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Equilibrium strategies and pricing for a congested system with opaque price and loss-averse customers
Abstract
In this paper, we study a service provider with opaque price strategy who sells services to loss-averse customers in a congestion-prone facility. The opaque price strategy means that the service provider discloses a price range of services to potential customers through social media. Loss aversion aims to characterize customers' reference-dependent preferences. We analyze a queueing game model with the help of an M/M/1 queueing system and adopt the cutoff structure characterized by reservation price to represent the loss-averse customers' joining decision. Then, we derive the loss-averse customer's reservation price and queueing strategies at equilibrium. In addition, the service provider sets the price to maximize her profit. We obtain some insights into how customers' degree of loss aversion affects the service provider's optimal price. For example, customers with a higher degree of loss aversion on the waiting time and the payment would push the service provider to decrease the optimal price. Customers with a higher degree of loss aversion about the service reward would urge the service provider to increase the optimal price. Furthermore, considering the attachment effect, we obtain the lowest possible expected profit of the service provider.
In social interaction, customers can observe the number of other customers receiving service and select queues with more customers. Simultaneously, customers in the queue anticipate waiting times and worry about utility loss. This study explores the impact of loss aversion psychology on customer queuing strategies, service provider pricing, and revenue in the context of social interaction. Firstly, we consider homogeneous customers and analyze the influence of loss aversion psychology on their queuing decisions and service choices in social interaction. Subsequently, we extend our investigation to heterogeneous customers, considering differences in customers’ sensitivity to social interaction. Social interaction and loss aversion are crucial in customer queuing decisions, affecting their perception of service utility and equilibrium decision-making. Social interaction and loss aversion also influence service providers’ revenue, necessitating tailored service pricing and strategies. This research provides profound insights into customer loss aversion behavior in the context of social interaction and offers practical service strategies for service providers.
In many service industries, information disclosure about the product can alleviate customers' loss aversion induced by uncertain product valuation. In this paper, we consider a single-server queueing system in which the manager who privately learns the valuation information discloses the valuation information strategically to loss-averse customers. We investigate the impact of the customers' loss aversion on the system's equilibrium arrival rate and the manager's optimal disclosure policy. We find that loss aversion restrains customers from joining the queue. Surprisingly, we find that there is no one disclosure policy that always prevails over other disclosure policies. Specifically, the full disclosure policy is optimal only when the valuation is large and the degree of loss aversion is moderate. The full non-disclosure policy is optimal when the degree of loss aversion is too large or too small, or the valuation is small. The threshold disclosure policy is optimal when the valuation and the degree of loss aversion are moderate. Furthermore, under the threshold disclosure policy, the increasing degree of loss aversion makes managers be more reluctant to disclose the valuation.
This paper proposes a theory of price discrimination based on consumer loss aversion. A seller offers a menu of bundles before a consumer learns his willingness to pay, and the consumer experiences gain–loss utility with reference to his prior (rational) expectations about contingent consumption. With binary consumer types, the seller finds it optimal to abandon screening under an intermediate range of loss aversion if the low willingness-to-pay consumer is sufficiently likely. We also identify sufficient conditions under which partial or full pooling dominates screening with a continuum of types. Our predictions are consistent with several observed practices of price discrimination.
An M/M/1 retrial queue is studied for local area network applications in which the server will take delayed vacations upon
completion of a service. An announced price charged by the server is imposed on the customers who decide to enter the system
for service. These customers have to make decisions upon arrival to balk or to enter the system for service based on their
expected payoff and available information about the status of the server. On the other hand, the server faces the revenue
maximization issue once the customers' Nash equilibrium strategies are obtained for any announced price. We further study
how the social planner, whose goal is to maximize the overall welfare of the customers and the server, determines the price
to charge a joining customer. Finally, the sensitivity of the solution of revenue optimization and the socially optimal price
as well as the balking behaviour of customers with respect to some main indicators of the system are explored via numerical
experiments.
