.As we shall show, equilibrium assignment with waiting times has a strong connection to unit-demand auctions [7, 1], and such a connection leads to our approximation result. One naturalquestion is whether this connection can be used in dynamic setting to show that the system willremain in the patient-optimal equilibrium as the population’s preferences slowly shift over time.A related question is whether it is possible to approximate optimal welfare in equilibrium if thegovernment only knows the approximate distribution of patient types in the population. Anotherrelated question is whether one can design mechanisms for our setting such that the patients haveincentives to truthfully reveal their valuations, so that the government does not need to know thesevaluations to begin with. A similar question is whether the government can elicit the hospitals’true costs via some mechanisms —given the existence of rent in healthcare, finding true costs andpaying hospitals accordingly would be helpful in reducing the government’s expenses.The study of waiting times as a rationing mechanism is closely related to the study of ordealmechanisms [2], where other tools (e.g. excessive bureaucracy) are used in place of waiting timesto reduce demand to the supply level . These may be used in settings where queues are not anoption such as school choice. Developing computational mechanism design tools for these settingsis a very interesting direction of study.Our third result looks beyond equilibrium solutions. We give evidence that equilibrium solutionsare in fact dominated in many cases. One immediate implication is that giving the governmentpower to restrict choice may in fact improve overall welfare. While this is perhaps not surprising,choice restriction may be very difficult or politically infeasible to implement in practice, due to thefact that patients have an inherent preference for choice [23].There are important indirect ways, however, in which the government may influence choice. Oneof them is through release (or non-release) of quality of care information about providers. The topicof quality of care information is important both in theory and in practice. In the United States,for example, Medicare has started to publicly release hospital performance information as part ofits pay-for-performance push [17]. The effect performance reporting has on provider incentives has Note that in medicine not all ordeals are necessarily dead-weight loss. For example, the famous (and highly-demanded) Shouldice hernia clinic in Ontario, Canada requires its patients to lose weight before being admitted fora surgery [15]. Most clinics do not place such a requirement. patient incentives. Inasmuch as quality information influences patients’ choices, it may actually cause harm in thecontext of allocation using waiting times. Consider a scenario where there are two hospitals, a goodone H g and a bad one H b . All patients prefer the good hospital over the bad by the same amount,but they do not know which is which. As a result, both hospitals will receive half the patients,and waiting time will be zero. If the government reveals that H g is the good hospital through itsquality-of-care disclosure, then all patients will prefer H g over H b by the same amount ∆. Unless H g has enough slots for everybody, the waiting time there will have to be ∆, which completelyburns social welfare and makes all patients worse-off than when they were ignorant. In effect, beforethe quality disclosure, uninformed patients implemented the randomized assignment – through freechoice. Once the quality information was disclosed, the game moved to the equilibrium solution.Our results and the discussion above suggest that in some cases a population of more informedpatients will experience higher waiting times and lower overall utility than uninformed patients.This suggests an unfortunate potential side effect of information disclosure in cases where allocationis done by waiting times. Such a side effect deserves further study since, at the moment, qualityinformation release is regarded as an absolute good. Understanding the optimal structure of infor-mation released to the patients in terms of overall welfare (as well as provider-side incentives) is animportant and interesting direction of study. The role of waiting time can be studied either from the supply side, namely, how waiting timesinteract with the hospitals’ incentives, or from the demand side, namely, how they interact withthe patients’ incentives. In [26] the authors give a thorough analysis of existing policies on reducingwaiting times by affecting the incentives of either side. Our model focuses on the demand side, andbelow we discuss some other works that also focus on this side.The authors of [11] study quality and waiting times with the existence of ex post moral hazard.They assume that the patients are ex ante identical, and that the treatment has objective qualitylevels with which both the valuations and the costs are monotonically increasing. But notice that ifthe patients are identical, rationing by waiting times is bounded to burn a lot of social welfare sinceat equilibrium every patient has to be treated in the same way —as elaborated in our results. Inour model the patients’ valuations can be arbitrarily associated with different hospitals, reflecting subjective views they may have, and the hospitals’ costs can also be arbitrary and do not necessarilyreflect their real quality.In [10, 12] the authors study the effect of waiting time prioritization on social welfare. Theyconsider a single waiting list (or in our language, a single hospital), and the patients are prioritizedand may face different waiting times in the same list. In our model different hospitals may havedifferent waiting times, but we do not discriminate the patients, and at the same hospital everybodyfaces the same waiting time. In [6] the authors give experimental evidence on the effect of expandingpatient choice of providers on waiting times. In their theoretical model, there are two hospitalsand the patients can freely go to the one with shorter waiting time. Thus the patients do not havesubjective preferences over hospitals, and waiting time is the only parameter affecting their choices.Moreover, the authors of [9] study the relationship between waiting times and coinsurance, with asingle hospital and a single representative consumer. In [5] the authors show that in special market structures the consumers may benefit from their uncertainty aboutthe product valuation. But the model is very different.
6n [19] the author studies resource allocation where the consumers wait for the stochastic arrivalof the items. Differently from our model and the models discussed above, in this work waiting timedoes not burn social welfare, as the total waiting time of the consumers is always the time forenough items to arrive. There are two different types of items to be allocated, and also two types ofconsumers, respectively preferring one type of items. A consumer can decide whether he wants totake the arriving item or to continue waiting for his preferred type. The social welfare of the systemis measured by the probability that a consumer is matched to his preferred type. Although this is avery different model from ours, it is worth mentioning that the author provides a truthful queuingpolicy which is optimal. As we have discussed in Section 1.2, it would be interesting to design atruthful mechanism in our model from which the government can elicit the patients’ valuations.Finally, in none of the works mentioned above is the insurance/resource provider’s budgetconstraint considered as a parameter affecting waiting times and social welfare.
Now let us be formal about our model. The
Provision-after-Wait problem studies how toprovide a single healthcare service to a population of patients, and is specified by the followingparameters. • The set of hospitals is { H , . . . , H k } . • For each i ∈ [ k ], the cost of H i serving one patient is c i ∈ Z + , where Z + is the set ofnon-negative integers. • The set of patients is { P , . . . , P m } . • For each i ∈ [ k ] and j ∈ [ m ], the value of patient P j for hospital H i is v ij ∈ Z + . • An assignment of the patients to the hospitals is a triple ( w, h, λ ), where w = ( w , . . . , w k ) ∈ ( Z + ) k is the waiting time vector of the hospitals, h : [ m ] → [ k ] is the assignment function , and λ = ( λ , . . . , λ k ) ∈ { , . . . , m } k with P i ∈ [ k ] λ i = m is the quota vector , such that | h − ( i ) | = λ i for each i ∈ [ k ].According to such an assignment, patient P j will receive the service at hospital H h ( j ) afterwaiting time w h ( j ) . • A patient P j ’s utility under assignment ( w, h, λ ) is u j ( w, h, λ ) , v h ( j ) j − w h ( j ) , that is, quasi-linear in the waiting time.The social welfare of this assignment is SW ( w, h, λ ) , P j ∈ [ m ] u j ( w, h, λ ). • The government has budget B ∈ Z + , and an assignment ( w, h, λ ) is feasible if P i ∈ [ k ] λ i · c i ≤ B .For the problem to be interesting, we assume that mc min ≤ B < mc max , where c min and c max are respectively the minimum and the maximum cost of the hospitals. Remark 1.
The hospitals’ costs, the patients’ valuations, and the waiting times are assumed to beintegers without loss of generality. As long as they have finite description, we can always chooseproper units so that all of them are integers.
Remark 2.
The quota vector of an assignment can be inferred from the assignment function andthus is redundant. We define it explicitly to ease the discussion of our main results.
7e would like to emphasize that, in the healthcare literature waiting time is recognized as atool to ration supply by driving down demand. As such, it does not depend on the congestion atthe hospitals, but rather on the patients’ “willingness to wait”. In our model, the waiting times aredecided by the government according to its budget and the patients’ values. Even if a hospital’sreal capacity (namely, the maximum number of patients it is able to handle, which is typicallyassumed to be large enough ) is bigger than the number of patients going there, the patients maystill have to wait for certain amount of time, because letting them wait for any shorter will resultin more patients demanding that hospital than the government can afford. This is demonstratedby the following example.Assume there are two hospitals, H and H , with costs $500 and $3,000 respectively. Thereare three patients, valuing H for 10, 7, 3 respectively, and all valuing H for 0. The governmenthas budget $6,000. Assume that H is capable of handling all three patients immediately. Yet,if the government lets H be saturated and sends all three patients there, the total cost will be$9,000, which is unaffordable. It is clear that the government can afford only one patient at H .Thus at equilibrium the waiting time at H must be 7, and only the patient who is willing to waitfor 10 will actually be served there. Notice that this patient has to wait even though there is nocongestion at all, because of the budget constraint.Since in reality the government may not be able or willing to force a patient to go to a hospitalassigned to him, it must ensure that wherever it wants that patient to go is indeed the best hospitalfor him, given the waiting times. Accordingly, we have the following definition. Definition 1.
