1 Introduction
Volunteers in the U.S. provide around billion hours of free labor annually. However, roughly of volunteers become disengaged the following year, representing a loss of approximately billion in economic value as well as a significant challenge for the sustainability of organizations relying on volunteerism (National Service 2015, Independent Sector 2018). Lack of retention partially stems from overutilization as well as the mismatch between a volunteer’s preferences and the opportunities presented to her (Locke et al. 2003, Brudney and Meijs 2009). The emergence of online volunteer crowdsourcing platforms presents a unique opportunity to design datadriven volunteer management tools that cater to volunteers’ heterogeneous preferences. In the present work, we move toward this goal by taking an algorithmic approach to designing nudging mechanisms commonly used to encourage volunteers to perform tasks. This work is motivated by our collaboration with a nonprofit platform, called Food Rescue U.S. (FRUS), that recovers food from local businesses and donates it to nonprofit agencies by crowdsourcing the transportation to volunteers. In the following, we provide background on FRUS and highlight the challenge it faces when making volunteer nudging decisions. Further, we offer insights into volunteer behavior by analyzing FRUS data from different locations. Then, we list a summary of our contributions. FRUS: A Crowdsourcing Platform for Food Recovery: FRUS is a leading online platform that simultaneously addresses the societal problems of food waste and hunger. Over 60 million tons of food go to waste in the U.S. each year, while 37 million people—including 6 million children—live in foodinsecure households. This mismatch is driven in part by the cost of lastmile transportation required to recover perishable donated food from local restaurants and grocery stores. FRUS has empowered donors by connecting them to local agencies and enabling free delivery through its dedicated volunteer base. Currently, it operates in tens of locations across different states, and so far it has recovered over 50 million pounds of food. On FRUS, a volunteering task—which is referred to as a rescue—involves transporting a prearranged, perishable food donation from a donor to a local agency. Scheduled donations are often recurring and they are posted on the FRUS app in advance. While around of rescues are claimed organically by volunteers before the day of the rescue, around remain unclaimed on the last day.^{1}^{1}1Here, by organic, we mean volunteers sign up for those rescues without the platform’s involvement. In that case, to encourage volunteers to claim the rescue, FRUS notifies a subset of volunteers with the hope that at least one of them responds positively. However, based on our interviews with the platform’s local managers, FRUS faces a challenge when deciding whom to notify: on the one hand, it aims to minimize the probability of a missed rescue—which is achievable by notifying more volunteers.^{2}^{2}2According to our analysis of FRUS data, a missed rescue increases the probability of donor dropout by a factor of approximately 2.5. On the other hand, it wants to avoid excessive notifications because that may reduce volunteer engagement.^{3}^{3}3We remark that FRUS’s current practice in many locations is to notify a volunteer at most once a week. Further, we note that FRUS is hesitant to demand prompt responses from volunteers, which renders the option of sequentially notifying volunteers impractical. Understanding volunteer behavior can help resolve the aforementioned tradeoff: if volunteers have preferences for certain rescues, then FRUS should mainly notify them for those tasks. Our analysis of two years of data indeed indicates that volunteer preferences are fairly consistent. To highlight this, in Figure 1 we visualize the first three principal components for characteristics of rescues completed by the most active volunteers in two FRUS locations. Each color represents a different volunteer and the size of each circle is proportional to the frequency with which the volunteer completes a rescue of that type. For instance, more than 90% of the rescues completed by the red volunteer in Location (a), as shown in Figure 0(a), are clustered within a cube whose volume is less than one tenth of the PCA component range. As evident from these plots, volunteers tend to claim rescues that have similar characteristics, reflecting their geographical and time preferences.^{4}^{4}4Characteristics or features of a rescue includes its origindestination location, day, time, etc. Our interviews and empirical findings raise a key question that motivates our work: facing such volunteer behavior, how should a volunteerbased online platform, such as FRUS, design an effective notification system for timesensitive tasks?
