Event ticket pricing with capacity constraints and price restrictions

Abstract

Motivated by ticket pricing challenges facing live event managers, we study how to maximize revenue when setting prices for multiple ticket categories with interdependent demand, realistic capacity constraints, and pricing restrictions. We define this decision problem as the Event Ticket Pricing (ETP) problem and formulate it as a constrained nonlinear optimization model. To examine how the nature of product differentiation (i.e., differentiation across ticket categories) affects seat portfolio pricing and revenue outcomes, we analyze the ETP problem under two demand specifications: vertically and horizontally differentiated ticket categories. For each case, we isolate the role of different constraints by developing and solving a sequence of problems with progressively added restrictions. This approach enables us to identify structural properties that either fully characterize the optimal solution or guide the development of efficient algorithms. Under vertical differentiation, we prove that the optimal solution has a sold-out threshold structure, with only high-quality categories sold out. The treatment of partially sold products depends on the active constraints. Sales vanish under capacity constraints alone but become positive once pricing restrictions are introduced, producing solutions distinct from the unconstrained benchmark. Under horizontal differentiation, the threshold structure persists, but the addition of constraints breaks the well-known “equal markup” rule, yielding prices that are non-increasing in quality. Finally, we study how flexible adjustments in seat capacities reshape optimal outcomes. These results underscore the importance of incorporating realistic constraints and offer practical guidance for live event planners.

1. Introduction

Live entertainment events such as musical performances, theatrical presentations, and sports constitute a large and influential industry. In 2025, the global revenue for the concert and live music sector alone is estimated to reach $38 billion and is projected to surpass $50 billion in 2030. ¹ Since most revenues come from ticket sales, ticket pricing is a key consideration for organizers. To optimize revenues, live event managers are increasingly resorting to data and analytics, with industry experts estimating typical uplifts of 5%–15% through improved pricing. ²

Our work on event ticket pricing (ETP) is motivated by a problem facing the box office of a live music venue in a large metropolitan area. The box office recognizes that customers’ preferences depend not only on the price, but also on non-price characteristics such as the visual/auditory experience and access to concessions/restrooms. Hence, the quality of the experience and demand vary by seat location. Based on this fact, the box office groups seats with similar viewing and participation experiences into a single ticket category and offers all the seats within a category at the same price. It is important to note, however, that individual customers may perceive the value of a seat in a ticket category differently, which suggests a diversity of opinions on “ideal” seat locations. For instance, in terms of the viewing experience, customers may uniformly prefer a location with a direct viewing angle. In contrast, customers may have idiosyncratic preferences for seats, as in the case where rear seats are favored by connoisseurs in audio concerts. Thus, the box office’s approach to pricing must explicitly incorporate customer preferences and anticipate their impact on prices and revenue. Finally, since ticket categories are natural substitutes, the price of one category affects not only its own demand but also that of comparable categories. Hence, prices must be jointly determined across categories.

In addition, the box office is concerned with several practical considerations that are vital in this context. The foremost one for the box office is to ensure affordable access. Audiences and journalists typically use measures such as the average ticket price and the lowest available ticket price to gauge the affordability of events. ³ Ticket prices of live concerts are closely monitored by fans and, if perceived as unaffordable, can trigger outrage and backlash against both the artist and the venue. ⁴ Performing artists themselves are also unhappy about overly high ticket prices and often seek to ensure access for committed fans. As such, a high average price, especially when reflected across multiple ticket categories, can damage customer goodwill and create reputational concerns, which may in turn impact future demand. Thus, the box office manager must ensure that both the lowest ticket price and the average ticket price across categories do not exceed predetermined upper limits. The box office that motivated our work faced such affordable-access mandates as well. Moreover, the box office is attuned to the impact that prices have on space considerations. Because the allocation of seats to categories precedes the sales of tickets, the prices cannot be set too low lest demand should exceed capacity. Together, these considerations define the box office’s decision problem as one of maximizing revenues by choosing a price for each ticket category while satisfying capacity constraints and price restrictions, and incorporating the impact that both prices and non-price factors have on the demand. We refer to this problem as the Event Ticket Pricing (ETP) problem.

Although the revenue management literature is extensive, most existing studies have focused on unconstrained or lightly constrained settings, and to our knowledge, no prior work systematically incorporates the practical constraints of the ETP problem. As such, there is a need for rigorous analysis of how such practical constraints affect optimal prices and revenues. We fill this gap by modeling and solving the ETP problem as a static, constrained, multi-product price optimization. We consider static, rather than dynamic, pricing for two reasons. First, since event ticket prices are keenly monitored by fans, the price changes brought by dynamic pricing may be viewed as unfair and akin to price gouging. ⁵ Second, for live shows performed by popular artists, high volumes of tickets are usually sold out in seconds, leaving little time for price adjustments. Ticketmaster, a leading online ticketing platform, indicates that ticket prices are typically set first as fixed base prices, after which dynamic pricing may be applied to only a limited subset of premium tickets; for example, just 12% of tickets for Springsteen’s concert in 2022 were dynamically priced while the rest were sold at fixed prices. ⁶ Accordingly, ticket pricing is often determined in two layers: a baseline of fixed prices set in advance, and dynamic adjustments applied to a subset of premium inventory. Our model focuses on the baseline pricing structure, which forms the foundation for subsequent dynamic pricing.

Therefore, there are many instances where static pricing is appropriate, and our model applies in these contexts. We focus on two models of the ETP problem, corresponding to vertical and horizontal product differentiation, which are commonly considered in revenue management literature with a similar application context (e.g., Arslan et al., 2022; Budish and Bhave, 2023). When the ticket categories are vertically differentiated, customers’ purchasing behavior is solely steered by their heterogeneous sensitivity to quality; when the ticket categories are horizontally differentiated, customer valuation shares the same quality sensitivity but reflects idiosyncratic preferences. We adopt a step-by-step approach to analyzing and solving these two models by adding the constraints to the unconstrained price optimization model one at a time. Progressively tightening the model in this manner not only reveals insights about the solution structure and guides us in solving the ETP problem, but also isolates the impact of each constraint, thereby clarifying the trade-offs managers face. We summarize the main findings below.

First, we consider the case where ticket categories are vertically differentiated. We show that the ETP is a convex optimization problem if and only if the random sensitivity to quality is uniformly distributed. While this condition is sufficient for the unconstrained problem, it also becomes necessary with capacity constraints, highlighting the critical role of constraints in optimal pricing. With only the capacity constraints, the optimal solution exhibits a sold-out threshold structure, that is, categories above a certain threshold are fully sold out. The threshold depends on the aggregate capacities of those categories. Notably, the capacitated problem prioritizes higher-quality seats, often leaving lower-quality ones unsold, potentially undermining affordable access. When an average price restriction is added, the sold-out threshold structure remains optimal, and the threshold is at least as high as in the capacity-only case. However, all categories may now generate sales, with identical sales across the partially sold categories. The average price cap curtails the box office’s ability to set high prices for premium seats, compelling it to sell lower-quality tickets. Hence, more of the market is served compared to the unconstrained and capacity-only models. Finally, incorporating an upper bound on the price of the lowest-quality category, in addition to the capacity and average price restrictions, we design an iterative polynomial-time algorithm by recursively reducing the problem to previous cases without the ceiling. With this upper bound constraint, lower-quality tickets sell more and may also sell out. Thus, the resulting optimal solution features two sold-out thresholds, with both the highest- and lowest-quality categories sold out.

Next, we consider the case with horizontally differentiated ticket categories. Customers’ choice in this case is well captured by the multinomial logit (MNL) model (e.g., Arslan et al., 2022; Li and Huh, 2011). With only capacity constraints, the optimal solution again features a sold-out threshold structure, and prices of the sold-out categories decrease in quality. For categories with leftover capacity, however, prices remain identical. We add an average price restriction, show that this constraint is quasi-convex, and derive necessary and sufficient optimality conditions. With the addition of this constraint, optimal sales still exhibit a sold-out threshold structure, but now all categories, regardless of sales, have distinct prices that decrease in quality. Finally, we introduce an upper bound on the lowest-quality category and show that it yields an equivalent convex form. Therefore, the previous optimality conditions still apply. However, deriving even simple structural results becomes analytically challenging. Hence, to gain insight, we study a relaxed case where the average price restriction is non-binding. Then, the sold-out threshold structure and monotonicity of prices among sold-out categories remain, but the lowest-quality category is priced strictly below the (identical) price of the partially sold ones. Overall, these results highlight how practical constraints reshape the structure of the optimal solution to the ETP problem. In particular, the constraints fundamentally alter pricing patterns, producing outcomes that deviate sharply from the unconstrained benchmark, including the well-known “equal markup” property (Anderson and De Palma, 1992).

We also summarize and compare the structural results obtained under the vertically and horizontally differentiated demand models. This comparison highlights how differences in demand structure lead to distinct outcomes, while also revealing important similarities across the two settings. In addition, our numerical experiments demonstrate that our approaches are computationally tractable and scale well in realistic problem settings. Together, these results sharpen the theoretical insights and support the practicality and scalability of the proposed methods.

Lastly, while the above findings characterize optimal prices for a fixed set of ticket categories and capacities, box offices may adjust category configurations in practice. To assess the value of such flexibility, we examine two scenarios. First, we consider changes to the category set: closing removes the lowest-quality category, whereas adding introduces a new highest-quality category. In the vertical case, closing has little effect, while adding increases revenue and changes the sold-out threshold by at most one category. In the horizontal case, closing increases per-category revenue, whereas adding reduces sales of partially sold categories. Second, we study capacity reallocation, where a small number of seats is shifted between categories while the optimal threshold remains unchanged. Although sales and prices respond differently across the two demand models, total revenue always increases. Overall, these results show that flexibility in category design can improve revenue, but its effects depend on the demand structure.

The rest of the article is organized as follows. Section 2 reviews the extant literature. Section 3 formulates the problem. We investigate the ETP problem for the vertical case in Section 4 and for the horizontal case in Section 5. Section 6 compares the results under the two demand models and discusses the implications. Finally, Section 7 discusses some additional considerations and Section 8 concludes the article. All proofs and Supplemental Material are in the E-Companion.

2. Literature review

Our paper is primarily related to the literature on multi-product price optimization. We review relevant work according to how demand is modeled. Since one of our demand models reflects customers’ heterogeneous valuation of non-price attributes, our paper builds on research in pricing with quality-differentiated products. Originating from the classical framework of vertical product differentiation (Tirole, 1988), this demand model has been widely adopted in economics and marketing (see, e.g., Choi and Shin, 1992; Moorthy, 1988). In the revenue management literature, vertically differentiated demand has been used to study dynamic pricing (Akçay et al., 2010; Stamatopoulos and Tzamos, 2019), bundling (Banciu et al., 2010; Honhon and Pan, 2017), and assortment planning (Chen and Yang, 2019; Pan and Honhon, 2012). More related to our setting, Budish and Bhave (2023) analyze Ticketmaster’s auction design in a vertically differentiated event pricing context. In addition to the above works, two papers merit particular attention. First, Akçay et al. (2010) examine multi-product dynamic pricing under vertical and horizontal differentiation, underscoring the role of demand structure in optimal pricing. Our work is closely related in this regard, but we study a constrained optimization problem. Second, Banciu et al. (2010) analyze pricing for two capacity-constrained products and their bundle. While our work also addresses capacity, we consider more than two products and introduce price restrictions, highlighting new practical considerations. Moreover, the comparative analysis of constrained solutions relative to prior work emphasizes the importance of incorporating these considerations.

