Non-cooperative competition among revenue maximizing service providers with demand learning

This paper recognizes that in many decision environments in which revenue optimization is attempted, an actual demand curve and its parameters are generally unobservable. Herein, we describe the dynamics of demand as a continuous time differential equation based on an evolutionary game theory perspective. We then observe realized sales data to obtain estimates of parameters that govern the evolution of demand; these are refined on a discrete time scale. The resulting model takes the form of a differential variational inequality. We present an algorithm based on a gap function for the differential variational inequality and report its numerical performance for an example revenue optimization problem.     

Bundle pricing of inventories with stochastic demand

We consider a retailer selling a fixed inventory of two perishable products over a finite horizon. Assuming Poisson arrivals and a bivariate reservation price distribution, we determine the optimal product and bundle prices that maximize the expected revenue. Our results indicate that the performances of mixed bundling, pure bundling and unbundled sales strategies heavily depend on the parameters of the demand process and the initial inventory levels. Bundling appears to be most effective with negatively correlated reservation prices and high starting inventory levels. When the starting inventory levels are equal and in excess of average demand, most of the benefits of bundling can be achieved through pure bundling. However, the mixed bundling strategy dominates the other two when the starting inventory levels are not equal. We also observe that an incorrect modeling of the reservation prices may lead to significant losses. The model is extended to allow for price changes during the selling horizon. It is shown that offering price bundles mid-season may be more effective than changing individual product prices.     

Optimal worst-case pricing for a logit demand model with network effects

We consider optimal pricing problems for a product that experiences network effects. Given a price, the sales quantity of the product arises as an equilibrium, which may not be unique. In contrast to previous studies that take a best-case view when there are multiple equilibrium sales quantities, we maximize the seller’s revenue assuming that the worst-case equilibrium quantity will arise in response to a chosen price. We compare the best- and worst-case solutions, and provide asymptotic analysis of revenues.     

A continuous-time dynamic pricing model knowing the competitor’s pricing strategy

In this paper we consider a dynamic pricing model for a firm knowing that a competitor adopts a static pricing strategy. We establish a continuous time model to analyze the effect of dynamic pricing on the improvement in expected revenue in the duopoly. We assume that customers arrive to purchase tickets in accordance with a geometric Brownian motion. We derive an explicit closed-form expression for an optimal pricing policy to maximize the expected revenue. It is shown that when the competitor adopts a static pricing policy, dynamic pricing is not always effective in terms of maximizing expected revenue compared to a fixed pricing strategy. Moreover, we show that the size of the reduction in the expected revenue depends on the competitor’s pricing strategy. Numerical results are presented to illustrate the dynamic pricing policy.     

Dynamic pricing of multiple home delivery options

Online grocers accept delivery bookings and have to deliver groceries to consumers’ residences. Grocery stores operate on very thin margins. Therefore, a critical question that an online grocery store needs to address is the cost of home delivery operations. In this paper, we develop a Markov decision process-based pricing model that recognizes the need to balance utilization of delivery capacity by the grocer and the need to have the goods delivered at the most convenient time for the customer. The model dynamically adjusts delivery prices as customers arrive and make choices. The optimal prices have the following properties. First, the optimal prices are such that the online grocer gains the same expected payoff in the remaining booking horizon, regardless of the delivery option independently chosen by a consumer. Second, with unit order sizes, delivery prices can increase due to dynamic substitution effects as there is less time left in the booking horizon.     

Optimal dynamic pricing of inventories with stochastic demand and discounted criterion

We consider a continuous time dynamic pricing problem for selling a given number of items over a finite or infinite time horizon. The demand is price sensitive and follows a non-homogeneous Poisson process. We formulate this problem as to maximize the expected discounted revenue and obtain the structural properties of the optimal revenue function and optimal price policy by the Hamilton-Jacobi-Bellman (HJB) equation. Moreover, we study the impact of the discount rate on the optimal revenue function and the optimal price. Further, we extend the problem to the case with discounting and time-varying demand, the infinite time horizon problem. Numerical examples are used to illustrate our analytical results.     