Summary of Contributions: Motivated by our collaboration with FRUS, we (1) introduce the online volunteer notification problem which captures key features of volunteer labor consistent with the literature, (2) develop two online randomized policies that achieve constantfactor guarantees for the online volunteer notification problem, (3) establish upper bounds on the performance of any online policy, and (4) demonstrate the effectiveness of our policies by testing them on FRUS’s data from various locations across the U.S. Modeling the Platform’s Notification Problem: We introduce the online volunteer notification problem to model a platform’s notification decisions when utilizing volunteers to complete timesensitive tasks. There are three main considerations that the platform should take into account: (1) volunteers’ response to a notification is uncertain, (2) the platform cannot expect volunteers to respond promptly, and (3) if notified excessively, volunteers may suffer from notification fatigue. To include all these considerations in our model, we assume that when each task arrives, the platform simultaneously notifies a subset of volunteers in the hope that at least one responds positively. To model a volunteer’s adverse reaction toward excessive notifications, we assume that a volunteer can be in one of two possible states: active or inactive. In the former state, the volunteer pays attention to the platform’s notifications and responds positively with her taskspecific match probability, whereas in the latter state she ignores all notifications. Upon notification, an active volunteer will transition to the inactive state for a random interactivity period. Because these platforms usually require the recurring completion of similar tasks, they can use historical data to predict their future lastminute needs. For instance, FRUS usually receives donations from the same source on a weekly basis. We model this by assuming that tasks belong to a given set of types and they arrive according to a (timevarying) distribution. The platform makes online decisions aiming to maximize the number of completed tasks knowing the arrival rates, match probabilities, and the interactivity time distribution, but without observing the state of each volunteer. Developing Online Policies:
We develop two randomized policies that are based on ex ante fractional solutions that can be computed in polynomial time. In order to assess the performance of our policies, we use a linear program benchmark whose optimal value serves as an upper bound on the value of a clairvoyant solution which knows the sequence of arrivals a priori as well as the state of volunteers at each time (see Program (LP), Proposition
3 and Definition 3). We remark that the platform’s objective—maximizing the number of completed tasks—jointly depends on the response of all volunteers and exhibits diminishing returns. For example, if the platform notifies two active volunteers and about a task , then the probability of completion would be where and are the match probabilities of the pairs and, respectively. This objective function presents two challenges: (1) an ex ante solution based on upper bounding such an objective function by a piecewise linear one can be ineffective in practice, and (2) jointly analyzing volunteers’ contribution for an online policy while keeping track of the joint distribution of their states (active or inactive) is prohibitively difficult. We address the former challenge by computing ex ante solutions that “better” approximate the true objective function as opposed to only relying on the LP solution (see Programs (
AA) and (SQ) and Proposition 5). We overcome the latter one by assuming an artificial priority among volunteers which allows us to decouple their contributions (see Definition 11 and Lemma 12). Attempting to follow the fractional ex ante solution can result in poor performance since volunteers can become inactive at inopportune times (see Appendix 11.2). Therefore, in the design of our policies, we modify the ex ante solution to account for inactivity while guaranteeing a constantfactor competitive ratio. Our first policy, the ScaledDown Notification (SDN) Policy, relies on a prori computing the probability that a volunteer is active when following this policy. Equipped with these probabilities, the SDN policy notifies each volunteer such that the joint probability that a volunteer is active and notified is proportional to the ex ante solution (see Algorithm 1 and the preceding discussion). On the other hand, our second policy, the Sparse Notification (SN) Policy, relies on solving a sequence of Dynamic Programs (DPs)—one for each volunteer—to resolve the tradeoff between notifying a volunteer now and saving her for future tasks. We solve the DPs in order of volunteers’ artificial priorities, and each subsequent DP is formulated based on the previous solutions (see Algorithm 2 and the preceding discussion). Our policies are parameterized by the minimum discrete hazard rate (MDHR) of the interactivity time distribution, which serves as a sufficient condition for the level of “activeness” of volunteers (see Definition 5 and the following discussion). We analyze the competitive ratios of both policies as functions of the MDHR. Interestingly, both policies achieve the same competitive ratio (see Theorems 1 and 2). However, the SN policy demonstrates significantly better performance in practice (as shown and discussed in Section 6). The analysis of both policies relies on decomposing the problem into individual contributions based on our (artificial) priority scheme. Further, the analysis of SDN relies on proving that the probability of being active can be computed in advance and in polynomial time (see Section 4.2 and Appendix 9.3). The analysis of SN crucially uses the dualfitting framework of Alaei et al. (2012) and it relies on formulating a linear program along with its dual to place a lower bound on the optimal value of each volunteer’s DP (see Section 4.3 and Appendix 9.7). Upper Bound on Online Policies: In order to gain insight into the limitation of online policies when compared to our benchmark, we develop an upper bound on the achievable competitive ratio of any online policy. Like our policies, the upper bound is parameterized by the MDHR (see Theorem 5). As a consequence, the gap between the achievable upper bound and our lower bound (attained through our policies) depends on the MDHR (see Figure 2). When it is small but positive, the gap is fairly small; however, the gap grows as the MDHR increases. Our upper bound relies on analyzing two instances, one of which provides a relatively tight upper bound when the MDHR is small. Testing on FRUS Data:In order to illustrate the effectiveness of our modeling approach and our policies in practice, we evaluate the performance of our policies by testing them on FRUS’s data from different locations. In Section 6, we describe how we estimate model primitives and construct problem instances. Then we numerically show the superior perform of our policies when compared to strategies that resemble the current practice at different locations. The rest of the paper is organized as follows. In Section