Our paper also relates to choice-based pricing models of horizontal differentiation. Researchers have employed MNL and its extensions, such as nested logit (NL), paired combinatorial logit (PCL), and mixed logit to model demand and derive optimal prices (Dong et al., 2009; Li and Huh, 2011; Li and Webster, 2017; Li et al., 2019). We focus on the MNL model in event ticketing, where ticket category quality is central. A related work is Li et al. (2020), who study product-line design where price and quality interact in customer utility, and propose a single-decision-variable approach for joint optimization. While this literature is extensive, it typically assumes unconstrained problems. Work on assortment optimization has examined cardinality or space constraints (e.g., Désir et al., 2022; Gallego and Topaloglu, 2014; Rusmevichientong et al., 2010; Wang, 2012), but unlike these studies, we impose seat capacity limits for each ticket category.

Thus, from both theoretical and practical perspectives, there remains a need to understand constrained price optimization. Three studies are particularly relevant. Keller (2013) investigates constrained price optimization under logit demand, establishing sufficient conditions for concavity but solving only via approximation. Shao and Kleywegt (2020) address a multi-attribute problem with resource and bound constraints, reformulating it as a convex conic program and solving numerically. Arslan et al. (2022) provide an empirical study of price optimization in sports with multiple channels and heterogeneous customers, incorporating constraints in practice but without analytical characterization of optimality or solution structure.

3. Problem formulation

Consider a box office manager who offers multiple ticket categories that are substitutable products ⁷ but vary in quality. The manager’s goal is to maximize revenues by setting appropriate prices for each category. Formally, suppose there are $N$ ticket categories with differentiated quality levels represented by the totality of all non-price attributes for the event, denoted by $q_1>q_2>\cdots >q_N>0$ . The box office manager decides prices $\mathbf{p}=(p_1,p_2,\dots ,p_N)$ for each category. While dynamic pricing has attracted considerable attention in the literature, static pricing remains the dominant practice in event ticketing. As previously mentioned, for many popular live events, the majority of tickets are sold almost immediately after release, leaving little room for price adjustments during the sales horizon. Moreover, frequent price changes can be perceived by fans as unfair, undermining trust in the seller. These observations underscore the relevance of studying static pricing, which is both practically important and managerially appropriate. Thus, given our intent to understand and derive insights for practicing managers, we hope that our work can serve as a foundation to address the more complex problem of dynamic pricing.

The underlying consumer choice model stems from a linear random utility framework; specifically, we assume that, from purchasing product $j$ , a consumer derives utility $u_j=\theta q_j-p_j+\mu {\zeta }_j$ . There are two random parts in the above utility function. Corresponding to the idea of vertical product differentiation, each consumer has a random component $\theta$ that represents the sensitivity to product quality. Similarly, to model horizontal product differentiation, the random term ${\zeta }_j$ captures each consumer’s idiosyncratic utility when purchasing product $j$ , and the parameter $\mu \ge 0$ measures the strength of the idiosyncrasy. This consumer choice model is commonly known as the mixed logit model (McFadden and Train, 2000). Despite its fitness for a wide range of practical problems, the mixed logit model’s analytical intractability prevents us from deriving any useful insights. To circumvent this difficulty and, more importantly, to understand the role of product differentiation in the ETP problem, we take a similar approach followed by Akçay et al. (2010) and examine two specific cases of the mixed logit model, namely, the vertical and horizontal product differentiation cases. We elaborate on these cases next.

Vertically differentiated demand. In this case, we focus on the random quality sensitivity $\theta$ and assume that $\mu =0$ . Let $F(\cdot )$ be the distribution function of $\theta$ with support $[0,1]$ that is at least twice differentiable. Then, the demand for product $j$ , ${\alpha }_j^V(\mathbf{p})$ , can be written as follows:

$${\alpha }_j^V(\mathbf{p})=\{\begin{array}{l}1-F\left(\frac{p_1-p_2}{q_1-q_2}\right),j=1\\ F\left(\frac{p_{j-1}-p_j}{q_{j-1}-q_j}\right)-F\left(\frac{p_j-p_{j+1}}{q_j-q_{j+1}}\right),\mathrm{\forall }1<j<N\\ F\left(\frac{p_{N-1}-p_N}{q_{N-1}-q_N}\right)-F\left(\frac{p_N}{q_N}\right),j=N\end{array}$$

(1)

Note that the demand function (1) depends critically on the distribution of $\theta$ . Moreover, the cross-price elasticity only exists for products whose quality levels are adjacent to each other.

Horizontally differentiated demand. Suppose that $\theta$ is a positive constant, ${\zeta }_j$ follows the standard Gumbel distribution, and $\mu$ is normalized to 1. Then, the consumer choice model reduces to the well-known MNL model, with the demand function being:

$${\alpha }_j^H(\mathbf{p})=\frac{e^{\theta q_j-p_j}}{1+\sum _{i=1}^N{e}^{\theta q_i-p_i}},1\le j\le N$$

(2)

In contrast to the vertically differentiated model, here the demand function of product $j$ is related to the prices of all products. Although the products are still quality differentiated, the utilities derived from them cannot be ordered universally.

We now formulate the price optimization problem and incorporate constraints. First, since defining ticket categories precedes the pricing decision, each category $j$ has limited capacity. Let ${\beta }_j$ be the scaled capacity limit for product $j$ (e.g., ${\beta }_j=C_j/\mathrm{\Lambda }$ , where $\mathrm{\Lambda }$ is the total market size and $C_j$ is the total number of tickets in category $j$ ). We assume that $0\le {\beta }_1\le {\beta }_2\le \cdots \le {\beta }_N$ , which is consistent with observed practice in the live entertainment industry. ⁸ As such, the assumption ensures analytical tractability without significantly sacrificing practical relevance.

Second, we require the average price ⁹ of all products not to exceed a predetermined threshold, which we denote as $\overline{P}$ . In addition to the affordability consideration, the average price constraint can also be motivated as a safeguard against overly extreme price prescriptions across ticket categories (see, e.g., Guan and Mišić, 2025; Mišić, 2020: for this common effect in multiproduct pricing settings), thereby improving the credibility and generalizability of the resulting pricing recommendations. Besides, by preventing prices from becoming uniformly too high across categories, the firm may reduce customer backlash and protect future demand, a consideration consistent with related work on repeated interactions and customer-oriented revenue management (see, e.g., Adelman and Mersereau, 2013; Calmon et al., 2021; Sumida et al., 2021). Third, the price of the lowest quality product, $p_N$ , is bounded above by an exogenous ceiling price $U_N$ . A lower $U_N$ permits a greater level of access. In practice, managers set these thresholds based on judgment to avoid unpalatable deviations from historical prices. We name the three constraints as (Cap), (AveP), and (CeilN), respectively. Then, the ETP problem is as follows:

$$\begin{array}{rl}\mathbf{[}\mathbf{E}\mathbf{T}\mathbf{P}\mathbf{]}\underset{\mathbf{p}}{max}& R(\mathbf{p})=\sum _{j=1}^N{\alpha }_j(\mathbf{p})p_j\\ \text{s.t.}& {\alpha }_j\le {\beta }_j,\mathrm{\forall }1\le j\le N;(\mathbf{C}\mathbf{a}\mathbf{p})\\ & \frac{1}{N}\sum _{j=1}^N{p}_j\le \overline{P};(\mathbf{A}\mathbf{v}\mathbf{e}\mathbf{P})\\ & p_N\le U_N.(\mathbf{C}\mathbf{e}\mathbf{i}\mathbf{l}\mathbf{N})\end{array}$$

4. Pricing under vertically differentiated demand

Consider the ETP problem for a vertically differentiated set of ticket categories, where the demand function is given by (1). From the problem formulation, we can see that the structure of the problem hinges on the distribution of the quality sensitivity parameter $\theta$ . While the distribution can be general, we would ideally want the resulting problem to be a convex program for tractability. For the unconstrained problem, the convexity can be achieved under mild conditions; for the constrained problem, however, the convexity is guaranteed only with specific conditions.

Lemma 1

Consider the vertical product differentiation setting where the demand function is given by ${\alpha }_j^V(\mathbf{p})$ . The revenue function $R(\mathbf{p})$ is concave in $\mathbf{p}$ if the following condition holds:

$$x{F}^{{}^{″}}(x)+2F^{{}^{\prime }}(x)\ge 0,\mathrm{\forall }x\in [0,1]$$

(3)

Furthermore, for product $j=1,2,\dots ,N$ , the demand function ${\alpha }_j^V(\mathbf{p})$ is convex if and only if $F(x)=x$ , that is, the quality sensitivity $\theta$ is uniformly distributed over $[0,1]$ .

Condition (3) stated in Lemma 1 is a sufficient condition, and clearly the uniform distribution satisfies it. Here, for the constrained problem to be convex, the constraint functions also need to be convex. The price restrictions are already linear, and Lemma 1 implies that ${\alpha }_j^V(\mathbf{p})-{\beta }_j$ is convex if and only if $F(x)=x$ . Consistent with the literature (e.g., Akçay et al., 2010), we hereafter assume uniform distribution for $\theta$ , which yields a convex program.

To analytically investigate the [ETP] problem in the vertical products setting, we build progressively constrained models. We define [ETP-v0] as the unconstrained pricing problem for vertical products; similarly, let problems [ETP-v1], [ETP-v2], and [ETP-v3] be the pricing problems with (Cap), with (Cap) and (AveP), and with (Cap), (AveP), and (CeilN), respectively. In this section, we write $\mathit{\alpha }$ instead of ${\mathit{\alpha }}^V$ for brevity. Additionally, let ${\alpha }_{N+1}:=1-\sum _{j=1}^N{\alpha }_j$ be the no-purchase probability. Now, we sequentially investigate problems [ETP-v $x$ ], $x=0,1,2,3$ .

