Static Pricing: Universal Guarantees for Reusable Resources
Columbia Business School, New York, NY, 10027, firstname.lastname@example.org
Adam N. Elmachtoub
Department of Industrial Engineering and Operations Research & Data Science Institute, Columbia University, New York, NY, 10027, email@example.com
Department of Industrial Engineering and Operations Research, Columbia University, New York, NY, 10027, firstname.lastname@example.org
We consider a fundamental pricing model in which a fixed number of units of a reusable resource are used to serve customers. Customers arrive to the system according to a stochastic process and upon arrival decide to purchase or not the service depending on their willingness to pay and the current price. The service time during which the resource is used by the customer is stochastic and the firm may incur a service cost. This model represents various markets for reusable resources such as cloud computing, shared vehicles, rotable parts, and hotel rooms. In the present paper, we analyze this pricing problem when the firm attempts to maximize a weighted combination of three central metrics: profit, market share, and service level. Under Poisson arrivals, exponential service times, and standard assumptions on the willingness to pay distribution, we establish series of results that characterize the performance of static pricing in such environments. In particular, while an optimal policy is fully dynamic in such a context, we prove that a static pricing policy simultaneously guarantees 78.9% of the profit, market share, and service level from the optimal policy. Notably this result holds for any service rate and number of units the firm operates. In the special case where there are two units and the induced demand demand is linear, we also prove that the static policy guarantees 95.5% of the profit from the optimal policy. Our numerical findings on a large testbed of instances suggest that the latter result is quite indicative of the profit obtained by the static pricing policy across all parameters.
Key words: reusable resources, dynamic pricing, static pricing, approximation algorithm
In many service industries, the same resource to serve one customer can be used to serve future customers once the initial service is completed. This type of resource is commonly referred to as a reusable resource. For instance, in the hotel or car rental industry, a fixed number of rooms or vehicles are available to accommodate customers. Each customer uses one of these resources for some number of days until check out or return, after which it is free to be used by another customer. In a related example, many offices, campuses, or apartment buildings offer a pool of bicycles or vehicles to be rented or shared by their members, and units are always returned to their origin after being used by a member. Another example of a reusable resource is cloud computing servers, which can be used by customers to complete jobs after which they become available for processing new jobs. Finally, another interesting example of a reusable resource arises in the repair industry for aircraft, trains, and other large machinery. Specifically, there are several parts that are known as rotable, meaning that when they break, the customer exchanges the broken part for a working part with the repair agent. When the repair agent receives the broken part, it is “utilized” for some time as it is being repaired, after which it becomes available again to service potential future customers.
All of the examples above share several important features which we shall incorporate into our model. First, the number of units available of each resource is fixed (over appropriate time horizons), as acquiring more capacity or units involves significant investments. Second, when a resource is used, the service time is generally stochastic and varies across customers. Third, customers are price-sensitive and in turn the demand rate in each of these applications can be controlled by the price (which can be a one-time fee or an hourly/daily fee to the customer). Fourth, there is a cost incurred by the service provider associated with the usage of a unit (e.g., in terms of cleaning, maintenance, or repair). Finally, in each of these settings it is highly unusual for a customer to wait for service. That is, if all units of the resource are occupied, the customer is typically lost.
In all of the settings above, the seller may have multiple objectives. The profit rate is clearly a fundamental objective for any service provider, but typically such providers also focus significantly on their market share and service level, i.e., the probability that an arriving customer finds a resource available. The latter two metrics are driven by the long term objectives of maintaining a prominent position in the market and ensuring that consumers find the service reliable.
In such environments, an optimal policy will be highly dynamic in general, adjusting its prices often, as a function of the supply conditions. The question the present paper aims to address is the following. What is the performance of a simple static pricing (one price) policy in such environments? This question has dual practical and theoretical motivations. On the one hand, in practice, dynamic pricing may not be feasible when prices need to be published in a catalog upfront or may be undesirable due to the negative perception by customers. On the other hand, the existing literature in dynamic pricing has argued for particular objectives that static pricing yields near-optimal performance in large-scale systems (see literature review). How robust is such an insight for a combination of objectives and for arbitrary scales? (While the scale for cloud computing may be large, it is often small as well, e.g., rotable spare parts.) In particular, in the present paper, we aim to derive universal performance guarantees for static pricing with respect to the profit, market share, and service level objectives, with an optimal dynamic pricing strategy serving as the benchmark. In particular, we aim to provide results on the strength of static pricing that hold across all possible parameter regimes and scales.