Assignment ( w, h, λ ) is an equilibrium assignment if: (1) it is feasible, (2) for each j ∈ [ m ] we have u j ( w, h, λ ) ≥ , and (3) for each j ∈ [ m ] and i ∈ [ k ] we have u j ( w, h, λ ) ≥ v ij − w i . Assignment ( w, h, λ ) is an optimal equilibrium assignment if: (1) it is an equilibrium assign-ment, and (2) for any other equilibrium assignment ( λ ′ , w ′ , h ′ ) , SW ( w, h, λ ) ≥ SW ( w ′ , h ′ , λ ′ ) . The social welfare of optimal equilibrium assignments is denoted by SW OEA . As we are interested in the (existence and) computation of optimal equilibrium assignments, weassume that the government has precise knowledge about the cost of each hospital. We may alsoassume that the government knows each patient’s valuation for each hospital, but we do not need it.In fact, it is enough for the government to know the “distribution” of the k -dimensional valuationvectors of the patients, namely, the fraction of the patients having each particular valuation vector.(How to obtain such information is an interesting mechanism design as well as learning problem.)Once it computes w in the optimal solution, the assignment function h will be automaticallyimplemented by the patients going to their favorite hospitals , and the government need not knowwhere each patient is going. It is easy to introduce the hospitals’ real capacities as additional parameters into our model, and require that ahospital’s quota in an assignment does not exceed its real capacity. But doing so does not make the problem anymore interesting —the optimization problem is even harder, and all our results remain true. Thus we simply assumethat the real capacities are large enough. In reality, the cheap “hospital” may in fact be a cheap service such as a CT scan, while the expensive one mayin fact be an expensive service such as an MRI. A patient is willing to get either one of them, with different values. Each patient can easily compute which hospital maximizes his utility, given that he knows the hospitals’ waitingtimes and his own valuations. If there are more than one favorite hospitals for a patient, we assume that he goes tothe cheapest one, so that the budget constraint is satisfied. H and H withcosts B − B >> P and P . The valuation vector( v , v , v , v ) is either (10 , , ,
6) or (10 , , , ,
0) while in the latter it’s (4 , We begin with two easy observations about our model, as a warm-up.The first observation is that, if the patients have unanimous preferences, namely, v ij = v ij ′ foreach i ∈ [ k ] and each j , j ′ ∈ [ m ], then no equilibrium assignment can improve the social welfareof the following trivial one: order the hospitals according to the patients’ valuations decreasingly,find the first hospital H i such that mc i ≤ B , and assign all patients to H i with w i = 0 and w i ′ = max i ′′ ∈ [ k ] v i ′′ for any i ′ = i . Indeed, for any equilibrium assignment ( w, h, λ ) we have v h ( j ) j − w h ( j ) = v h ( j ′ ) j − w h ( j ′ ) for each j, j ′ ∈ [ m ]. Letting i ∗ = argmin i : h − ( i ) = ∅ c i , λ ′ be such that λ ′ i ∗ = m and λ ′ i = 0 for all other i , h ′ be such that h ′ ( j ) = i ∗ for all j , we have that ( w, h ′ , λ ′ ) isanother equilibrium assignment with the same social welfare as ( w, h, λ ). Thus it suffices to lookfor an optimal equilibrium assignment that sends all patients to the same hospital. This is alsointuitive: if the patients are all the same, then at equilibrium the government must make themequally happy, and it can do so by treating them in the same way.Another observation is that, even if the government only cares about meeting the budget con-straint in expectation, and is allowed to assign each patient to several hospitals probabilistically(with the total probability summing up to 1), the optimal social welfare it can get in expectationwill just be the same as the optimal one obtained by deterministic assignments. This is so because,at equilibrium, all the hospitals to which a patient P j is assigned with positive probability mustyield the same utility for him. Thus assigning P j deterministically to the one with the smallestcost leads to another equilibrium assignment with the same social welfare and still meeting thebudget constraint. Accordingly, to maximize social welfare it suffices to consider only deterministicassignments .The following theorem shows that even the optimal deterministic assignments are hard to findin general. Theorem 1.
Finding optimal equilibrium assignments is
N P -hard.Proof.
The reduction is from the knapsack problem, which is well known to be
N P -hard. In thisproblem there are k items, a , . . . , a k , and each a i has value v i and cost c i . We are also given abudget B , and the goal is to select a subset of items so as to maximize their total value whilekeeping their total cost less than or equal to B .We can transform this problem to a Provision-after-Wait problem with k + 1 hospitals and k patients. Each hospital H i with 1 ≤ i ≤ k has cost c i , and each patient P i has value v i for H i and 0 for all others. Hospital H k +1 has cost 0 and is valued 0 by all patients. The government hasbudget B .Given an equilibrium assignment ( w, h, λ ) to the Provision-after-Wait problem, we canconstruct a solution to the knapsack problem with total value equal to SW ( w, h, λ ) —the set A = { i : h ( i ) = i } is such a solution. Indeed, without loss of generality we can assume h ( i ) = k + 19henever h ( i ) = i . By the definition of equilibrium assignments, we can also assume w k +1 = 0, w i = v i if h ( i ) = k + 1, and w i = 0 otherwise. Thus SW ( w, h, λ ) = P i ∈ A v i , which is the totalvalue of A in the knapsack problem. As the total cost of ( w, h, λ ) is P i ∈ A c i ≤ B , the set A meetsthe budget constraint in the knapsack problem.It is easy to see that the other direction is also true, that is, given a solution A ⊆ [ k ] to theknapsack problem, we can construct an equilibrium assignment ( w, h, λ ) for the Provision-after-Wait problem whose social welfare equals the total value of A .Accordingly, an optimal equilibrium assignment to Provision-after-Wait corresponds to anoptimal solution to knapsack.
Remark 3.
The NP-hardness of the knapsack problem comes from the need for integrality. Itsfractional version can be easily solved using a greedy bang-per-buck approach. But this is not thecase in our problem. Indeed, as we have noted, given a fractional equilibrium assignment we canconstruct a deterministic equilibrium assignment with the same social welfare. Thus for our problemthe fractional version is as hard as the integral version.
Although the optimization problem is hard when both the numbers of patients and hospitals arelarge, in practice we expect the number of hospitals to be small, and it makes sense to solve theproblem efficiently in this case.An easy observation is that optimal equilibrium assignments can be found in time O ( m k poly( m, k )).Indeed, there are at most m k possible assignment functions h : [ m ] → [ k ]. For each h and the cor-responding quota vector λ satisfying P i ∈ [ k ] c i λ i ≤ B , the total value of the patients are fixed, andthus maximizing social welfare is equivalent to minimizing total waiting time. Accordingly, thebest equilibrium waiting time vector given h and λ can be found using the linear program below(or one can prove that no feasible waiting time vector exists at equilibrium).min w X i ∈ [ k ] w i λ i s.t. ∀ j ∈ [ m ] , i ∈ [ k ] , v h ( j ) j − w h ( j ) ≥ v ij − w i . We then choose h such that the corresponding equilibrium assignment ( w, h, λ ) maximizes socialwelfare.Given the above observation, we are interested in replacing the m k part with a better bound.As we shall show, if the government is willing to violate its budget constraint by an arbitrarilysmall fraction, then the problem can be solved much more efficiently. Definition 2.
Let ǫ be a positive constant. An assignment ( w, h, λ ) is an equilibrium assignmentwith ǫ -deficit if it is an equilibrium assignment with the feasibility condition replaced by the followingcondition: P i ∈ [ k ] λ i c i ≤ (1 + ǫ ) B . We shall construct an algorithm that, in time O (log k ǫ m · (1 + ǫ ) m ), finds an equilibriumassignment with ǫ -deficit whose social welfare is at least SW OEA , the social welfare of the optimalequilibrium assignments with budget B . To do so, we first establish a strong connection betweenthe Provision-after-Wait problem and the well-studied problem of unit-demand auctions (see,e.g., [7, 1, 3, 8]). 10 .1 A connection between the Provision-After-Wait problem and unit-demandauctions
A unit-demand auction is specified by n goods (perhaps including identical ones), m buyers, andthe values v ij of each buyer j ∈ [ m ] for each good i ∈ [ n ]. The goal is to find an equilibriumallocation and prices, where each buyer gets the good that maximizes his utility given the prices.If we consider the patients in the Provision-after-Wait problem as buyers who want to buyhospital services using waiting times, our setting looks a lot like a unit-demand auction. Exceptone thing: in our setting the set of goods for sale is unknown. It is natural to consider the k hospitals as k goods, but each one of them has to have certain amount of identical copies, as eachhospital may serve more than one patients. One cannot simply model the hospitals as k goods with m copies each, as then the resulted auction will give each patient his favorite hospital with zerowaiting time, and the budget constraint may be broken.Notice that, if we were given the quota vector λ in the optimal equilibrium solution of the Provision-after-Wait problem, then we can consider each hospital H i as λ i copies of identicalgoods, and we have a well defined unit-demand auction. Every equilibrium solution to this auctionleads to an assignment function h and a waiting time vector w , such that ( w, h, λ ) is an equilibriumassignment to the original Provision-after-Wait problem. In particular, the budget constraintis satisfied automatically, since we started with a quota vector that meets the budget constraint.In general, for any quota vector λ such that P i λ i ≥ m , the problem of finding equilibriumassignments with respect to λ reduces to finding equilibrium prices and allocations in unit-demandauctions where each hospital H i corresponds to λ i identical goods. If λ meets the budget constraint,namely, P i c i λ i ≤ B , then the resulting equilibrium assignment meets the budget constraint.It is well known that a unit-demand auction always has equilibrium prices and allocations,which can be found by the Hungarian method [18]. The only caution is that, for a hospital to havea well-defined waiting time, the prices of its corresponding goods in the unit-demand auction mustbe all the same. Fortunately, as will become clear in Section 4.2, at equilibrium identical goodsmust always have the same price, although this is not explicitly required.Therefore for each quota vector λ , whether it meets the budget constraint or not, there existsan equilibrium assignment with respect to λ . Following the result of [1], the optimal equilibriumassignment with respect to λ can be computed efficiently, and this will lead to our algorithm forapproximating the optimal equilibrium solution of the Provision-after-Wait problem. Our algorithm uses that of [1] for unit-demand auctions as a black box, therefore we first recalltheir result (while using our notation to help establish the connection with our results).