4.1. Unconstrained optimization of vertically differentiated categories ([ETP-v0])

A direct corollary following Theorem 3 by Akçay et al. (2010) is that the optimal prices are given by $p_j^{*}=\frac{q_j}{2},1\le j\le N$ . Moreover, the optimal sales are ${\alpha }_1^{*}=\frac{1}{2}$ and ${\alpha }_j^{*}=0$ for $j=2,\dots ,N$ ; and, therefore, the no purchase probability ${\alpha }_{N+1}^{*}=\frac{1}{2}$ . Indeed, assuming unlimited products available for sale, the optimal strategy is to extract the value from the highest quality product alone, because the lower quality products are not as profitable. In practice, however, this solution is not implementable because of the capacity constraint (Cap) on ticket categories. Moreover, pricing for the highest quality category in this manner may not satisfy the accessibility mandate facing the event manager. The constrained problems we consider next aim to address these drawbacks.

4.2. Optimization with capacity constraints ([ETP-v1])

Let us consider the following constrained price optimization:

$$[\mathbf{E}\mathbf{T}\mathbf{P}\mathbf{-}\mathbf{v}\mathbf{1}]\underset{\mathbf{p}}{max}\{\sum _{j=1}^N{\alpha }_j(\mathbf{p})p_j|{\alpha }_j(\mathbf{p})\le {\beta }_j,\mathrm{\forall }1\le j\le N\}$$

To solve problem [ETP-v1], we adopt the idea of variable transformation. Note that the demand function (1) with $F(x)=x$ is a system of linear equations ${\mathit{\alpha }}^T=M{\mathbf{p}}^T+(1,0,\dots ,0{)}^T$ , where $M$ is an $N\times N$ matrix. Since we can show that $M$ is invertible, a one-to-one correspondence between the price vector and the demand vector is guaranteed. Specifically, we can write the price vector as a function of the demand vector and the no-purchase probability; that is, $p_j=p_j(\mathit{\alpha },{\alpha }_{N+1})$ , $j=1,2,\dots ,N$ . Note that the one-to-one mapping still holds here due to the relationship ${\alpha }_{N+1}=1-({\alpha }_1+{\alpha }_2+\cdots +{\alpha }_N)$ . Therefore, by treating the demand vector $\mathit{\alpha }$ as the decision variable, we can obtain a convex program with linear constraints, which is relatively easier to solve. Define the partial sum of the capacity limits as $K(s)=\sum _{j=1}^s{\beta }_j,s=1,2,\dots ,N$ . Then, our analysis of the KKT conditions for this case yields the following results.

Proposition 1

Consider problem [ETP-v1]. The optimal sales ${\mathit{\alpha }}^{*}$ can be obtained in the following three cases.

1. (1)

If $K(1)>\frac{1}{2}$ , then (Cap) is redundant and the problem reduces to [ETP-v0].

(2)

If $K(N)\le \frac{1}{2}$ , then ${\alpha }_j^{*}={\beta }_j$ for all $j=1,2,\dots ,N$ .

(3)

If $K(1)\le \frac{1}{2}<K(N)$ , then there exists a threshold

$$s_1=min\{K(s+1)\ge \frac{1}{2}|s=1,2,\dots ,N-1\}$$

such that ${\alpha }_j^{*}={\beta }_j$ for $j\le s_1$ , ${\alpha }_{s_1+1}^{*}=\frac{1}{2}-K(s_1)$ , and all remaining products (if any) have zero sales.

For a price $p$ , we define a product as unsold if ${\alpha }_j(p)=0$ , as sold-out if ${\alpha }_j(p)={\beta }_j$ , and as partially sold if ${\alpha }_j(p)<{\beta }_j$ . As such, Proposition 1 characterizes conditions that help us identify the three classes of products. In the first case, the box office has enough capacity for the highest quality product to serve half of the market, just as in the unconstrained problem [ETP-v0], and the second case occurs when the capacities of all the categories are so limited that together they cannot serve even half of the market, resulting in all products being sold out. Interestingly, in the third case, there exists a threshold $s_1$ such that the ticket categories $1,\dots ,s_1$ are sold out. We refer to this as the sold-out threshold structure of the optimal solution. Moreover, the next product $s_1+1$ is partially sold such that exactly half of the market is covered, leaving all other products unsold. It is worth mentioning that our result is in line with Akçay et al. (2010), where the aggregated inventories determine the optimal prices, and only the part of the inventories (analogous to the partial sum of capacity $K(s)$ ) that equals the potential demand has a positive value.

In addition to revealing the structural properties of [ETP-v1], Proposition 1 also produces an efficient algorithm to find the optimal solution. Specifically, the procedure of identifying the optimality includes (1) computing the cumulative sums $K(s)$ for $s=1,\dots ,N$ , which requires $O(N)$ time, and (2) looking for the cutoff threshold $s_1$ , which is $O(N)$ in the worst case. Hence, Proposition 1 can be implemented by a linear-time algorithm.

As for the optimal prices, it is straightforward to find the optimal price vector for the first two cases. For the third case, by noting that ${\alpha }_{N+1}=\frac{1}{2}$ , we can obtain the optimal price $p_j^{*}=q_j/2$ for the partially sold and unsold products, that is, $j=s_1+1,\dots ,N$ . For the sold-out products, that is, the first $s_1$ products, we can recursively find their prices, which can be written as follows:

$$p_j^{*}=\frac{q_{s_1+1}}{2}+\sum _{n=j}^{s_1}(1-K(n))Q_n,j=1,\dots ,s_1$$

(4)

In the above, $Q_j:=q_j-q_{j+1}$ for $j=1,\dots ,N-1$ and $Q_N:=q_N$ . It is interesting to note that for every sold-out product, its price is affected by the capacity limits of sold-out products and not by those of the partially sold and unsold products. However, all the partially sold products’ prices are independent of any capacity information. It is also noteworthy that the partial sum of the capacity limits ( $K(s)$ ) plays a vital role in the optimal solution. While the box office wants to extract most of the value from higher quality products, that goal is constrained by the sub-aggregate capacity, $K(s)$ , at quality level $s\le s_1$ . As a result, the box office can only extract as much value from high quality products as their capacity limits will allow.

4.3. Optimization with capacity constraints and average price restriction ([ETP-v2])

Next, we augment [ETP-v1] with average price restriction (AveP) to obtain

$$\begin{array}{rl}& \mathbf{[}\mathbf{E}\mathbf{T}\mathbf{P}\mathbf{-}\mathbf{v}\mathbf{2}\mathbf{]}\\ & \underset{\mathbf{p}}{max}\{\sum _{j=1}^N{\alpha }_j(\mathbf{p})p_j|{\alpha }_j(\mathbf{p})\le {\beta }_j,\mathrm{\forall }1\le j\le N;\sum _{j=1}^N{p}_j\le N\overline{P}\}\end{array}$$

The extra restriction (a linear inequality in price) leads to a complex form in terms of demand vector $\mathit{\alpha }$ , which makes this problem more difficult than [ETP-v1]. Nevertheless, we are still able to identify useful properties of the optimal solution to [ETP-v2].

Lemma 2

Suppose there exists a feasible solution to the problem [ETP-v2]. Then, at optimality, there exists an $s$ ( $1\le s\le N$ ) such that only products $1,2,\dots ,s$ are sold out. Moreover, if $s\le N-1$ , then the optimal prices of the sold-out products can be expressed as a function of the optimal price of product $s+1$ in the following way:

$$p_j^{*}=p_{s+1}^{*}+\sum _{n=j}^s(1-K(n))Q_n,1\le j\le s$$

(5)

Lemma 2 offers an important insight into the optimal solution to problem [ETP-v2]: there exists a threshold structure in terms of product sales. Then, we may leverage this structure to simplify the $N$ -dimensional problem [ETP-v2] and recast it as a one-dimensional threshold search. In turn, this threshold will allow us to fully specify the optimal solution. In particular, if the threshold $s$ turns out to be $N$ , then all products are sold out, and we can easily obtain the optimal prices. If the threshold $s<N$ , then we can reduce the problem to a lower-dimensional sub-problem with fewer constraints. Specifically, since the sales of the sold-out products are fully characterized by their capacity limits ( ${\alpha }_j={\beta }_j$ for $j=1,\dots ,s$ ), and their prices can be derived from $p_{s+1}^{*}$ , we can focus on the remaining $N-s$ products. Because these are partially sold or unsold products, their capacity constraints become redundant. In addition, the average price restriction can be rewritten by substituting (5) into (AveP) $s$ times. Then, we can formulate a sub-problem where only products $s+1,\dots ,N$ exist in the offered collection:

$$\mathbf{[}\mathbf{E}\mathbf{T}\mathbf{P}\mathbf{-}\mathbf{v}\mathbf{2}\mathbf{(}\mathbf{s}\mathbf{)}\mathbf{]}\underset{p_{s+1},\dots ,p_N}{max}\{\sum _{j=s+1}^N{\tilde{\alpha }}_j{p}_j|s{p}_{s+1}+\sum _{j=s+1}^N{p}_j\le N{\overline{P}}^{\prime }\}$$

Here, the demand function ${\tilde{\alpha }}_j$ is simply the $(N-s)$ -product equivalence of the $N$ -product demand function (1). Moreover, the price constraint in the above sub-problem is derived from the original constraint (AveP) and equation (5); therefore, we have ${\overline{P}}^{\prime }=\overline{P}-\sum _{n=1}^{s}n(1-K(n))Q_n/N$ .

Compared to [ETP-v2], the sub-problem [ETP-v2( $s$ )] has fewer decision variables and only one constraint. Moreover, we can verify that [ETP-v2( $s$ )] is a convex program. With these insights, we can develop an approach to solve [ETP-v2]. Let $\lambda \ge 0$ be the Lagrange multiplier associated with the price constraint in [ETP-v2( $s$ )]. Then, the KKT conditions ensure the complementary slackness: $\lambda (2N{\overline{P}}^{\prime }-\sum _{n=s+1}^N{Q}_n(n-n^2\lambda ))=0$ . If this constraint is non-binding, then (AveP) in problem [ETP-v2] is redundant, which reduces [ETP-v2] to [ETP-v1]. Otherwise, we can solve the sub-problem [ETP-v2( $s$ )] based on the multiplier $\lambda$ , which is given by the following equation:

$$\lambda =\frac{\sum _{n=s+1}^{N}n{Q}_n-2N{\overline{P}}^{\prime }}{\sum _{n=s+1}^N{n}^2{Q}_n}$$

(6)

Lemma 3

Given an $s=1,\dots ,N-1$ , let $\lambda$ be defined by (6) above.

1. (1)

If $\sum _{n=s+1}^{N}n{Q}_n\le 2N{\overline{P}}^{\prime }$ , then (AveP) is redundant and [ETP-v2] becomes [ETP-v1].