We study a noncooperative game problem for queueing control in the Cognitive Radio (CR) system where selfish Secondary User's (SU) data packets (a.k.a. "customers" in this work) are served by a CR base station (a.k.a. "server"). The scenario is modeled as an M/M/1 queueing game with server breakdowns where each customer wants to optimize their benefit in a selfish distributed manner. We first show that the game has an inefficient unique Nash Equilibrium (NE). In order to improve the outcome efficiency, we propose an appropriate admission fee that can be easily implemented at the server. We then show that the social welfare at the equilibrium point can be coincided the social welfare of the socially optimal strategy.
We analyze a dynamic pricing problem where consumer's purchase decisions are a ected by representative past prices, summarized in a reference price. We propose a new, behaviorally motivated reference price mechanism, based on the peak-end memory model proposed by Kahneman et al. (1993). Speci cally, we assume that consumers' reference price is a weighted average of the lowest and last price. Gain or loss perceptions with respect to this reference price a ect consumer purchase decisions in the spirit of prospect theory, resulting in a non-smooth demand function.We investigate how these behavioral patterns in consumer anchoring and decision processes a ect the optimal dynamic pricing policy of the rm. In contrast with previous literature, we show that peak-end anchoring leads to a range of optimal constant pricing policies even with loss neutral buyers. This range becomes wider if consumers are loss averse. In general, we show that skimming or penetration strategies are optimal, i.e. the transient pricing policy is monotone, and converges to a steady state, which depends on the initial price perception. The value of the steady state price decreases, the more consumers are sensitive to price changes, and the more they anchor on the lowest price.
It takes time to process purchases and as a result a queue of customers may form. The pricing and service rate decissions of a monopolist who must take this into account are characterized. We find that an increase in the average number of customers arriving in the market either has no effect on the monopoly price, or else causes the monopolist to reduce the price in the short run. In the long run the monopolist will increase the service rate and raise the price. When customer preferences are linear the equilibrium is socially efficient. When preferences are not linear equilibrium will not normally be socially efficient.
We modify the Salop (1979) model of price competition with differentiated products by assuming that consumers are loss averse relative to a reference point given by their recent expectations about the purchase. Consumers' sensitivity to losses in money increases the price responsiveness of demand—and hence the intensity of competition—at higher relative to lower market prices, reducing or eliminating price variation both within and between products. When firms face common stochastic costs, in any symmetric equilibrium the markup is strictly decreasing in cost. Even when firms face different cost distributions, we identify conditions under which a focal-price equilibrium (where firms always charge the same "focal" price) exists, and conditions under which any equilibrium is focal. (JEL D11 , D43, D81, L13)
This paper studies the impact of loss aversion on queue configurations in a doubled‐ended queueing system such as taxi stations. We assume that there exist two types of demands/passengers, namely, short‐haul and long‐haul, with long‐haul demand offering a larger monetary reward for drivers. Loss‐averse drivers decide to either join or balk based on their overall utility. We first consider the passenger‐side system to be a loss system; that is, an arriving passenger balks if there is no taxi waiting for her. We derive the system performance of two queue configurations, i.e., a pooled queue and two dedicated queues. By comparing the total passenger boarding rates, we find that the pooled queue is not always preferable. From the drivers’ perspective, the pooled queue generates a rather stable waiting time and hence mitigates their loss aversion in the waiting time dimension; nevertheless, it creates uncertainty in the reward dimension and thus brings about potential loss. We also compare the two configurations regarding the boarding rate of each type of passengers. We show that dedicated queues can achieve a Pareto improvement; that is, they can generate a higher boarding rate for each type of passenger when drivers are sufficiently loss averse in the reward dimension and not so in the waiting time dimension. Nevertheless, dedicated queues may decrease both passenger boarding rates. We then extend our setting by incorporating passengers’ joining/balking decisions and find that our managerial insights derived from the lost‐demand system still hold.
