Equilibrium in Queues Under Unknown Service Times and Service Value

In the operations research literature, the queue joining probability is monotonic decreasing in the queue length; the longer the queue, the fewer consumers join. Recent academic and empirical evidence indicates that queue-joining probabilities may not always be decreasing in the queue length. We provide a simple explanation for these nonmonotonic queue-joining strategies by relaxing the informational assumptions in Naor\u27s model. Instead of imposing that the expected service time and service value are common knowledge, we assume that they are unknown to consumers, but positively correlated. Under such informational assumptions, the posterior expected waiting cost and service value increase in the observed queue length. As a consequence, we show that queue-joining equilibria may emerge for which the joining probability increases locally in the queue length. We refer to these as “sputtering equilibria.” We discuss when and why such sputtering equilibria exist for discrete as well as continuously distributed priors on the expected service time (with positively correlated service value)

Selling to Conspicuous Consumers: Pricing, Production, and Sourcing Decisions

Consumers often purchase goods that are “hard to find” to conspicuously display their exclusivity and social status. Firms that produce such conspicuously consumed goods such as designer apparel, fashion goods, jewelry, etc., often face challenges in making optimal pricing and production decisions. Such firms are confronted with precipitous trade-off between high sales volume and high margins, because of the highly uncertain market demand, strategic consumer behavior, and the display of conspicuous consumption. In this paper, we propose a model that addresses pricing and production decisions for a firm, using the rational expectations framework. We show that, in equilibrium, firms may offer high availability of goods despite the presence of conspicuous consumption. We show that scarcity strategies are harder to adopt as demand variability increases, and we provide conditions under which scarcity strategies could be successfully adopted to improve profits. Finally, to credibly commit to scarcity strategy, we show that firms can adopt sourcing strategies, such as sourcing from an expensive production location/supplier or using expensive raw materials

Blind Queues: The Impact of Consumer Beliefs on Revenues and Congestion

In many service settings, customers have to join the queue without being fully aware of the parameters of the service provider (e.g., customers at checkout counters may not know the true service rate before joining). In such “blind queues,” customers make their joining/balking decisions based on limited information about the service provider’s operational parameters (from past service experiences, reviews, etc.) and queue lengths. We analyze a firm serving customers making decisions under arbitrary beliefs about the service parameters in an observable queue for a service with a known price. By proposing an ordering for the balking threshold distributions in the customer population, we are able to compare the effects of customer beliefs on the queue. We show that, although revealing the service information to customers improves revenues under certain conditions, it may destroy consumer welfare or social welfare. Given a market size, consumer welfare can be significantly reduced when a fast server announces its true service parameter. When revenue is higher under some beliefs, one would expect the congestion to also be higher because more customers join, but we show that congestion may not necessarily increase

Joining Longer Queues: Information Externalities in Queue Choice

A classic example that illustrates how observed customer behavior impacts other customers\u27 decisions is the selection of a restaurant whose quality is uncertain. Customers often choose the busier restaurant, inferring that other customers in that restaurant know something that they do not. In an environment with random arrival and service times, customer behavior is reflected in the lengths of the queues that form at the individual servers. Therefore, queue lengths could signal two factors—potentially higher arrivals to the server or potentially slower service at the server. In this paper, we focus on both factors when customers\u27 waiting costs are negligible. This allows us to understand how information externalities due to congestion impact customers\u27 service choice behavior. In our model, based on private information about both the service-quality and queue-length information, customers decide which queue to join. When the service rates are the same and known, we confirm that it may be rational to ignore private information and purchase from the service provider with the longer queue when only one additional customer is present in the longer queue. We find that, due to the information externalities contained in queue lengths, there exist cycles during which one service firm is thriving whereas the other is not. Which service provider is thriving depends on luck; i.e., it is determined by the private signal of the customer arriving when both service providers are idle. These phenomena continue to hold when each service facility has multiple servers, or when a facility may go out of business when it cannot attract customers for a certain amount of time. Finally, we find that when the service rates are unknown but are negatively correlated with service values, our results are strengthened; long queues are now doubly informative. The market share of the high-quality firm is higher when there is service rate uncertainty, and it increases as the service rate decreases. When the service rates are positively correlated with unknown service values, long queues become less informative and customers might even join shorter queues

