Optimizing Service Operations With Price- and Density-Dependent Demand: A Copula-Based Approach

Abstract

Exclusivity is a key component of certain service offerings such as elite members-only social or country clubs. Indeed, such services may only appeal to customers if they limit the crowd density, that is, the maximal number of customers granted access to the service. We consider a service provider, who sets its crowd density cap in addition to its price. Customers are heterogeneous in both their willingness to pay (WTP) for the service and their density tolerance. Each customer purchases the service only if the price is below her WTP and the density is below her tolerance. Importantly, customers’ WTP and density tolerance may be statistically dependent, and we develop a novel, copula-based framework to model the dependence structure of these two dimensions of heterogeneity. We analytically characterize the provider’s optimal price and density decisions. As long as the customer population is not too severely sensitive to density: (i) the provider optimally serves all segments of the market and sets the price and density so that the price, not the density, is the determining factor for customers (from a particular subpopulation) on the margin between purchasing and not and (ii) as customers’ valuations and density tolerances become more positively dependent, the provider earns higher revenue and optimally increases its service density. By contrast, the optimal price is quasi-convex but not necessarily increasing. On the other hand, with severely density-sensitive customers, it may be optimal to partially cover the market and to make density the determining factor in customer decisions. Our findings offer prescriptive guidelines for service operations in the presence of density-sensitive demand and also provide an explanation for the failed versus successful practices of prominent private clubs.

1. Introduction

Members-only private clubs, a fast-growing service industry, offer a unique experience for customers who crave privacy and exclusivity (Kodé, 2024). These clubs strategically limit the general public’s access to their service, typically through invitation-only membership at an extravagant price (Kendall, 2008). Prominent examples abound and include private social clubs such as Soho House (Burton, 2017) and member-exclusive country clubs such as the (private ski resort) Yellowstone Club (Tong, 2022) and Augusta National Golf Club (Buteau and Paskin, 2015). With high-profile members such as Bill Gates and Warren Buffett at Yellowstone Club and Augusta National, and celebrities such as Kendall Jenner and Harry and Meghan, Duke and Duchess of Sussex, at Soho House (Neate, 2024), privacy and exclusive access to service are naturally paramount for these clubs.

To curate an image of exclusivity, service providers often deliberately cap the crowd density, that is, the maximal number of customers granted access to their service facilities. Yellowstone Club, for example, states that it is “limited to 864 residential properties to protect exclusivity and exceptional, highly personalized service” (Yellowstone Club, 2016). On the other hand, although Soho House may aspire to achieve buzz without busyness, it recently posted a more than $100million annual loss (Neate, 2024). In late 2023, the club announced that it was “closing the doors to new members across our houses based in London, New York, and Los Angeles” (Sperance, 2023), and in a letter to members, founder Nick Jones noted the importance of ensuring that the clubs “don’t feel too busy” (Baldwin, 2024). Prominent investors even publicly questioned Soho House’s future “viability as a public company” in early 2024. In addition to its “persistent non-profitability,” their report notes that Soho House’s “rapid expansion … raises worries about the potential dilution of the exclusive experience” and that “overcrowding decreases member satisfaction, potentially eroding the unique ambiance” (GlassHouse Research, 2024). In other words, Soho House has struggled to achieve enough volume to be profitable without jeopardizing its aura of velvet-rope exclusivity and privacy.

Just as a recent New York Times article (Kodé, 2024) argued, “while people who join private clubs often seek intimacy, personalized treatment and a feeling of exclusivity, the clubs usually seek profitability and increased membership.” Service providers such as the above-mentioned private clubs must now assess their customers’ sensitivity to the crowd density, namely their willingness to accept the provider’s announced density cap, in addition to their willingness to pay (WTP) for the service. That is, the service provider must choose at what density cap to operate in addition to setting the price. For instance, after extending access to its lounges through a credit card partnership, Delta Air Lines “has become the poster child for overcrowded airport lounges” (Potter, 2022). Such overcrowdedness forced Delta to revise its access policy to allow only flyers of certain membership status tiers to gain access within a limited 3-h window before their departure, among other changes (Delta Airlines, 2024). However, the airline has had to backpedal on some of the changes to its access policy due to the significant backlash by its customers (Rubio, 2023). As illustrated by the woes of both Delta and Soho House, the service density decision involves a subtle tradeoff between how much crowdedness customers can tolerate and how much they are willing to pay for the service.

Moreover, different customer groups likely also differ in their density tolerance and WTP, which can exhibit a nontrivial relationship. For example, business travelers, who have higher WTP, prefer a less crowded and quieter airport lounge to get work done, while family travelers, who are typically budget-constrained, may not be as sensitive to the crowd density (O’Shea and Etzel, 2024). This may suggest a negative correlation between the WTP and the density tolerance. However, in other cases, the correlation may be positive. For instance, for a major live sporting event or marquee concert, devotees of the sport or musical artist should have a higher WTP for a ticket. These same devotees are often willing to tolerate large crowds in order to see their favorite team or artist in person, ¹ whereas a casual fan, by comparison, will likely have both lower WTP and lower crowd density tolerance. This may suggest a positive correlation. Such a correlation between the WTP and the sensitivity to the provider’s announced density limit can be a major consideration for service providers when making their pricing and other operational decisions. To further illustrate, consider Alaska Airlines, which adopted an unusual policy during the COVID-19 pandemic. It blocked middle seats in Premium Class—which has extra legroom and can be purchased for an extra fee—but not in the rest of the plane (Griff, 2021). Alaska seems to have judged that customers in different cabins have different density preferences, on top of their differences in WTP. Specifically, since it blocked middle seats only in Premium Class, Alaska may have determined that the passengers willing to pay for Premium Class tended to be more sensitive to the crowd density limit and that their demand would be more elastic to a change in the middle-seat-blocking policy than would demand for Main Cabin seats.

Also related to COVID-19, it is worth noting that customer density sensitivity arises beyond just the desire for exclusivity. Many service operations such as dine-in restaurants (Rubin and Maddaus, 2020) and public transportation (BTS, 2020; Kaufman et al., 2020) ground to a halt during the outbreak of the pandemic. Upon reopening, service providers did and continue to wrestle with a vexing problem: how to successfully operate a service business with customers who have become notably sensitive to crowd density due to health and safety concerns. For example, in contrast to other U.S. airlines (Honig, 2021), Delta Air Lines continued voluntarily blocking middle seats on its aircraft until 1 May 2021 (Pallini, 2021) to give its customers “more peace of mind” (Delta, 2021).

Motivated by these practical settings, our main research question asks how a profit-maximizing service provider should price and set its density cap in the presence of customers’ heterogenous density tolerance (i.e., their maximum acceptable density) and, in particular, its dependence with their WTP. In the above examples, the density tolerance is likely to be “rigid” for each customer in that it is a hard threshold above which she is unwilling to accept the service. Indeed, extremely low density is the ultimate determinant for celebrities or jet-setters, who value privacy and exclusivity the most, to join elite private social or country clubs. Referring in a New York Times interview to the celebrity membership of his ultra-exclusive San Vicente Bungalows (SVB) in Los Angeles— which includes, for example, Taylor Swift and Elon Musk (Kay, 2023)—Jeffrey Klein notes that “Privacy is the new luxury” and describes the club as “an oasis” (Trebay, 2019). The article observes that “Mr. Klein understood better than most that the one craving that famous people cannot satisfy is to get their anonymity back,” and SVB allows them to do so. Maintaining this aura of pampered privacy is critical to this and similar clubs’ success, and if a privacy-seeking celebrity member resigns from SVB because management increases the density cap, then a reduced price will likely be completely ineffective at wooing her back. In other words, an individual customer deems WTP and density tolerance as nonsubstitutable: each customer will make the purchase if and only if both (i) the price is below her valuation and (ii) the density cap is below her tolerance. Demand in this context may not be adequately modeled in a conventional way, whereby the service provider can lower the price to induce customers to accept the service at a higher density. In particular, as a novel feature in this work, we model customers’ preference heterogeneity and the interdependence between density tolerance and WTP through a parametric family of copulas that naturally captures a continuum of dependence structures including three polar dependence structures: both perfect positive and negative dependence as well as independence.