We consider a service system in which customers are loss averse toward both price and delay attributes. That is, customers compare these two attributes with their rational expectations of outcomes, with losses being more painful than equal-sized gains are pleasant. We first study customers’ equilibrium queueing strategies. We find that, unlike the traditional case in which loss aversion is not considered, there may exist three equilibrium strategies, one of which is preferred in the sense that customers’ utility is highest at this equilibrium. We then investigate the optimal pricing problem for a monopoly server and find that loss aversion polarizes queues, making long queues even longer and short queues even shorter. Furthermore, loss aversion toward the delay attribute drives the optimal price down, whereas loss aversion toward the price attribute drives it up. We also find that profit- and welfare-maximizing prices are not the same in a monopoly market. Finally, we consider pricing competition in a symmetric duopoly market and find that the conclusions depend on the size of the service capacity relative to the market size. For fast servers, there exists a unique symmetric price equilibrium. Under certain conditions, the effect of loss aversion on waiting time drives the price down, whereas that on the monetary term drives it up. For moderate-speed servers, there also exists a unique symmetric equilibrium. However, the effect of loss aversion on the two attributes works in reverse compared with that in the fast server case. For slow servers, we show that a symmetric equilibrium may not exist, and we numerically find that there may exist two asymmetric equilibria. Interestingly, with loss-averse customers, a firm can obtain a higher profit in a duopoly market than in a monopoly market.
The online appendix is available at https://doi.org/10.1287/opre.2017.1702 .
Many studies in operations management started to explicitly model customer behavior. However, it is typically assumed that customers are fully rational decision-makers and maximize their utility perfectly. Recently, modeling customer bounded rationality has been gaining increasing attention and interest. This paper summarizes various approaches of modeling customer bounded rationality, surveys how they are applied to relevant operations management settings, and presents the new insights obtained. We also suggest future research opportunities in this important area.
We present a new partial equilibrium theory of price adjustment, based on consumer loss aversion. In line with prospect theory, the consumers’ perceived utility losses from price increases are weighted more heavily than the perceived utility gains from price decreases of equal magnitude. Price changes are evaluated relative to an endogenous reference price, which depends on the consumers’ rational price expectations from the recent past. By implication, demand responses are more elastic for price increases than for price decreases and thus firms face a downward-sloping demand curve that is kinked at the consumers’ reference price. Firms adjust their prices flexibly in response to variations in this demand curve, in the context of an otherwise standard dynamic neoclassical model of monopolistic competition. The resulting theory of price adjustment is starkly at variance with past theories. We find that - in line with the empirical evidence - prices are more sluggish upwards than downwards in response to temporary demand shocks, while they are more sluggish downwards than upwards in response to permanent demand shocks. The degree of these asymmetries, in turn, depends on the size of the shock.
This paper describes a two-echelon supply chain model with two manufacturers and one common retailer. Two types of complementary prod ucts are produced by two manufacturers, and the common retailer buys prod ucts separately using a reservation price and bundles them for sale. The de mands of manufacturers and retailer are assumed to be stochastic in nature. When the retailer orders for products, any one of manufacturers agrees to allow those products, and the rest of the manufacturers have to provide the same amount. The profits of two manufacturers and the retailer are maximized by using Stackelberg game policy. By applying a game theoretical approach, several analytical solutions are obtained. For some cases, this model obtains quasi-closed-form solutions, for others, it finds closed-form solutions. Some numerical examples, sensitivity analysis, managerial insights, and graphical illustrations are given to illustrate the model.
To maximize a firm's profit over a finite planning horizon, we develop a dynamic optimization model by considering loss aversion when making pricing and inventory decisions. We estimate customer demand through a choice model, which incorporates reference price, utility function and customer loss aversion. Our model forms the core of the expert system for decision support. Through a sequence of Bellman equations, we find that the firm's profit is a concave function of price and inventory, and we solve the model optimally. The profit is positively correlated with the reference price, and the price and inventory decisions are non-monotonic functions of loss aversion intensity. Our results shed new light on pricing and inventory management with customer behavior in a multi-period system. Through various theorem developments, we are able to identify the optimal inventory level and the corresponding price. Numerical examples are provided to illustrate and validate the model and to derive managerial insights. To show the potential significance, we demonstrate how a dynamic programming model yields good results with customer loss aversion under realistic customer behavior assumptions. Our system can improve the efficiency of decision making and provide better customer service.