Herding in Queues with Waiting Costs: Rationality and Regret

We study how consumers with waiting cost disutility choose between two congested services of unknown service value. Consumers observe an imperfect private signal indicating which service facility may provide better service value as well as the queue lengths at the service facilities before making their choice. If more consumers choose the same service facility because of their private information, longer queues will form at that facility and indicate higher quality. On the other hand, a long queue also implies more waiting time. We characterize the equilibrium queue-joining behavior of arriving consumers and the extent of their learning from the queue information in the presence of such positive and negative externalities. We find that when the arrival rates are low, utility-maximizing rational consumers herd and join the longer queue, ignoring any contrary private information. We show that even when consumers treat queues as independently evolving, herd behavior persists with consumers joining longer queues above a threshold queue difference. However, if the consumers seek to minimize ex post regret when making their decisions, herd behavior may be dampened

Now or Later: A Simple Policy for Effective Dual Sourcing in Capacitated Systems

We examine a possibly capacitated, periodically reviewed, single-stage inventory system where replenishment can be obtained either through a regular fixed lead time channel, or, for a premium, via a channel with a smaller fixed lead time. We consider the case when the unsatisfied demands are backordered over an infinite horizon, introducing the easily implementable, yet informationally rich dual-index policy. We show very general separability results for the optimal parameter values, providing a simulation-based optimization procedure that exploits these separability properties to calculate the optimal inventory parameters within seconds. We explore the performance of the dual-index policy under stationary demands as well as capacitated production environments, demonstrating when the dual-sourcing option is most valuable. We find that the optimal dual-index policy mimics the behavior of the complex, globally optimal state-dependent policy found via dynamic programming: the dual-index policy is nearly optimal (within 1% or 2%) for the majority of cases, and significantly outperforms single sourcing (up to 50% better). Our results on optimal dual-index parameters are generic, extending to a variety of complex and realistic scenarios such as nonstationary demand, random yields, demand spikes, and supply disruptions

Revenue Management with Strategic Customers: Last-Minute Selling and Opaque Selling

Companies in a variety of industries (e.g., airlines, hotels, theaters) often use last-minute sales to dispose of unsold capacity. Although this may generate incremental revenues in the short term, the long-term consequences of such a strategy are not immediately obvious: More discounted last-minute tickets may lead to more consumers anticipating the discount and delaying the purchase rather than buying at the regular (higher) prices, hence potentially reducing revenues for the company. To mitigate such behavior, many service providers have turned to opaque intermediaries, such as Hotwire.com, that hide many descriptive attributes of the service (e.g., departure times for airline tickets) so that the buyer cannot fully predict the ultimate service provider. Using a stylized economic model, this paper attempts to explain and compare the benefits of last-minute sales directly to consumers versus through an opaque intermediary. We utilize the notion of rational expectations to model consumer purchasing decisions: Consumers make early purchase decisions based on expectations regarding future availability, and these expectations are correct in equilibrium. We show that direct last-minute sales are preferred over selling through an opaque intermediary when consumer valuations for travel are high or there is little service differentiation between competing service providers, or both; otherwise, opaque selling dominates. Moreover, contrary to the usual belief that such sales are purely mechanisms for disposal of unused capacity, we show that opaque selling becomes more preferred over direct last-minute selling as the probability of having high demand increases. When firms randomize between opaque selling and last-minute selling strategies, they are increasingly likely to choose the opaque selling strategy as the probability of high demand increases. When firms with unequal capacities use the opaque selling strategy, consumers know more clearly where the opaque ticket is from and the efficacy of opaque selling decreases

Multi-Attribute Loss Aversion and Reference Dependence: Evidence from the Performing Arts Industry

We study the prevalence of multiattribute loss aversion and reference effects in a revenue management setting based on data of individual-level purchases over a series of concert performances. The reference dependence that drives consumer choice is not only based on the price but also on observed sales (as a fraction of the seating capacity) during their past visits. We find that consumers suffer from loss aversion on both prices and seats sold: consumers incur significant utility loss when prices are above their references or when the actual seat sales are lower than their references. We suggest pricing policies that can address consumer decisions driven by such reference dependence and loss aversion

Pricing Theater Seats: The Value of Price Commitment and Monotone Discounting

We examine the value of price commitment in a non-profit organization using individual-level purchases over a series of concert performances. To decide on a pricing policy, the performing arts organization must be able to accurately measure when each ticket will be sold and what type of audience will purchase the tickets for each performance. We use a competing hazards framework to model the timing of ticket purchases when customer segments differ in their valuations and arrival times. We show that the customer purchase likelihoods change based on the prices observed earlier in the season. Hence, price commitment can aid in improving sales, revenues, and customer visits. In particular, we show that price commitment to a decreasing monotone discount policy can improve the revenues in the range 2.1%–6.7% per concert