To that end, we anchor our analysis around the following prototypical model. A service provider manages a pool of a single type of reusable resource. The firm uses the reusable resources to deliver service to customers over an infinite horizon. Customers arrive according to a Poisson process in which the rate depends on the price set by the firm. We make the standard assumption that the revenue rate is concave in the arrival rate. Upon arrival, a customer seizes one unit of the resources for an exponentially distributed random amount of time and pays a fee (which could depend on the realized duration of usage or be fixed in advance). The unit of resource occupied by a customer becomes available to others after service completion. The firm may also incur some cost of service. The goal of the firm is to decide on the optimal pricing policy to maximize a combination of three different objectives: profit rate, market share, and service level.
The main contributions of the present paper lie in deriving universal performance guarantees for static pricing and can be summarized as follows.
We establish that for any combination of the three objectives, profit rate, market share, and service level, there exists a static pricing policy which can achieve at least 78.9% of the value of each objective under the optimal dynamic pricing policy. This result holds for any capacity size, market size, and service rate. Our proof relies on constructing an explicit static policy, characterizing a lower bound in terms of the stock-in probabilities of our policy and the optimal policy, and analyzing this ratio using a change of variables.
We consider a special case where the service provider is a profit maximizer, there are two units of the reusable resources, and the demand rate is linear in the price. We prove in this case that the static policy achieves at least 95.5% of the optimal profit from dynamic pricing. This result holds for any market size and service rate. Moreover, this exact scenario arises frequently in practice in the context of providing (very expensive) rotable spare parts.
We complement the theoretical results above with numerical experiments over a broad test bed. These illustrate that the performance of static pricing is in general even better. Furthermore, for profit maximization, we find the performance of static pricing is always above 97.5% of that obtained by an optimal dynamic pricing policy, indicating the robustness of the insights derived beyond the exact conditions assumed in the theorems.
To the best of our knowledge, these are the first universal guarantees derived for static pricing for this class of problems. Furthermore, the bounds derived highlight the very high performance of static pricing.
We next provide an overview of the literature on the effectiveness of static pricing policies in the context of perishable inventory, queuing systems, and reusable resources. We note that although a server in a queueing system is indeed a reusable resource, these systems typically allow for customers to wait for service. In contrast, the reusable resources literature assumes that customers are immediately lost if no units are available.
The dynamic pricing literature has had an extensive focus on the context of perishable resources, where there is a finite time horizon to consume a finite number of units of one or more resources (see den Boer (2015) for a recent survey). The seminal work of Gallego and Van Ryzin (1994) shows that if the revenue function is concave, a static pricing policy loses at most , where is the number of units and represents the expected number of sales under the myopic price. The authors also show a universal guarantee of for any parameter regime, with both results relying on a concavity assumption on the revenue rate (see also Ma et al. (2018)). Ma et al. (2018) recently generalize these results for the same model without the concavity assumption, and also provide non-adaptive pricing policies for assortment optimization and non-stationary demand settings with constant factor performance guarantees. Chen et al. (2018) showed that the guarantee and asymptotic optimality for static pricing also holds in the presence of strategic customers. Gallego et al. (2008) establish conditions for when static pricing is optimal in the presence of strategic customers. The value of static over dynamic pricing policies has also been considered in models with inventory cost and replenishment considerations, such as those in Federgruen and Heching (1999), Chen et al. (2006), Yin and Rajaram (2007), Chen et al. (2010). Related to static pricing policies are policies with limited price changes, such as those considered in Feng and Gallego (1995), Bitran and Mondschein (1997), Netessine (2006), Çelik et al. (2009), Chen et al. (2015), Cheung et al. (2017). Note that in our model there are no inventory costs, and inventory can be repeatedly reused over an infinite time horizon.
