An Efficient Algorithm for Dynamic Pricing Using a Graphical Representation

A firm that accounts for consumer behavior sets the selling price of a product considering the reference price of consumers. In the literature, a reference price is usually modeled as depending on past selling prices. That is, past selling prices implicitly constrain the current selling price of a product. In this article, the author explicitly measures this constraint with an optimal control framework. He works on the structural properties of a general demand function, which depends on both selling and reference prices. Analytical results prove the following claims. Adjusting reference prices effects increase the price elasticity of demand, the demand function becoming flatter. Thus, the reference price effect weakens the market power of the firm. Also, the reference price effect constitutes a main driver of the dynamics of the selling price. But contrary to intuition, selling price dynamics does not systematically imitate reference price dynamics.

:Using a novel scanner data set, I study price-setting decisions of major retailers in an emerging market economy. Pricing dynamics are heterogeneous across retail chains. Heterogeneity is especially pronounced in the case of posted (as opposed to reference) prices. Furthermore, retail chains appear to set prices in a centralized fashion: most barcode/store-level prices coincide with the intrachain modal price. The relationship between reference and chain-wide prices reveals that deviations from reference prices cannot be solely attributed to shocks to local market conditions. I find strong evidence of synchronization of price changes across stores within chains but weaker evidence of synchronization across retail chains.

We consider a Cournot oligopoly game, where firms produce an homogenous good and the demand and cost functions are nonlinear. These features make the classical best reply solution difficult to be obtained, even if players have full information about their environment. We propose two different kinds of repeated games based on a lower degree of rationality of the firms, on a reduced information set and reduced computational capabilities. The first adjustment mechanism is called “Local Monopolistic Approximation” (LMA). First firms get the correct local estimate of the demand function and then they use such estimate in a linear approximation of the demand function where the effects of the competitors’ outputs are ignored. On the basis of this subjective demand function they solve their profit maximization problem. By using the second adjustment process, that belongs to a class of adaptive mechanisms known in the literature as “Gradient Dynamics” (GD), firms do not solve any optimization problem, but they adjust their production in the direction indicated by their (correct) estimate of the marginal profit. Both these repeated games may converge to a Cournot–Nash equilibrium, i.e. to the equilibrium of the best reply dynamics. We compare the properties of the two different dynamical systems that describe the time evolution of the oligopoly games under the two adjustment mechanisms, and we analyze the conditions that lead to non-convergence and complex dynamic behaviors. The paper extends the results of other authors that consider similar adjustment processes assuming linear cost functions or linear demand functions.

A substantial body of research evidence has now accumulated in the reference price literature. One stream of research has identified the antecedents of reference price and has assessed their effects through experimentation. Others have calibrated a variety of reference price models on panel data and reported the effects on brand choice and other purchase decisions. In this article, the authors review the published literature on reference price in both the behavioral and modeling streams. They offer an integrative framework to review prior research on (1) the formation of reference price, (2) retrieval and use of reference price, and (3) influences of reference price on various purchase decisions and evaluations. In doing so, the authors examine the influences of consumers' prior purchase history and contextual moderators, such as specific purchase occasions, promotional environment of the store, and product category characteristics. They summarize the key findings, identify the unresolved issues, and offer an agenda for further research, which includes a set of testable propositions. They also identify the methodological challenges that face reference price research. In the concluding section, the authors discuss the managerial implications of reference price research.

The importance of closed-loop supply chains has been widely recognized both in academic communities and in industrial sectors. This paper starts from the traditional supply chains and the new self-supply chain of GREE to extract realistic problems, to mainly investigating two noncooperative dynamic pricing policies in a dual-channel closed-loop supply chain consisting of a manufacturer and a retailer. Then, it studies the influence of different channel power structures on dynamic decisions and their complexities. Furthermore, the reference price affects the purchase decisions of consumers. Therefore, the model takes into account the influence of reference price of the market demands. Results show that the manufacturer who opens up a direct channel can make a huge profit in the game. In the dynamic game evolution process, the game leader is in a more advantageous position when the system is in a stable region; once entering into the bifurcating region or chaotic region, the game follower needs to adjust his price to follow the leader’s decision in order to make a profit. In addition, the system’s stable region becomes smaller when the market demand becomes more sensitive to the difference between the reference price and the actual price. In this model, if the manufacturer acts as a leader, he is in a more advantageous position when the market is sensitive to channel competition in the stable stage while the result is opposite in the unstable stage.

We analyze a finite-horizon dynamic pricing model in which demand at each period depends on not only the current price but also past prices through reference prices. A unique feature but also a significant challenge in this model is the asymmetry in reference price effect, which implies that the underlying optimization problem is nonsmooth and no standard optimization methods can be applied. We identify a few key structural properties of the problem, which enable us to develop strongly polynomial-time algorithms to compute the optimal prices for several plausible scenarios. We complement our exact algorithms by proposing an approximation heuristic and provide an upper bound on the optimal objective value. Finally, we conduct numerical experiments to study the optimal price path and demonstrate the value of dynamic pricing when demands are seasonal. We further compare numerically one of the exact algorithms with the heuristic and offer managerial suggestions. This paper was accepted by Yinyu Ye, optimization.

Considering customers' bounded-rationality purchasing behavior, this paper focuses on dynamic pricing decision with a static reference price in revenue management situations. Firstly, it defines the concept of multiplicative reference effect, and describes the characteristics of the reference effect and the demand function with multiplicative reference effect. Secondly, it constructs the optimal control model with continuous time and continuous prices. By solving the Hamilton- Jacobi equation, it presents the optimal price paths. Finally, it compares the pricing strategies with the GVR model without reference effect under different circumstances through simulation experiments. The results show that the stronger the reference effect is, the more it influences the optimal prices and total revenue. The optimal prices and revenue increase with the reference price. The markup policy is effective in high demand situations and the markdown policy is effective in low demand situations. In the situations with low demand and strong reference effect, the enterprises should adopt demand-oriented strategies. In other situations, they should adopt price-oriented strategies.