(2)

If $\sum _{n=s+1}^{N}n{Q}_n>2N{\overline{P}}^{\prime }$ , then sub-problem [ETP-v2( $s$ )] can be solved as follows. The optimal demand of product $s+1$ is ${\alpha }_{s+1}^{*}=\frac{1}{2}-K(s)+\frac{s+1}{2}\lambda$ , and the optimal demand of each remaining product (if any) is $\frac{\lambda }{2}$ . The optimal prices are given by the following equation:

$$p_j^{*}=\frac{q_j}{2}-\frac{\lambda }{2}\sum _{n=j}^{N}n{Q}_n,j=s+1,\dots ,N$$

When $\lambda >0$ , the average price constraint is binding. In this case, the lower quality products ( $j>s$ ) have positive demand and, since the no-purchase probability ${\alpha }_{N+1}=(1-N\lambda )/2<1/2$ , the market coverage increases to more than half. Recall that without (AveP), the box office manager could charge a premium in the high quality ticket categories and only half of the customers are served; by contrast, a binding (AveP) (meaning that the manager’s pricing power is limited) would help more customers get access to the tickets. Thus, constraint (AveP) addresses one of the drawbacks of the unconstrained solution by ensuring affordability and helping the box office manage public perceptions regarding the cost of access and align with accessibility goals.

To solve problem [ETP-v2], we need to find the threshold of the sold-out products. First, based on Lemma 2, we begin with the all-sold-out scenario and evaluate the box office’s revenue. Next, we construct a threshold-type solution that is feasible and yields a higher revenue. We proceed in this manner until we reach a situation where the next solution does not satisfy the average price constraint. Then, we define the $s$ -sold-out demand vector as ${\mathit{\alpha }}^{(s)}:=({\beta }_1,\dots ,{\beta }_s,0,\dots ,0)$ , and then the resulting no-purchase probability is ${\alpha }_{N+1}^{(s)}=1-K(s)$ . We focus on the total price given the $s$ -sold-out demand vector,

$$T(s):=\sum _{j=1}^N{p}_j({\mathit{\alpha }}^{(s)},{\alpha }_{N+1}^{(s)})$$

Clearly, the total price $T(s)$ is a decreasing function in $s$ ; that is, $T(1)\ge T(2)\ge \cdots \ge T(N)$ . Hence, linear search is applicable here whenever feasible. In addition, recall that $s_1$ is the critical value for problem [ETP-v1]. Hence, being the optimum of a less constrained problem, $s_1$ serves as a lower bound for the optimal threshold of problem [ETP-v2]. Using that threshold, we could solve the [ETP-v2] by reducing it to a smaller sub-problem.

Proposition 2

Problem [ETP-v2] can be solved as follows:

1. (1)

If $T(N)>N\overline{P}$ , then (AveP) is always violated and there is no feasible solution.

(2)

If $T(N)<N\overline{P}<T(N-1)$ , all products are sold out except $N$ , that is, ${\alpha }_j^{*}={\beta }_j$ for $j=1,\dots ,N-1$ ; otherwise, if $T(N)=N\overline{P}$ , then all products are sold out.

(3)

If $T(N-1)\le N\overline{P}$ , then there exists a threshold

$$s_2=max\{s_1,min\{T(s+1)\le N\overline{P}|s=1,2,\dots ,N-1\}\},$$

such that ${\alpha }_j^{*}={\beta }_j$ for $j=1,2,\dots ,s_2$ ; moreover, the remaining demand vector $({\alpha }_{s_2+1}^{*},\dots ,{\alpha }_N^{*})$ can be obtained by solving the sub-problem [ETP-v2( $s_2$ )] as prescribed by Lemma 3.

Proposition 2 considers three cases. The first case is infeasible: since $T(N)$ corresponds to all products being sold out (i.e., prices are already low), if $T(N)>N\overline{P}$ , prices cannot be reduced further, and [ETP-v2] has no feasible solution. In the second case, all products are sold out at optimality if $T(N)=N\overline{P}$ ; but if $T(N)<N\overline{P}<T(N-1)$ , the revenue could be enhanced by setting a slightly higher price for product $N$ , resulting in all but the last product being sold out.

In the third case, we find the threshold $s_2<N$ and invoke sub-problem [ETP-v2( $s_2$ )] to solve for the optimal prices. Note that $s_1\le s_2$ ; that is, adding an average price restriction may help the box office sell out more ticket categories. While this may not be profitable for the box office, it benefits the consumers as more seats are offered at lower prices. Moreover, the solution of [ETP-v2( $s_2$ )] shows that the optimal prices of all products are affected by the capacity limits of the sold-out ones, which is in contrast with problem [ETP-v1], where the capacities of the sold-out products impact just their own prices. As such, the average price restriction accentuates and spreads the impact of the capacity constraints of the sold-out products to the pricing decisions of all products. Together, Proposition 2 simplifies a constrained, multi-dimensional optimization problem into a one-dimensional search. To implement the linear search of the threshold $s_2$ , one must first compute the price sum $T(s)=\sum _{j=1}^N{p}_j({\alpha }^{(s)})$ for $s=1,\dots ,N$ , which requires $N$ calls of $O(N)$ operations, leading to $O(N^2)$ time complexity. Since no other step takes a longer time, the procedure that implements Proposition 2 to solve [ETP-v2] has overall time complexity $O(N^2)$ .

4.4. Optimization with all three constraint classes ([ETP-v3])

Augmenting [ETP-v2] with the constraint (CeilN)  defines

$$\begin{array}{rl}\mathbf{[}\mathbf{E}\mathbf{T}\mathbf{P}\mathbf{-}\mathbf{v}\mathbf{3}\mathbf{]}\underset{\mathbf{p}}{max}& \{\sum _{j=1}^N{\alpha }_j(\mathbf{p})p_j|{\alpha }_j(\mathbf{p})\le {\beta }_j,\mathrm{\forall }1\le j\le N;\\ & \sum _{j=1}^N{p}_j\le N\overline{P};p_N\le U_N\}\end{array}$$

Despite its simple structure, (CeilN) can cause a considerable structural change to the optimal solution of problem [ETP-v2]. To tackle this difficulty, we recursively construct sub-problems with fewer decision variables: given an $s=N-1,N-2,\dots ,1$ , we define an $s$ -dimensional sub-problem:

$$\begin{array}{rl}\mathbf{[}\mathbf{E}\mathbf{T}\mathbf{P}\mathbf{-}\mathbf{v}\mathbf{3}\mathbf{(}\mathbf{s}\mathbf{)}\mathbf{]}\underset{p_1,\dots ,p_s}{max}& \{\sum _{j=1}^s{\overline{\alpha }}_j(\mathbf{p})p_j|{\overline{\alpha }}_j(\mathbf{p})\le {\beta }_j,\mathrm{\forall }1\le j\le s;\\ & \sum _{j=1}^s{p}_j\le N\overline{P}-\sum _{n=s+1}^{N}n{U}_n\}\end{array}$$

Note that problem [ETP-v3( $s$ )] is essentially problem [ETP-v2] with $s$ products. However, the demand function and the average price restriction are modified. In particular, the quality of product $j$ is modified to be ${\overline{q}}_j=q_j-q_{s+1}$ ( $j=1,2,\dots ,s$ ). Additionally, the demand function ${\overline{\alpha }}_j$ is modified from (1) by using quality ${\overline{q}}_j$ and letting product $s$ be the lowest quality product. Lastly, the average price restriction is transformed by adding an extra term that is a function of $(U_N,U_{N-1},\dots ,U_{s+1})$ where $U_j$ can be calculated iteratively:

$$U_j=Q_j(U_{j+1}/Q_{j+1}+{\beta }_{j+1}),s\le j\le N-1$$

(7)

Hence, the upper bound on the price of the lowest quality product has a ripple effect across the categories of all products, because the impact of price may pass along to adjacent products. Based on this insight, we develop an easily implemented algorithm to solve problem [ETP-v3].

• Step 1: Initialization. Given the quality levels $q_1,\dots ,q_N$ and the capacity limits ${\beta }_1,\dots ,{\beta }_N$ , solve the corresponding [ETP-v2] to obtain the optimal price vector $(p_1^{*},\dots ,p_N^{*})$ .

• (b)

  Else, let $p_N^{*}=U_N$ and go to Step 2.

Step 2: Iteration. For $s=N-1,N-2,\dots ,1$ , do the following:

• (a)

Update the quality levels $q_1,\dots ,q_s$ such that $q_j\to q_j-q_{s+1}$ .

(b)

Generate the $(U_N,\dots ,U_{s+1})$ sequence based on equation (7).

(c)

Use these parameters to construct a sub-problem [ETP-v3( $s$ )].

(d)

Solve [ETP-v3( $s$ )] to obtain $(p_1^{*},\dots ,p_s^{*})$ .

1. -

If $p_s^{*}\le U_s$ , then update $p_j^{*}\to p_j^{*}+\sum _{n=s+1}^N{U}_n$ for $j=1,\dots ,s$ ; and go to Step 3.

-

Else, let $p_s^{*}=U_s$ and proceed to the next iteration.

Step 3: Conclusion. Let $p_j^{*}=\sum _{n=j}^N{U}_n$ for $s+1\le j\le N$ and concatenate the prices obtained in the above procedure to form the price vector ${\mathbf{p}}^{*}$ . Return it as the optimal solution.

In the above algorithm, we assume that all problems are feasible; otherwise, if any of the problems is infeasible, then so is problem [ETP-v3]. In terms of complexity, our proposed algorithm consists of a one-dimensional iteration (Step 2), and each loop involves solving a sub-problem [ETP-v3( $s$ )] (essentially [ETP-v2] with $s$ variables). Since solving [ETP-v2] takes $O(N^2)$ time, our algorithm has an overall complexity $\sum _{s=1}^{N-1}O(s^2)=O(N^3)$ , which is computationally efficient. The next proposition formally establishes the validity of the above algorithm.

Proposition 3

Consider problem [ETP-v3]. The above algorithm produces an optimal price vector ${\mathbf{p}}^{*}$ . Moreover, if the algorithm ever enters into Step 2 and stops at $s<N-1$ , there exist two thresholds $1\le s_3\le s_3^{\prime }\le N$ such that ${\alpha }_j^{*}={\beta }_j$ for $1\le j\le s_3$ or $s_3^{\prime }\le j\le N$ .

Compared to the optimal solution of the problem [ETP-v2], we see two important effects of (CeilN) on the ETP problem. First, under the vertical structure, even though the restriction applies directly only to the lowest quality product, its effect propagates through all of the ticket categories in an increasing order of quality. Hence, every product effectively faces an implicit price upper bound. Second, there exists a new sold-out threshold $s_3^{\prime }$ where the sales of the last $N-s_3^{\prime }+1$ products reach their capacity limits. Indeed, when $U_N$ is small enough, all $p_j$ ( $j\ge s_3^{\prime }$ ) are determined by relatively low ceiling prices $U_j$ ( $j\ge s_3^{\prime }$ ). As a result, the ticket categories with the lowest appeal attract enough demand to sell out. Additionally, as with [ETP-v2], the optimal prices in [ETP-v3] are affected by the capacities of the sold-out products. Applying Proposition 2 to sub-problem [ETP-v3( $s$ )] with $s=s_3^{\prime }-1$ , we see that the prices of products $j<s_3^{\prime }$ are determined by the capacities of sold-out products $j\le s_3$ and the modified average price, which depends on the capacities of sold-out products $j\ge s_3^{\prime }$ . Thus, the prices of the high quality sold-out products are determined by the capacity limits of all sold-out products. For the low quality sold-out products $j\ge s_3^{\prime }$ , however, their optimal prices are independent of the capacities of the sold-out products at the higher end.