2, we review the related literature. In Section 3, we formally introduce the online volunteer notification problem as well as the benchmark and the measure of competitive ratio. Section 4 is the main algorithmic section of the paper and is devoted to describing and analyzing our two online policies. In Section 5, we present our upper bound on the achievable competitive ratio of any online policy. In Section 6, we revisit the FRUS application and demonstrate the effectiveness of our policies by testing them on the platform’s data from various locations. Section 7 concludes the paper. For the sake of brevity, we only include proof ideas in the main text. A detailed proof of each statement is provided in the referenced appendix.2 Related Work
Our work relates to and contributes to several streams of literature. Volunteer Operations and Staffing: Due to the differences between volunteer and traditional labor as highlighted in Sampson (2006), managing a volunteer workforce provides unique challenges and opportunities that have been studied in the literature using various methodologies (Lacetera et al. 2014, Ata et al. 2016, Sönmez et al. 2016, McElfresh et al. 2019, Urrea et al. 2019). One key operational challenge is the uncertainty in both volunteer labor supply and demand. Using an elegant queuing model, Ata et al. (2016) study the problem of volunteer staffing with an application to gleaning organizations. Our approach to modeling volunteer behavior (specifically, assuming that notifying an active volunteer triggers a random period of inactivity) bears some resemblance to the approach taken in Ata et al. (2016). In a novel recent work, McElfresh et al. (2019) studies the problem of matching blood donors to donation centers, assuming that donors have preferences (over centers) and constraints on the frequency of receiving notifications. Using a stochastic matching policy, they demonstrate strong numerical performance relative to various benchmarks. There are some similarities between our modeling approach and the approach used in McElfresh et al. (2019), but we highlight three key differences. (1) While their work focuses on the numerical evaluation of policies, we theoretically analyze the performance of our policies and provide an upper bound on the performance achievable by any online policy, as stated in Theorems 1, 2, and 5. (2) We model volunteers’ adverse reactions to excessive notifications in a general form by considering an arbitrary interactivity time distribution. (3) We parameterize our achievable upper and lower bounds by the minimum discrete hazard rate of that distribution. Crowdsourcing Platforms: Reflecting the growth of online technologies, there is a burgeoning literature on the operations of crowdsourcing platforms (see e.g. Karger et al. (2014), Hu et al. (2015), Alaei et al. (2016), Papanastasiou et al. (2018)). Our work adds to the growing collection of papers that focus specifically on nonprofit crowdsourcing platforms, with applications as varied as educational crowdfunding (Song et al. 2018), disaster response (Han et al. 2019), and smallholder supply chains (de Zegher et al. 2018, de Zegher and Lo 2019). Nonprofits often use crowdsourcing in the absence of monetary incentives; in such settings, successful crowdsourcing relies on efficient utilization and engagement of participants. We contribute to this literature by designing online policies for effectively notifying volunteers while avoiding overutilization. Online Matching and Prophet Inequalities: Abstracting away from the motivating application, our work is related to the stream of papers on online stochastic matching, prophet matching inequalities, and online allocation of reusable resources. Given the scope of this literature, we highlight only recent advances and kindly refer the interested reader to Mehta et al. (2013) for an informative survey. A standard approach is to design online policies based on an offline solution (see e.g. Feldman et al. (2009), Manshadi et al. (2012), Jaillet and Lu (2014), Wang et al. (2018), Stein et al. (2019)) and to compare the performance of these policies to a benchmark such as the clairvoyant solution described in Golrezaei et al. (2014). Our work builds on this approach by applying techniques from prophet matching inequalities and the magician’s problem (Alaei et al. 2012, Alaei 2014). Most similar to our work are Dickerson et al. (2018) and Feng et al. (2019), which both consider settings with unitcapacity reusable resources (see also Gong et al. (2019) and Rusmevichientong et al. (2020) which mainly focus on largecapacity settings). The former designs an adaptive policy to address an online stochastic matching problem, while the latter considers an online assortment optimization problem. We highlight three key differences between our work and these papers. (1) In our work, the platform’s objective function is nonlinear. Despite that, we only consider offline solutions that can be computed in polynomial time (as opposed to relying on an oracle). (2) Volunteers—which represent the resources in our setting—can become unavailable without being matched (i.e., just through notification). (3) We develop parameterized lower and upper bounds based on the minimum discrete hazard rate of the usage duration. Such an approach enables us to gain insight into the impact of the characteristics of the usage duration distribution on the achievable bounds.