We analyze the joint inventory and pricing decisions of a firm when demand depends on not only the current selling price but also a memory-based reference price and customers are loss averse. The presence of reference price effect leads to a nonconcave one-period expected revenue in price and reference price. We introduce a transformation technique that allows us to prove under some mild assumptions the optimality of a reference-price-dependent base-stock list-price policy, which is characterized by a base-stock level and a target reference price. In addition, we show that the target reference price is increasing in the reference price, but except in the loss-neutral case, the base-stock level is not monotone in the reference price. We also show that in the steady state of the model with the reference price effect, the optimal price is lower while the optimal base-stock level is higher than their counterparts in the model without the reference price effect.
With the rapidly growing demand for the cloud services, a need for efficient methods to trade computing resources increases. Commonly used fixed-price model is not always the best approach for trading cloud resources, because of its inflexible and static nature. Market-based trading shows promise for more efficient resource allocation and pricing in the cloud. However, most of the existing mechanisms ignore the seller's costs of providing the resources. In order to address it, we design a single-sided market mechanism for trading virtual machine instances in the cloud, where the cloud provider can express the reservation prices for traded services. We prove that the proposed mechanism is truthful, i.e. the buyers do not have an incentive to lie about their true valuation of the services. We perform extensive experiments in order to investigate the impact of the reserve price on the market outcome. Our experiments show that the proposed mechanism yields near-optimal allocations and has a low execution time.
We study a newsvendor who sells a perishable asset over repeated periods to consumers with a given consumption valuation for the product. The market size in each period is random, following a stationary distribution. Consumers are loss averse with stochastic reference points that represent their beliefs about possible price and product availability. Given the distribution of reference points, they choose purchase plans to maximize their expected total utility, including gain-loss utility, before visiting the store, and follow the plans in the store. In anticipation of consumers’ purchase plans, in each period, before demand uncertainty resolves, the firm chooses an initial order quantity. After the uncertainty resolves, the firm chooses a contingent price depending on the demand realization, with the option of clearing inventory by charging a sale price, and otherwise, posting a full price. Over repeated periods, the interaction of the firm’s operational decisions about ordering and contingent pricing and the consumers’ purchase actions results in a distribution of reference points, and, in equilibrium, this distribution is consistent with consumers’ beliefs. Under this framework of endogenized reference points, we fully characterize the firm’s optimal inventory and contingent pricing policies. We identify conditions under which the firm’s expected price and profit are increasing in the consumer loss aversion level. We also show that the firm can prefer demand variability over no-demand uncertainty. We obtain a set of insights into how consumers’ loss aversion affects the firm’s optimal operational policies that are in stark contrast to those obtained in classic newsvendor models. As examples, the optimal full price increases in the initial order quantity; and the optimal full price decreases, while the optimal sales frequency increases, in the procurement cost.
When facing heterogeneous customers, how should a service firm make its pricing decision to maximize revenue? If discrimination is allowed, then priority schemes and differentiated pricing are often used. In many applications, however, the firm cannot or is not allowed to set discriminatory prices, e.g., list price in retail stores, online shopping, and gas stations; thus a uniform price must be applied to all customers. This paper addresses the optimal uniform pricing problem of a service firm using a queueing system with two classes of customers. Our result shows that, the potential pool of customers plays a central role in the firm's optimal decision. Depending on the range of system parameters, which are determined explicitly by the primitive data, the firm's optimal strategy may choose to serve only one class of customers, a subset of a class of customers, or a combination of different classes of customers. In addition, the optimal price is in general not monotonic with respect to the potential market sizes because their changes may lead to a major shift in the firm's decision on which customer class to serve. However, unless such a shift occurs, it is weakly decreasing in the potential market sizes. This article is protected by copyright. All rights reserved.