Reference pricing systems for prescription drugs are usually implemented with the aim of curbing public expenditure with pharmaceuticals, induce drug substitution from branded to generic drugs, and enhance competition. In these systems, patients co-pay the difference between the drug's pharmacy retail price and the health system reimbursement level. Relying on a detailed product-level panel dataset of prescription drugs sold in Portuguese retail pharmacies, from 2016 to 2019, we evaluate pharmaceutical firms' pricing decisions for branded and generic drugs, as well as consumers' reaction to price changes. In particular, we exploit the variation induced by a policy change, which decreased reference prices for 36% of the drug groups in our sample. Results from difference-in-differences analyses show that, despite the reference price decrease, affected firms increased their prices-particularly for off-patent branded products. Such reaction from firms resulted in an increase in the co-payment paid by patients. Such price effects caused a 17% decline on branded drugs' consumption, with significant heterogeneity across therapeutics. Estimates suggest that NHS reimbursement savings were mainly achieved through higher co-payments paid by patients. Additionally, pharmaceutical firms' reaction to the reference price decrease was contrary to what was expected, suggesting underlying competitive dynamics which should be considered prior to policy changes.

Single-source yogurt data are used to examine whether both internal and external reference prices affect purchase decisions. Internal reference prices are memory-resident prices based on actual, fair, or other price concepts. External reference prices are observed stimuli, such as “regular prices,” that stores may display along with a sale price for comparability. Discrete choice models with variables representing the two types of reference prices are estimated, and both types of variables are found to have significant effects on purchase probabilities. This suggests that consumers may use multiple reference points in evaluating price in purchase decisions. In addition, a model using an indicator of a sale price discount explains purchase probabilities as well as one that models the actual discount, which suggests that consumers may be reacting to the indication of savings rather than to the amount of the discount.

Evaluation of the efficacy of a short course of technical clinical skills to change performance in general practice. Subjects were self-selected general practitioners (n=59) who were unaware of the study design. They were assigned to the intervention group (n=31) or control group (n=28) according to their preference for course date. The course covered four different technical clinical skills (shoulder injection technique, PAP-smear, laboratory examination of vaginal discharge, ophthalmoscopic control in diabetes mellitus). Main outcome measures used were pre- and post-training scores on a knowledge test of skills (60 multiple choice items), and pre- and post-training performance of procedures in practice using a log-diary covering 20 days. University of Maastricht, The Netherlands. Self-selected general practitioners. Competence, as measured by the knowledge test of skills, improved significantly as a result of the training and skills test scores were satisfactory after training. A significant effect on performance in practice was found for PAP-smear and shoulder injection technique, whereas no effect could be demonstrated for examination of vaginal discharge and ophthalmoscopic control in diabetes mellitus. A good degree of competence is a necessary but not always sufficient condition for a physician to change his performance in practice. While some skills training seems adequate to bring about desired changes, for other skills more complex interventions are probably needed.

Price as an Indicator of Quality: Implications for Utility and Demand Functions

Due to its agility and mobility, the unmanned aerial vehicle (UAV) is a promising technology to provide high-quality mobile services (e.g., fast Internet access, edge computing, and local caching) to ground users. Major Internet Service Providers (ISPs) want to enable UAV-provided services (UPS) to improve and enrich the current mobile services for additional profit. This profit-maximization problem is not easy as the UAV has limited energy storage and needs to fly closely to serve users, requiring an optimal energy allocation for balancing both hovering time and service capacity. When hovering in a hotspot, how the UAV should dynamically price its capacity-limited UPS according to randomly arriving users with private service valuations is another question. We prove that the UAV should ask for a higher price if the leftover hovering time is longer or its service capacity is smaller, and its expected profit approaches to that under complete user information if the hovering time is sufficiently large. As the hotspot's user occurrence rate increases, a shorter hovering time or a larger service capacity should be allocated. Finally, when the UAV faces multiple hotspot candidates with different user occurrence rates and flying distances, we prove that it is optimal to deploy the UAV to serve a single hotspot. With multiple UAVs, however, this result can be reversed with UAVs' forking deployment to different hotspots.

Optimal dynamic pricing for deteriorating items with reference price effects when inventories stimulate demand

Promotions are a critical decision for supermarket managers, who must decide the price promotions for a large number of items. Retailers often use promotions to boost the sales of the different items by leveraging the cross-item effects. We formulate the promotion optimization problem for multiple items as a nonlinear integer program. Our formulation includes several business rules as constraints. Our demand models can be estimated from data and capture the postpromotion dip effect and cross-item effects (substitution and complementarity). Because demand functions are typically nonlinear, the exact formulation is intractable. To address this issue, we propose a general class of integer programming approximations. For demand models with additive cross-item effects, we prove that it is sufficient to account for unilateral and pairwise contributions and derive parametric bounds on the performance of the approximation. We also show that the unconstrained problem can be solved efficiently via a linear program when items are substitutable and the price set has two values. For more general cases, we develop efficient rounding schemes to obtain an integer solution. We conclude by testing our method on realistic instances and convey the potential practical impact for retailers. This paper was accepted by Yinyu Ye, optimization.

Distributed dynamic pricing of multiple perishable products using multi-agent reinforcement learning