For a wide range of distributional parameters that exclude only the most severely density-sensitive customer populations, we provide the complete characterization of the provider’s optimal strategy, which always takes a “full-coverage, price-active” structure. A “full-coverage” strategy induces positive demand from all three clusters of polar dependence structure, despite this requiring it to accommodate the demanding customers from the negatively dependent cluster, who have high WTP only if their density tolerance is very low (and vice versa). Under a “price-active” strategy, the demand from the cluster of perfect positive dependence is determined only by the price and not by the density. In other words, for those lucrative customers who have both high valuation and high-density tolerance, it is better for the provider to price some of them out (with a high price) rather than to crowd some out (with a high density).

Furthermore, we find that as the two preference dimensions become more positively dependent, both the optimal density and revenue increase, but not necessarily the optimal price, which can be increasing, decreasing, or first decreasing followed by increasing. We characterize the parametric condition for these distinct price movements, which we show is driven by the provider’s intention to maintain the balance between the demand and the density. In particular, the optimal price is shown to be higher than the price that the provider should charge when customers are insensitive to the service density (i.e., have only a one-dimensional preference for WTP as in the classical model). The positive dependence between the customer valuation and density tolerance exerts a direct positive and an indirect negative effect on the optimal price. For a given price–density pair, more positive dependence between these two dimensions enhances the demand, calling for a higher price to rebalance the demand with the density if the latter is fixed. Besides this direct positive effect, however, more positive dependence also incentivizes the provider to increase the density cap, lowering the demand if the price remains unchanged. Again, to rebalance the demand with the density, the provider now has the incentive to lower the price, resulting in an indirect negative effect. When customers’ density tolerance is more evenly distributed (e.g., follows a uniform distribution), the direct positive effect dominates the indirect negative effect, leading to an increasing optimal price as the two preference dimensions become more positively dependent. On the other hand, when customers’ density tolerance is sufficiently concentrated on the higher level (i.e., a left-skewed density tolerance distribution), the indirect negative effect is dominant, and hence the monotonicity of the optimal price is reversed (i.e., decreasing as the dependence becomes more positive). Naturally, the optimal price exhibits nonmonotone behavior in between these two regimes.

The managerial implications of our findings abound. In general, we provide guidelines for service providers on how to make price decisions and set the density limit when facing density-sensitive customers with dependent valuations and density tolerances. In reality, it is likely that the extremely density-sensitive customers are not the majority in the market, suggesting a more uniform or left-skewed distribution of the population density tolerance where our above-mentioned results apply. Indeed, as an illustrative example from another context with density sensitivity, we leverage a real dataset (on train travel decisions during COVID-19 in the Netherlands) to numerically validate the left-skewness of the density tolerance distribution. For exclusivity-driven service businesses, the customers’ density tolerance will likely feature similar left-skewness as the most zealous pursuers of extreme exclusivity such as celebrities or other ultra-wealthy individuals are a small minority of the population. Furthermore, those individuals typically also exhibit higher WTP, implying a negative dependence between valuation and density tolerance. Accordingly, our results suggest that the operator of an exclusive country club or social club will optimally operate with a very low-density cap and charge a high price. These findings offer a potential explanation for the struggles of Soho House in recent times and possibly also for Delta’s troubles with its Sky Club. If the density cap is not set low enough, then too many of the customers with very high WTP—who, under negative dependence, are likely to have very low density tolerance—will deem the service not sufficiently exclusive and will decline to purchase. Our prescriptive guidance for these service providers would thus be to lower the density cap and raise the price. Indeed, a number of prominent private clubs are known for the notorious gatekeeping strategy of imposing a hard (low) density cap and charging an exorbitant membership fee; according to our results, this aligns with the optimal strategy given a customer base that exhibits negative dependence between valuation and density tolerance. For instance, Yellowstone Club has an explicit cap of 864 resident members, and “there’s a long waitlist to buy a $22 million condo at the Yellowstone Club, the only way to become a member at the ultra-exclusive private ski resort” (Tong, 2022). Similarly, The Madison Club, situated within an exclusive private residential community in the resort town of La Quinta, CA (near Palm Springs), reportedly has limited its membership to only 225 and charges a one-time initiation fee of more than $200,000 with annual membership dues of $46,000. ² On the other end of the spectrum, patrons of public nightclubs and bars may exhibit less negative dependence between their WTP and density tolerance, which may even be positively dependent. Think of two types of club-goers in this case: those “singles” who are seeking a romantic relationship or to expand their social network, and those “couples” who are already in a relationship and have less desire to socialize with strangers. While overcrowding (i.e., high density) is unpleasant for both types of customers, singles, compared with couples, are likely to have higher WTP for the service and have a higher tolerance for the crowd density due to their needs to socialize. As such, the club-goers’ WTP is likely positively associated with density tolerance. ³ For these service settings, our results suggest that the service providers should be less exclusive and charge lower prices (than those where customers exhibit more negative dependence between WTP and density tolerance), consistent with our anecdotal observations.

Finally, we characterize the optimal policy for the remaining range of distributional parameters, which reflects a severely density-sensitive population. In this regime, the full-coverage, price-active strategy is not always optimal. Instead, it can be optimal to only partially cover the market without serving the cluster with negative dependence, and/or to use density instead of price to regulate the demand from the cluster with positive dependence. In particular, partial coverage is optimal only when customers are extremely density-sensitive. Under either full or partial coverage, density (respectively, price) should be the active lever when customers are highly (respectively, relatively less) sensitive to the density cap.

2. Literature Review

As elaborated below, our work speaks and contributes to multiple research streams.

As discussed in Section 1, density sensitivity is exemplified by luxury services. In economics, there is a small stream of literature on the theory of clubs (e.g., swimming pools) initiated by Buchanan (1965) and continued by Scotchmer (1985) and others (see Sandler and Tschirhart, 1997 for a survey). This literature examines a competitive setting where customers are typically homogeneous and their utility depends additionally on the service’s congestion level. In contrast, we model customer heterogeneity both in their WTP and their sensitivity to density and explore their dependence structure.

Operations literature has a long history of studying queuing-based service systems. Recently, the approach has been applied to various applications including infection-averse customers (Hassin et al., 2023) and time-based pricing (Tang et al., 2023). More broadly, in queues with strategic customers, dating back to Naor (1969), the majority of studies model delay-averse customers, who exhibit avoid-the-crowd behavior and hence prefer to join services with shorter queue length. In such studies, valuation and delay sensitivity tend to be modeled as independent from each other. Importantly, although delay sensitivity seems to resemble our notion of density sensitivity, two important distinctions exist. First, the delay sensitivity in queuing-based models refers to customers’ waiting time, while our density sensitivity refers to the simultaneous crowdedness present in a service setting. Second, preferences for time and money are substitutable in the delay-sensitive models by monetizing the waiting time as a cost subtracted from the valuation of the service, whereas our model explicitly separates the WTP and density sensitivity in a nonsubstitutable way and treats them as two heterogeneous preference dimension with a rich dependence structure.

Multidimensional preferences, though understudied in service settings, have been featured prominently in the literature of product line design, whereby products exhibit multiple quality traits. Representative works include Chen (2001), Lacourbe et al. (2009), Lauga and Ofek (2011), Liu and Shuai (2019), Lin et al. (2020), and Rashkova and Dong (2022). Common in this stream of research is the substitutability between different quality dimensions in that consumers’ utility along one quality dimension can be compensated by another dimension. Furthermore, the interdependence between different heterogeneous preference dimensions has not yet been explored. By contrast, we consider a customer’s “rigid” preferences, whereby a purchase is made only if both dimensions of her preferences, namely the price and density, are satisfied. In addition, we investigate the effects of their dependence structure on the service provider’s optimal strategies.