Definition 3. A unit-demand auction , or simply an auction in this paper, is a triple ( g, m, v ) ,where the set of goods is { , , . . . , g } , the set of bidders is { , , . . . , m } , and v is the valuationmatrix , that is, a g × m matrix of non-negative integers. Each v ij denotes the valuation of bidder j for good i .Given an auction ( g, m, v ) , a matching is a triple ( u, p, µ ) , where u = ( u , . . . , u m ) ∈ ( Z + ) m is the utility vector , p = ( p , . . . , p g ) ∈ ( Z + ) g is the price vector , and µ ⊆ [ g ] × [ m ] is a set ofgood-bidder pairs such that no bidder and no good occur in more than one pair. Bidders and goodsthat do not appear in any pair in µ are unmatched . Although equilibrium assignments can be efficiently computed given λ , the problem of deciding the “correct” λ makes the Provision-after-Wait problem hard, even in very special cases, as shown in Section 3. efinition 4. Given an auction ( g, m, v ) , a matching ( u, p, µ ) is weakly feasible if for each ( i, j ) ∈ µ we have u j = v ij − p i , and for each unmatched bidder j we have u j = 0 .A matching ( u, p, µ ) is feasible if it is weakly feasible and for each unmatched good i we have p i = 0 .A matching ( u, p, µ ) is stable if for each ( i, j ) ∈ [ g ] × [ m ] we have u j ≥ v ij − p i .A matching ( u ∗ , p ∗ , µ ∗ ) is bidder-optimal if: (1) it is stable and feasible, and (2) for everymatching ( u, p, µ ) that is stable and weakly feasible, and for every bidder j , we have u ∗ j ≥ u j . In [1] the authors construct an algorithm,
StableMatch , which, given an auction ( g, m, v ),outputs a bidder-optimal matching ( u ∗ , p ∗ , µ ∗ ) in time O ( mg ).Notice that the original definitions in [1] have for each good-bidder pair a reserve price and amaximum price. In our model we do not need them, so the definitions above are more succinctthan the original ones. In fact, as pointed out by [1], with maximum prices, there may be nobidder-optimal matching. But without them such a matching always exists, as shown by [7].Notice also that [1] does not distinguish between weak feasibility and feasibility. But it is easyto see that their algorithm and its analysis still apply under our definitions. We shall use these twonotions when analyzing our algorithm.Next we establish two properties for the matching ( u ∗ , p ∗ , µ ∗ ) output by StableMatch . • Property 1. If g ≥ m , then without loss of generality we can assume that ( u ∗ , p ∗ , µ ∗ ) has nounmatched bidder.Indeed, if there exists an unmatched bidder j , then there must exist an unmatched good i (since g ≥ m ). Since ( u ∗ , p ∗ , µ ∗ ) is bidder-optimal, we have u ∗ j = 0, p ∗ i = 0, and u ∗ j ≥ v ij − p ∗ i .Thus we have v ij = 0, and the matching ( u ∗ , p ∗ , µ ∗ ∪ { ( i, j ) } ) is another bidder-optimalmatching. • Property 2.
If two goods i, i ′ are identical, namely, v ij = v i ′ j for each bidder j , then p ∗ i = p ∗ i ′ .Indeed, if both goods are unmatched then p ∗ i = p ∗ i ′ = 0. Otherwise, say ( i, j ) ∈ µ ∗ . Bydefinition, u ∗ j = v ij − p ∗ i ≥ v i ′ j − p ∗ i ′ . As v ij = v i ′ j , we have p ∗ i ≤ p ∗ i ′ . If i ′ is unmatched then p ∗ i ′ = 0, implying p ∗ i = 0. If ( i ′ , j ′ ) ∈ µ ∗ then similarly we have p ∗ i ′ ≤ p ∗ i , and thus p ∗ i = p ∗ i ′ again. Now we are ready to construct our algorithm for approximating optimal equilibrium assignments.The algorithm takes as input the number of patients m , the number of hospitals k , the hospitals’costs c , . . . , c k , the patients’ valuations v ij ’s for the hospitals, the budget B , and a small constant ǫ >
0. Letting ( w, h, λ ) be an optimal equilibrium assignment, the algorithm works by guessing λ , constructing a multi-unit auction based on the guessed vector, computing the bidder-optimalmatching using StableMatch , and extracting the waiting time vector and the assignment functionfrom the matching.More precisely, let L , ⌈ log ǫ m ⌉ , C ,
0, and C ℓ , ⌊ (1 + ǫ ) ℓ ⌋ for each ℓ = 1 , . . . , L . Thealgorithm examines all the vectors ˆ λ = (ˆ λ , . . . , ˆ λ k ) ∈ { C , C , . . . , C L } k one by one, say lexico-graphically.If P i ∈ [ k ] ˆ λ i [ m, (1 + ǫ ) m ] or if P i ∈ [ k ] ˆ λ i c i > (1 + ǫ ) B , the algorithm disregards this vectorand moves to the next. Otherwise it constructs an auction ( g, m, ˆ v ) as follows. The set of patientscorresponds to the set of bidders; each hospital H i corresponds to ˆ λ i copies of identical goods H i , . . . , H i ˆ λ i , thus g = P i ∈ [ k ] ˆ λ i ; the valuation matrix ˆ v has rows indexed by { ir : i ∈ [ k ] , r ∈ [ˆ λ i ] } ,columns indexed by [ m ], and for each j ∈ [ m ], i ∈ [ k ], and r ∈ [ˆ λ i ], ˆ v ir,j = v ij .12he algorithm then runs StableMatch with input ( g, m, ˆ v ) to generate the bidder-optimalmatching ( u ∗ , p ∗ , µ ∗ ), and extracts the waiting time vector ˆ w and the assignment function ˆ h asfollows. For each hospital H i , let ˆ w i = p ∗ i . For each patient P j , let H ir be the unique good towhich P j is matched (by Property 1 in Section 4.2 such a good always exists) according to µ ∗ , andlet ˆ h ( j ) = i . The triple ( ˆ w, ˆ h, ˆ λ ) may not be an assignment as P i ∈ [ k ] ˆ λ i may be larger than m , butthere is a unique quota vector ˆ λ ′ such that ( ˆ w, ˆ h, ˆ λ ′ ) is an assignment.The algorithm computes the social welfare of the assignment ( ˆ w, ˆ h, ˆ λ ′ ) for each ˆ λ that is notdisregarded, and output the assignment ( w ∗ , h ∗ , λ ∗ ) with the maximum social welfare.We prove the following theorem. Theorem 2.
Our algorithm runs in time O (log k ǫ m · m ) , and outputs an equilibrium assignmentwith ǫ -deficit, ( w ∗ , h ∗ , λ ∗ ) , such that SW ( w ∗ , h ∗ , λ ∗ ) ≥ SW OEA .Proof.