Preface. 1. Introduction. 2. Observable Queues. 3. Unobservable Queues. 4. Priorities. 5. Reneging and Jockeying. 6. Schedules and Retrials. 7. Competition Among Servers. 8. Service Rate Decisions. Index.
Consider a system that is modeled as an M/M/1 queueing system with multiple user classes. Each class is characterized by its delay cost per unit of time, its expected service time and its demand function. This paper derives a pricing mechanism which is optimal and incentive- compatible in the sense that the arrival rates and execution priorities jointly maximize the expected net value of the system while being determined, on a decentralized basis, by individual users. A closed-form expression for the resulting price structure is presented and studied.
Consumer reservation price is a key concept in marketing and economics. Theoretically, this concept has been instrumental in studying consumer purchase decisions,competitive pricing strategies,and welfare economics. Managerially,knowledge of consumer reservation prices is critical for implementing many pricing tactics such as bundling,tar get promotions,nonlinear pricing,and one-to-one pricing,and for assessing the impact of marketing strategy on demand. Despite the practical and theoretical importance of this concept, its measurement at the individual level in a practical setting proves elusive.
We propose a conjoint-based approach to estimate consumer-level reservation prices. This approach integrates the preference estimation of traditional conjoint with the economic theory of consumer choice. This integration augments the capability of traditional conjoint such that consumers' reservation prices for a product can be derived directly from the individuallevel estimates of conjoint coefficients. With this augmentation,we can model a consumer's decision of not only which product to buy,but also whether to buy at all in a category. Thus, we can simulate simultaneously three effects that a change in price or the introduction of a new product may generate in a market: the customer switching effect,the cannibalization effect,and the market expansion effect. We show in a pilot application how this approach can aid product and pricing decisions. We also demonstrate the predictive validity of our approach using data from a commercial study of automobile batteries.
We study the customers' Nash equilibrium behavior in a single server observable queue with Poisson arrivals and general service times. Each customer takes a single decision upon arrival: to join or not to join. Furthermore, future regrets are not allowed. The customers are homogenous with respect to their linear waiting cost and the reward associated with service completion. The cost of joining depends on the behavior of the other customers present, which naturally forms a strategic game. We present a recursive algorithm for computing the (possibly mixed) Nash equilibrium strategy. The algorithm's output is queue-dependent joining probabilities. We demonstrate that depending on the service distribution, this equilibrium is not necessarily unique. Also, we show that depending on the service time distribution, either the 'avoid the crowd' phenomenon or the 'follow the crowd' phenomenon may hold.
We consider a revenue management model for pricing a product line with several customer segments under the assumption that
customers’ product choices are determined entirely by their reservation prices. We highlight key mathematical properties of
the maximum utility model and formulate it as a mixed-integer programming problem, design heuristics and valid cuts. We further
present extensions of the models to deal with various practical issues arising in applications. Our computational experiments
with real data from the tourism sector as well as with the randomly generated data show the effectiveness of our approach.
KeywordsRevenue management–Mixed-integer programming–Product pricing
We study a problem of dynamic pricing faced by a vendor with limited inventory, uncertain
about demand, aiming to maximize expected discounted revenue over an infinite time horizon.
The vendor learns from purchase data, so his strategy must take into account the impact of
price on both revenue and future observations. We focus on a model in which customers arrive
according to a Poisson process of uncertain rate, each with an independent, identically
distributed reservation price. Upon arrival, a customer purchases a unit of inventory if and only if his reservation price equals or exceeds the vendor’s prevailing price.
We study a two-period intertemporal pricing game in a single-server service system with forward-looking strategic customers who make their purchase decision based on current information and anticipated future gains. Subgame perfect Nash equilibrium (SPNE) prices are derived. A comparison between revenue-maximizing equilibrium prices and welfare-maximizing equilibrium prices is conducted and the impact on the system’s performance of misunderstanding customers’ type is evaluated.