5. Pricing under horizontally differentiated demand

We now study the price optimization problem where products are horizontally differentiated, which is captured by the well-known MNL model, to incorporate customers’ idiosyncratic preferences that often manifest in purchasing behavior. The demand function of each ticket category is given by equation (2); and we write $\mathit{\alpha }$ instead of ${\mathit{\alpha }}^H$ in this section for brevity. Note that all customers are assumed to have an identical price sensitivity (equaling one) in the utility function, so that the concavity of the revenue function is protected. In the following, we progressively add the three constraints sequentially to construct and study problems [ETP-h $x$ ], $x=0,1,2,3$ .

5.1. Unconstrained optimization of horizontally differentiated categories ([ETP-h0])

We solve the unconstrained problem [ETP-h0] as a benchmark using the price-demand variable transformation (see Bitran and Caldentey, 2003) derived from the demand function:

$$p_j(\mathit{\alpha })=\theta q_j+\log (1-\sum _{i=1}^N{\alpha }_i)-\log {\alpha }_j,j=1,2,\dots ,N$$

(8)

Expressing revenue as a function of the demand vector yields a concave maximization, allowing the problem to be solved via first-order conditions. Then, the maximized revenue $R^{*}$ and the optimal prices $p_j^{*}$ are given by $R^{*}=W(\sum _{j=1}^N\exp (\theta q_j-1))$ and $p_j^{*}=R^{*}+1$ ( $j=1,2,\dots ,N$ ), where $W(z)$ is the Lambert $W$ function, that is, the real solution $w$ to the equation $z=w{e}^w$ for $z\ge 0$ . Observe that the optimal prices for all ticket categories are equal, known as the “equal markup” property (see, e.g., Anderson and De Palma, 1992; Dong et al., 2009) (prices equal margins here since production cost is assumed to be zero). Next, we investigate the impact of capacity constraints and price restrictions on this optimality structure of the pricing problem.

5.2. Optimization with the capacity constraints ([ETP-h1])

Consider the price optimization [ETP-h0] augmented with the capacity constraints (Cap). Using the one-to-one mapping (8), we rewrite the resulting optimization problem using the demand vector $\mathit{\alpha }$ as the decision variable:

$$\mathbf{[}\mathbf{E}\mathbf{T}\mathbf{P}\mathbf{-}\mathbf{h}\mathbf{1}\mathbf{]}\underset{\mathit{\alpha }}{max}\{\sum _{j=1}^N{\alpha }_j{p}_j(\mathit{\alpha })|{\alpha }_j\le {\beta }_j,\mathrm{\forall }1\le j\le N\}$$

With a concave objective function and linear constraints, the problem is convex, and we can use the KKT conditions to compute the globally optimal solution. From the KKT conditions, we deduce an interesting threshold structure as characterized by the following lemma.

Lemma 4

In the solution ${\mathit{\alpha }}^{*}$ to the KKT conditions of problem [ETP-h1], if product $j$ is sold out, then product $i$ with $i\le j$ must also be sold out.

Therefore, the optimal solution to problem [ETP-h1] is well-structured. In fact, the optimal demands feature a certain threshold such that all ticket categories with quality above that threshold are sold out at optimality. As a result, we can narrow the search for the optimal solution down to the sold-out threshold type. Thus, we only need to characterize the optimal demands given a specific threshold. In other words, for any threshold, we want to compute the corresponding optimal demand vector and maximal revenue. Recall the partial sum of capacity limits $K(s)=\sum _{j=1}^s{\beta }_j$ ( $s=1,2,\dots ,N$ ). Then, given an $s=0,1,\dots ,N$ , the following algorithm will produce a solution $\mathit{\alpha }(s)$ with the maximized revenue $R(s)$ :

1. (1)

If $s=0$ , then $\mathit{\alpha }(s)$ equals the optimal solution to the problem [ETP-h0].

(2)

If $s\ge 1$ , then ${\alpha }_j(s)={\beta }_j$ for $j=1,\dots ,s$ . Furthermore, if $s<N$ , then for $j=s+1,\dots ,N$ ,

$${\alpha }_j(s)=\frac{W(Q)(1-K(s){)}^2}{1+W(Q)(1-K(s))}\frac{e^{\theta (q_j-q_N)}}{\sum _{i=s+1}^N{e}^{\theta (q_i-q_N)}}$$

(9)

where $Q=\sum _{i=s+1}^N{e}^{\theta q_i}f(1-K(s))$ and $f(x)=x^{-1}{e}^{-x^{-1}}$ .

(3)

Lastly, if $s<N$ and ${\alpha }_{s+1}(s)>{\beta }_{s+1}$ , then $R(s)=0$ ; otherwise, substituting $\mathit{\alpha }(s)$ into the objective function, we obtain the total revenue:

$$R(s)=\sum _{j=1}^N{\alpha }_j(s)p_j(\mathit{\alpha }(s))$$

(10)

Interestingly, every product $j=s+1,\dots ,N$ is partially sold as its demand ${\alpha }_j$ is not zero, which is a contrast to the vertical products case (see Proposition 1). Rather, its demand is correlated with the aggregate capacity of all the sold-out products $K(s)$ . Furthermore, since the demand given by (9) can be shown to decrease in $K$ (keeping $s$ unchanged), a larger capacity of sold-out products will negatively affect the demand for every partially sold product. In addition, the optimal sales of the partially sold products also depend on their quality levels $q_{s+1},\dots ,q_N$ . In particular, we have ${\alpha }_j>{\alpha }_{j+1}$ , indicating that the product with higher quality has larger sales. Therefore, product quality has a considerable impact on product substitutions and, in turn, influences product sales. Note that the condition ${\alpha }_{s+1}(s)\le {\beta }_{s+1}$ guarantees that the capacity constraints are met for all products $s+1,\dots ,N$ , because ${\alpha }_j>{\alpha }_{j+1}$ and ${\beta }_j\le {\beta }_{j+1}$ .

Now, once the optimal total revenue is expressed as a function of the threshold $s$ , the [ETP-h1] can be solved via a one-dimensional search. Note that computing the revenue aggregation $R(s)$ for each $s=1,\dots ,N$ requires several vector operations on length- $N$ arrays, which takes $O(N)$ time; hence, the overall complexity for the algorithm that implements the above procedure and identifies the optimal solution is $O(N^2)$ . Next, we uncover interesting properties of the optimal prices based on the KKT conditions.

Proposition 4

Consider problem [ETP-h1]. The optimal sold-out threshold $s_1$ ¹⁰ is given by the following equation:

$$s_1\in \arg max\{R(s)\mid s=0,1,\dots ,N\}$$

where $R(s)$ is the total revenue function (10). Hence, the optimal demand is ${\mathit{\alpha }}^{*}=\mathit{\alpha }(s_1)$ . Moreover, the optimal prices have the following property:

1. (1)

   If $s_1=0$ , then $p_1^{*}=\cdots =p_N^{*}$ .

2. (2)

   If $s_1=N$ , then $p_1^{*}>\cdots >p_N^{*}$ .

3. (3)

   If $0<s_1<N$ , then $p_1^{*}>\cdots >p_{s_1}^{*}>p_{s_1+1}^{*}=\cdots =p_N^{*}$ .

First, unlike Proposition 1 in the vertical product case, Proposition 4 reveals that the sold-out threshold $s_1$ cannot be predetermined. Rather, it can only be searched by comparing the resulting optimal revenues. Nevertheless, such a one-dimensional search is computationally efficient. Second, in contrast to problem [ETP-h0], the equal-markup result does not hold for problem [ETP-h1]. Instead, the optimal prices have the monotonicity property, which is aligned with practice: the prices of the sold-out products can be ordered based on their quality levels, that is, higher quality products are priced higher. On the other hand, the partially sold products, although having non-zero sales that increase in quality, are still valued identically at the same price. Hence, the capacity constraints (Cap), which are commonly seen in practice, have a strong impact on the structure of the optimal solution, and do so in a manner that contrasts with insights that stem from unconstrained models.

5.3. Optimization with capacity constraints and average price restriction ([ETP-h2])

Problem [ETP-h2] includes the average price restriction (AveP) in addition to the capacity constraints. To solve this model, we treat the demand vector as the decision variable and write the problem via the relationship described in (9).

$$\begin{array}{rl}\mathbf{[}\mathbf{E}\mathbf{T}\mathbf{P}\mathbf{-}\mathbf{h}\mathbf{2}\mathbf{]}\underset{\mathit{\alpha }}{max}& \{\sum _{j=1}^N{\alpha }_j{p}_j(\mathit{\alpha })|{\alpha }_j\le {\beta }_j,\mathrm{\forall }1\le j\le N;\\ & \sum _{j=1}^N{p}_j(\mathit{\alpha })\le N\overline{P}\}\end{array}$$

The main difficulty introduced by the constraint (AveP) is that the price function $p_j(\mathit{\alpha })$ is neither convex nor concave in $\mathit{\alpha }$ . However, we can show that the sum of all prices is quasi-convex in the demand $\mathit{\alpha }$ . As such, it is still a convex program, and we can solve for optimality via the KKT conditions. Note that, even when considering the weighted average price constraint, we can still show the same result; see proof in E-Companion EC.1 for details. As Lemma 5 shows, the sold-out threshold structure holds in this case as well.

Lemma 5

The KKT conditions are necessary and sufficient for the optimality of problem [ETP-h2]. Moreover, the optimal demand vector has a sold-out threshold structure; that is, if product $j$ is sold out, then product $i$ with $i\le j$ must also be sold out.

The threshold-type optimal solution described in Lemma 5 not only facilitates the algorithm for the optimality search, but also helps us characterize the optimal prices and demands of the products. Although we cannot specify the threshold or present the optimal solution in closed form, our analysis uncovers several useful insights concerning the optimality structure.

Proposition 5

Let $s_2$ ( $0\le s_2\le N$ ) be the optimal sold-out threshold for problem [ETP-h2]. Then, $s_2\ge s_1$ . Moreover, unless the problem is equivalent to the unconstrained case, the optimal prices have the strict monotonicity property $p_1^{*}>p_2^{*}>\cdots >p_N^{*}$ ; and for the optimal demands, ${\alpha }_j^{*}={\beta }_j$ for the sold-out products $j\le s_2$ and, if $s_2<N$ , ${\alpha }_{s_2+1}^{*}>\cdots >{\alpha }_N^{*}$ for the remaining products.