3 Model
In this section, we formally introduce the online volunteer notification problem that a volunteerbased crowdsourcing platform faces when deciding whom to notify for a task. As part of the problem definition, we highlight the platform’s objective as well as the tradeoff it faces due to the volunteers’ adverse reactions to excessive notifications and the uncertainty in future tasks. Further, we define the measure of competitive ratio and establish a benchmark against which we compare the performance of any online policy. The online volunteer notification problem consists of a set of volunteers, denoted by , and a set of task types, denoted by .^{5}^{5}5For FRUS, a task represents a scheduled rescue (food donation) which has not been claimed in advance. Volunteers (resp. tasks) are indexed from to (resp. ). Over time steps, the platform solicits volunteers to complete a sequence of tasks. In particular, in each time step , a task of type arrives with known probability . Without loss of generality, we assume at most one task arrives in each time step. Said differently, we assume and with probability , no task arrives. Whenever a task arrives, the platform can notify volunteers. However, excessively notifying a volunteer may lead her to suffer from notification fatigue. To model this behavior in a general form, we assume that a volunteer can be in two possible states: active or inactive. In the former state, the volunteer pays attention to the platform’s notifications, whereas in the latter state, she is inattentive. Initially, each volunteer is active. However, upon being notified she transitions to the inactive state and will only become active again in periods, where is independently drawn from a known interactivity time distribution denoted by . Mathematically,
. To capture the minimum rate at which volunteers transition from inactive to active, we define the minimum discrete hazard rate of the interactivity time distribution as follows: [Minimum Discrete Hazard Rate] For a probability distribution
, the minimum discrete hazard rate (MDHR) is given by , where denotes the corresponding CDF.^{6}^{6}6By convention, if the fraction is , we define it to be equal to 1. Note that a large value ofis a sufficient condition to ensure that volunteers’ activity level is high. For example, if
is a geometric distribution,
is the same as its success probability. However, a small value of does not imply inactive volunteers: if , i.e., if the interactivity times are deterministic and equal to 2 periods, then but volunteers are quite active. Before proceeding, we point out that similar modeling assumptions have been made in previous work. In particular, Ata et al. (2016) models volunteer staffing for gleaning and assumes once a volunteer is utilized, she will go into a random repose period. Similarly, McElfresh et al. (2019) focuses on blood donation and puts a constraint on the frequency with which a volunteer can be notified, which is equivalent to assuming a deterministic interactivity time. The latter strategy is also practiced in many FRUS locations. When a donation arrives, the platform observes the donation type and must immediately and irrevocably notify a subset of volunteers.^{7}^{7}7As explained in the introduction, the platform cannot expect a prompt response from volunteers and therefore sequential notification is impractical. If an active volunteer is notified about a task , she will respond with match probability , independently from all other volunteers. Thus the arriving task is completed if at least one notified volunteer responds. If task arrives at time and if the the subset of volunteers that are both notified and active is given by , then the task will be completed with probability . We highlight that this probability is monotone and submodular with respect to the set . In Section 6, we describe how can be estimated accurately in the FRUS setting by using historical data. As mentioned earlier, all volunteers are initially active. The platform knows the arrival rates , the match probabilities , and the interactivity time distribution , but it does not observe volunteers’ states. For any instance of the online volunteer notification problem where ,^{8}^{8}8For ease of notation, for any , we use to refer to the set . the platform’s goal is to employ an online policy that maximizes the expected number of completed tasks. In order to evaluate an online policy, we compare its performance to that of a clairvoyant solution that knows the entire sequence of arrivals in advance as well as volunteers’ states in each period. However, the clairvoyant solution does not know before notifying a volunteer how long her period of inactivity will be. Two observations enable us to upper bound the clairvoyant solution with a polynomiallysolvable program. First, note that if the clairvoyant solution notifies a subset of volunteers about task at time , the probability of completing isIn words, we can upper bound the success probability of a subset with a piecewiselinear function that is the minimum of the expected total number of volunteer responses and . Second, recall that the clairvoyant solution only notifies active volunteers and does not know how long those notified volunteers will remain inactive. As a consequence, we can upper bound the clairvoyant solution via the following program which we denote by (LP):
(LP)  
s.t.  (1)  
(2) 
With a slight abuse of terminology, we refer to this program with a piecewiselinear objective as (LP) because it can be expressed as a linear program by adding a constraint which ensures the linearity of the objective function. The decision variables represent the probability of notifying volunteer when a task of type arrives at time . Constraint (1) ensures that is a valid probability. Constraint (2) places limits on the frequency with which volunteers can be notified according to the interactivity time distribution. In particular, the clairvoyant solution will only notify an active volunteer who will then become inactive for a random number of periods. Thus, in expectation the clairvoyant solution must meet constraint (2). For ease of reference, in the following, we define the set of all feasible solutions to (LP). Such a definition proves helpful in the rest of the paper. [Feasible Set] For any , if and only if it satisfies constraints (1) and (2). The following proposition, which we prove in Appendix 8.1, establishes the relationship between the clairvoyant solution and : [Upper Bound on the Clairvoyant Solution] For any instance of the online volunteer notification problem, is an upper bound on its clairvoyant solution. In light of Proposition 3, we use as a benchmark against which we compare the performance of any policy. Consequently, we define the competitive ratio of an online policy as follows: [Competitive Ratio] An online policy is competitive for the online volunteer notification problem if for any instance , we have: , where represents the expected number of completed tasks by the online policy for instance . We will use the competitive ratio as a way to quantify the performance of an online policy. For each of our two policies (presented in the following section), the competitive ratio is parameterized by the MDHR, , and it improves as increases.
4 Online Policies
In this section, we present and analyze two policies for the online volunteer notification problem. Both policies are randomized and rely on a fractional solution we compute ex ante using the instance primitives. Thus, we begin this section by introducing the ex ante solution in Section 4.1. We then proceed to describe our algorithms and analyze their competitive ratios in Sections 4.2 and 4.3.