A competitive dynamic pricing model when demand is interdependent over time

In this study, we contribute to the dynamic pricing literature by developing a finite horizon model for two firms offering substitutable and nonperishable products with different quality levels. Customers can purchase and store the products, even if they do not need them at the time, in order to use them in future. The stockpile of the products generated by customers affects the demand in future periods. Therefore, the demand for each product not only is a function of prices and quality levels, but also of the products’ stockpile levels. In addition, the stockpile levels change the customers’ consumption behavior; more product in a stockpile leads to more consumption. Therefore, we address not only the price and demand relationship but also the stockpiling and consumption relationship in a competitive environment.     

Optimal static pricing for a service facility with holding costs

We study a service facility modelled as a single-server queueing system with Poisson arrivals and limited or unlimited buffer size. In systems with unlimited buffer size, the service times have general distributions, whereas in finite buffered systems service times are exponentially distributed. Arriving customers enter if there is room in the facility and if they are willing to pay the posted price. The same price is charged to all customers at all times (static pricing). The service provider is charged a holding cost proportional to the time that the customers spend in the system. We demonstrate that there is a unique optimal price that maximizes the long-run average profit per unit time. We also investigate how optimal prices vary as system parameters change. Finally, we consider buffer size as an additional decision variable and show that there is an optimal buffer size level that maximizes profit.     

Two-period dynamic versus fixed-ratio pricing in a capacity constrained duopoly

This paper analyzes the impact of dynamic and fixed-ratio pricing policies on firm profits and equilibrium prices under competition. Firms that have equal inventories of perfectly substitutable and perishable products compete for customer segments that demand the product at different times. In each period, customers first purchase from the low price firm and then from the high price firm up to their inventories, provided the prices are lower than the maximum they are willing to pay. The main conclusions of this paper are as follows: although dynamic pricing is a more sophisticated policy than fixed-ratio pricing, it may lead to decreased equilibrium profits; under both pricing policies, one firm assumes the role of a low-cost high-output firm while the other assumes the role of a high-cost low-output firm; and, the supply demand ratio has more impact on the outcome of the competition than the heterogeneity in consumer reservation prices.     

Open-loop policies in Bayesian dynamic pricing: Some counter-intuitive observations and insights

In this paper, we show some counter-intuitive observations regarding the open-loop policies in a Bayesian dynamic pricing problem. Specifically, while the common intuition that a policy incorporating more information performs better continues to hold under ample inventory for sale, it breaks down in the case where inventory is limited. This can be explained by the unique feature of selling a limited amount of inventory: information updating may stop prematurely when inventory is depleted, rendering certain demand information being inconsequential.     

Combined pricing and supply chain operations under price-dependent stochastic demand

In this study, we determined product prices and designed an integrated supply chain operations plan that maximized a manufacturer’s expected profit. The computational results of this study revealed that as the variance of the demand distribution increases, a manufacturer will increase its inventory to levels that are greater than the anticipated demand to prevent the potential loss of sales and will simultaneously raise product prices to obtain a greater profit. In the cost minimization approach, the manufacturer may earn the highest possible profits, as determined by the profit optimization approach, only if this firm precisely forecasts the mean market demand for its products. Greater inaccuracies in this forecast will produce lower levels of expected profit.     

Markdown money contracts for perishable goods with clearance pricing

It is common in practice that retailers liquidate unsold perishable goods via clearance pricing. Markdown money is frequently used between manufacturers and retailers in such a supply chain setting. It is a form of rebate from a manufacturer to subsidize a retailer’s clearance pricing after the regular season. Two forms of markdown money are percent markdown money, in which the markdown money is limited to only a certain percentage of the retail price markdown, and quantity markdown money, which is essentially a buyback contract or returns policy with a rebate credit paid to the retailer for each unsold unit after the regular season. We show both forms of markdown money contracts can coordinate the supply chain and we discuss their strengths and limitations.     