The running time of the algorithm can be immediately seen. Indeed, if a vector ˆ λ is notdisregarded, then it takes O ( mg ) = O ( m ) time to construct the auction as g ∈ [ m, (1 + ǫ ) m ], O ( mg ) = O ( m ) time to run StableMatch , and O ( m ) time to extract the assignment. Ac-cordingly, it takes O ( m ) time to examine a single vector ˆ λ , and there are O (log k ǫ m ) vectors intotal.The remaining part of the theorem follows from the two lemmas below. Lemma 1. ( w ∗ , h ∗ , λ ∗ ) is an equilibrium assignment with ǫ -deficit.Proof. In fact, we show that for each vector ˆ λ that is not disregarded, the extracted assignment( ˆ w, ˆ h, ˆ λ ′ ) is an equilibrium assignment with ǫ -deficit. To see why this is true, first notice that P i ∈ [ k ] ˆ λ i c i ≤ (1 + ǫ ) B by the construction of the algorithm, thus X i ∈ [ k ] ˆ λ ′ i c i ≤ X i ∈ [ k ] ˆ λ i c i ≤ (1 + ǫ ) B. (1)Second, for each j ∈ [ m ], letting H ˆ h ( j ) r be the good matched to P j according to µ ∗ , we have u j ( ˆ w, ˆ h, ˆ λ ′ ) = v ˆ h ( j ) j − ˆ w ˆ h ( j ) = ˆ v ˆ h ( j ) r,j − p ∗ ˆ h ( j )1 = ˆ v ˆ h ( j ) r,j − p ∗ ˆ h ( j ) r = u ∗ j ≥ , (2)where the third equality is because of Property 2 in Section 4.2 (in particular, H ˆ h ( j )1 and H ˆ h ( j ) r are identical goods, and p ∗ ˆ h ( j )1 = p ∗ ˆ h ( j ) r ), and the other equalities/inequality are by definition.Third, since ( u ∗ , p ∗ , µ ∗ ) is a bidder-optimal matching for auction ( g, m, ˆ v ), we have that for each j ∈ [ m ], i ∈ [ k ], and r ∈ [ˆ λ i ], u ∗ j ≥ ˆ v ir,j − p ∗ ir = v ij − p ∗ i = v ij − ˆ w i , and thus u j ( ˆ w, ˆ h, ˆ λ ′ ) = u ∗ j ≥ v ij − ˆ w i . (3)Equations 1, 2, and 3 together imply that every ( ˆ w, ˆ h, ˆ λ ′ ) is an equilibrium assignment with ǫ -deficit, and so is ( w ∗ , h ∗ , λ ∗ ). Lemma 2. SW ( w ∗ , h ∗ , λ ∗ ) ≥ SW OEA . roof. To see why this is true, arbitrarily fix an optimal equilibrium assignment ( w, h, λ ). Noticethat for each hospital H i , there exists a “good guess” ˆ λ i ∈ { C , . . . , C L } such that λ i ≤ ˆ λ i ≤ (1 + ǫ ) λ i . Since λ satisfies P i ∈ [ k ] λ i = m and P i ∈ [ k ] λ i c i ≤ B , the vector ˆ λ = (ˆ λ , . . . , ˆ λ k ) satisfies X i ∈ [ k ] ˆ λ i ∈ [ m, (1 + ǫ ) m ] and X i ∈ [ k ] ˆ λ i c i ≤ (1 + ǫ ) B. Thus it won’t be disregarded by the algorithm. Let ( g, m, ˆ v ) be the auction constructed fromˆ λ , ( u ∗ , p ∗ , µ ∗ ) the output of StableMatch under input ( g, m, ˆ v ), and ( ˆ w, ˆ h, ˆ λ ′ ) the assignmentextracted from ( u ∗ , p ∗ , µ ∗ ). Following the same reasoning as in Equation 2, we have that for each j ∈ [ m ], u j ( ˆ w, ˆ h, ˆ λ ′ ) = u ∗ j . Thus SW ( ˆ w, ˆ h, ˆ λ ′ ) = X j ∈ [ m ] u ∗ j . (4)From ( w, h, λ ), we construct a matching ( u, p, µ ) for the auction ( g, m, ˆ v ) as follows. For eachbidder j , we have u j = v h ( j ) j − w h ( j ) ; for each good H ir with i ∈ [ k ] and r ∈ [ˆ λ i ], we have p ir = w i ;and for each hospital H i , letting j ≤ j ≤ · · · ≤ j λ i be the patients assigned to H i by h , we have µ = { ( j r , ir ) : i ∈ [ k ] , r ∈ [ λ i ] } .It is easy to verify that the so constructed ( u, p, µ ) is stable and weakly feasible, thus by theoptimality of u ∗ we have that for each j ∈ [ m ], u ∗ j ≥ u j . (5)Moreover, for the same reason as Equation 4, we have SW ( w, h, λ ) = X j ∈ [ m ] u j . (6)Equations 4, 5, and 6 together imply SW ( ˆ w, ˆ h, ˆ λ ′ ) ≥ SW ( w, h, λ ) = SW OEA as we want to show.In sum, Theorem 2 holds.
Remark.
By running our algorithm with input budget B/ (1 + ǫ ), we obtain an assignment whosebudget is at most B and whose social welfare is at least the optimal social welfare with budget B/ (1 + ǫ ). However, this social welfare may be much smaller than the optimal social welfare withbudget B . That is why we insist on having a deficit instead of meeting the budget constraintstrictly. Next we study the dynamics between hospitals and patients. As we shall consider continuouschanges of waiting times, below the patients’ valuations and the waiting times can be any non-negative reals, not necessarily integers.We show that in our model, when the patients’ valuationsare in some generic position, the only thing the government needs to enforce is the amount of moneyit is willing to pay to each hospital, which can be equivalently enforced by the quota vector. Giventhe quotas, the optimal waiting times and the optimal social welfare will emerge endogenously fromthe dynamics. 14 .1 The uniqueness of the optimal equilibrium
We start by defining the generic position of the patients and studying the structure of the optimalequilibrium under it. Following [3], we have the following.
Definition 5.
The patients { P , . . . , P m } with valuations ( v ij ) i ∈ [ k ] ,j ∈ [ m ] are independent if, theredo not exist two different subsets S and T of the multiset { v ij : i ∈ [ k ] , j ∈ [ m ] } such that, both S and T contains positive numbers and P v ∈ S v = P v ′ ∈ T v ′ . Notice that the above definition of independent patients is weaker than the typical definition ofgeneric position, which rules out any relevant equality relation among the valuations. Notice alsothat it is easy to perturb the numbers in the proof of Theorem 1 so that the resulted Provision-after-Wait problem is generic. Thus the optimization problem is still NP-hard in the generic case.But our results below apply to any λ , which may be obtained via approximation algorithms orheuristics.Let λ be a quota vector with P i ∈ [ k ] λ i ≥ m . Recall that given λ , the Provision-after-Wait problem reduces to a unit-demand auction. Thus following [25, 7], among all equilibrium waitingtime vectors with respect to λ , there is a unique one that simultaneously minimizes the waitingtime at each hospital and maximizes the utility of each patient. Denoting this minimum waitingtime vector by ¯ w , we prove the following theorem. Theorem 3.
Assuming the patients are independent, there is a unique equilibrium assignment withrespect to λ and ¯ w . Moreover, denoting this equilibrium by ( ¯ w, ¯ h, λ ) , we have that min i ∈ [ k ] ¯ w i = 0 ,and that at this equilibrium every hospital with positive waiting time is saturated, namely, | ¯ h − ( i ) | = λ i whenever ¯ w i > .Proof. Without loss of generality, we assume λ i > i ∈ [ k ]. Consider the demand graph G given ¯ w , that is, a bipartite graph with k nodes on one side for the hospitals and m nodes onthe other side for the patients. For each i ∈ [ k ] and j ∈ [ m ], the edge ( i, j ) is in G if and only if H i maximizes P j ’s utility, namely, v ij − ¯ w i = max i ′ ∈ [ k ] v i ′ j − ¯ w i ′ . By definition, any equilibriumassignment must assigns each patient P j to an adjacent hospital H i . Thus it suffices to show thatwithin each connected component of G there is only one equilibrium assignment. We start byproving the following claim. Claim 1.
There is no cycle in G .Proof. For the sake of contradiction, assume there exists a (necessarily even-length) cycle( i , j , i , j , . . . , i ℓ , j ℓ , i ), where i r ’s are hospitals and j r ’s are patients. By the construction of G ,we have that for each r ∈ [ ℓ ], both H i r and H i r +1 maximize P j r ’s utility, with ℓ + 1 defined to be1. Thus v i r j r − ¯ w i r = v i r +1 j r − ¯ w i r +1 . Summing all ℓ equations together, we have X r ∈ [ ℓ ] ( v i r j r − ¯ w i r ) = X r ∈ [ ℓ ] (cid:0) v i r +1 j r − ¯ w i r +1 (cid:1) , therefore, X r ∈ [ ℓ ] v i r j r − X r ∈ [ ℓ ] ¯ w i r = X r ∈ [ ℓ ] v i r +1 j r − X r ∈ [ ℓ ] ¯ w i r +1 . Notice that we do not require that λ satisfies the budget constraint, and our results apply to such λ s as well. Notice that this is the waiting time vector computed by the
StableMatch algorithm of [1]. P r ∈ [ ℓ ] ¯ w i r = P r ∈ [ ℓ ] ¯ w i r +1 , we have X r ∈ [ ℓ ] v i r j r = X r ∈ [ ℓ ] v i r +1 j r . Accordingly, we have found two different subsets { v i r j r : r ∈ [ ℓ ] } and { v i r +1 j r : r ∈ [ ℓ ] } that sum upto the same value, contradicting the hypothesis that the patients are independent.Following Claim 1, the connected components of G are all trees. Similarly, we have the following: Claim 2.
Each connected component of G contains at most one hospital with waiting time 0.Proof. Again for the sake of contradiction, assume there is a connected component with two differenthospitals H i and H i ′ such that ¯ w i = ¯ w i ′ = 0. Accordingly, there is a path ( i , j , i , j , . . . , i ℓ ) where i r ’s are hospitals and j r ’s are patients, such that i = i and i ℓ = i ′ . Similar to the proof of Claim1, for each r < ℓ , we have v i r j r − ¯ w i r = v i r +1 j r − ¯ w i r +1 . Summing all ℓ − ℓ − X r =1 v i r j r − ℓ − X r =1 ¯ w i r = ℓ − X r =1 v i r +1 j r − ℓ − X r =1 ¯ w i r +1 . As ¯ w i = ¯ w i ℓ = 0, the above equation implies ℓ − X r =1 v i r j r − ℓ − X r =2 ¯ w i r = ℓ − X r =1 v i r +1 j r − ℓ − X r =2 ¯ w i r , and thus ℓ − X r =1 v i r j r = ℓ − X r =1 v i r +1 j r , again contradicting the hypothesis that the patients are independent.Claim 2 and the following claim together imply that each connected component of G has exactlyone hospital with waiting time 0. Claim 3.