4.1 Ex Ante Solution
As stated in Section 1, both of our online policies rely on an ex ante solution which we denote by . Given our benchmark, we focus our attention on solutions that are feasible in (LP), i.e., (see Definition 3). Clearly, —the solution to (LP) in Section 3—is a potential ex ante solution. However, in practice, such a solution can prove ineffective because it does not take into account the diminishing returns of notifying an additional volunteer about a task. As a result, it may ignore some tasks while notifying an excessive number of volunteers about others (e.g., see Appendix 11.1). Given any
, for a moment, suppose volunteers are always active. Then if we notify each volunteer independently according to
, the expected number of completed tasks would be:^{9}^{9}9Since a task can only be completed if one arrives, we limit all sums to task types indexed from to .(3) 
Because is the optimal solution of a piecewiselinear objective, it ignores the submodularity in .^{10}^{10}10We remark that we design our online policies such that they achieve a constant factor of as defined in (3). In light of this intuition, we introduce two other candidates that can be computed in polynomial time. First, we aim to find the feasible point that maximizes . We denote this optimization problem by (AA) which stand for Always Active. Even though AA is hard (Bian et al. 2017), simple polynomialtime algorithms such as the variant of the FrankWolfe algorithm described below (proposed in Bian et al. (2017)) are known to work well in practice. The algorithm iteratively maximizes a linearization of and returns a convex combination of feasible solutions, which therefore must be feasible. We denote the output of this algorithm by and use it as another candidate for the ex ante solution.
(4) 
For any , the term in (4) represents the probability that under the indexbased priority scheme, volunteer is the lowestindexed volunteer to respond positively to a notification about task at time . Further, this term only depends on the fractional solution of volunteers with lower index than . In addition, if we treat as fixed for , then is linear in . In light of these observations, we define our last candidate as the solution of a sequence of linear programs in which volunteers maximize their individual contributions in the order of their priority. This is summarized in the program (SQ). For from to :
(SQ)  
For a given volunteer , the program (SQ) uses the solutions from previous iterations, i.e., for . As a result, this solution takes into account the diminishing returns from notifying multiple volunteers. We denote the solution to these sequential LPs as . Finally, we remark that the above decoupling idea proves helpful in both designing and analyzing our online policies. Having three candidates, we define
(5) 
The following proposition establishes a lower bound on based on the benchmark . [Lower Bound on Ex Ante Solution] For defined in (5),
The above worst case ratio is achieved by the ratio of to , and it is tight. However, we stress that and can provide significant improvements. A simple example illustrating this point can be found in Appendix 11.1, while a full proof of Proposition 5 can be found in Appendix 9.2. When testing our policies on FRUS data (as detailed in Section 6), we find that using instead of results in an average improvement of 5% up to maximum of 23%. We conclude this section by noting that an online policy which directly follows (i.e., a policy that at time , upon arrival of , notifies volunteer independently with probability ) does not achieve a good competitive ratio. This stems from the fact that “respects” the inactivity period of volunteers only in expectation. Consequently, it is possible that volunteers are inactive when highvalue tasks arrive (e.g. tasks where the match probability is close to ) because they were notified earlier (according to ) for lowvalue tasks. We present an illustrative example in Appendix 11.2. Therefore, we develop two policies based on two different modifications of the ex ante solution: (1) properly scaling it down and (2) sparsifying it. The former guides our first policy which we call the scaleddown notification policy, whereas the latter guides our second policy, referred to as the sparse notification policy. These policies are described and analyzed in the next two sections, respectively.