Proposition 5 reveals important structural results about the optimal solution to problem [ETP-h2]. First, similar to the vertical differentiation case, the thresholds for [ETP-h1] and [ETP-h2] are comparable, and we have $s_1\le s_2$ . Suppose that we search for the sold-out threshold in decreasing order from the all-sold-out scenario. Then, a lower sold-out threshold indicates higher prices for certain products, and they may violate the average price restriction, at which time we terminate the search. If (AveP) is not violated even when the threshold is down to $s_1$ , then the average price restriction is redundant, and $s_1$ is the optimal threshold.

Second, unless the problem is equivalent to its unconstrained counterpart, there exists a monotonicity property such that higher quality products charge higher prices. Therefore, we show that the average price restriction can differentiate the pricing of the partially sold products as well, further breaking down the equal-markup structure observed in the unconstrained scenario. Additionally, while the sales of the sold-out products are known, the demands of the remaining products can be ordered such that the higher quality products attract more customers. Such a partial order among products’ sales implies that the manager must prioritize and monitor the high quality partially sold products, as they can generate higher revenue. Although the sales of the sold-out products decrease in quality and the sales of partially sold products increase in quality, the two monotonic chains are not comparable, because either ${\beta }_{s_2}$ or ${\alpha }_{s_2+1}^{*}$ could be larger.

5.4. Optimization with all three constraint classes ([ETP-h3])

Next, we consider the optimization problem with all of the constraints. As in Section 5.3, we write the problem using $\mathit{\alpha }$ as the decision variable based on the one-to-one mapping (8) as follows:

$$\begin{array}{rl}\mathbf{[}\mathbf{E}\mathbf{T}\mathbf{P}\mathbf{-}\mathbf{h}\mathbf{3}\mathbf{]}\underset{\mathit{\alpha }}{max}& \{\sum _{j=1}^N{\alpha }_j{p}_j(\mathit{\alpha })|{\alpha }_j\le {\beta }_j,\mathrm{\forall }1\le j\le N;\\ & \sum _{j=1}^N{p}_j(\mathit{\alpha })\le N\overline{P};p_N(\mathit{\alpha })\le U_N\}\end{array}$$

For the constraint (CeilN), we can write its equivalence as a linear inequality in $\mathit{\alpha }$ . Hence, just as in Lemma 5, the KKT conditions are still necessary and sufficient for problem [ETP-h3]. However, the specific KKT conditions in this case are not amenable to analysis, and we are unable to generate useful structural results. Therefore, we will focus on a special case where the constraint (AveP) is not binding. Consider the problem

$$\begin{array}{rl}& \mathbf{[}\mathbf{E}\mathbf{T}\mathbf{P}\mathbf{-}\mathbf{h}{\mathbf{3}}^{\prime }\mathbf{]}\\ & \underset{\mathit{\alpha }}{max}\{\sum _{j=1}^N{\alpha }_j{p}_j(\mathit{\alpha })|{\alpha }_j\le {\beta }_j,\mathrm{\forall }1\le j\le N;p_N(\mathit{\alpha })\le U_N\}\end{array}$$

Solving the above problem analytically may help us better understand the solution to [ETP-h3]. Without the average price restriction, problem [ETP-h3’] is a convex optimization problem. We can therefore solve it by the KKT condition and the solution actually has a closed form. Moreover, similar to the previous scenarios, the optimal sales of all ticket categories exhibit a sold-out threshold structure. Furthermore, when the constraint (CeilN) is binding, we only need to conduct a one-dimensional search for the optimal threshold from $0$ to $N-1$ . For a given threshold $s$ , we give the optimal solution in the lemma below.

Lemma 6

Consider problem [ETP-h3’] and assume that the constraint (CeilN) is binding (otherwise the problem reduces to [ETP-h1]). Then, its optimal solution has a sold-out threshold structure. Given the threshold $s=0,1,\dots ,N-1$ , the sales/prices are as follows:

1. (1)

If $s=N-1$ , then ${\alpha }_j^{*}={\beta }_j$ for $j=1,\dots ,N-1$ and ${\alpha }_N^{*}=\frac{1-\sum _{i=1}^{N-1}{\beta }_i}{1+e^{U_N-\theta q_N}}$ .

(2)

If $s=1,\dots ,N-2$ , then the optimal demand ${\alpha }_j^{*}={\beta }_j$ for $j=1,\dots ,s$ ; ${\alpha }_j^{*}={\alpha }_N^{*}\exp (-\omega +\theta (q_j-q_N))$ for $j=s+1,\dots ,N-1$ ; and ${\alpha }_N^{*}=[e^{(U_N-\theta q_N)}(U_N+\omega )+\omega {]}^{-1}$ , where $\omega >0$ is uniquely determined by the following equation:

$$\begin{array}{rl}& (1-K(s))(e^{(U_N-\theta q_N)}(U_N+\omega )+\omega )\\ & =1+\sum _{j=s+1}^{N-1}{e}^{-\omega +\theta (q_j-q_N)}+e^{(U_N-\theta q_N)}\end{array}$$

(3)

If $s=0$ , then the optimal price $p_j^{*}=U_N+\omega$ for $j=1,\dots ,N-1$ and $p_N^{*}=U_N$ , where $\omega$ is determined by the above equation with $K(0)=0$ .

The above lemma gives the sales/prices of the products for any sold-out threshold $s$ . However, we only focus on the feasible thresholds, which are the ones that correspond to feasible sales $\mathit{\alpha }$ . Moreover, although we only provide either the sales or the prices in the lemma, note that we may easily derive the full optimal solution using the optimality condition as well as the one-to-one mapping (9) between the price vector and demand vector. For example, in case (2) above, since (CeilN) is binding, we have $p_N^{*}=U_N$ . Then, for the sold-out products, subtracting (9) for product $j$ and $N$ , we have $p_j^{*}=U_N+\theta (q_j-q_N)+\log {\alpha }_N^{*}-\log {\beta }_j$ for $j=1,\dots ,s$ . For the remaining products (if any) $j=s,\dots ,N-1$ , we can use optimality conditions to obtain $p_j^{*}=U_N+\omega$ . In fact, we can perform similar operations in cases (1) and (3) to compute the full optimal solution. It is worth noting that the optimal prices of all products are influenced by the upper bound $U_N$ ; however, for sales, only the demands of the partially sold products depend on $U_N$ . Hence, the ceiling constraint on the price of the lowest quality ticket category propagates across all the other ticket categories.

From part (2) of Lemma 6, an important factor in deciding the optimal solution to [ETP-h3’] is $\omega =\tilde{\omega }/{\alpha }_N^{*}$ , where $\tilde{\omega }$ is the multiplier of the constraint (CeilN) in the KKT conditions; i.e., $\tilde{\omega }(U_N-p_N)=0$ . As a result, all the partially sold products have the same demand structure at optimality, which can be expressed by the optimal demand of product $N$ , the quality gap, and $\omega$ . Lastly, we compare the optimal ticket price across categories in the next proposition.

Proposition 6

Suppose that (CeilN) is binding. Problem [ETP-h3’] can be optimally solved by searching the feasible threshold $s_3\in \{0,1,\dots ,N-1\}$ . The optimal solution is given by Lemma 6. Moreover, we have $p_1^{*}>\cdots p_{s_3}^{*}>p_{s_3+1}^{*}=\cdots =p_{N-1}^{*}>p_N^{*}$ .

Although the impact of $U_N$ ripples through the optimal prices of all products, its impact on the equal-markup property manifests only with the last product. Furthermore, the optimal prices have the monotonicity property, which is similar to what we observed for [ETP-h1] (see Proposition 4). Specifically, the prices first decrease with quality levels and then stay the same, but the lowest quality product has an even lower price. Additionally, we have ${\alpha }_{s_3+1}>\cdots >{\alpha }_{N-1}$ , meaning that the lower quality products are generally sold less than the higher quality ones. Furthermore, this statement holds true even for the last product, i.e., ${\alpha }_{N-1}>{\alpha }_N$ , if $\omega <\theta (q_{N-1}-q_N)$ at optimality.

6. Results comparison and discussions

In this section, we summarize and compare the structural results obtained for the vertically and horizontally differentiated demand models in the ETP problem. While earlier sections analyze each model separately, placing their results side-by-side offers several advantages: it reveals how differences in demand structure lead to distinct sales patterns and pricing rules, and highlights many shared structural features. The comparison thereby sharpens theoretical contrasts and helps practitioners anticipate how consumer behavior and the resulting model choice affect implementation. We also discuss key implications, including how practical constraints shape the optimal solution, why identifying the correct demand model is critical, and the computational efficiency of our algorithms. These insights underscore the managerial relevance and robustness of our approach.

6.1. Comparison of structural results

We begin our comparison of the vertically and horizontally differentiated demand models by summarizing their key structural properties. While the detailed derivations of these results have been presented in the preceding sections, here, to facilitate easy comparison, we list the results for each type of product differentiation across three progressively constrained models in Table 1.

Table 1.

Structural results comparison between demand models: Optimal sales and prices.

		(Cap) only	(Cap) and (AveP)	(Cap), (AveP), and (CeilN) ${}^{\mathrm{a}}$
Vertical	Sales	$\mathrm{\exists }{s}_1^V$ , ${\alpha }_j={\beta }_j$ for $j\le s_1^V$ , ${\alpha }_{s_1^V+1}=1/2-K(s_1^V)$ . All other partially sold products have zero sales.	$\mathrm{\exists }{s}_2^V\ge s_1^V$ , ${\alpha }_j={\beta }_j$ for $j\le s_2^V$ . Partially sold products may have positive sales.	$\mathrm{\exists }{s}_3^V$ and $s_3^{V^{\prime }}$ (two thresholds), ${\alpha }_j={\beta }_j$ for $j\le s_3^V$ and $j\ge s_3^{V^{\prime }}$ . Partially sold products may have positive sales.
	Prices	Given by equation (5). Prices decrease in quality. Soldout products’ capacity limits only affect their own prices.	Given by Lemma 3 and (6). Prices decrease in quality. Soldout products’ capacity limits affect all prices.	Given by our algorithm. Prices are decreasing in quality.
Horizontal	Sales	$\mathrm{\exists }{s}_1^H$ , ${\alpha }_j={\beta }_j$ for $j\le s_1^H$ . Partially sold products have positive sales that decrease in quality.	$\mathrm{\exists }{s}_2^H\ge s_1^H$ , ${\alpha }_j={\beta }_j$ for $j\le s_2^H$ . Partially sold products’ sales are positive, and decrease in quality.	$\mathrm{\exists }{s}_3^H$ , ${\alpha }_j={\beta }_j$ for $j\le s_3^H$ . Partially sold products’ sales are positive and decrease in quality, except the last one.
	Prices	Prices weakly decrease in quality; all partially sold products have the same price.	Prices strictly decrease in quality.	Prices weakly decrease in quality. Partially sold products’ prices are identical except for the last one.