There is also an extensive literature on dynamic pricing in the presence of reusable resources. Paschalidis and Tsitsiklis (2000) provide numerical results showing the promise of static pricing in multi-class systems. Ata and Shneorson (2006) studied the dynamic pricing of an M/M/1 service system where the objective is welfare maximization, and numerically show it can have significant gains over static pricing. Maglaras and Zeevi (2005) showed in a service system, the revenue generated by the fluid-optimal prices are near optimal when the capacity and market potential are both large. Closest to our formulation is the work of Gans and Savin (2007) who study dynamic pricing to maximize the expected profit for rentals. Their model considers discounted rewards with a discrete price ladder, although with multiple customer types. They show the near-optimality of static pricing in highly utilized rental systems where both the offered load and system capacity are large. In a related model where the objective is revenue maximization, with observable queues, Kim and Randhawa (2017) quantify more precisely the asymptotic value of dynamic pricing in large systems and prove conditions under which a two price policy is almost as good as a dynamic pricing policy. Banerjee et al. (2015) provide a queueing analysis of a ride-share platform where the customers are modeled as servers, and show that a static price is asymptotically equal to a dynamic price policy for large-scale systems, although dynamic pricing is more robust to modeling error. To the best of our knowledge, all of the previously mentioned results quantifying the gap between static pricing and dynamic pricing hold asymptotically when the scale of the system is large. In contrast, our results provide universal guarantees that do not rely on the scale of the system. Recently, Banerjee et al. (2016) consider a general network of a single type of resource where prices control the rates between nodes, and prove a guarantee of for possibly multiple objectives but zero service times, where is the number of nodes and is the number of units. With non-zero service times, as we consider in our paper, a looser guarantee can be provided only when . Our paper provides a guarantee for any number of units, but does not consider a general network.
Various related studies focus on dynamic heuristics, multiple types of reusable resources, and other levers beyond pricing. Lei and Jasin (2018) studies the dynamic pricing problem in a setting with deterministic service times and describe policies that are asymptotically optimal in the regime where demand and resource capacity are both large. Variants of the assortment optimization problem have been considered in *rusmevichientong2017dynamic, *Owen2017price and *Goyal2018online with various universal approximation guarantees. The results in the first two papers can be extended to allow for dynamic pricing with discrete price points. Levi and Radovanović (2010) use a linear programming approach to design admission control policies for such systems, which is a special case of dynamic pricing where a resource is either priced at a nominal price or at infinity. Their admission control policy is asymptotically optimal, and also provides universal guarantee of . *chen2017revenue and Chen and Shi (2018) consider generalizations of this model that permit advanced reservations and strategic customers, respectively.
The paper is organized as follows. In Section id1, we describe the model along with structural properties of the optimal policy. In Section id1, we prove the 78.9% performance guarantee of static pricing under our multi-objective setting for any parameter range. We then refine our guarantee to 95.5% in Section id1 for the special case of profit maximization with two units under linear demand. We conduct numerical experiments in Section id1 that show that the actual performance of static pricing is even stronger than our guarantee, and that such performance still holds when the exact assumptions of the theoretical results do not hold.
In this section, we first describe a general model for pricing a reusable resource. We then describe the various performance criteria the decision-maker focuses on, followed by several important properties of the optimal dynamic pricing policy.
We consider a model in which a service provider has a fixed number of identical, non-perishable units of a reusable resource that are sold to price-sensitive customers. The total number of units of the reusable resource that the provider has is . At any point in time, each unit of the resource is either available for sale or occupied. Note that an occupied unit can be interpreted as a customer using the unit in the cloud computing and travel examples, or being repaired in the rotable spare parts example from Section id1.
Customers arrive to the system over time according to a Poisson process with rate . Each customer has an i.i.d. willingness-to-pay drawn from a valuation distribution . When a customer arrives, the provider offers a unit at some price , and a customer decides to purchase usage of the resource if their willingness to pay exceeds . We denote by the effective arrival rate at price . When a customer decides to purchase usage, one unit is then occupied for a random amount of time that follows an exponential distribution with mean . We assume that the usage times are i.i.d. across customers and independent of the customer valuations.
While a unit is being occupied, the firm cannot sell that unit until it is returned to the system, i.e., a customer finishes using the unit or the provider finishes repairing the unit. The firm incurs a cost to service one customer, which may be a cleaning, maintenance, or repair cost. Any customer that arrives when all units are occupied are lost, regardless of the current price being offered. This assumption is largely motivated by the fact that in most of our applications the customers are seeking immediate service, and would naturally seek out a competitor if the provider has no units available.