4.2 ScaledDown Notification Policy
In this section, we present our scaleddown notification (SDN) policy which is a nonadaptive randomized policy that independently notifies volunteers according to a predetermined set of probabilities based on .^{13}^{13}13Some of the ideas used in our SDN policy are similar to the adaptive algorithm of Dickerson et al. (2018). The policy relies on the following ideas: (1) Fixing a policy, suppose we can compute the ex ante probability that any volunteer is active at time when following that policy. Let us denote such an ex ante probability by . Then if arrives at time , we notify with probability where . As a result, she will be active and notified with probability . (2) If she was the only notified volunteer, then her probability of completing this task would be simply . Even though this is not the case, using the indexbased priority scheme and the contribution decoupling idea in Lemma 12, we can show her contribution will be proportional to . (3) Consequently, we would like to set as large as possible. However, cannot be larger than since notification probabilities cannot exceed . Thus in the design of the policy, we find the largest feasible , which we prove to be where is the MDHR of the interactivity time distribution (see Definition 5). The formal definition of our policy is presented in Algorithm 1. In the rest of this section, we analyze the competitive ratio of the SDN policy. Our main result is the following theorem:
[Competitive Ratio of the ScaledDown Notification Policy] Suppose the MDHR of the interactivity time distribution is . Then the scaleddown notification policy, defined in Algorithm 1, is competitive. We remark that Theorem 1 implies that the competitive ratio of our policy improves as increases. However, a larger value of does not imply that the probability of notification is uniformly larger. If increases, the ex ante solution as well as the ex ante probability of being active will also change, both of which affect the notification probability. The proof of Theorem 1 builds on the ideas described above and consists of several steps. First, in the following lemma, we prove that for any and , defined in Algorithm 1 is indeed the probability that is active at time and is at least .^{14}^{14}14We also highlight that computing for all and can be done in polynomial time. [Volunteer’s Active State Probability] For the SDN policy defined in Algorithm 1, let represent the event that volunteer is active in period . Then for all and all , . Further, . We prove this lemma via total induction, relying critically on the fact that (see Definition 3). The full proof can be found in Appendix 9.3. Next, utilizing the indexbased priority scheme (in Definition 11) and the contribution decoupling idea (in Lemma 12), we lower bound the contribution of each volunteer according to their priority in the following lemma: [Volunteer PriorityBased Contribution under the SDN Policy] Under the indexbased priority scheme (in Definition 11) and the SDN policy, for any , the contribution of volunteer , i.e., the expected number of tasks she completes, is at least , with defined in (4). To prove Lemma 14, we first show that a volunteer responds to a notification about task in period with probability . We then place an upper bound on the probability that a volunteer with a smaller index also responds to a notification about that same task. A full proof can be found in Appendix 9.4. The last steps in the proof of Theorem 1 are to compare the aggregate contribution of volunteers with the benchmark utilizing Lemma 12 and Proposition 5. The detailed proof of Theorem 1 is presented in Appendix 9.5.
4.3 Sparse Notification Policy
In this section, we present our second policy, the sparse notification (SN) policy, which relies on a different modification of the ex ante solution. Before describing the policy, we briefly discuss our motivation for designing a second policy. Though simple and intuitive, the SDN policy only relies on the ex ante solution to resolve the tradeoff between the immediate reward of notifying a volunteer and saving her for a future arrival. To see this, note that even in the last period , the SDN policy follows a scaleddown version of . To more accurately resolve this tradeoff, in designing the SN policy, we utilize the ex ante solution and the indexbased priority scheme (see Definition 11) to formulate a sequence of onedimensional DPs whose optimal value will serve as a lower bound on the contribution of each volunteer according to her priority (as shown in Lemma 2). The solution of the these DPs is a sparsified version of the ex ante solution . Namely, let us denote as the solution of the sequence of DPs. For any , , and , is either or . Equipped with , which we compute in advance, the SN policy simply follows in the online phase. Our DP formulation and its analysis follows the framework developed in Alaei et al. (2012) and Alaei (2014), which is also used in Feng et al. (2019). Next we describe the DP formulation. Consider volunteer and suppose we have already solved the first DPs. Thus we have . Let us denote the valuetogo of the DP at time by . Clearly . We set ’s reward at time for task to be
(6) 
The actions available when task arrives at time are to notify with probability or to not notify . Thus when deciding on the optimal action (which can be either or ), we compare the (current and future) reward of notifying now to the reward of saving her for the next period. Formally,
(7) 
The term in the indicator on the left hand side is the reward of notifying in the current period , which consists of two parts: (1) the immediate reward we get from notifying —which will make her inactive for periods—and (2) the future reward once she becomes active again. The right hand side within the indicator simply represents the reward when is not notified and remains active in period . Given (6), (7), and , we can iteratively compute as follows:
(8) 
The formal definition of our policy is presented in Algorithm 2. In the rest of this section, we analyze the competitive ratio of the SN policy. Our main result is the following theorem:
[Competitive Ratio of the Sparse Notification Policy] Suppose the MDHR of the interactivity time distribution is . Then the sparse notification policy, defined in Algorithm 2, is competitive. A few remarks are in order: (1) The competitive ratio of our two policies are identical, implying that in the worst case they guarantee the same performance. However, for practical instances, the SN policy performs significantly better (as shown by our test results in Section 6). Intuitively, this is because the design of the SN policy explicitly aims to optimally resolve the tradeoff between notifying a volunteer now or keeping her active for later based on . On the other hand, the design of the SDN policy only aims to proportionally follow . As a result, the SDN policy’s numerical performance is not substantially better than its worstcase guarantee. On the other hand, the SN policy can perform much better than its worstcase guarantee (see Appendix 11.3 for an illustrative example). (2) Similar to the SDN policy, the competitive ratio of the SN policy improves when increases. However, the design of the SN policy does not directly make use of . The proof of Theorem 2 consists of two main lemmas. First, in the following lemma, we lower bound the contribution of each volunteer by : [Volunteer PriorityBased Contribution under the SN Policy] Under the indexbased priority scheme (in Definition 11) and the SN policy, the contribution of volunteer , i.e., the expected number of tasks she completes, is at least , where is defined in (8). The proof of this lemma follows from the DP formulation as well as the observation that for any , , and , the probability that a higherpriority volunteer completes the task is upper bounded by . A full proof can be found in Appendix 9.6. The second main step of the proof is to compare to the benchmark . In order to do so, we follow the dualfitting approach of Alaei et al. (2012). In particular, given the interactivity time distribution, we set up a linear program to find the “worst” possible combination of perstage rewards that give rise to the minimum possible value of . Finding the optimal solution to this LP proves to be difficult. Instead we find a feasible solution to its dual, which enables us to lower bound . The LP and its dual are presented in Table 1. In the LP formulation, the first two sets of constraints follow from the DP definition. Note that the value of will crucially depend on , e.g., if for all , , and , then . This motivates the final constraint, which provides a constant against which we can compare . The following lemma establishes a lower bound on .