${}^{\mathrm{a}}$ For the horizontal case, this column refers to the special case where (AveP) is non-binding.

Table 1 organizes the comparison along two dimensions. The columns correspond to different problem settings, starting from the basic capacity-constrained problem and progressively adding the average price restriction (AveP) and then the upper bound constraint on the lowest-quality ticket price (CeilN). The rows correspond to the two demand models, with separate entries for the optimal sales pattern and the corresponding price structure. Within each cell, we summarize the structural results derived earlier in the paper in concise form.

From this side-by-side layout, several features are easy to observe. In all cases (except [ETP-v3)], the optimal sales pattern follows a threshold policy, that is, tickets above a certain quality threshold sell out, while lower-quality tickets may have partial or no sales, depending on the cases. For problem [ETP-v3], there is an additional threshold where products with lower quality are also sold out. For the optimal prices, they generally decrease with quality, although the strength of this monotonicity varies with both the demand model and the active constraints. For example, while the vertical model always preserves decreasing prices regardless of the constraints, while strict monotonicity is guaranteed in the horizontal model only under (Cap) and (AveP). Such patterns set the stage for the numerical illustrations and discussions that follow.

To corroborate the structural comparisons above, we present a numerical example with $N=10$ products under both demand models. This example mirrors the settings summarized in Table 1 and visualizes how the optimal sales and prices evolve as additional constraints become active. For the problem parameters, we generate the product quality levels $q_j$ ( $j=1,\dots ,10$ ) that is evenly spaced in descending order and choose the corresponding capacity limits ${\beta }_j$ in ascending order. For the vertical model, when the corresponding constraint is present we set the average-price cap to $\overline{P}=0.25$ and the lowest-quality price ceiling to $U_N=0.025$ . For the horizontal model, we fix $\theta =0.5$ , and (when applicable) set $\overline{P}=1$ and $U_N=2$ . Panels (a) to (c) of Figure 1 report the three vertical problems, and panels (d) to (f) report the horizontal counterparts. In each panel, we plot the optimal price (black dashed) for every product together with its realized sales (blue solid) and the capacity limit (red dotted); moreover, sales and capacity are measured on the left vertical axis, and prices on the right.

Figure 1.

Numerical confirmation of structural results under both demand models: (a) problem [ETP-v1], (b) problem [ETP-v2], (c) problem [ETP-v3], (d) problem [ETP-h1], (e) problem [ETP-h2], and (f) problem [ETP-h3’].

Each panel, therefore, displays three objects: (i) optimal prices across qualities, (ii) realized sales, and (iii) capacity limits for each category. The numerical illustration allows direct identification of sold-out versus partially sold (or unsold) products and the induced threshold structure. Moving left to right within each row, added constraints that reshape these curves in ways consistent with the analysis. For instance, in the vertical case [ETP-v1], the sales curve (blue) coincides with capacity up to a threshold and then drops to zero, whereas in [ETP-v2], it tapers off with positive (but lower-than-capacity) sales beyond a higher threshold; in [ETP-v3], the “sold-out at both ends” pattern appears, with partial sales in the middle. On prices, the vertical panels maintain a decreasing profile in quality throughout. In the horizontal row, [ETP-h1] exhibits a flat price segment across partially sold products, [ETP-h2] shows prices strictly decreasing in quality, and in [ETP-h3’], prices are weakly decreasing with a visibly lower last point due to the $U_N$ ceiling constraint. These visual patterns align with the structural summaries in Table 1.

6.2. Discussion of implications

Both the analytical results in Table 1 and numerical confirmation in Figure 1 point to several managerial implications. In the following, we discuss them around three themes. First (and central to our motivation), practical constraints would effectively reshape both the sales thresholds and the price profiles. Second, the underlying demand model matters: Vertical and horizontal specifications produce systematically different structural behaviors, so a correct demand model is critical to solving the ETP. Third, we assess computational efficiency and scalability, showing that the proposed algorithms solve large instances quickly under both the demand model and in line with the theoretical complexity established earlier.

6.2.1. Impact of practical constraints

Across both demand models, the capacity-only problem provides a clean baseline: The optimal sales pattern obeys a threshold rule, and prices are non-increasing in quality. In the vertical case, this yields a sold-out prefix with strictly decreasing prices; in the horizontal case, partially sold products share a common price, producing a weakly decreasing trend. These commonalities highlight capacity as the primary local driver of sales and price at the baseline.

After introducing (AveP) (moving from the first to the second column of Table 1 and from panels (a)/(d) to (b)/(e) in Figure 1), two universal effects follow. First, the sold-out set weakly expands (the threshold weakly increases), because lowering the average price reallocates demand toward higher-quality items until their capacities bind. Second, prices of partially sold products align more tightly with the binding constraints: in the vertical model, a sold-out product’s capacity now influences all prices (not just its own), and in the horizontal model, the plateau disappears and prices become strictly decreasing. Partially sold products typically retain positive sales in both models, reflecting the global nature of the average-price cap.

Adding a lowest-quality price ceiling (third column; panels (c)/(f)) introduces a bottom-end corner that further reshapes the structure. Under both demand models, the ceiling lowers the tail price and can trigger additional sold-out categories. In the vertical model, this manifests as two thresholds: high- and low-quality products sell out, with partial sales in the middle, while prices remain decreasing. In the horizontal model, prices revert to weakly decreasing, with all partially sold products priced identically except the last one, which is bounded by the ceiling. Similarly, the sales threshold persists, and tail sales remain (weakly) decreasing in quality except for the last one.

Taken together, these patterns show that practical constraints can (i) shift thresholds, (ii) alter how the capacity limit of each product binds, and (iii) change price monotonicity from strict to weak (or vice versa). Such impacts represent fundamental changes that directly affect decisions; they determine which ticket classes sell out, how prices across products are interlinked, and whether the resulting solution is feasible and revenue-maximizing.

6.2.2. Importance of correct demand model

The comparison in Table 1 and Figure 1 underscores that the choice of demand model has a direct bearing on both sales patterns and pricing decisions. The contrasting differences in solution structures imply that the underlying demand specification and parameter estimation cannot be treated as interchangeable. Knowing the quality levels and whether demand is primarily vertical or horizontal is essential for predicting which ticket categories sell out, how prices evolve with quality, and what managerial insights can be drawn.

The importance of demand model correctness has three aspects. The first is model estimation from real data. In our setting, this process involves three steps. First, the quality levels of products must be quantified. This can be done empirically using a combination of objective characteristics (e.g., seat location or sightline) and subjective measures (e.g., willingness-to-pay surveys). Second, it is important to determine whether products are vertically or horizontally differentiated. In practice, this can be inferred from pricing, demand, and customer behavior data. Third, once the nature of differentiation is established, model parameters can be estimated. For the vertical case, this means estimating the distribution parameters of $\theta$ , while for the horizontal case (MNL) it involves estimating the value of $\theta$ from observed sales and pricing data. For the sake of brevity, we defer the details of estimation procedures to the E-Companion EC.2, but emphasize that reliable model estimation is the first step in ensuring the chosen model matches reality.

The second aspect of correctness is model specification. Here, the risk of model misspecification arises if the analyst assumes the wrong structural form. Therefore, we conduct numerical experiments to assess the robustness of our approach under mild misspecification. Our results indicate that when capacity constraints remain satisfied, revenue loss from misspecification is typically small, suggesting that the constrained optimization framework is reasonably robust. However, if misspecification leads to capacity violations, the resulting solution may become unreliable. These findings, with supporting numerical evidence presented in E-Companion EC.3, reinforce the need for careful model validation before implementing pricing decisions in practice.

The third aspect is model interpolation. Even when the two benchmark models are well understood, actual demand may lie somewhere between the two. Thus, an important question is whether the two extreme demand specifications adequately capture pricing behavior in practice. To examine this issue, we consider a parameterized mixed logit choice model, numerically solve the corresponding price optimization, and compare the optimal prices with those obtained from interpolation of the two extremes. The results, reported in E-Companion EC.4, help clarify whether the two benchmark cases together provide a useful approximation to more a general demand model.

6.2.3. Computational efficiency and scalability

Based on the identified structural properties, we have proposed computationally efficient pricing procedures for all three problems with vertically and horizontally differentiated demand and the capacity constraint. Theoretical analysis in earlier sections established that those problems can be solved in polynomial time by our pricing procedures. To complement this analysis, we now provide numerical evidence on scalability by examining average runtimes across different problem sizes. We focus on [ETP-v3] and [ETP-h1] here in this discussion. Recall that the algorithm for [ETP-v3] has worst case time $O(N^3)$ whereas the algorithm for [ETP-h1] has worst case time $O(N^2)$ . In our numerical test, for $N=\{100,500,1000,5000,10,000\}$ , we randomly generate $M=100$ instances (except for the case of $N=10,000$ , where $M=50$ ) of each problem and report the average runtime and its standard deviation in Table 2.

Table 2.

Runtime statistics of proposed algorithms for different problem sizes.

	$N=$ 100	$N=$ 500	$N=$ 1000	$N=$ 5000	$N=$ 10,000
[ETP-v3]	0.014 (0.002)	0.344 (0.015)	2.257 (0.042)	276.43 (11.24)	2270.33 (88.91)
[ETP-h1]	0.004 (0.001)	0.020 (0.003)	0.050 (0.006)	0.522 (0.041)	1.643 (0.092)

Notes: Numbers in the table are average (standard deviation); all times are in seconds.

The runtime statistics confirm that both algorithms scale polynomially with problem size, though in markedly different ways. For [ETP-h1], runtimes remain very small (even at $N=10,000$ the average runtime is under 2 seconds), consistent with its $O(N^2)$ complexity and the efficiency of closed-form updates under the MNL structure. By contrast, [ETP-v3] becomes considerably slower as $N$ grows, reflecting its higher $O(N^3)$ complexity and the need to recursively construct prices and solve nested subproblems. In our experiments, we also chose parameters so that all three constraints bind in [ETP-v3], which typically forces multiple iterations, further increasing runtime. As a result, solving [ETP-v3] is slower in absolute terms and grows faster in relative terms than solving [ETP-h1], even though their theoretical complexities differ by only one polynomial degree. Still, for [ETP-v3], instances with $N=1000$ are solved within seconds and those with $N=10,000$ within tens of minutes on a standard computer, demonstrating practical feasibility. Moreover, the standard deviations are consistently small relative to the means, indicating stable performance across random instances. Overall, both algorithms exhibit clear polynomial growth yet remain computationally tractable for large-scale applications, underscoring the practicality of the proposed pricing procedures.

7. Additional considerations

While our earlier analysis assumes fixed number of ticket categories and capacities, in practice, the box office may have some flexibility to adjust them. We examine two such scenarios: (i) closing the lowest-quality category or adding a new highest-quality one, and (ii) reallocating a small amount of capacity from a lower- to a higher-quality category. Because both extensions focus on capacity planning, we restrict our analysis to problems [ETP-v1] and [ETP-h1] with only the (Cap) constraint.