To model such dependence structure between heterogeneous preferences, we leverage the notion of copulas (see Nelsen, 2007 for a general introduction), whose applications in operations management seem to be limited, apart from a few notable studies. Wang and Dyer (2012) develop a copulas-based framework to model dependent uncertainties in a decision tree. They illustrate their methods using the elliptical family (e.g., normal copulas and $t$ -copulas) as well as the Archimedean family (e.g., Clayton copulas, Gumbel copulas, and Frank copulas). We focus on the Fréchet family of copulas (see, e.g., Dhaene et al., 2002a,b; Embrechts et al., 2002; Yang et al., 2006, for its applications) but also verify the robustness of our findings on Archimedean copulas. Clemen and Reilly (1999) study the use of copulas to model dependence among variables in a decision analysis context and illustrates their use in conducting risk analysis for an airline. Related, Bagchi and Paul (2014) apply copulas to the optimal allocation of screening resources in airport security. Motivated by practical concerns in agribusiness (Bansal et al., 2017), Bansal and Wang (2019) use copulas to approximate joint distributions for the uncertainty in subjective expert judgments of the means and standard deviations of random variables. From a decision-theoretic perspective, Abbas (2009) proposes the concept of utility copulas (as inspired by the Sklar copulas from multivariate statistics) to build a general framework to model preferences over-dependent attributes. Our framework is based on the classical notion of Sklar copulas as we intend to incorporate preference heterogeneity and its dependence structure. In addition, our focus is on the service provider’s decision problem rather than consumers’.

3. Model

We consider a market with a monopolistic service provider (e.g., a members-only social or country club, serving a specific area). The provider must determine a price $p\ge 0$ and a density level $c\ge 0$ , where $c$ specifies the number of customers the provider can serve simultaneously (the provider’s decision is a cap on the density, which in principle could be realized either below or equal to the cap). Motivated by the service contexts discussed in the introduction, we consider a setting where the provider has already incurred the fixed cost associated with offering up to an established maximum density, and hence the provider can choose any density $c$ up to the established maximum without incurring any additional cost. For example, the cost of upkeep of an existing golf course at a country club is relatively inelastic to the number of members playing the course. (Similarly, as discussed in the introduction, the cost of operating a given commercial flight is not very sensitive to whether the middle seats are occupied.) For a specified density and price, the actual units sold (the sales) are the minimum of the density and the demand at that density and price.

The customer population consists of a continuum of unit mass. First, each customer has a valuation for the service, represented by a random variable $V$ (following the convention in the literature, we denote all random variables in upper case and their realized values as well as other model parameters in lower case). The valuation is essentially the maximum price that a customer is willing to pay for the service. Second, each customer possesses a certain density tolerance, namely the density at the service facility below which she is willing to accept the service and above which she will decline the service. For a private social or country club, the density tolerance would be the number of members above which a prospective member deems the service to be not exclusive or private enough and will decline to join. On the other hand, in an airplane or movie theater with density-sensitive customers who fear for their safety, a customer’s density tolerance would reflect the maximum proportion of seats occupied for which the customer would be willing to board the plane or sit in the theater. We denote this quantity by another random variable $S$ . In short, valuation determines whether the customer is willing to pay the service price set by the provider, but density tolerance determines whether the customer is willing to accept the service at the provider’s chosen density. Finally, we normalize the supports of both $V$ and $S$ (thus also the price $p$ and density $c$ ) to be between 0 and 1. This normalization is without loss of generality relative to support from zero to any finite upper bound because the rigid preferences allow us to rescale the units without translating between them, unlike in the case of substitutable preferences. Thus, our qualitative insights and comparisons also apply to market size or with a maximum WTP that is larger or smaller than 1, and the numerical solutions (optimal price, density, and revenue) for different unit scales can be obtained as scalar multiples of our solutions.

As a novel feature in our setting, customers demonstrate rigid preferences in that they will purchase the service if and only if the price does not exceed their valuation of the service (i.e., $V\ge p$ ) and the provider’s maximum density does not exceed their tolerance (i.e., $S\ge c$ ). This specification captures the setting where customers’ purchase decisions are primarily driven by the provider’s announced density limit rather than the realized density, that is, the actual number of customers who decide to purchase the service. The realized density is difficult for customers to observe or infer prior to consuming the service due to the lack of information. This is particularly apt for exclusivity-driven services where perceived exclusivity is indeed created by the provider’s announced density cap (Yellowstone Club, 2016); for a pandemic setting, it reflects customers’ conservatism and their most pessimistic estimate of the realized density. Accordingly, the demand is given by

$$D(p,c)=\mathsf{P}(V\ge p,S\ge c).$$

(1)

Even though price and density appear to operate similarly from the customer’s point of view, there is an important difference from the provider’s perspective. To wit, the price enters the objective function of the firm by itself as well as through demand, but the density enters into the firm’s objective only through the demand. The effects on the firm of using density versus price to regulate the demand are thus different, and we will show that which one is used in the optimal strategy depends on the density tolerance distribution.

Model Dependence Structure via Copula

The above-mentioned characteristics may not be independent of each other. For instance, customers with higher WTP may tend to have lower density tolerance, or vice versa. We are interested in the role that the dependence structure between $V$ and $S$ plays in determining the provider’s price and density. To model the wide range of possible dependence structures, we use copulas. By Sklar’s theorem (Nelsen, 2007: Theorem 2.10.9), any joint probability distribution can be represented by a copula, which is a tool to model a multivariate distribution that cleanly separates the marginal distributions from the dependence structure. The copula offers a general, flexible approach to model dependence among random variables. Specifically, in our setting with two random variables, a copula can be defined as follows. Let $F$ and $G$ denote the continuous, strictly increasing marginal cumulative distribution functions (CDFs) of customers’ valuation $V$ and density tolerance $S$ , respectively (when convenient, and with a slight abuse of terminology, we will at times use the word “distribution” to refer to $F$ or $G$ ). A copula in two variables is defined as a bivariate CDF $C(u_1,u_2)$ whose marginal distributions are both standard uniform. A copula $C$ induces a joint distribution for $V$ and $S$ through

$$\mathsf{P}(V\le v,S\le s)=C(F(v),G(s)).$$

As noted, Sklar’s theorem implies that any bivariate joint distribution can be represented by the above equation with an appropriately chosen copula $C$ .

We now express the demand function in (1) for a given copula $C$ that specifies the dependence structure between valuation $V$ and density tolerance $S$ . Applying the inclusion–exclusion principle, for a chosen price $p$ and density $c$ , we have

$$\begin{array}{rl}D(p,c)& =\mathsf{P}(V\ge p,S\ge c)\\ & =1-F(p)-G(c)+\mathsf{P}(V\le p,S\le c)\\ & =1-F(p)-G(c)+C(F(p),G(c)).\end{array}$$

(2)

It is straightforward to verify that $D(p,c)$ is nonincreasing in each of its arguments. Throughout, we make the additional assumption that $D(p,c)$ is strictly decreasing in $c$ for each $p\in [0,1)$ and in $p$ for each $c\in [0,1)$ .

Provider’s Problem

Given the provider’s decision $(p,c)$ and the induced demand $D(p,c)$ given in (6), the actual service units sold is equal to $min\{c,D(p,c)\}$ . The provider’s revenue is then given by

$$R(p,c)=pmin\{D(p,c),c\},$$

(3)

and its optimization problem is

$$\underset{p,c\in [0,1]}{max}R(p,c).$$

(4)

We denote the service provider’s optimal price and density by $p^{*}$ and $c^{*}$ .

3.1. Preliminary Result

We now establish a crucial property of the provider’s density decision and the induced demand, which allows us to further simplify the provider’s problem. The following lemma demonstrates that it is optimal for the provider to induce demand equal to its chosen density.

Lemma 1 (Balance density and demand)

The provider’s optimal strategy $(p^{*},c^{*})$ must satisfy $c^{*}=D(p^{*},c^{*})$ .