[Lower Bounding the Dynamic Program] Under the indexbased priority scheme (see Definition 11), for any and volunteer , we have where is defined in (4). The proof of Lemma 1 (presented in Appendix 9.7) amounts to confirming that setting and defining all other dual variables such that the constraints hold with equality is a feasible solution to (Dual). Given Lemma 1, we complete the proof of Theorem 2 by applying Lemma 12 and Proposition 5. The complete proof is presented in Appendix 9.8.
5 Upper Bound on Competitive Ratio
In this section, we provide an upper bound on the competitive ratio of any online policy in the online volunteer notification problem. Like the lower bound achieved by our policies in Section 4, the upper bound is parameterized by the MDHR of the interactivity time distribution, . The main result of this section is the following theorem: [Upper Bound on Achievable Competitive Ratio] Suppose the MDHR of the interactivity time distribution is where . Then no online algorithm can achieve a competitive ratio greater than , where for
(9) 
and for , we have .^{16}^{16}16We remark that the condition imposed on when is added for ease of presentation of the theorem statement as well as its proof. Relaxing the aforementioned condition amounts to modifying the second term in by rounding any up to the closest and slightly modifying the instance in the proof. We omit these details for the sake of brevity. Figure 2 provides a summary of our lower and upper bounds on the achievable competitive ratio for the online volunteer notification problem as a function of . We make the following observations based on the theorem and accompanying plot: (1) the upper bound applies to all policies, even those that cannot be computed in polynomial time, (2) both the upper and lower bounds improve as increases, and (3) the competitive ratio of our online policies are fairly close to the upper bound when is small but positive. However, the gap grows for larger values of .
The proof of Theorem 5 relies on analyzing the following two instances, each giving one of the terms in the definition of as shown in (9). Instance attains the minimum when whereas Instance attains it when . Instance : Suppose , , , and , where . The arrival probabilities are given by and , where . The volunteer match probabilities are given by and . The left panel of Figure 3 visualizes Instance . The following lemma—which we prove in Appendix 10.1—states that no online policy can complete more than a fraction of . [Upper Bound for Instance ] In instance , The expected number of completed tasks under any online policy is at most . Before proceeding to the second instance, we make two remarks: (1) If , the above instance is equivalent to the canonical instance used in the prophet inequality to establish an upper bound of (see, e.g., Hill and Kertz (1992)). (2) The term in the competitive ratio of both policies corresponds to the gap between (defined in (3)) and the benchmark , whereas the corresponds to the gap between the performance of our online policy and due to the loss in the online phase.In Instance , there is only one volunteer and consequently . Therefore, Instance shows that the lower bound achieved in the online phase of our policies is tight, as they both attain at least . The construction of our second instance is more delicate as it aims to find an instance for which both the loss in the offline phase (i.e., the gap between and ) and the loss in the online phase (i.e., the gap between the performance of the online policy and ) are large. Instance : Suppose , , , and is the geometric distribution with parameter , e.g. . The arrival probabilities are given by and for . The volunteers are homogeneous with for all . The right panel of Figure 3 visualizes Instance . The following lemma—which is proven in Appendix 10.2—states that no online policy can complete more than a fraction of . [Upper Bound for Instance ] In instance , the expected number of completed tasks under any online policy is at most . The proof of this lemma involves three steps: (1) placing a lower bound on by finding a feasible solution, (2) establishing that always notifying every volunteer is the best online policy, and (3) assessing the performance of this policy relative to . A full proof can be found in Appendix 10.2.
6 Evaluating Policy Performance on FRUS Data
In this section, we use data from FRUS to evaluate the performance of the two online policies described in Section 4. First, we briefly explain how we use data to determine the model primitives. Then we exhibit the superior performance of our policies compared to policies that resemble the strategies used at various FRUS locations. Estimating model primitives: As explained in Section 3, in order to define an instance of the online volunteer notification problem, we must determine the match probabilities, i.e., ; the arrival rates of tasks, i.e., ; and the interactivity time distribution .

Match probabilities: As evidenced in Figure 1, volunteer preferences over tasks are heterogeneous and predictable. To come up with estimates
for each FRUS location, we first create a feature vector for each task. We then build a
Nearest Neighbors classification model, tuning the parameter using crossvalidation. The AUCs of such classification models range between 0.89 and 0.95 across tested locations. 