7.1. Closing and adding ticket categories

In our motivating setting, the box office manager could make small adjustments to ticket categories to improve profitability. One option, closing, eliminates the lowest-quality category, changing the product set to $j=1,\dots ,N-1$ . Another option, adding, introduces a new highest-quality category, so that the set becomes $j=0,1,\dots ,N$ , where product 0 has quality $q_0>q_1$ and capacity ${\beta }_0<{\beta }_1$ . We use index 0 for the new product to preserve the existing product indexes.

Impact on the optimal solution of [ETP-v1]. For vertically differentiated products, lower-quality categories may have no demand (see Proposition 1), so closing the last product typically has no effect. However, adding a new highest-quality product does affect the optimal solution.

Proposition 7

Let $s_1$ be the sold-out threshold index before the change.

1. (1)

Closing: If $s_1<N-1$ or $s_1=N-1$ with $K(N-1)=\frac{1}{2}$ , then closing has no effect on the optimal solution. If $s_1>N-1$ or $s_1=N-1$ with $K(N-1)<\frac{1}{2}$ , then all products are sold out; moreover, their optimal prices and the maximized revenue will decrease after closing.

(2)

Adding: When adding product 0 to the ticket categories, the optimal revenue will increase and the index of the optimal sold-out threshold will either stay the same or decrease by 1; if it stays the same, then the optimal prices of all existing products will (weakly) decrease.

Consistent with intuition, Proposition 7 shows that closing generally has little effect since product $N$ almost never sells. The exception arises when capacity is limited, and the sold-out threshold is high, in which case closing product $N$ reduces revenue. By contrast, adding product 0 draws sales from lower-quality categories and lowers their prices, making product 0 the main revenue generator. Despite these price reductions, total revenue strictly increases. Depending on $q_0$ , the former sold-out product ( $s_1$ ) may either remain sold out or have leftover capacity, shifting the threshold to $s_1-1$ . Notably, this means a previously sold-out product may now have excess capacity, but no previously unsold category would be influenced by adding.

Impact on the optimal solution of [ETP-h1]. For horizontally differentiated products, the demand function is more complex, and analytical results are harder to derive. A key difficulty is that the optimal sold-out threshold may shift arbitrarily, making prices and demands difficult to compare, especially when a new product is added. To gain useful insights, we therefore focus on the case where the threshold remains unchanged after adjustments.

Proposition 8

Consider ticket category adjustments for problem [ETP-h1]. Suppose the sold-out threshold index, $s_1<N$ , remains the same before and after the change.

• (i)

Closing: When closing product $N$ , for the sold-out products, their optimal prices increase; but for the partially sold products, their optimal prices decrease and optimal demands increase. Moreover, the optimal revenue associated with each product $j=1,\dots ,N-1$ becomes higher.

(ii)

Adding: When adding product 0 to the ticket categories, the optimal demands for each of the partially sold products ( $j=s_1+1,\dots ,N$ ) will decrease.

In the horizontal case, customers who would have purchased product $N$ switch to other categories after closing, unlike in the vertical case. This substitution explains Proposition 8(i): high-quality, sold-out products command higher prices, while low-quality products drop in price to attract more sales. As a result, revenues for individual products increase, though the overall revenue may rise or fall depending on the closed category. Adding a new product, by contrast, is analytically more complex, as prices and revenues are difficult to compare before and after. Still, from the closed-form solution (9), demand for partially sold products ( $j=s_1+1,\dots ,N$ ) decreases in the total capacity of sold-out products. Thus, with the threshold unchanged, adding ${\beta }_0$ new product takes sales away from the lower-quality, partially sold categories. Our finding here is inline with observations from practice—adding premium sections would be more beneficial for event managers.

7.2. Reallocating capacities between ticket categories

Although capacity planning is typically a long-term decision, the box office may have flexibility to reallocate seats between categories to expand the capacity of more profitable tickets. For example, stadium sections can be rearranged across adjacent areas. Since the highest quality category often generates the most revenue, we focus on reallocating capacity from product $r>1$ to product 1. Given the optimal sold-out threshold, we consider two cases: reallocating excess supply from a partially sold category or converting seats from another sold-out category. We analyze these scenarios separately for problems [ETP-v1] and [ETP-h1], assuming the reallocated amount is small so that the sold-out threshold remains unchanged. ¹¹

Impact on the optimal solution of [ETP-v1]. We consider the capacity reallocation of amount $ϵ>0$ from product $r$ to product 1 and examine its influences on the optimal solution.

Proposition 9

Consider problem [ETP-v1] and assume that the capacity of $ϵ>0$ is removed from product $r>1$ and reallocated to product 1. Furthermore, assume that the optimal sold-out threshold $s_1$ is unchanged and satisfies $1\le s_1\le N-1$ . Then we have the following:

• (i)

Suppose $r>s_1$ . All optimal demands remain the same except product 1, whose demand increases by $ϵ$ . All optimal prices remain the same except products $j=1,\dots ,s_1$ , whose prices decrease. The optimal total revenue increases.

(ii)

Suppose $1<r\le s_1$ . The optimal demand of product 1 increases by $ϵ$ and that of product $r$ decreases by $ϵ$ ; all others remain the same. All optimal prices remain the same except products $j=1,\dots ,r-1$ , whose prices decrease. The optimal total revenue increases.

Since the sold-out threshold is unchanged and the reallocation is small, the impact on sales is straightforward. All products maintain the same demand except products 1 and $r$ . The effect on prices is more nuanced. All sold-out (higher quality) products with $j<r$ see lower prices, while the rest remain unchanged. This is because the price of a sold-out product is negatively correlated with the aggregate capacity of higher-quality products. Hence, in Proposition 9(ii), even though products $j=r,\dots ,s_1$ are sold out, their prices remain the same since the aggregate capacity is unaffected. Importantly, total revenue always increases regardless of the reallocation source. In fact, all products except product 1 experience unchanged or lower revenue, while product 1’s revenue rises, underscoring its central role as the primary revenue driver across categories.

Impact on the optimal solution of [ETP-h1]. For the horizontal differentiation case, we conduct a similar study on the impact of capacity reallocation.

Proposition 10

Consider problem [ETP-h1] and assume that the capacity of $ϵ>0$ is removed from product $r>1$ and reallocated to product 1. Furthermore, assume that the optimal sold-out threshold $s_1$ is unchanged and satisfies $1\le s_1\le N-1$ . Then we have the following:

• (i)

Suppose $r>s_1$ . The optimal demand of product 1 increases by $ϵ$ , the optimal demand of product $j=s_1+1,\dots ,N$ decreases, and the no-purchase probability ${\alpha }_{N+1}^{*}$ increases; all other demands are the same. The optimal price of the product $1$ decreases whereas those of all other products increase. The optimal total revenue increases.

(ii)

Suppose $1<r\le s_1$ . The optimal demand of product 1 increases by $ϵ$ and that of product $r$ decreases by $ϵ$ . The optimal price of product 1 decreases and that of product $r$ increases. The demands and prices of all other products are the same. The optimal total revenue increases.

In Proposition 10(i), reallocating excess supply from a low-quality partially sold product to product 1 increases the sales of the latter but reduces that of the former. The total sales reduction exceeds $ϵ$ because the no-purchase probability rises and market coverage shrinks. Prices adjust accordingly—product 1’s price decreases while all others increase. Overall, the total revenue rises, consistent with intuition. Proposition 10(ii) considers reallocating $ϵ$ capacity from a sold-out product $r$ to product 1. Only products 1 and $r$ are affected by such a conversion between sold-out products: Product 1 gains sales but at a lower price, while product $r$ loses sales but charges a higher price. The net effect on revenue favors the higher-quality product, with the increase approximately $ϵ\theta (q_1-q_r)+ϵ\log \frac{{\beta }_r}{{\beta }_1}$ for small $ϵ$ . Thus, the revenue gain is linear in $ϵ$ and grows with the quality gap. Notably, this result generalizes to reallocation from any product $i$ to $j<i\le s_1$ .

8. Concluding remarks

Motivated by an application in the live entertainment industry, we study a multi-product pricing problem under vertical and horizontal product differentiation. Unlike existing research, our model incorporates both capacity constraints and price restrictions. The former captures the limited supply of seats in a venue, while the latter imposes an upper bound on the average ticket price and on the price of the lowest-quality product to ensure affordability and customer satisfaction. Incorporating these constraints is a major contribution of our work, as they are not only essential in practice but also fundamentally alter the structure of optimal solutions compared to unconstrained settings. Our analysis provides characterizations of optimal policies under both demand models, enabling managers to assess the impact of each constraint and better design pricing strategies.

In the vertical case, we show that the feasible region is convex if and only if consumers’ valuations for quality follow a uniform distribution. Under this assumption, the optimal solution exhibits a sold-out threshold structure under both capacity constraints and the average price restriction, that is, only higher-quality products are fully sold. With an additional upper bound on the lowest-quality price, a second threshold emerges in which the lowest-quality products also sell out alongside the highest-quality ones. In the horizontal case, we prove that the sold-out threshold structure remains optimal. A key departure from the literature is that the classic equal-markup result no longer holds under constraints. With capacity limits alone, sold-out products have prices that decrease with quality, while partially sold products share a common price. Finally, adding the average price restriction enforces strictly decreasing prices across all categories.

Overall, our paper makes three contributions. Modeling-wise, we develop a unified framework that captures both vertical and horizontal consumer choice contexts, enabling the study of constrained revenue management across industries. Analytically, we characterize optimal solutions despite the problem’s complexity, uncovering structural properties that isolate the role of each constraint. Managerially, we show that capacity and price restrictions substantially reshape optimal pricing relative to unconstrained benchmarks, with direct implications for revenue and accessibility.

To conclude, we outline three directions for future research that address the limitations of our model. First, a promising avenue is to examine dynamic pricing strategies, such as staggered release of ticket blocks over time, and compare their performance with static pricing. Such extensions would require a different modeling framework but could yield valuable insights into the interaction of dynamic and static paradigms. Second, incorporating stock-out substitution would add considerable analytical complexity to the ETP problem, but it remains an important direction, and future studies may build on the structural results established here. Third, while our horizontal differentiation model supports empirical research based on the MNL framework (e.g., Arslan et al., 2022), future work could investigate more general logit models under practical constraints.

Supplemental Material

sj-pdf-1-pao-10.1177_10591478261449015 - Supplemental material for Event ticket pricing with capacity constraints and price restrictions

Supplemental material, sj-pdf-1-pao-10.1177_10591478261449015 for Event ticket pricing with capacity constraints and price restrictions by Yunke Li, Xin Geng and Harihara Prasad Natarajan in Production and Operations Management