We assume that there is a one-to-one correspondence between prices and effective arrival rates so that has a unique inverse, denoted by . Therefore, one can view the effective arrival rate as the decision variable. The firm dynamically determines a target effective arrival rate which can be realized with the corresponding price . From an analysis perspective, the effective arrival rate is more convenient to work with. We shall make the standard assumption in the revenue management literature that the profit rate function is concave in .
The set of admissible policies, , is the set of non-anticipating policies, i.e., policies such that the effective arrival rate at time , , may only depend on events up to . We shall also be interested in the class of static policies, , that simply fix a single arrival rate (price) at every time .
One natural metric when selling the reusable resources is the expected profit rate. Fix a pricing policy and let denote the corresponding effective arrival rate at time . Let denote the corresponding arrival process. Note that the latter is a non-stationary Poisson process with intensity . Let denote the number of on-hand units at time . The long-run average profit rate is given by
For simplicity in the exposition of the paper, we assume is a one time fee a user needs to pay for the service. Note that the analysis presented also generalizes to the case when a user’s payment depends on usage time, i.e., it is equivalent to charge a one-time price that is simply the price per time unit multiplied by the expected usage time.
While the firm wants to maximize its profit, it may also want to keep a certain level of market share, i.e., the expected number of units sold as well as a certain service level. The market share objective is directly aligned with maximizing the number of sales sold, while the service level objective is measured by the fraction of time at least one unit is available. These two objectives can be represented as
Note that there is a trade-off between the various metrics and the optimal solution for one objective will generally be sub-optimal for another. For instance, maximizing the service level corresponds to setting a static price as large as possible, while maximizing market share corresponds to setting a static price of zero. Clearly neither price will result in any profit at all.
In order to take the different objectives into account simultaneously, we assume the firm maximizes a weighted combination of the objectives,
where are the weights placed on each objective by the service provider. Without loss of generality, we assume . We let denote the long-run value under the optimal policy, and is thus defined by
We denote by an optimal policy. Similarly, we let denote the long-run value under the optimal static policy, and is thus defined by
In the present paper, we focus on universal performance guarantees for static pricing. In particular, we shall focus on the worst-case performance of the optimal static pricing policy in comparison to the optimal dynamic policy. That is, we seek to characterize the maximum possible loss over all possible instances of our model. Formally, we let denote the family of instances
In turn, we aim to provide a universal lower bound on
which is the ratio between the objectives under the optimal static and dynamic pricing policies. In fact, we shall show that our bound applies to the corresponding ratios of each of the three objectives.
We shall now characterize the structure of an optimal solution to the dynamic pricing problem stated in Equation (3). Given the Poisson assumption on arrivals and the exponential assumption on service times, without loss of optimality, one may focus on stationary policies that update the price only at changes in the number of units on-hand. As we shall see, this allows us to provide closed form expressions for the steady state distribution and objectives under a particular policy.
A stationary policy may be represented by arrival rates . When the provider has only units available, the price is set to . Note that the static policy is a special case where . Furthermore, the system can now be modeled as a birth-death process where each state represents the number of units available. The transition rate from state to is for . The transition rate from to is for . A standard calculation for computing the steady state probabilities, yields that
Using the steady state probabilities, we may express our three objectives simply as
Let us denote by the effective arrival rate in state under the optimal policy, and by the steady-state probabilities of being in state under the optimal policy. In Lemma 1, we show a fundamental property that effective arrival rates are decreasing as the number of units available increases. Moreover, all such arrival rates do not exceed the myopic rate , which yields the highest possible instantaneous objective rate.
Let be the optimal arrival rate when the on-hand inventory level is . Let denote the myopic arrival rate where . Then
The proof of Lemma 1 can be found in the id1. Notice that the result presented in Lemma 1 shares the same structural property as presented in Theorem 1 in Gans and Savin (2007) where the objective is only profit maximization in a discounted reward setting. We extend the analysis to a long-run average reward setting with multiple objectives and prove that monotonicity of optimal prices (and rates) still holds. We will make use of this property in the subsequent analysis.
We next investigate the performance of static pricing and present our first main result.