Each connected component of G has at least one hospital with waiting time 0.Proof. By contradiction. Assume there is a component C such that ¯ w i > H i in C . Let ǫ = min H i ∈ C ¯ w i . Notice that for each P j not in C , by definition, the best utility that j can get from hospitals in C is strictly less than u maxj , the best utility that j can get from his favorite hospital. Let ǫ = min P j C (cid:20) u maxj − max H i ∈ C ( v ij − ¯ w i ) (cid:21) . We have ǫ > ǫ >
0. Let ǫ = min { ǫ ,ǫ } , w ′ i = ¯ w i − ǫ for each H i ∈ C , and w ′ = ( ¯ w − C , w ′ C ).That is, w ′ is ¯ w with all waiting times of hospitals in C reduced by ǫ . As ǫ < ǫ , w ′ is a validwaiting time vector. 16otice that for any equilibrium assignment ( ¯ w, h, λ ), the assignment ( w ′ , h, λ ) is still an equi-librium. Indeed, when the waiting time vector changes from ¯ w to w ′ , for each patient P j , his utilityat every hospital H i ∈ C increases by ǫ , and his utility at every other hospital remains the same.For P j C , ǫ < ǫ , and thus the best utility j gets from C is still smaller than u maxj , which is j ’sutility at H h ( j ) C . For P j ∈ C , we have H h ( j ) ∈ C as well, and H h ( j ) still maximizes j ’s utilityafter the increase.Accordingly, w ′ is another equilibrium waiting time vector. But w ′ i < ¯ w i for each H i ∈ C and w ′ i = ¯ w i for each H i C , contradicting the hypothesis that ¯ w minimizes the waiting time of eachhospital among all equilibrium waiting time vectors. Therefore Claim 3 holds.Following Claims 1, 2, and 3, each connected component C can be considered as a tree rootedat the unique hospital with waiting time 0, with hospitals and patients alternating along each path.Based on this structure, we show that there is only one way of assigning the patients to the hospitalsat equilibrium in C . To do so, we need the following: Claim 4.
For each hospital H i ∈ C with ¯ w i > , the degree of H i in G is strictly larger than itsquota λ i . The proof is similar to that of Claim 3: if the degree of some H i ∈ C is at most λ i , then wecan find a proper value ǫ ∈ (0 , ¯ w i ) such that the vector w ′ , ( ¯ w − i , ¯ w i − ǫ ) is still an equilibriumwaiting time vector. Indeed, with properly chosen ǫ , for every equilibrium ( ¯ w, h, λ ), let h ′ be theassignment such that h ′ ( j ) = i if P j is adjacent to H i (this is doable because the degree of H i is atmost λ i ), and h ′ ( j ) = h ( j ) otherwise. Then ( w ′ , h ′ , λ ) is another equilibrium. But this contradictsthe hypothesis that ¯ w minimizes the waiting time of each hospital among all equilibrium waitingtime vectors. The formal analysis is omitted.Following Claim 4, we have that the leaves of tree C are all patients. Indeed, if there is a hospitalwith degree 1 and positive waiting time, then its quota is 0, contradicting our original assumptionthat all hospitals have positive quotas. Accordingly, at every equilibrium, every patient at a leafmust be assigned to his preceding hospital, as this is the only one maximizing his utility. Letting H i be a non-root hospital whose descendants are all leaves, we have that the number of descendantsof H i , denoted by d i , is at most λ i , otherwise no equilibrium exists. As ¯ w i >
0, by Claim 4 wehave that the degree of H i is strictly larger than λ i , which implies d i ≥ λ i . Accordingly, H i usesup all its quota to serve its descendants, and the patient P j preceding H i must be assigned to hispreceding hospital.Repeating the above reasoning in a bottom-up way along the tree, we have that there is onlyone way of assigning the patients to hospitals at equilibrium with respect to λ and ¯ w , that is,each patient is assigned to his predecessor in G , and every hospital with positive waiting time issaturated by its descendants. Thus Theorem 3 holds.By definition, the equilibrium ( ¯ w, ¯ h, λ ) maximizes social welfare with respect to λ , thus it isreasonable to assume that this is the equilibrium that the government aims to implement. We now show that given λ , the waiting time vector ¯ w will endogenously emerge from the dynamicsbetween hospitals and patients, and so will ¯ h . We consider a continuous-time dynamics, wherethe patient population arrives continuously and uniformly along time. In such a dynamics, thequota-vector λ represents the service rate of the hospitals that the government is willing to pay17or. Namely, for each hospital H i , the total number of patients paid by the government in any timeinterval ( t , t ) is at most λ i ( t − t ). The set of patients in previous sections, { P , . . . , P m } with valuations ( v ij ) i ∈ [ k ] ,j ∈ [ m ] , now rep-resents the set of types of the arriving patients. That is, although the patient population goes toinfinity, there are only finitely many types of them. Every type has arrival rate
1: by any time t ,the number of patients that have arrived is mt , where t of them are of type P (i.e., with valuation( v j , . . . , v kj )), and another t of them are of type P , etc. We say that the patient population is independent if { P , . . . , P m } is independent. Notice that in general there may be different P j and P j ′ with the same valuation, and the number of patients of a particular type by time t may belarger than t . But when the population is independent, any different P j and P j ′ must have differentvaluations, and indeed represent different types. Below we consider independent population.Let w ( t ) , ( w ( t ) , . . . , w k ( t )) be the non-negative waiting time vector of the hospitals at time t , such that w (0) = (0 , . . . , P j arriving at time t chooses a hospital H i maximizing his utility given w ( t ), and will be served there at time t + w i ( t ). To break tiesconsistently throughout time, we impose a partial ordering over the hospitals, according to theirpositions in the demand graph G with respect to ¯ w . In particular, if H i and H i ′ are in the sameconnected component of G and H i precedes H i ′ , then at any time t and for any patient of type P j whose utility is maximized at both H i and H i ′ given w ( t ), we assume that P j does not choose H i . If H i and H i ′ are in different connected components, then P j can choose one arbitrarily, oreven split the population of this type arbitrarily between H i and H i ′ , as indicated by the definitionbelow. Definition 6.
For any i, j, t , the demand rate of P j for H i at time t , denoted by d ij ( t ) , is a numberin [0 , such that, • P i ∈ [ k ] d ij ( t ) = 1 for all j , • d ij ( t ) > only if H i maximizes P j ’s utility at time t , and there is no other hospital H i ′ preceded by H i in the same connected component of G that does so.The demand rate for H i at time t is d i ( t ) , P j ∈ [ m ] d ij ( t ) . The fractional values of the d ij ’s indicate how the patients of the same type will split betweenall hospitals maximizing their utilities. For example, d ij ( t ) = 1 / P j will choose H i . Notice that we donot completely specify how the patients should make their decisions when there are ties, and yetour results hold no matter how these ties are broken.Because the patients arrive continuously under a constant rate, their effect on the waiting timesat any point of time is infinitesimal, and w ( t ) is continuous. By definition, within an arbitrarilysmall time interval ( t, t + δ ), the number of patients choosing H i is d i ( t ) δ . Since the number ofpatients served by H i in time δ is λ i δ , the waiting time will not change if d i ( t ) = λ i (i.e., if thedemand rate matches the service rate), and will change by d i ( t ) δ − λ i δλ i otherwise, unless w i ( t ) = 0and d i ( t ) < λ i , in which case w i ( t + δ ) will remain 0. That is, w i ( t + δ ) − w i ( t ) = ( (cid:16) d i ( t ) λ i − (cid:17) δ if w i ( t ) > d i ( t ) ≥ λ i , . (7) The budget constraint B now represents the spending rate of the government: the total amount of money thegovernment can afford by time t is Bt . But as already said, our conclusion in this section holds even when λ doesnot satisfy the budget constraint. Thus we shall not talk about the budget constraint in the remaining part of thissection. So the patients are served in a first-in-first-out queue. i ∈ [ k ] the right derivative of w i ( t ) is d + w i ( t ) dt = ( lim δ → w i ( t + δ ) − w i ( t ) δ = d i ( t ) λ i − w i ( t ) > d i ( t ) ≥ λ i , . (8)Notice that for particular tie-breaking rules, the function d i ( t ) may not be continuous, and thus w i ( t ) may not be differentiable. But we can always define its right derivative as above.We say that w ( t ) is at most ¯ w , written as w ( t ) ≤ ¯ w , if w i ( t ) ≤ ¯ w i for each i ∈ [ k ]. Moreover, wesay that w ( t ) is smaller than ¯ w , written as w ( t ) < ¯ w , if the above inequality holds for some i ∈ [ k ].The following two theorems show that the dynamics will always converge to ¯ w in finite time, andwill never exceed ¯ w before converging. Theorem 4.