To understand this result, we note that for a fixed price, demand is decreasing in the density $c$ . Starting from the full density (i.e., $c=1$ ), the demand will be zero for any price because density tolerances are continuously distributed over $[0,1]$ . If the provider decreases the density from 1, then its demand will increase, and so will its sales, which are the minimum of demand and density. As long as the density is strictly larger than the demand, further decreasing density will continue to increase sales, which will also increase revenue since the price is kept constant. Once density and demand are equal, to decrease density further would continue to increase demand, but it would now decrease sales because the sales are the minimum of demand and density. Thus, it is always optimal to exactly balance demand and density. ⁴

Intuitively, the lower the membership cap is at a private club, the more private and exclusive the club is and thus the more people would desire to join, but by definition, the membership cap would keep the club from admitting all of these potential members. Similarly, in a pandemic, it might be that 80% of potential customers would feel safe flying on an airplane or visiting a movie theater that is operating at 20% density, but at 20% density the airplane or theater cannot serve all of these customers. In either case, it is optimal for the provider to lower the density just enough that it can serve exactly as many customers as wish to purchase; otherwise, it is either under or over-serving the addressable market. The density level that achieves this outcome will be different for different prices, as demand decreases in price. Using Lemma 1, we essentially reduce the provider’s problem that originally contained two decision variables into a problem with only one decision variable (we find it convenient to use the density $c$ as our decision variable where the price $p(c)$ is set to achieve $c=D(p,c)$ , but a similar approach could be followed with $p$ and $c(p)$ ). In the following sections, we will apply this result to help us solve for the provider’s optimal policy.

3.2. The Fréchet Copula

In this paper, we will primarily focus on a particular family of copulas—the Fréchet copula—to model the dependence between valuation $V$ and density tolerance $S$ . It is a well-established property that any bivariate copula $C$ with marginal distributions $F(p)$ and $G(c)$ is bounded between the Fréchet–Hoeffding bounds representing two extremes of dependence (Nelsen, 2007: Theorem 2.2.3). One represents perfect positive dependence given by

$$M(F(v),G(s))=min\{F(v),G(s)\},$$

and the other represents perfect negative dependence given by

$$W(F(v),G(s))=max\{F(v)+G(s)-1,0\}.$$

Then, the copula $C$ is bounded between these, that is, $W(F(v),G(s))\le C(F(v),G(s))\le M(F(v),G(s))$ . If copula $C$ coincides with $M$ (respectively, $W$ ), then $V$ is almost surely an increasing (respectively, decreasing) function of $S$ (Nelsen, 2007: Section 2.5). If $V$ and $S$ are completely independent, the corresponding copula $C$ can be represented by $\mathrm{\Pi }(F(v),G(s))=F(v)G(s)$ .

Fréchet’s (1958) family of copulas is defined as a convex combination of the three copulas $M(F(v),G(s))$ , $W(F(v),G(s))$ , and $\mathrm{\Pi }(F(v),G(s))$ . Formally, a Fréchet copula can be expressed as

$$\begin{array}{rl}& C(F(v),G(s))\\ & =\alpha M(F(v),G(s))\\ & +(1-\lambda )\mathrm{\Pi }(F(v),G(s))+(\lambda -\alpha )W(F(v),G(s)),\end{array}$$

(5)

for two parameters $\alpha$ and $\lambda$ such that $0\le \alpha \le \lambda \le 1$ . These parameters characterize the dependence structure intrinsic to the copula itself. Importantly, for the Fréchet copula, the commonly used rank correlation measures Spearman’s $\rho$ and Kendall’s $\tau$ are uniquely determined by $\alpha$ and $\lambda$ (regardless of the marginal distributions), and both are increasing in $\alpha$ for given $\lambda$ (see, e.g., Nelsen, 2007: Examples 5.3 and 5.6). Accordingly, a situation in practice with strong negative dependence, such as the airport lounge example from the introduction, would correspond to a very low value of $\alpha$ . On the other hand, a practical situation with positive dependence, such as the concert/sporting event example from the introduction, would correspond to a high value of $\alpha$ .

We focus on the Fréchet copula for three main reasons. First, the Fréchet copula is intuitive and simple to interpret. It is essentially a weighted average of three polar dependence relationships, namely perfect positive dependence, perfect negative dependence, and independence, including all three polar dependence relationships as special cases. The weights are determined by the two parameters $\alpha$ and $\lambda$ , with a lower value of $\lambda$ indicating more independence between customer valuation $V$ and density tolerance $S$ and a higher (respectively, lower) value of $\alpha \in (0,\lambda )$ indicating more positive (negative) dependence. (For ease of exposition, we will subsequently consider nontrivial mixtures of the three polar dependences with $0<\alpha <\lambda <1$ .) The demand function (2) under the Fréchet copula is

$$\begin{array}{rl}D(p,c)& =\mathsf{P}(V\ge p,S\ge c)\\ & =\alpha min\{1-F(p),1-G(c)\}\\ & +(1-\lambda )(1-F(p))(1-G(c))\\ & +(\lambda -\alpha )max\{1-F(p)-G(c),0\}.\end{array}$$

(6)

With this parameterization, the customer population can be visualized as if it consists of three clusters: $\alpha$ proportion of customers exhibit perfect positive dependence between valuation and density tolerance, $1-\lambda$ proportion of customers have independent valuation and density tolerance, and $\lambda -\alpha$ proportion of customers exhibit perfect negative dependence between these two preference dimensions. However, we emphasize that these “sub-populations” are used only for interpretation purposes: customers do not necessarily come from these three literal sub-populations.

Second, the parameterized form of the Fréchet family offers a simple and tractable way to conduct sensitivity analysis, based on which we will draw important managerial and policy implications. As the Fréchet copula is completely determined by parameters $\alpha$ and $\lambda$ (given the marginal distributions $F$ and $G$ ), this allows us to trace the optimal price and density decisions and the optimal revenue as the dependence structure changes by maneuvering the values of these two parameters, albeit with highly nontrivial and intricate analysis. Such comparative static results would be difficult to obtain for copulas of other forms. Nonetheless, we numerically verify the robustness of our results for other families of copulas in Section 7. ⁵

Last but not least, the Fréchet family of copulas proves to be able to approximate any bivariate copula in a unique way and the error bound can be estimated (Yang et al., 2006). More precisely, Yang et al. (2006) showed that any bivariate copula can be decomposed uniquely as a convex combination of a Fréchet copula and an indecomposable copula, which can be approximated again by a Fréchet copula.

4. Solving the Provider’s Problem

We now proceed to solve the provider’s optimization problem. We consider a wide range of distributions for the density tolerance, focusing on power distributions $G(c)=c^k$ for $k>0$ . We begin in Section 4.1 by introducing the benchmark problem in which customers are sensitive only to the price but not to the density; this benchmark is meaningful in practice to compare against settings where customers do not exhibit density sensitivity. Then, in Section 4.2, we consider the provider’s problem for density tolerances that follow a power distribution with $k\ge 1$ , which covers the most plausible degrees of customer sensitivity to density. This leaves only severely density-sensitive customer populations ( $k<1$ ), which we study in Section 5. Throughout the remainder of the paper, we consider customer valuations that follow a standard uniform distribution, that is, $F(p)=p$ . In Appendix A.2 of E-Companion, we relax this assumption and the power distribution assumption on $G$ , and we show that our main findings generalize to any $F$ and $G$ that satisfy appropriate conditions.

4.1. Benchmark Problem Without Density Sensitivity

To see the impact of customer density sensitivity on the provider’s optimal decisions, we introduce a benchmark case in which customers are completely insensitive to the density level (i.e., the marginal distribution $G$ becomes degenerate with a probability mass concentrated at 1). This benchmark is useful in practice to compare outcomes with versus without density sensitivity. Without density sensitivity, in the benchmark, the demand only depends on the price and reduces to $D^{\circ }(p)=1-F(p)$ . In this case, for a given price $p$ , any density that falls between $1-F(p)$ and 1 is optimal for the provider. Without loss of optimality, we simply set the density equal to the actual demand, that is, $1-F(p)$ . Therefore, the provider’s benchmark problem is

$$\underset{p\in [0,1]}{max}{R}^{\circ }(p)=p(1-F(p)).$$

(7)

We denote the optimal price in the benchmark problem (7) by $p^{\circ }$ . The price $p^{\circ }$ implies an optimal density equal to $c^{\circ }=1-F(p^{\circ })$ in the benchmark problem.