There exists a static pricing policy that guarantees at least of the profit rate, market share, and service level of the optimal dynamic pricing policy. Equivalently,
Theorem 1 provides a strong guarantee: there exists a static price that nearly approximates the performance of an optimal dynamic pricing policy. Specifically, this price guarantees that the profit rate, market share, and service level are at least of the corresponding values under the dynamic pricing policy. Of course, a direct consequence of Theorem 1 is that the optimal single price will have an overall objective of at least of the objective under the optimal dynamic pricing policy as well. It is important to note that our result makes no assumption on the number of units in the system, demand rate, or service rate. This is in stark contrast to the previous literature which require the system usage and capacity to be large to provide theoretical guarantees.
It is worthwhile noting that our proof is constructive and exhibits a particular static price that yields such performance. The static price behind our major finding is constructed using the optimal policy, which we denote by . Recall that are the arrival rates under the optimal policy and are the steady-state probabilities. The single price is simply chosen so that the corresponding arrival rate, , is the same as the expected arrival rate under the optimal policy when units are available. More specifically, the static arrival rate is selected so that
Our proof, that we detail in the next subsection exploits this structure, together with the structure of the underlying birth and death process, to derive a universal guarantee.
The proof is organized around two main steps. In the first step, we exploit the concavity of the revenue rate function (in the quantity space) to establish that for each of the three objectives, the ratio of the performances under the static and optimal policies is at least the ratio of the corresponding service levels. The second step bounds the ratio of the service levels by by enumerating several cases, with each case proven using elementary calculus. A key component of this step is a change of variables from demand rates to the product of demand rates. Both steps fundamentally exploit the explicit construction of in Eq. (9). With some abuse of notation, we index quantities with to denote these under the static policy induced by this static rate.
Step 1. In the first step, we lower bound each of , , and by . Note that by Eq. (7), the lower bound is exact for the service level ratio, i.e.,
For the profit ratio by the ratio, we have
The first and second inequalities follow from algebra. The inequality follows from the fact that the function is concave in and applying Jensen’s inequality to a random variable that takes value with probability for . Note that the expected value of this random variable is exactly by Eq. (9).
Step 2. To find the lower bound of the ratio of stock-in probabilities, we define a set of auxiliary notation which will be useful in our subsequent analysis. We define for and for . For clarity, note that . We also define and .
Using the steady-state probabilities derived in Section id1 and the definition of , the service levels of the static and dynamic policies can be written as
From the above, it is clear that the ratio of the service levels may be written as a function of . We call this function . Formally,
We next develop a uniform lower bound on by developing separate bounds for the cases where is small () or large ().
Step 2a. We prove the lower bound for the cases where is at most 3. When , then and therefore . When , we have
where the inequality follows by matching terms and looking at the minimum ratio. When , the numerator of is
while the denominator of is
By matching terms in the numerator and denominator, it is then clear that
Step 2b. Next, we consider the case where . While the function ratio is difficult to analyze directly, we will derive a lower bound on , which we denote by , which will be amenable to analysis. The bound can be derived simply by observing that
Next, we derive a lower bound on though two subcases, depending on the ratio of to . We first establish in Lemma 2 (proved in the Appendix) that the partial derivative with respect to the first argument is non-negative as long as .
Fix . Fix and suppose , then
Fix . For all , we have
If , then there is no guarantee on the derivative, but one may directly establish a uniform lower bound on as articulated in Lemma 4 (proved in the Appendix).
Fix and suppose , then
Combining both cases, We conclude that . This completes the proof of Theorem 1.
We present an example which shows that the lower bound of in Theorem 1 can be tight for a family of instances. That is, we shall describe instances in which the static policy we construct, , achieves exactly a fraction of the optimal dynamic policy. Namely, we shall fix , , , , to be arbitrarily close to 0, and .
Since , then the objective is to maximize the service level, that is
The service level is always bounded above by 1, and hence it is clear that the policy is optimal since
Now let us consider the static policy which we construct according to Eq. (9). Recall from Section id1 that the performance of the static pricing policy with respect to the service level and market share objectives is , where . In addition, the performance of the static pricing policy with respect to the profit rate is also because is linear in if , which makes the Jensen’s inequality tight in the derivation of Eq. (12). Since , then the ratio becomes
Since and , we have and as , and hence
In this section, we seek to focus more deeply on the profit maximization objective corresponding to . This objective is central in the literature and we aim to understand to what extent can our guarantee from Section id1 be improved.