When the patient population is independent we have that:(1) w ( t ) ≤ ¯ w for any t ≥ ;(2) if w ( t ) = ¯ w then d + w i ( t ) dt = 0 for any i ∈ [ k ] ; and(3) if w ( t ) < ¯ w then there exists i ∈ [ k ] such that d + w i ( t ) dt > .Proof. To prove Statement (1), it suffices to show that whenever w ( t ) ≤ ¯ w and w i ( t ) = ¯ w i forsome i , we have d i ( t ) ≤ λ i and thus w i ( t ) will not increase. Since | ¯ h − ( i ) | ≤ λ i by the definition ofequilibrium ( ¯ w, ¯ h, λ ), it suffices to show d i ( t ) ≤ | ¯ h − ( i ) | , or equivalently, to show that if ¯ h ( j ) = i then d ij ( t ) = 0 . To do so, arbitrarily fix a type P j such that ¯ h ( j ) = i . If v ij − w i ( t ) < max i ′ v i ′ j − w i ′ ( t ) thencertainly P j does not choose H i given w ( t ), and d ij ( t ) = 0. Assume now v ij − w i ( t ) = max i ′ v i ′ j − w i ′ ( t ) . Notice that v ij − w i ( t ) = v ij − ¯ w i ≤ v ¯ h ( j ) j − ¯ w ¯ h ( j ) ≤ v ¯ h ( j ) j − w ¯ h ( j ) ( t ) ≤ max i ′ v i ′ j − w i ′ ( t ) , where the equality is because w i ( t ) = ¯ w i , the first and the last inequalities are by definition, andthe second is because w ¯ h ( j ) ( t ) ≤ ¯ w ¯ h ( j ) by hypothesis. Thus we have v ij − w i ( t ) = v ij − ¯ w i = v ¯ h ( j ) j − ¯ w ¯ h ( j ) = v ¯ h ( j ) j − w ¯ h ( j ) ( t ) = max i ′ v i ′ j − w i ′ ( t ) . The second equality implies that both H i and H ¯ h ( j ) are adjacent to P j in the demand graph G according to ¯ w , and thus it must be the case that H ¯ h ( j ) precedes P j and P j precedes H i in G . Thelast equality implies that H ¯ h ( j ) also maximizes the utility of P j given w ( t ), and thus P j will notchoose H i according to our tie-breaking rule, namely, d ij ( t ) = 0.Accordingly, d i ( t ) ≤ | ¯ h − ( i ) | ≤ λ i , and Statement (1) holds.Statement (2) simply follows from the fact that, when w ( t ) = ¯ w , the patients choose theirhospitals according to the unique equilibrium ( ¯ w, ¯ h, λ ), and thus d i ( t ) = | ¯ h − ( i ) | = λ i for every i such that ¯ w i >
0, and d i ( t ) = | ¯ h − ( i ) | ≤ λ i for every i such that ¯ w i = 0.19o prove Statement (3), it suffices to show that when w ( t ) < ¯ w , there exists some hospital H i with d i ( t ) > λ i . For the sake of contradiction, assume d i ( t ) ≤ λ i ∀ i . We shall construct a newdemand vector ( d ′ ij ( t )) i ∈ [ k ] ,j ∈ [ m ] such that d ′ ij ( t ) ∈ { , } ∀ i, j, and d ′ i ( t ) , X j d ′ ij ( t ) ≤ λ i ∀ i. To do so, consider the demand graph G ( t ) with respect to w ( t ). For each H i and P j , d ij ( t ) > H i and P j are adjacent in G ( t ). Since the patient population is independent, G ( t ) isa forest with hospitals and patients alternating along each path, as in the proof of Theorem 3.The construction starts from the graph G ( t ), processes and removes its nodes step by step andin a bottom-up fashion, and assigns patients to hospitals in a greedy way. To be more precise, weinitialize the following intermediate variables: d ′ ij ( t ) = 0 ∀ i, j , and λ ′ i = λ i ∀ i . At any time of theconstruction, for each H i , λ ′ i is an integer and denotes H i ’s remaining quota, after some patientshave been assigned to it. It will be invariant that d ′ i ( t ) + λ ′ i = λ i ∀ i, d i ( t ) ≤ λ ′ i ∀ i, and X i ∈ [ k ] d ij ( t ) = 1 ∀ P j in the graph . (9)Notice that Equation (9) trivially holds at the beginning.In each step of the construction, from the remaining graph, we choose a leaf with the longestpath from its root. We distinguish two cases. Case 1.
The chosen leaf is a patient, say P j ∗ .This is the simpler case. Letting the unique adjacent hospital be H i ∗ , we have d ij ∗ ( t ) = 0 ∀ i = i ∗ , and d i ∗ j ∗ ( t ) = 1 ≤ d i ∗ ( t ) ≤ λ ′ i ∗ . Set d ′ i ∗ j ∗ ( t ) = 1, d i ∗ j ∗ ( t ) = 0, and λ ′ i ∗ = λ ′ i ∗ −
1, and remove P j ∗ from the graph. That is, P j ∗ is assigned to H i ∗ and occupies 1 quota there. Notice that the invariance remains. Indeed, d ′ i ∗ ( t ) increases by 1 and λ ′ i ∗ decreases by 1, both d i ∗ ( t ) and λ ′ i ∗ decrease by 1, and everythingelse remains unchanged. Case 2.
The chosen leaf is a hospital, say H i ∗ .This is the more complicated case. Letting the unique adjacent patient be P j ∗ , we have0 ≤ d i ∗ j ∗ ( t ) = d i ∗ ( t ) ≤ λ ′ i ∗ . If λ ′ i ∗ ≥ H i ∗ still has quota for one more patient), then set d ′ i ∗ j ∗ ( t ) = 1, d ij ∗ ( t ) = 0 ∀ i , and λ ′ i ∗ = λ ′ i ∗ −
1. Remove P j ∗ and its children (which are all leaves) from the graph. Thatis, P j ∗ is assigned to H i ∗ , and for any other hospital H i with P j ∗ the only adjacent patient,no patient will be assigned to it any more. Notice that the invariance remains. Indeed, d ′ i ∗ ( t )increases by 1, λ ′ i ∗ decreases by 1, d i ∗ ( t ) = d i ∗ j ∗ ( t ) = 0, λ ′ i ∗ is non-negative, and for any i = i ∗ , d i ( t ) either decreases or remains unchanged. Everything else remains unchanged.If λ ′ i ∗ = 0, then d i ∗ j ∗ ( t ) = d i ∗ ( t ) = 0 by Equation (9). That is, no remaining patient wants H i ∗ . We simply remove H i ∗ from the graph, keeping the invariance.Notice that we finish processing all nodes after at most m + k steps. In the end, all the d ′ ij ( t )’s areeither 0 or 1, and d ′ i ( t ) ≤ λ i ∀ i . Accordingly, the d ′ ij ( t )’s correspond to an equilibrium assignmentwith waiting time w ( t ), contradicting the fact that ¯ w is the minimum equilibrium waiting timevector with respect to λ .Therefore Statement (3) holds. 20etting M SW = P j ∈ [ m ] max i ∈ [ k ] v ij and λ max = max i ∈ [ k ] λ i , we have the following theorem. Theorem 5.
When the patient population is independent, the dynamics converges to ¯ w in time atmost kλ max M SW .Proof.