4.2. Provider’s Optimal Solution Under Intermediate or Mild Density Sensitivity

We now solve the provider’s problem under a broad range of distributions for customer density tolerance. Specifically, we focus on a power distribution, that is, $G(c)=c^k$ with $k\ge 1$ (we will later consider the case of severely density-sensitive customers, i.e., $k<1$ ). (For $k=1$ , the power distribution corresponds to the standard uniform distribution.) As $k$ increases, the power distribution becomes larger in the usual stochastic order with the probability mass shifting to higher values of the density tolerance; that is, the customers become less sensitive to the density. In the extreme when $k\to \mathrm{\infty }$ , the power distribution becomes degenerate with all the probability mass concentrated at 1, in which case customers become completely insensitive to density.

We first show that it is optimal for the provider to follow a certain strategy, as evidenced by the next lemma.

Lemma 2 (Full-coverage, price-active strategy)

For density tolerance with the CDF $G(c)=c^k,k\ge 1$ , the optimal price $p^{*}$ and density $c^{*}$ satisfy $G(c^{*})\le F(p^{*})\le 1-G(c^{*})$ .

A key to understanding Lemma 2 is the impact of the inequalities on the service demand. First, the right inequality in the lemma implies that it is optimal to set price and density to satisfy $F(p)+G(c)\le 1$ . By equation (6), this implies a nonnegative (strictly positive, if the inequality holds strictly) demand is induced from customers whose valuation and density tolerance are negatively dependent. This cluster of customers is the most difficult for the service provider to attract because those willing to tolerate higher density are not willing to pay high prices. Lemma 2, however, informs us that it is optimal not to completely exclude such customers and the provider should set its price and density levels to obtain a full coverage of the market. The opposite of a full-coverage strategy is a partial-coverage strategy, in which the negatively dependent cluster does not purchase at all. When the partial-coverage strategy is optimal, it can be verified that the conditional joint distribution for the valuation and density tolerance for the purchasing customers (i.e., the joint distribution whose CDF is given by $\mathsf{P}(V\le p,S\le c|V\ge p^{*},S\ge c^{*})$ ) is positively dependent. Specifically, the rank-correlation measure Kendall’s $\tau$ —see, for example, Nelsen (2007: Section 5.1)—can be verified to be positive.

Regarding the left inequality, we recall from (6) that the demand from customers with perfectly positive dependence between their valuation and density tolerance is determined by $min\{1-F(p),1-G(c)\}$ and hence can be regulated by either the density $c$ or the price $p$ . Lemma 2 finds it optimal to set $G(c)\le F(p)$ , suggesting that the price acts to regulate the demand from this cluster of customers (i.e., $min\{1-F(p),1-G(c)\}=1-F(p)$ ). Thus, we refer to it as a price-active strategy, with the understanding that the density still plays a role in determining the demand from the negatively dependent and independent clusters.

Using Lemma 1, we can reduce the problem to a single-variable optimization over the density $c$ , with the price $p(c)$ set to achieve $c=D(p,c)$ . By solving this reduced problem, we obtain a key result, namely the complete solution to the provider’s optimization, which applies over the entire space of distributions considered in this section.

Proposition 1 (Optimal strategy)

For density tolerance with the CDF $G(c)=c^k,k\ge 1$ , the provider’s optimal density $c^{*}\in (0,1)$ is given by the unique solution in $c$ to

$$\begin{array}{rl}& 1-2c+(1-\alpha )(1-\lambda )c^{2k}-(k-2)(1-\lambda )c^{k+1}\\ & +c^k(\alpha +\alpha k-k\lambda +\lambda -2)=0.\end{array}$$

(8)

The optimal price is given by ⁶

$$p^{*}=\frac{1-(1-\alpha )(c^{*}{)}^k-c^{*}}{1-(1-\lambda )(c^{*}{)}^k},$$

(9)

and the optimal revenue is $R(p^{*},c^{*})=p^{*}{c}^{*}$ .

This result allows us to compare the provider’s optimal price and density with the optimal solutions of the benchmark problem (7). The following corollary reveals that the provider sets a higher price and a lower density when customers are density-sensitive.

Corollary 1 (Comparison with benchmark)

For density tolerance with the CDF $G(c)=c^k,k\ge 1$ , the optimal price is higher than the benchmark price, and the optimal density is lower than the benchmark density, that is, $p^{*}\ge p\circ$ and $c^{*}\le c\circ$ .

It is perhaps intuitive that the provider should operate at a lower density when customers are density-sensitive than in the benchmark case, that is, $c^{*}\le c^{\circ }$ . However, it is less obvious that $p^{*}\ge p^{\circ }$ . We can understand why by looking closer at the demand function. For any $c>0$ and $p<1$ , the demand is strictly less with density sensitivity than in the benchmark problem. For a privacy- or exclusivity-based service such as a private club, this decrease in demand stems from the desire for an “elevated” or “bespoke” experience catering to a privileged few. In a related but slightly distinct sense, for an airport lounge, it reflects customers’ preference for peace and quiet, personal space, and less competition for limited resources such as comfortable chairs and charging outlets. In addition to changing the density, the service provider can also modify the price to adjust to density-sensitive customers. One option is to lower the price—relative to the benchmark $p^{\circ }$ —to recover some customers. The other is to increase the price to earn more revenue per customer. It is not clear a priori which option leads to higher total revenue.

Corollary 1 reveals that in the face of customer density sensitivity, a provider starting from the benchmark price should increase the price and serve fewer (but more lucrative) customers. This is partially because the preference dimensions are not substitutable from the customer’s perspective. In an alternative model where congestion was translatable into monetary units, any and all customers could be recovered by sufficiently lowering the price. By contrast, in our setting, lowering the price (without lowering the density) cannot recover the customers for whom the density is unacceptable. Since a lower price can only attract those who were priced out but were willing to accept the service at the chosen density, it is better to raise the price above the benchmark and serve fewer, more lucrative customers. This finding is also consistent with anecdotal evidence from practice. As we have previously described, the prices charged by elite private clubs are often exorbitant, and this phenomenon is especially apparent when compared to their “public” counterparts (e.g., public restaurants and public golf courses); this aligns with our finding, since the latter serve a customer population that can be considered closer to the benchmark without density sensitivity (relative to customers of private clubs).

Next, we prove the monotonicity properties of several key quantities.

Proposition 2 (Monotonicity)

For density tolerance with the CDF $G(c)=c^k,k\ge 1$ , the optimal density, sales, and revenue are all increasing in the degree of positive dependence $\alpha <\lambda$ for fixed $\lambda$ .

Figure 1 plots the optimal price, density, and revenue as functions of $\alpha$ for different values of $\lambda$ in the special case of uniform $G$ (i.e., $k=1$ ). It can be easily seen that the optimal density and the optimal revenue increase monotonically in $\alpha$ (as does the price in this case, but not in general: see Proposition 3 and Figure 2 below). The intuition behind the monotonicity of the optimal revenue and optimal density is as follows. As the WTP and density tolerance become more positively dependent, the fraction of customers with high realizations for both quantities increases, and therefore the demand will increase for all $(p,c)$ pairs (although the sales will only weakly increase because the demand could exceed the density). For $\alpha ={\alpha }_1$ , consider the optimal price–density pair $(p_1^{*},c_1^{*})$ and the corresponding revenue. At this price and density, for $\alpha ={\alpha }_2>{\alpha }_1$ , the sales will be weakly higher, and thus the revenue will also be weakly higher than for $\alpha ={\alpha }_1$ , implying that the optimal revenue must be increasing in $\alpha$ . Moreover, because the demand will be higher for each price–density pair, the service provider can afford to set a higher density and still achieve $c=D(p,c)$ ; hence, the optimal density (and optimal sales since the two must be equal at optimality by Lemma 1) is also increasing in $\alpha$ .

Figure 1.

Provider’s optimal solution versus $\alpha$ for $\lambda =0.9$ with uniform $F$ and $G$ (i.e., $k=1$ ): (a) price, (b) density, and (c) revenue.