Fix , any rate and linear demand function . Let denote a revenue maximizing dynamic policy. Then there exists a static pricing policy such that
This result establishes that for profit maximization, a simple static pricing policy guarantees more than 95.5% of an optimal dynamic pricing policy. This is a much higher guarantee than for the general multi-objective case. In particular, for profit maximization, there is extremely limited value in dynamic pricing.
We note that due to the technical difficulty of the analysis, our result is limited to the case with only 2 units (), and when the demand is linear and are linear), a common assumption in both the literature and practice. However, in Section id1, we will see numerically that the level of guarantee above is valid beyond the case and linear demand. In fact, our computational results shows that the 95.5% lower bound holds across all possible instances considered in this paper.
The proof of Theorem 2 is again constructive in that it exhibits a particular static policy with such a guarantee. This policy is the same as the one presented in Eq (9). The proof relies on lower bounding the ratio of the service levels, which is indeed a lower bound on the profit ratio as seen in Eq. (12). Then, the first order conditions of the profit maximization objective are used to impose constraints on the worst-case arrival rates of an optimal policy, which allows us to find a tighter bound on the ratio of the service levels.
Proof. Let be the effective arrival rates under the optimal policy for profit maximization, and be the corresponding optimal prices. Let and . For the static policy, let be defined according to (9). Since , by Eq. (12) we have
Next, we show that is increasing in and decreasing in by simply looking at the first partial derivatives. Taking derivatives of w.r.t and gives
To see that the partial derivative w.r.t. is non-negative, it is sufficient to show that , which follows from the fact that , established in Lemma 1. To see that the partial derivative w.r.t. is non-positive, observe that all terms in the numerator are negative.
The remainder of the proof proceeds by dividing the analysis in two cases: if is above or below . When , then in this case
since is increasing in and decreasing in .
For the remainder of the proof we consider the case where . In this case, we leverage the first-order optimality conditions of the problem to show in Lemma 5 that and must be within a provable quantity of one another. This constraint then allows us to tighten the lower bound on . Denote . Notice that since the demand is linear, then . Define , and now we are ready to state the bounds on and .
Let . Then
By Lemma 5 and the fact that is non-decreasing in we have that
Therefore, minimizing is equivalent to maximizing , for which we provide an upper bound in Lemma 6.
If , then . Furthermore,
In this section, we conduct a set of numerical experiments to test the performance of the static pricing policy. We consider three types of demand functions: linear, exponential, and logistic. For a linear demand curve, we assume it takes the form ; for an exponential demand curve, we assume it follows ; for the logistic demand curve, we assume it is where is the inflection point. Notice that in all three demand curves, the maximum demand rate is set to be when the price is set to 0.
For each value of , we randomly generate the mean usage time uniformly in ; is randomly generated uniformly between 0.1 and 5; is randomly generated uniformly between 0.5 and 10; is randomly generated uniformly between [0,20]. We assume that the average service cost is 0, i.e., . We generate 1,000 different instances of inputs and calculate the profit rate under the optimal dynamic pricing policy, the constructed static price policy according to Eq. (9), and the best static price policy . We report the worst case of and for each capacity level . The results are summarized in Table 1.
As one can observe, the performance of static pricing is generally higher than 97.5%. When , we observe the worst case to be 99.53% in the case of linear demand, which is even higher than the 95.5% guarantee proven in Theorem 2. We also note that this very high performance of static prices continues to hold when we depart from the exact assumptions of Theorem 2, for general values of and for exponential and logistic demand curves.
In general, the worst case performance of static pricing (either the best static price or the price we construct) does not happen when . However, the ratio of the profit rate achieved by the static policy and the optimal profit rate appears to be relatively independent of the value of . Of course, as approaches infinity, the worst case ratio indeed converges to 1.
In addition, one may observe that the performance of the static price policy we constructed in the proofs is very close to the performance of the best static price for profit maximization. The difference of the worst case performance between the two static prices is usually less than 0.2%.
Using a similar testbed, we also conducted numerical experiments for the multi-objective case. We use the linear demand model in the numerical experiment and randomly generate the values of ’s uniformly at random. The rest of the experiment settings are the same as described before. We calculate the worst case performance of our constructed static price compared to the total objective as well as for the three performance metrics. The results are presented in Table 2.