Similar to the Hungarian method (see, e.g., [8]), we consider the following potential function: P ( t ) , X i ∈ [ k ] λ i w i ( t ) + X j ∈ [ m ] u j ( t ) , where u j ( t ) , max i ∈ [ k ] ( v ij − w i ( t )). Since w i ( t ) is continuous for each i ∈ [ k ], u j ( t ) is continuousfor each j ∈ [ m ], and P ( t ) is continuous as well.By Theorem 3 we have min i ∈ [ k ] ¯ w i = 0. By Theorem 4 we have that before the dynamicsconverges, (0 , . . . , ≤ w ( t ) < ¯ w for any t , and thus min i ∈ [ k ] w i ( t ) = min i ∈ [ k ] ¯ w i = 0. Accordingly, u j ( t ) ≥ P j , and P ( t ) ≥
0. As P (0) = M SW to begin with, it suffices to prove that P ( t )strictly decreases, and the local decreasing rate is at least 1 / ( kλ max ).To do so, notice that P ( t ) = X i λ i w i ( t ) + X j X i d ij ( t )( v ij − w i ( t ))= X i λ i w i ( t ) − X i ( X j d ij ( t )) w i ( t ) + X i,j d ij ( t ) v ij = X i ( λ i − d i ( t )) w i ( t ) + X i,j d ij ( t ) v ij . Thus for any arbitrarily small δ >
0, by definition we have P ( t + δ ) − P ( t )= X i ( λ i − d i ( t + δ )) w i ( t + δ ) − ( λ i − d i ( t )) w i ( t ) + X i,j d ij ( t + δ ) v ij − X i,j d ij ( t ) v ij = X i ( w i ( t + δ ) − w i ( t ))( λ i − d i ( t )) − X i w i ( t + δ ) d i ( t + δ ) + X i w i ( t + δ ) d i ( t )+ X i,j d ij ( t + δ ) v ij − X i,j d ij ( t ) v ij = X i ( w i ( t + δ ) − w i ( t ))( λ i − d i ( t )) + X i,j d ij ( t + δ ) v ij − X i,j d ij ( t + δ ) w i ( t + δ ) − X i,j d ij ( t ) v ij + X i,j d ij ( t ) w i ( t + δ )= X i ( w i ( t + δ ) − w i ( t ))( λ i − d i ( t )) + X i,j ( d ij ( t + δ ) − d ij ( t ))( v ij − w i ( t + δ )) . Again since w ( t ) is continuous, we have lim δ → v ij − w i ( t + δ ) = v ij − w i ( t ). Since the patients onlychoose hospitals that maximize their utilities, for any i, j such that v ij − w i ( t ) < u j ( t ), we have v ij − w i ( t + δ ) < u j ( t ) for arbitrarily small δ , and thus d ij ( t ) = d ij ( t + δ ) = 0. That is, for each P j , X i : v ij − w i ( t )= u j ( t ) d ij ( t ) = X i : v ij − w i ( t )= u j ( t ) d ij ( t + δ ) = 1 . δ → P ( t + δ ) − P ( t ) δ = − X i : w i ( t ) > d i ( t ) ≥ λ i ( d i ( t ) − λ i ) λ i + X j u j ( t ) lim δ → P i : v ij − w i ( t )= u j ( t ) ( d ij ( t + δ ) − d ij ( t )) δ = − X i : w i ( t ) > d i ( t ) ≥ λ i ( d i ( t ) − λ i ) λ i + X j u j ( t ) lim δ → − δ = − X i : w i ( t ) > d i ( t ) ≥ λ i ( d i ( t ) − λ i ) λ i . (10)To upper-bound the last part of Equation (10), consider the set of hospitals B , { i : ¯ w i − w i ( t ) = max i ′ ∈ [ k ] ¯ w i ′ − w i ′ ( t ) } . As w ( t ) < ¯ w before the dynamics converges, there exists i such that ¯ w i − w i ( t ) >
0. Thus for any i with ¯ w i = 0, i / ∈ B . By Theorem 3, | ¯ h − ( i ) | = λ i ∀ i ∈ B. For any patient j with ¯ h ( j ) ∈ B , we have P i ∈ B d ij ( t ) = 1, because when the waiting times changefrom ¯ w to w ( t ) the utilities of j at hospitals in B become strictly more advantageous against hisutilities at hospitals not in B . Thus X j :¯ h ( j ) ∈ B X i ∈ B d ij ( t ) = X j :¯ h ( j ) ∈ B X i ∈ B | ¯ h − ( i ) | = X i ∈ B λ i . Let BP be the set of patients j such that ¯ h ( j ) / ∈ B and j is adjacent to a hospital in B in thedemand graph of ¯ w ( BP for “boundary patients”). Notice that BP = ∅ as B = [ k ]. For any j ∈ BP , we again have P i ∈ B d ij ( t ) = 1, for a similar reason as before —that is, at ¯ w patient j isindifferent between the best hospital for him in B and the best for him not in B , and from ¯ w to w ( t ) the hospitals in B become strictly more advantageous. Accordingly, X i ∈ B d i ( t ) ≥ X j :¯ h ( j ) ∈ B X i ∈ B d ij ( t ) + X j ∈ BP X i ∈ B d ij ( t ) = X i ∈ B λ i + X j ∈ BP ≥ X i ∈ B λ i + 1 . Let B ′ , { i ∈ B | d i ( t ) ≥ λ i } . Note that P i ∈ B \ B ′ λ i ≥ P i ∈ B \ B ′ d i ( t ), and therefore X i ∈ B ′ d i ( t ) ≥ X i ∈ B ′ λ i + 1 . Thus we have by the concavity of the x function and Jensen’s inequality:lim δ → P ( t + δ ) − P ( t ) δ = − X i : w i ( t ) > d i ( t ) ≥ λ i ( d i ( t ) − λ i ) λ i ≤ − X i ∈ B ′ ( d i ( t ) − λ i ) λ max = − | B ′ | λ max · | B ′ | · X i ∈ B ′ ( d i ( t ) − λ i ) ≤ − | B ′ | λ max · (cid:18) P i ∈ B ′ ( d i ( t ) − λ i ) | B ′ | (cid:19) ≤ − | B ′ | λ max · (cid:18) | B ′ | (cid:19) ≤ − kλ max , (11)22or any time t before the dynamics converges.Letting T = 2 kλ max M SW and assuming that the dynamics does not converge before time T ,we now show that P ( T ) = 0 and thus the dynamics must converge at time T . For any t < T , byInequality (11) and our hypothesis, there exists δ ( t ) such that for all δ ∈ (0 , δ ( t )), P ( t + δ ) − P ( t ) ≤− δ/ (2 kλ max ). Assume P ( T ) >
0, and let t ∗ , sup { t : t ≤ T, P ( t ) − P (0) ≤ − t/ (2 kλ max ) } . As P ( t ) is continuous, we have P ( t ∗ ) − P (0) ≤ − t ∗ / (2 kλ max ). Thus t ∗ < T , as P ( T ) − P (0) > − M SW = − T / (2 kλ max ). Accordingly, there exists δ ∈ (0 , T − t ∗ ) such that P ( t ∗ + δ ) − P ( t ∗ ) ≤− δ/ (2 kλ max ). Letting t ′ = t ∗ + δ , we have t ∗ < t ′ ≤ T and P ( t ′ ) − P (0) = P ( t ∗ + δ ) − P ( t ∗ ) + P ( t ∗ ) − P (0) ≤ − ( t ∗ + δ ) / (2 kλ max ) = − t ′ / (2 kλ max ) , contradicting the definition of t ∗ .Therefore P ( T ) = 0, and the dynamics converges to ¯ w in time at most T , as desired. Remark 4.
Although the potential function used in the above proof is similar to that used in theHungarian method for unit-demand auctions, the analysis is different. For example, the potentialfunction in the latter measure the total price paid at each time step, while ours measures the“budgeted” total waiting time P i λ i w i ( t ) , which can be very different from the total waiting time.Moreover, in the latter the prices of the goods for sale never go down, making the analysis mucheasier. While in our dynamics the waiting times may go up and down, depending on the demands. Although waiting time is widely used to ration demand in economic settings, it may burn a lotof social welfare, since the time waited is not beneficial to anybody. Therefore in this section, westudy different allocation schemes in healthcare and give evidence that the government can avoidthe welfare-burning effect of waiting times by limiting the choices available to the patients. Inparticular, we show that the randomized assignment is actually optimal in terms of social welfarein many cases.Following our discussion in Section 1, we consider the case of two hospitals, a “good” one H and a “bad” one H , with costs c > c . As already said, whoever prefers H can be directlyassigned there and we do not consider them in our setting any more. The patients preferring H are indexed by the interval [0 , x is associated with a value v ( x ), indicating howlong he is willing to wait at H to be treated there instead of H . We assume that the patientshave been renamed and normalized, so that v ( x ) is non-decreasing and v (0) = 0. Since the numberof patients is infinite, we talk about the cost density c i ( x ) of each hospital, rather than the cost forserving a single patient. Without loss of generality, c ( x ) ≡ c ( x ) ≡
0. The government hasbudget B ∈ (0 , B fraction of the patients can be served at H . Thegovernment’s goal is to maximize the expected social welfare subject to the requirement that thebudget constraint is satisfied in expectation.In the randomized assignment, the government assigns each patient to H with probability p and waiting time 0. The budget constraint gives Z pc ( x ) dx = p = B, and the corresponding social welfare, denoted by SW r , is SW r = Z pv ( x ) dx = B Z v ( x ) dx. (12)23elow we compare this social welfare with that of lotteries. Definition 7. A contract is a pair ( p, w ) , where p ∈ [0 , is the probability of assigning a patientto H , and w ≥ is the waiting time for that patient at H .A lottery consists of a set of contracts, denoted by the domain D ⊆ [0 , of the probabilities,and the waiting time function w ( p ) defined over D . Given a contract C = ( p, w ) for patient x , the expected utility of x is u ( x, C ) = p · ( v ( x ) − w ) . Given a lottery L = ( D, w ( p )), each patient x chooses the contract C ( x ) = ( p ( x ) , w ( p ( x ))) maxi-mizing his expected utility. Namely, for each p ∈ D , u ( x, C ( x )) ≥ u ( x, ( p, w ( p ))) . If there are more than one values of p that maximize the expected utility of x , we assume that p ( x )is the smallest one, so that the cost of serving patient x is minimized. Notice that p ( x ) depends on x only indirectly, via the function v ( x ): indeed, p ( x ) = p ( x ′ ) whenever v ( x ) = v ( x ′ ). Thus we canwrite p ( x ) as p ( v ( x )).As an example, the randomized assignment is a lottery with D = { B } and w ( B ) = 0. Asanother example, any equilibrium assignment is also a lottery, with D = [0 ,
1] and w ( p ) alwaysequal to the waiting time of H specified by the equilibrium. Indeed, for every patient x , thecontract maximizing his expected utility is to go to the hospital assigned by the equilibrium withprobability 1.Without loss of generality, we assume that D is a subinterval of [0 , a, b ]. Indeed,if a patient can choose between ( p , w ( p )) and ( p , w ( p )) according to the lottery, then by usinga “mixed strategy” he can choose to be assigned to H with any probability p = αp + (1 − α ) p with α ∈ [0 , αp w ( p ) + (1 − α ) p w ( p ).Also without loss of generality, we assume that the patients’ expected waiting time function p · w ( p ) is convex, and thus differentiable almost everywhere. Indeed, for any contracts C =( p , w ( p )), C = ( p , w ( p )), and C = ( p, w ( p )) with p = αp + (1 − α ) p for some α ∈ [0 , p · w ( p ) > αp w ( p ) + (1 − α ) p w ( p ), then a patient is always better off by mixing between C and C instead of choosing C . Thus we may simply assume that p · w ( p ) ≤ αp w ( p ) + (1 − α ) p w ( p ). The social welfare and the budget constraint are naturally defined for lotteries, as follows.