Figure 2.

Optimal price versus $\alpha$ for $\lambda =0.5$ ( $\underset{\_}{k}\approx 1.035$ , $\overline{k}\approx 1.414$ ) with $G(c)=c^k$ : (a) $k=1.03$ , (b) $k=1.26$ , and (c) $k=1.43$ .

Intuitively, an increase in $\alpha$ suggests a more positive dependence between customers’ valuation and density tolerance, thus increasing the demand for a given price–density pair. If the true positive dependence parameter is ${\alpha }^{\prime }+ϵ$ , and if the provider sets the price and density at their optimal values based on a positive dependence parameter ${\alpha }^{\prime }$ , then his revenue will be the same as the optimal revenue for ${\alpha }^{\prime }$ , but the demand will exceed the set density. From this price–density pair, the provider can increase his revenue by slightly increasing the density and keeping the price fixed; this will achieve more sales at the same price.

From the axis scales in Figure 1, we also see that the optimal price is relatively less sensitive to changes in the dependence parameters, while the optimal density cap appears sensitive. This suggests that, in practice, the right choice for the provider’s density cap critically depends on an accurate measurement of the dependence parameters in the customer population. We also note that the optimal price is smooth in $\alpha$ . See Appendix F of E-Companion for other examples with different fixed $\lambda$ values, as well as plots with changing $\lambda$ for fixed $\alpha$ . We additionally observe there that the optimal density is particularly sensitive to the value of the positive dependence parameter $\alpha$ , and less so to the independence parameter $\lambda$ .

Proposition 2 shows the monotonicity of the optimal density and revenue but does not offer a conclusion about the optimal price. Indeed, the optimal price may not be monotonic and may demonstrate novel behavior; we characterize the possibilities in the next proposition.

Proposition 3 (Price comparative statics)

For density tolerance with the CDF $G(c)=c^k,k\ge 1$ , the optimal price $p^{*}$ is quasi-convex in $\alpha$ ; that is, it can either increase, decrease, or first decrease then increase as $\alpha$ increases. In particular, there exist two thresholds $\underset{\_}{k},\overline{k}\in [1,2]$ with $\underset{\_}{k}\le \overline{k}$ such that $p^{*}$ is increasing in $\alpha$ for any $k\le \underset{\_}{k}$ and decreasing in $\alpha$ for any $k\ge \overline{k}$ .

Proposition 3 reveals that as customer valuation and density tolerance become more positively dependent, the behavior of the optimal price depends on the density tolerance distribution $G$ , as illustrated in Figure 2. To better understand this phenomenon, it is helpful to consider the derivative of the optimal price. By (EC.25 of E-Companion) in the proof of Proposition 3, the effect of an increase in $\alpha$ on the optimal price can be decomposed into two parts as ⁷

$$\begin{array}{r}\frac{\mathrm{d}{p}^{*}}{\mathrm{d}\alpha }=\underset{\text{positive effect}}{\underbrace{\frac{(c^{*}{)}^k}{1-(1-\lambda )(c^{*}{)}^k}}}+\frac{\mathrm{d}{c}^{*}}{\mathrm{d}\alpha }\underset{\text{negative effect}}{\underbrace{\frac{k(c^{*}-1)-c^{*}}{c^{*}(k+1-(1-\lambda )(c^{*}{)}^k)}}}.\end{array}$$

(10)

Specifically, an increase in $\alpha$ has a direct positive effect on the optimal price because demand increases with $\alpha$ for any given price–density pair, and hence, all else being equal, a higher price is needed to bring the demand down so as to maintain $c=D(p,c)$ (see Lemma 1). However, there is also an indirect negative effect. Lemma 1 implies that it is always optimal to induce $c=D(p,c)$ . Because the optimal density increases with $\alpha$ , keeping this balance for higher $\alpha$ will tend to pull the price down, all else being equal. So, the net change in the optimal price depends on which of these two effects dominates.

When $k$ is smaller and close to 1 (i.e., $k\le \underset{\_}{k}$ ), the optimal price increases with $\alpha$ (see Figures 1(a) and 2(a)). In this case, customers are sufficiently heterogeneous in their density sensitivity, so an increase in $\alpha$ has a larger direct impact on demand and hence the positive effect dominates. On the other hand, for sufficiently large $k$ (i.e., $k\ge \overline{k}$ ), the density sensitivity distribution becomes much more skewed and concentrated more on the higher tolerance level, so the demand is less sensitive to the density around the optimal solution. Thus, the direct effect is weakened and the indirect negative effect dominates, reversing the monotonicity of the optimal price with respect to $\alpha$ . During the transition phase (i.e., $\underset{\_}{k}<k<\overline{k}$ ), the optimal price exhibits nonmonotonic behavior, namely first decreases and then increases in $\alpha$ .

Combining Propositions 2 and 3, we note that as the demand becomes more negatively dependent (i.e., $\alpha$ decreases), the provider would prefer to restrict the service capacity and raise the price (when consumers are not extremely density sensitive, i.e., $k\ge \overline{k}$ ). These findings resonate with the anecdotal case of the exclusive private clubs such as the previously mentioned Augusta National and Yellowstone Club, and other elite private clubs such as Cypress Point Club in California and Seminole Golf Club in Florida, with only 250 and 300 members, respectively (Dobson, 2021).

5. Severe Density Sensitivity

In this section, we consider customer populations exhibiting severe density sensitivity, represented by distributions $G(c)=c^k$ with $k<1$ . As $k$ decreases, the probability mass of the power distribution shifts toward smaller values of the density tolerance, and customers become more sensitive to the density. In particular, as $k{\displaystyle \to 0}$ , the power distribution becomes degenerate with the probability mass concentrated at $0$ , that is, customers become completely density-intolerant. At the other extreme, that is, as $k{\displaystyle \to 1}$ , the power distribution approaches the standard uniform distribution.

For a severely density-sensitive customer population, the optimal strategy is not always full-coverage, price-active; that is, the optimal price and density do not necessarily satisfy $G(c)\le F(p)$ and $G(c)+F(p)\le 1$ . Based on the result of $min\{1-F(p),1-G(c)\}$ and $max\{0,1-F(p)-G(c)\}$ in the demand function (6), the provider has four possible qualitatively different strategies. We recall that, under price and density levels satisfying $G(c)+F(p)\le 1$ , the provider achieves full coverage of the market by serving all three clusters of customers with independence, perfect positive and negative dependence between the valuation and density tolerance. Conversely, if $G(c)+F(p)\ge 1$ , then the provider has only partial coverage, since the demand from the cluster with perfect negative dependence reduces to zero. Regarding the cluster with perfect positive dependence (represented by $min\{1-F(p),1-G(c)\}$ ), we recall that the price is active in regulating the demand from this cluster if $G(c)\le F(p)$ ; otherwise, the density would be active in doing so. Therefore, the provider’s price and density decisions $(p,c)$ can be categorized into four distinct strategies:

Full-coverage, price-active ( $\mathrm{FP}$ ), if $G(c)+F(p)\le 1$ and $G(c)\le F(p)$ .

Full-coverage, density-active ( $\mathrm{FD}$ ), if $G(c)+F(p)\le 1$ and $G(c)\ge F(p)$ .

Partial-coverage, price-active ( $\mathrm{PP}$ ), if $G(c)+F(p)\ge 1$ and $G(c)\le F(p)$ .

Partial-coverage, density-active ( $\mathrm{PD}$ ), if $G(c)+F(p)\ge 1$ and $G(c)\ge F(p)$ .

When both price and density are active in regulating the demand from the cluster with perfect positive dependence, that is, $G(c)=F(p)$ , we refer to the strategy as $\mathrm{FB}$ or $\mathrm{PB}$ depending on whether the coverage is full or partial. On the other hand, when $G(c)+F(p)=1$ , the full coverage in effect degenerates to partial coverage, as the demand from the cluster with perfect negative dependence reduces to zero. ⁸ Our solution strategy is to first identify the optimal price and density decisions under full and partial coverage. Then, we compare their corresponding optimal revenues to determine the overall optimal strategy for the provider, which we characterize in the proposition below.