Mathematical structure of a bilevel strategic pricing model

This paper is concerned with the characterization of optimal strategies for a service firm acting in an oligopolistic environment. The decision problem is formulated as a leader–follower game played on a transportation network, where the leader firm selects a revenue-maximizing price schedule that takes explicitly into account the rational behavior of the customers. In the context of our analysis, the follower’s problem is associated with a competitive network market involving non atomic customer groups. The resulting bilevel model can therefore be viewed as a model of product differentiation subject to structural network constraints.     

Periodic pricing and replenishment policy for continuously decaying inventory with multivariate demand

This paper deals with the joint decisions on pricing and replenishment schedule for a periodic review inventory system in which a replenishment order may be placed at the beginning of some or all of the periods. We consider a single product which is subject to continuous decay and a demand which is a function of price and time, without backlogging over a finite planning horizon. The proposed scheme may adjust periodically the selling price upward or downward that makes the pricing policy more responsive to structure changes in supply or demand. The problem is formulated as a dynamic programming model and solved by numerical search techniques. An extensive numerical study is conducted to attend qualitative insights into the structures of the proposed policy and its sensitivity with respect to major parameters. The numerical result shows that the solution generated by the periodic policy outperforms that by the fixed pricing policy in maximizing discount profit.     

The effect of market segmentation with demand leakage between market segments on a firm’s price and inventory decisions

In this paper, we address the simultaneous determination of price and inventory replenishment in a newsvendor setting when the firm faces demand from two or more market segments in which the firm can set different prices. We allow for demand leakage from higher-priced segments to lower-priced segments and assume that unsatisfied demand can be backlogged. We examine the case where the demands occur concurrently without priority and are met from a single inventory. We consider customer’s buy-down behavior explicitly by modeling demand leakage as a function of segment price differentiation, and characterize the structure of optimal inventory and pricing policies.     

Revenue management for low-cost providers

Low-cost providers have emerged as important players in many service industries, the most predominant being low-cost, or the so-called discount airlines. This paper presents models and results leading toward understanding the revenue management outlook for a discount pricing firm. A framework and model is formulated specifically for the airline industry, but is generalizable to low-cost providers in similar revenue management settings. We formulate an optimal pricing control model for a firm that must underprice to capture a segment of exogenous demand. Two specific model formulations are considered: a continuous deterministic version, and a discrete stochastic version. Structural results are derived for the deterministic case, providing insight into the general form of optimal underpricing policies. The stochastic results support the structural insight from the deterministic solution, and illuminate the effect of randomness on the underpricing policies.     

Dynamic pricing of limited inventories for multi-generation products

In this research, we consider a retailer selling products from two different generations, both with limited inventory over a predetermined selling horizon. Due to the spatial constraints or the popularity of a given product, the retailer may only display goods from one specific generation. If the transaction of the displayed item cannot be completed, the retailer may provide an alternative from another generation. We analyze two models - posted-pricing-first model and negotiation-first model. The former considers negotiation as being allowed on the price of the second product only and in the latter, only the price of the first product is negotiable. Our results show that the retailer can adopt both models effectively depending on the relative inventory levels of the products. In addition, the retailer is better off compared to the take-it-or-leave-it pricing when the inventory level of the negotiable product is high.     

Performance bounds on optimal fixed prices

We consider the problem of selling a fixed stock of items over a finite horizon when the buyers arrive following a Poisson process. We obtain a general lower bound on the performance of using a fixed price rather than dynamically adjusting the price. The bound is 63.21% for one unit of inventory, and it improves as the inventory increases. For the one-unit case, we also obtain tight bounds: 89.85% for the constant-elasticity and 96.93% for the linear price-response functions.     

A review of multi-product pricing models

The importance of good pricing strategies in business theory is clearly recognized, as can be seen from the huge volume of pricing research done over the years. What we attempt to do is to provide a general review of multi-product pricing models, focusing primarily on those where demands are explicitly dependent on prices. As the pricing decision may be made jointly with other economic parameters, we will not only review models that focus solely on pricing; we will also discuss models where pricing choices are made jointly with other decisions like production or distribution of resources.