Definition 8.
Given a lottery L = ([ a, b ] , w ( p )) and the contracts ( p ( x ) , w ( p ( x ))) chosen by thepatients x ∈ [0 , , letting u ( x ) , u ( x, ( p ( x ) , w ( p ( x ))) , the social welfare of L , denoted by SW L , is SW L = Z u ( x ) dx. Lottery L is feasible if the budget constraint is satisfied, namely, R p ( x ) dx = B . Notice that we require a feasible lottery to use up all the budget. This is again without anyloss of generality, since our theorem below implies that any lottery with cost B ′ < B is beaten bythe randomized assignment with budget B ′ , and thus by the one with budget B .We assume that the expected waiting time function pw ( p ) is piece-wise twice differentiable in p .Notice that, although assuming twice differentiability of pw ( p ) over the whole domain is too much, In general D can be a proper subset of [0 , ,
1] for thepatients to choose from. Notice that w ( p ) itself may not be convex. w ( p )’sfor different intervals of p , but inside each interval it uses a smooth w ( p ). Both the randomizedassignment and equilibrium assignments trivially satisfy this assumption.The following theorem shows that, when the distribution of the patients’ valuations accumulatestoward the higher-value side, the randomized assignment is optimal compared with any lottery.Since equilibrium assignments are special cases of lotteries, the randomized assignment is optimalcompared with them as well. Theorem 6.
For any concave valuation function v ( x ) and any feasible lottery L = ([ a, b ] , w ( p )) ,we have SW r ≥ SW L .Proof. As the choice of p ( x ) maximizes the utility of x , for any ∆ > x prefers contract C ( x ) = ( p ( x ) , w ( p ( x ))) to contract C ( x + ∆) = ( p ( x + ∆) , w ( p ( x + ∆))), and patient x + ∆ prefers C ( x + ∆) to C . That is, u ( x ) = p ( x )[ v ( x ) − w ( p ( x ))] ≥ p ( x + ∆)[ v ( x ) − w ( p ( x + ∆))] , and u ( x + ∆) = p ( x + ∆)[ v ( x + ∆) − w ( p ( x + ∆))] ≥ p ( x )[ v ( x + ∆) − w ( p ( x ))] . Accordingly, v ( x ) · ∆ p ( x ) ≤ ∆( p ( x ) · w ( p ( x ))) , and v ( x + ∆) · ∆ p ( x ) ≥ ∆( p ( x ) · w ( p ( x ))) . (13)As pw ( p ) is piece-wise twice differentiable, all the differential equations and statements madein this paragraph hold piece-wisely, and we shall not mention the piece-wiseness again and again.To begin with, letting ∆ → x omitted for conciseness) v = d ( pw ( p )) dp , (14)where the function on the right-hand side is well defined and differentiable in p . As p ( v ) is theinverse of Equation 14, it is differentiable in v . As v ( x ) is concave, it is differentiable in x almosteverywhere. Thus p ( x ) = p ( v ( x )) is differentiable in x . Accordingly, we have du ( x ) = dp · ( v − w ) + p · ( dv − dw ) = p · dv + v · dp − ( w · dp + p · dw )= p · dv + v · dp − d ( p · w ) = p · dv + v · dp − v · dp = p · dv. (15)(Notice that p ( v ) and p ( x ) may not be continuous functions, but we only need them to be “nice”piece-wisely.)Now putting all the pieces together and integrating both sides of Equation 15 over the wholedomain, we have u ( x ) = Z v ( x )0 p (ˆ v ) d ˆ v. (16)As v ( x ) is non-decreasing and concave, we have that v ′ ( x ) ≥ v ′ ( x ) is non-increasing.If there exists x < v ′ ( x ) = 0, then let x be the smallest number with v ′ ( x ) = 0;otherwise (i.e., v ( x ) is strictly increasing) let x = 1. We have that v ( x ) is strictly increasing on[0 , x ] and constant on [ x , v = v ( x ). Following Equation 16 the social welfare of lottery25 is SW L = Z u ( x ) dx = Z Z v ( x )0 p (ˆ v ) d ˆ vdx = Z x Z v ( x )0 p (ˆ v ) d ˆ vdx + Z x Z v p (ˆ v ) d ˆ vdx = Z v p (ˆ v ) Z x v − (ˆ v ) dx ! d ˆ v + Z v (cid:18) p (ˆ v ) Z x dx (cid:19) d ˆ v = Z v p (ˆ v ) · ( x − v − (ˆ v )) d ˆ v + Z v p (ˆ v ) · (1 − x ) d ˆ v = Z x p ( x )( x − x ) v ′ ( x ) dx + Z x p ( x )(1 − x ) v ′ ( x ) dx = Z x p ( x )(1 − x ) v ′ ( x ) dx. Similarly, the social welfare of the randomized assignment can be written as SW r = Z Bv ( x ) dx = Z Z v ( x )0 Bdvdx = Z x Z v ( x )0 Bdvdx + Z x Z v Bdvdx = Z v Z x v − (ˆ v ) Bdxd ˆ v + Z v Z x Bdxd ˆ v = Z v B ( x − v − (ˆ v )) d ˆ v + Z v B (1 − x ) d ˆ v = Z x B ( x − x ) v ′ ( x ) dx + Z x B (1 − x ) v ′ ( x ) dx = Z x B (1 − x ) v ′ ( x ) dx. To prove SW r − SW L ≥
0, below we first show that p ( x ) is non-decreasing. To do so, again noticethat p ( x ) maximizes the expected utility of x . Thus for any two patients x < x , we have u ( x ) = p ( x )( v ( x ) − w ( p ( x ))) ≥ p ( x )( v ( x ) − w ( p ( x )))and u ( x ) = p ( x )( v ( x ) − w ( p ( x ))) ≥ p ( x )( v ( x ) − w ( p ( x ))) . Thus p ( x )( v ( x ) − v ( x )) ≥ p ( x )( v ( x ) − v ( x )). If v ( x ) = v ( x ) then p ( x ) = p ( x ) (as wealready said, p ( x ) only depends on v ( x )), otherwise p ( x ) ≥ p ( x ). That is, the function p ( x ) isnon-decreasing.As L is feasible, we have R p ( x ) dx = B . Since v ( x ) is constant on [ x , p ( x ). Therefore p ( x ) ≥ B . Accordingly, there exists x B ∈ [0 , x ] such that p ( x ) ≤ B for all x < x B , and p ( x ) ≥ B for all x ≥ x B . Thus we have SW r − SW L = Z x ( B − p ( x ))(1 − x ) v ′ ( x ) dx = Z x B ( B − p ( x ))(1 − x ) v ′ ( x ) dx + Z x x B ( B − p ( x ))(1 − x ) v ′ ( x ) dx. Notice that the value of p ( x B ) does not affect the value of the integration, thus without loss ofgenerality we assume p ( x B ) = B .Again because v ′ ( x ) is non-negative and non-increasing, for any x ≤ x B , we have (1 − x ) v ′ ( x ) ≥ (1 − x B ) v ′ ( x B ) ≥
0. Because B − p ( x ) ≥ x ≤ x B , we have( B − p ( x ))(1 − x ) v ′ ( x ) ≥ ( B − p ( x ))(1 − x B ) v ′ ( x B ) . Similarly, for any x ≥ x B , we have 0 ≤ (1 − x ) v ′ ( x ) ≤ (1 − x B ) v ′ ( x B ) and B − p ( x ) ≤
0, whichagain implies ( B − p ( x ))(1 − x ) v ′ ( x ) ≥ ( B − p ( x ))(1 − x B ) v ′ ( x B ) . SW r − SW L ≥ Z x B ( B − p ( x ))(1 − x B ) v ′ ( x B ) dx + Z x x B ( B − p ( x ))(1 − x B ) v ′ ( x B ) dx = (1 − x B ) v ′ ( x B ) Z x ( B − p ( x )) dx. Following the budget constraint we have Z p ( x ) dx = Z x p ( x ) dx + p ( x )(1 − x ) = B = Z x Bdx + B (1 − x ) , and thus Z x ( B − p ( x )) dx = ( p ( x ) − B )(1 − x ) . Therefore SW r − SW L ≥ (1 − x B ) v ′ ( x B )( p ( x ) − B )(1 − x ) ≥ , where the second inequality is because x B ≤ v ′ ( x B ) ≥ p ( x ) ≥ B , and x ≤ Remark 5.
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