Arrival Rates: Recall that for FRUS, a task is a food rescue (donation) that remains available on the day of delivery. Most food rescues are repeated on a weekly cycle; therefore we define a type for each recurring rescue. Empirically, we observe a relationship between the last minute availability of a rescue of type and its status over the past six weeks (the correlation coefficient is between and across all tested locations). Therefore, we estimate as the proportion of times in the past six weeks that a rescue of type was a lastminute availability.

Interactivity time distribution: At FRUS, many site directors follow a policy of waiting at least a week before notifying the same volunteer about another lastminute food rescue. Consequently, we assume the interactivity time is deterministic and equal to seven days, e.g. .
In the following, we compare the performance of our online policies to strategies that simulate the current practice at various FRUS locations using instances constructed with data from two different locations as described above. First, we compare our policies against ‘notify1’ and ‘notify3’ policies that, respectively, notify one and three volunteer(s) chosen uniformly at random among “eligible” volunteers. Note that here a volunteer is eligible if she has not been notified for at least 6 days. The top panels of Figure 4 display the ratio between each policy and across 50 simulations. We highlight that the SN policy significantly outperforms all other policies. Further note that the SN policy’s performance far exceeds its competitive ratio of , as given in Theorem 2, while the SDN policy performs only slightly above its competitive ratio.^{17}^{17}17Part of why our policies outperform their competitive ratio is that in the FRUS locations studied, using as an ex ante solution improves on using by an average of , up to a maximum of . Next, we compare our policies against a ‘notifyall’ policy that sends a notification to all volunteers. This policy clearly does not respect the 7day gap between two successive notifications. Therefore, here we assume that the interactivity time distribution is geometric with an expected duration of 7 days. The bottom panels of Figure 4 display the ratio between each policy and across 50 simulations. Here, we also observe that the SN policy significantly outperforms all other policies as well as its worstcase guarantee.
7 Conclusion
In this paper, we take an algorithmic approach to a commonly faced challenge on volunteerbased crowdsourcing platforms: how to utilize volunteers for timesensitive tasks at the “right” pace while maximizing the number of completed tasks. We introduce the online volunteer notification problem to model volunteer behavior as well as the tradeoff that the platfrom faces in this online decision making process. We develop two online policies that achieve constantfactor guarantees parameterized by the MDHR of the volunteer interactivity time distribution, which gives insight into the impact of volunteers’ activity level. The guarantees provided by our policies are close to the upperbound we establish for the performance of any online policy. In this paper, we measure the performance of an online policy by comparing it to an LPbased benchmark which upper bounds a clairvoyant solution. From a theoretical perspective, considering other benchmarks (perhaps less strong) is an interesting future direction. This work is motivated by our collaboration with FRUS, a leading volunteerbased food recovery platform, analysis of whose data confirms that, by and large, volunteers have persistent preferences. Leveraging on historical data, we estimate the match probability between volunteertask pairs as well as the arrival rate of tasks. This enables us to test our policies on FRUS data from different locations and illustrate their effectiveness compared to common practice. From an applied perspective, studying the robustness of our policies as well as developing decision tools that can be integrated with the FRUS app are immediate next steps that we plan to pursue. Finding other platforms that can benefit from our work is another direction for future work.
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8 Proofs for Section 3
8.1 Proof of Proposition 3
To show that is an upper bound on the clairvoyant solution, we will construct a feasible solution based on the clairvoyant solution. We will then prove that the value of this solution is an upper bound on the value of the clairvoyant solution. Let us define the random realizations of interactivity times as , where is the interactivity time of volunteer if notified at time . In addition, we denote the random arrival sequence as , where is the arrival at time . Finally, suppose we have an indicator variable , which is equal to one if and only if the clairvoyant solution contacts volunteer at time when the arrival order is given by and the interactivity times are given by . Because the clairvoyant solution does not know until after time , cannot depend on for . For any volunteer , task , and time , we define
To show that (see Definition 3), we immediately note that , since we are summing indicator variables over probability distributions. We now need to show that constraint is met, namely that . Note that for a given sequence of arrivals and interactivity times given by , we must have
(10) 
This is because both and are indicator variables, and if both equal at time , then the volunteer must be inactive until after time . Since the clairvoyant solution only notifies active volunteers, if volunteer is inactive from until after , then for all . Thus, the sum from to of the product of these indicator variables cannot exceed . We now take a weighted sum over all possible arrival sequences and interactivity times:
(11)  
(12)  
(13)  
(14)  
(15) 
In line (12), we use the independence of and to rewrite the expected value of their product as the product of their expectations. We substitute in the expected value of in line (13). In line (14
), we use the law of total probability to sum over all possible arriving tasks in time
. We then substitute in the definition of in line (15). This proves that . It remains to be shown that exceeds the value of the clairvoyant solution. Let be the event that task arrives at time and is completed when following the clairvoyant solution. We must have . In addition, since a volunteer must respond in order to complete a task, we must haveCombining these two bounds and summing over all tasks and time periods, we see that the clairvoyant solution must be less than
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