Proposition 4 (Optimal strategy: Severe density sensitivity)

When the optimal strategy is full-coverage, there exist two thresholds $0<k_{\mathrm{FD}}<k_{\mathrm{FP}}\le 0$ such that it is (i) $\mathrm{FD}$ if $k\le k_{\mathrm{FD}}$ , (ii) $\mathrm{FB}$ if $k_{\mathrm{FD}}<k<k_{\mathrm{FP}}$ , and (iii) $\mathrm{FP}$ if $k\ge k_{\mathrm{FP}}$ . When the provider’s optimal strategy is partial-coverage, there exist two thresholds $0<k_{\mathrm{PD}}<k_{\mathrm{PP}}\le 1$ such that it is (i) $\mathrm{PD}$ if $k\le k_{\mathrm{PD}}$ , (ii) $\mathrm{PB}$ if $k_{\mathrm{PD}}<k<k_{\mathrm{PP}}$ , and (iii) $\mathrm{PP}$ if $k\ge k_{\mathrm{PP}}$ .

(The explicit characterization of the thresholds, whose values depend on $\alpha$ and $\lambda$ , and the optimal price and density are given in Lemmas EC.3 and EC.4 and Proposition EC.2 in Appendix C of E-Companion.)

Proposition 4 shows which lever (price vs. density) the full-coverage and partial-coverage strategy would activate in the optimum to regulate the demand from the customers with positively dependent preferences. In both cases, we find that density (respectively, price) should be the active lever when customers become highly (respectively, relatively less) sensitive to density, that is, $k$ is below a lower threshold $k_{\mathrm{FD}}$ or $k_{\mathrm{PD}}$ (respectively, above a higher threshold $k_{\mathrm{FP}}$ or $k_{\mathrm{PP}}$ ). The intuition is as follows. For highly density-sensitive customers (i.e., for $k$ sufficiently small), very few of them are willing to accept the service even if the provider sets a small density, that is, $G(c)$ will be relatively close to 1. Setting $F(p)>G(c)$ would further dampen the limited demand, so for small $k$ , the provider chooses to regulate the demand using the density by setting $F(p)\le G(c)$ . On the other hand, when $k$ is larger, that is, when the density tolerance distribution is more evenly spread, there are more customers with relatively high-density tolerance. With more customers willing to accept the service at higher densities, the potential demand is higher. Thus, the provider can set a higher price to earn more revenue per customer because setting $F(p)>G(c)$ can still achieve a healthy demand. In other words, a price-active strategy is optimal for larger $k$ . Naturally, in between these two polar cases, the density and price become equally effective and should be both activated to regulate the demand from the customers with positively dependent preferences.

While having characterized the optimal structure for both full- and partial-coverage strategies in the optimum, Proposition 4 remains agnostic about when each of these two strategies is optimal. The answer to this question depends on the comparison of their respective optimal revenues, which turns out to be analytically intractable. Nonetheless, our numerical experiment suggests that the full-coverage strategy identified for $k\ge 1$ continues to be optimal for $k<1$ above a threshold; the partial-coverage strategy becomes optimal only when $k$ falls below that threshold, that is, when customers become extremely density-sensitive. To visualize the provider’s optimal strategy, we illustrate it in the $(\alpha ,k)$ space, as shown by Figure 3, where the thick line partitions the $(\alpha ,k)$ space into a full-coverage region (upper left) and a partial-coverage region (lower right). Intuitively, when the customers become extremely sensitive to the density, the provider must set a low density in order to attract any customer. Alternatively, the provider prefers not to serve the negative-dependent cluster but only focuses on the more lucrative customers from the independent and positively dependent clusters. Indeed, as shown in Figure 3, the partial-coverage strategy is optimal for sufficiently large $\alpha$ .

Figure 3.

Optimal strategy in $(\alpha ,k)$ space with $\lambda =0.75$ . Note. The thresholds $k_{\mathrm{FP}}$ , $k_{\mathrm{FD}}$ , $k_{\mathrm{PP}}$ , and $k_{\mathrm{PD}}$ depicted in the figure are characterized analytically in Appendix C of E-Companion.

As our last analytical result of the section, the following proposition characterizes how the optimal price and density are affected by the positive dependence parameter $\alpha$ .

Proposition 5 (Effects of $\alpha$ on optimal density and price)

The optimal density is increasing in $\alpha$ under both full- and partial-coverage optimal strategies; the optimal price is increasing in $\alpha$ under all optimal strategies ( $\mathrm{FP},\mathrm{FD},\mathrm{PD},\mathrm{FB}\text{, and }\mathrm{PB}$ ) but is decreasing when the optimal strategy is $\mathrm{PP}$ .

Figure 4 illustrates the findings in Proposition 5. Consistent with our findings for $k=1$ , the optimal density is increasing in $\alpha$ under both the full-coverage and partial-coverage strategies (albeit with a discontinuity as the optimal strategy transits from full- to partial-coverage). As previously discussed, an increase in $\alpha$ implies higher demand for each price–density pair, which allows the provider to operate at a higher density. Also in line with our earlier findings, the optimal price is increasing in $\alpha$ except under strategy $\mathrm{PP}$ . Intuitively, the partial-coverage strategy eschews the negatively dependent cluster and limits the demand to independent and positively dependent clusters. Since price is the active lever for regulating demand under strategy PP, the provider thus lowers the price to attract more customers from the growing positively dependent cluster.

Figure 4.

Optimal density and price versus $\alpha$ for $\lambda =0.75$ and $k=0.45$ : (a) density and (b) price.

6. Illustrative Example: Train Travel in the COVID-19 Pandemic

Ideally, we would calibrate our model using real-world data on exclusive private clubs. However, it is unfortunately extremely difficult to gather information about such clubs (let alone granular data) because they have a strong incentive to gatekeep any details about their membership and operations due to the nature of the business model and the clientele served. Indeed, Kendall (2008: p. 48) emphasizes “how difficult it is to gain accurate information about the practices of private clubs” and states that “we know little about membership in private clubs throughout the United States.” Without access to data on private clubs, we resort to an illustrative example using data from a different setting where customers are also likely to exhibit density sensitivity: train travel during the COVID-19 pandemic. We use this data to numerically validate our distributional assumptions, calibrate the model parameters, and subsequently evaluate the provider’s optimal strategy. The dataset is drawn from Shelat et al. (2022), which conducted a choice experiment measuring the impact of COVID-19 risk perceptions on train travel decisions in the Netherlands.

We use their data to construct an empirical distribution for crowd density tolerance; see Appendix E.1 of E-Companion for details. We can then use this empirical distribution to estimate the power distribution parameter by the method of moments. Specifically, for a power distribution with parameter $k$ , the mean $\mu =k/(1+k)$ ; simple algebra then gives the method of moments estimator for $k$ as $k=\mu /(1+\mu )$ , which is the unique power parameter such that the mean of the corresponding power distribution matches the mean of the empirically constructed distribution. We find $\tilde{k}=1.635$ . Figure 5(a) depicts both the empirical CDF constructed from the data and the theoretical CDF of the estimated power distribution, revealing a close visual match (we also assess the goodness of fit of the estimated distribution with the Kolmogorov–Smirnov (K–S) test; again, see Appendix E.1 of E-Companion). Encouragingly, the best-fitting parameter ( $\tilde{k}=1.635$ ) falls comfortably within the parameter range ( $k\ge 1$ ) to which our main results from Section 4 apply. In other words, the density tolerance distribution among the survey respondents is slightly left-skewed, implying that only a relatively small fraction of the population has extremely low-density tolerance. Indeed, commuters of public transportation typically need to go to work to earn income, and hence the majority of them are likely to tolerate higher health risks.