We develop a model in which a profit-maximizing monopolist with uncertain cost of production sells to loss-averse, yet rational, consumers. We first introduce (portable) techniques for analyzing the demand of such consumers, and then investigate the monopolist’s pricing strategy. In contrast to the standard monopoly model, we find that in relatively stable environments, the firm chooses a discrete price distribution, endogenously giving rise to a kinked demand curve. A low price responsiveness of demand and a high frequency of purchase by consumers also promote such “price stickiness”. Whether or not the monopolist’s prices are sticky, markups follow a countercyclical pattern. Despite this tendency toward price stability, there are also circumstances in which a firm with unchanging cost offers random “sales ” to attract more demand at higher prices. Keywords: Reference-dependent utility, price stickiness, monopoly pricing, kinked demand curve, countercyclical markups, sales and promotions, (seemingly) predatory pricing. We thank Stefano DellaVigna, Chris Shannon, and especially Wei Li and Matt Rabin for numerous helpful discussions, and have benefitted from comments of seminar participants at CalTech, UC Berkeley, and the University
Much experimental evidence indicates that choice depends on the status quo or reference level: changes of reference point often lead to reversals of preference. The authors present a reference-dependent theory of consumer choice, which explains such effects by a deformation of indifference curves about the reference point. The central assumption of the theory is that losses and disadvantages have greater impact on preferences than gains and advantages. Implications of loss aversion for economic behavior are considered. Copyright 1991, the President and Fellows of Harvard College and the Massachusetts Institute of Technology.
We consider the impact of bundling products on retail merchandising. We consider two broad classes of retail products: basic and fashion. For these product classes, we develop models to calculate the optimal bundle prices, order quantities, and profits under bundling. We use this analysis to establish conditions and insights under which bundling is profitable. Our analysis confirms that bundling profitability depends on individual product demands, bundling costs, and the nature of the relationship between the demands of the products to be bundled. We also provide detailed numerical examples.
A queueing model--together with a cost structure--is presented, which envisages the imposition of tolls on newly arriving customers. It is shown that frequently this is a strategy which might lead to the attainment of social optimality.
We develop a model of reference-dependent preferences and loss aversion where “gain-loss utility” is derived from standard
“consumption utility” and the reference point is determined endogenously by the economic environment. We assume that a person's
reference point is her rational expectations held in the recent past about outcomes, which are determined in a personal equilibrium by the requirement that they must be consistent with optimal behavior given expectations. In deterministic environments,
choices maximize consumption utility, but gain-loss utility influences behavior when there is uncertainty. Applying the model
to consumer behavior, we show that willingness to pay for a good is increasing in the expected probability of purchase and
in the expected prices conditional on purchase. In within-day labor-supply decisions, a worker is less likely to continue
work if income earned thus far is unexpectedly high, but more likely to show up as well as continue work if expected income
is high.
Analysis of decision making under risk has been dominated by expected utility theory, which generally accounts for people's actions. Presents a critique of expected utility theory as a descriptive model of decision making under risk, and argues that common forms of utility theory are not adequate, and proposes an alternative theory of choice under risk called prospect theory. In expected utility theory, utilities of outcomes are weighted by their probabilities. Considers results of responses to various hypothetical decision situations under risk and shows results that violate the tenets of expected utility theory. People overweight outcomes considered certain, relative to outcomes that are merely probable, a situation called the "certainty effect." This effect contributes to risk aversion in choices involving sure gains, and to risk seeking in choices involving sure losses. In choices where gains are replaced by losses, the pattern is called the "reflection effect." People discard components shared by all prospects under consideration, a tendency called the "isolation effect." Also shows that in choice situations, preferences may be altered by different representations of probabilities. Develops an alternative theory of individual decision making under risk, called prospect theory, developed for simple prospects with monetary outcomes and stated probabilities, in which value is given to gains and losses (i.e., changes in wealth or welfare) rather than to final assets, and probabilities are replaced by decision weights. The theory has two phases. The editing phase organizes and reformulates the options to simplify later evaluation and choice. The edited prospects are evaluated and the highest value prospect chosen. Discusses and models this theory, and offers directions for extending prospect theory are offered. (TNM)
Pricing in queues without demand information, Manufacturing and Service Operations Management
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