Next, using the estimated power parameter, we compute the optimal decisions and revenue by varying the degree of positive dependence $\alpha$ between valuation and density tolerance. As shown in Figure 5(b), ⁹ the optimal density and optimal revenue are both increasing in $\alpha$ (consistent with Proposition 2), while the optimal price is decreasing in $\alpha$ (consistent with Proposition 3). Indeed, the estimated power $\tilde{k}=1.635$ is sufficiently large (above the threshold $\overline{k}$ identified in Proposition 3) so that the indirect negative effect is dominant (see the discussion following Proposition 3). In practice, higher positive dependence between commuters’ WTP and density tolerance indicates that people are more willing to go out (e.g., to work) and at the same time to tolerate a denser crowd on the train. This is likely to occur when the society weighs the economic and social benefits more than the health benefits. In this case, public transportation operators should lower the transit fee and relax the social distancing requirement in order to accommodate more commuters.

Figure 5.

Estimated density tolerance CDF, and optimal decisions and revenue versus $\alpha$ for $k=\tilde{k}=1.635$ : (a) empirical and theoretical CDFs and (b) optimal decisions and revenue for $\lambda =0.75$ . Note. CDF = cumulative distribution function.

Finally, for additional robustness tests, we consider, in Appendix E.2 of E-Companion, another dataset—from related work by Shelat et al. (2022)—that also measures traveler preferences during the COVID-19 pandemic. On this alternate dataset, we again find that the power family of distributions is a reasonable approximation for the customer density tolerance distribution. Although this setting is different from that of exclusive clubs, it is nevertheless encouraging to see that the power distribution family can be a reasonable choice for modeling density tolerance in a real-world setting.

Figure 6.

Optimal density versus $\theta$ for different copulas (higher $\theta$ means more positively dependent, uniform $F$ and $G$ ): (a) Clayton, (b) Gumbel, and (c) Frank.

Figure 7.

Optimal revenue versus $\theta$ for different copulas (higher $\theta$ means more positively dependent, uniform $F$ and $G$ ): (a) Clayton, (b) Gumbel, and (c) Frank.

7. Extension

In most of the above development, we have focused on the Fréchet family of copulas, and under this family, we analytically characterized the service provider’s optimal price and density decisions. However, other copulas exist, and they allow the modeling of additional forms of dependence, so in this section, we extend our study to other copulas to validate the robustness of our qualitative findings.

First, we prove a general result that extends the revenue monotonicity from Proposition 2 to a broad class of copulas under arbitrary marginal distributions for valuation and density tolerance.

Proposition 6 (Monotonicity for general copulas and marginals)

Suppose valuation and density tolerance are related according to a copula $C_{\theta }$ parameterized by a value $\theta$ , such that the demand is increasing in $\theta$ for a given price and density. Then the provider’s optimal revenue is increasing in $\theta$ .

Demand is increasing in a particular parameter is exemplified by the Fréchet case, where demand increases in $\alpha$ as valuation and density tolerance become more positively dependent. The above result shows that in any such case, the provider’s optimal revenue will be increasing in the positive dependence parameter; this is a highly general result that makes no other assumptions about the copula or the marginal distributions.

Next, we study our problem numerically under three different copulas: Clayton, Frank, and Gumbel. Each exhibits a different type of tail dependence (tail dependence reflects the frequency with which different types of extreme events occur jointly). For instance, the Clayton copula is characterized by asymmetric lower tail dependence, such that low realizations of both random variables are likely to occur jointly. We refer the reader to Wang and Dyer (2012: Table 2) for an in-depth treatment of these copulas, including complete formulas and structural properties.

The demand function under these different copulas is obtained by substituting each different copula $C$ into equation (2). Each of the three additional copulas has a parameter $\theta$ ; the allowable ranges of this parameter are different for the different copulas, but for all of them, higher values of $\theta$ mean more positive dependence (or less negative dependence for negative $\theta )$ . In Figure 6, we plot the optimal density as a function of $\theta$ for the Clayton, Frank, and Gumbel copulas. Consistent with Proposition 2, we observe that the optimal density increases as the valuation and density tolerance become more positively dependent. Also, since the sales are equal to the density at optimality, this also implies that the optimal sales increase with positive dependence, again consistent with Proposition 2.

Next, in Figure 7, we plot the optimal revenue versus $\theta$ for each copula. As expected from Proposition 6, and matching our earlier findings as well, the optimal revenue increases with $\theta$ for all three copulas. Finally, Figure 8 shows the optimal price versus $\theta$ for each copula. In our earlier analytical results, we found that the optimal price can sometimes be nonmonotonic in the positive dependence parameter. Similarly, in the case of Clayton, Gumbel, and Frank, we also find it to be nonmonotonic (albeit with different curvature from the Fréchet case, that is, increasing–decreasing instead of decreasing–increasing).

Figure 8.

Optimal price versus $\theta$ for different copulas (higher $\theta$ means more positively dependent, uniform $F$ and $G$ ): (a) Clayton, (b) Gumbel, and (c) Frank.

Overall, we thus find that our structural results for the Fréchet copula largely extend to more general families of copulas, even those exhibiting very different types of tail dependence.

8. Conclusion

Motivated by exclusivity- and privacy-based club operations, we have introduced a novel, copula-based framework for studying a service provider’s problem whose customers are sensitive to both the service price and the crowd density. A novel and crucial feature of our framework is the statistical dependence between customers’ valuation and density tolerance. Under this framework, we characterize the provider’s joint optimal price and density cap decisions in relation to the marginal density tolerance distribution and the degree of (positive or negative) dependence between the two customer attributes.

We find that a full-coverage, price-active strategy—in which the provider serves all segments of the market and activates the price (rather than the density) to regulate demand—is optimal for the service provider over a broad range of dependence structures and marginal density tolerance distributions. In particular, we demonstrate the nuanced interaction between the dependence structure and the marginal density tolerance distribution. As a result, while the optimal density cap and revenue are increasing as the valuation and density tolerance become more positively dependent, the optimal price may be nonmonotonic. For severely density-sensitive customer populations, we find that the full-coverage, price-active strategy may no longer be optimal. It may be optimal for the service provider to drop the demand cluster of negative dependence or/and activate the density to regulate the demand from the positively dependent cluster.

Our findings speak to some highly debated economic and social issues. In particular, our results shed light on how the interdependence between customers’ WTP and their tolerance for crowds can impact the service provider’s price and density decisions. Intuitively, more positive dependence benefits the service provider, who should set a higher density limit. Nonetheless, the effects on the price are more nuanced and may not be necessarily monotonic. When the extremely density-sensitive customers only constitute a small fraction of the population, likely the case in reality, the service provider should optimally lower its price as the customers’ WTP becomes more positively associated with their density tolerance. Our prescriptive guidelines thus provide an explanation for the failed versus successful practices of some prominent private clubs, offering a unique perspective to study exclusivity-driven operations.

Our incorporation of rigid multidimensional preferences both addresses a gap in the literature (which has typically assumed perfectly substitutable preferences) and allows us to employ a highly general model of dependence between the dimensions of valuation and density tolerance. While these are positive features of our work, the rigidity assumption does represent an extreme specification of preferences. In practice, customer preferences may fall somewhere between the extremes of perfectly substitutable and perfectly rigid; that said, a copula approach can still be applicable to model multidimensional heterogeneity of preferences in those cases, albeit potentially with the loss of analytical tractability. Along with this, our framework and results open multiple avenues for future work. We have assumed uniformly distributed WTP and a broader class of distributions for density tolerance, allowing more richness for the latter as a novel feature of our work. For more general WTP distributions, we believe that our main insights would likely continue to hold, but the analysis would be technically challenging if both $F(p)$ and $G(c)$ took general forms. More broadly, as one of the first works on service operations to employ copulas, we hope that our work can inspire further use of this oft-overlooked tool in related settings.

Supplemental Material

sj-pdf-1-pao-10.1177_10591478241302764 - Supplemental material for Optimizing Service Operations With Price- and Density-Dependent Demand: A Copula-Based Approach

Supplemental material, sj-pdf-1-pao-10.1177_10591478241302764 for Optimizing Service Operations With Price- and Density-Dependent Demand: A Copula-Based Approach by Andrew E Frazelle, Toghrul Rasulov and Shouqiang Wang in Production and Operations Management