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Competitive Pricing in a Multi‐Product Multi‐Attribute Environment

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Abstract

We address the problem of simultaneous pricing of a line of several products, both complementary products and substitutes, with a number of distinct price differentiation classes for each product (e.g., volume discounts, different distribution channels, and customer segments) in both monopolistic and oligopolistic settings. We provide a generic framework to tackle this problem, consider several families of demand models, and focus on a real-world case-study example. We propose an iterative relaxation algorithm, and state sufficient conditions for convergence of the algorithm. Using historical sales and price data from a retailer, we apply our solution algorithm to suggest optimal pricing, and report on numerical results.

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... In this paper, we solve dynamic pricing problems for large input specifications that are common in real-world systems, e.g., competition with dozens of firms, thousands of distinct products, multiple offer dimensions (e.g., product qualities, seller ratings, cf. [12]), and several financial constraints (e.g., holding costs, discounting). Existing re-pricing techniques cannot handle such scenarios efficiently and hence, force managers to limit the scope of pricing strategies, e.g., by using deterministic or highly stylized demand functions [22], monopoly settings, or by pricing less frequently. ...
... (11), or useT −t recursion steps to compute the specific value V t mod J (X t , ⃗ S t ), cf. (12), and obtain the associated offer price a t mod J (X t , ⃗ S t ). ...
... In case demand can be assumed to be independent of time the computational effort of Algorithm 4.1 via (11) or (12), can be even further reduced. Theorem 4.1. ...
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... In this paper, we solve dynamic pricing problems for large input specifications that are common in real-world systems, e.g., competition with dozens of firms, thousands of distinct products, multiple offer dimensions (e.g., product qualities, seller ratings, cf. [12]), and several financial constraints (e.g., holding costs, discounting). Existing re-pricing techniques cannot handle such scenarios efficiently and hence, force managers to limit the scope of pricing strategies, e.g., by using deterministic or highly stylized demand functions [22], monopoly settings, or by pricing less frequently. ...
... (11), or useT −t recursion steps to compute the specific value V t mod J (X t , ⃗ S t ), cf. (12), and obtain the associated offer price a t mod J (X t , ⃗ S t ). ...
... In case demand can be assumed to be independent of time the computational effort of Algorithm 4.1 via (11) or (12), can be even further reduced. Theorem 4.1. ...
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... As online marketplaces benefit from an increased number of retailers on their platforms, they typically support sellers to establish automated dynamic pricing systems (Kachani et al. 2010). 18 However, Schlosser and Richly (2019) claim that current dynamic pricing systems are not able to deal with the complexity of competitor-based pricing and therefore most often ignore competition altogether or solely rely on manually adjusted rule-based mechanics. ...
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Past reviews of studies concerning competitive pricing strategies lack a unifying approach to interdisciplinarily structure research across economics, marketing management, and operations. This academic void is especially unfortunate for online markets as they show much higher competitive dynamics compared to their offline counterparts. We review 132 articles on competitive posted goods pricing on either e-tail markets or markets in general. Our main contributions are (1) to develop an interdisciplinary framework structuring scholarly work on competitive pricing models and (2) to analyze in how far research on offline markets applies to online retail markets.
... Traditionally, for each of their offered products, merchants regularly request (cf. Amazon Marketplace Web Service 2019) the current market situation containing the competitors' offers (including price, quality, ratings, etc., see Kachani and Shmatov 2010). Often such requests occur several times a day for each of a seller's products. ...
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... However, a product's value manifests itself in various aspects. This principle suggests that a particular object, or a particular product, can be abstracted in several attributes (e.g., price, durability, comfort, delivery time and image), and their value is derived from the performance of the same (Meredith et al., 1994;Stonehouse and Snowdon, 2007;Kachani and Shmatov, 2011). ...
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... To address markets with multiple offer dimensions (cf. [41]) the demand learning component needs to be extended. In our setting, additional characteristic explanatory variables can be easily defined. ...
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... However, a product's value manifests itself in various aspects. This principle suggests that a particular object, or a particular product, can be abstracted in several attributes (e.g., price, durability, comfort, delivery time and image), and their value is derived from the performance of the same (Meredith et al., 1994;Stonehouse and Snowdon, 2007;Kachani and Shmatov, 2011). ...
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... In our model, P is a general function of our offer price a and a market situation denoted by s. The market situation s is a vector which includes all relevant observable quantities of interest, such as time t, the competitors' prices p, customer ratings v, product conditions, etc. (see, e.g., Kachani, Shmatov (2010)). The probability to sell exactly i items within the time span (t, t + 1) in a stable market situation s = (t, p, v, ...) is denoted by, t = 0, 1, 2, ..., T − 1, a ≥ 0, i = 0, 1, 2, ..., P t (i, a, s). ...
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... In our model, P is a general function of our offer price a and a market situation denoted by s. The market situation s is a vector which includes all relevant observable quantities of interest, such as time t, the competitors' prices p, customer ratings v, product conditions, etc. (see, e.g., Kachani, Shmatov (2010)). The probability to sell exactly i items within the time span (t, t + 1) in a stable market situation s = (t, p, v, ...) is denoted by, t = 0, 1, 2, ..., T − 1, a ≥ 0, i = 0, 1, 2, ..., P t (i, a, s). ...
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A central problem in marketing is: how should the firm position (reposition) and price a line of related (substitute) products in order to maximize profits (or welfare). We formulate this problem faced by a monopolist as a mathematical program, outline how to obtain the market data from a sample of customers, discuss what cost data are relevant, and suggest a heuristic algorithm to solve the problem. The output of the process is a list of products to offer, their prices, and the customer segments which purchase each product. While additional real world complexities, e.g., uncertainty about customer wants, product performance, and competitive response, are not modeled, we believe the system developed can serve as an important input into the decision process when new products are designed and priced. The methodology can be used as a part of a decision support system, where management specifies the number of products desired. The system suggests a few good solutions, together with the prices and customer segments served by each product. We use the standard assumption that the market is composed of different customer segments of various sizes, each containing homogeneous customers. Customers choose one brand only, the one that provides them with maximum value for the money. The firm faces both fixed and variable production and marketing costs for each product. Competition is either nonexistent, or assumed not to respond to the firm's moves. The information available to the firm is the sizes and preferences of the segments, based on a sample of customers, and the cost data. As an alternative to the traditional approach of estimating a parametric utility function, and aggregating customers into segments, we can also use the raw data as input, where each customer in the sample represents a segment. This, we believe, allows us to reduce the errors introduced in the process. Heuristics for solving the problem are suggested. The heuristics are evaluated on a set of simulated problems, and compared to the optimal solutions. The heuristics perform well when compared to all feasible solutions on a set of small simulated problems. We also discuss the application of the procedure to a ‘real life' sized problem.
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This paper investigates the impact of reference price effects on retailer price promotions and describes why these effects can make promoting profitable. First, we analyze the profit impact of reference price effects generated by a single period of promotion. The promotion can increase profit if the gain that these effects create in the promotion period outweighs the loss they create in future periods. We then describe how retailers can estimate the optimal strategy of recurring promotions that maximizes profits from reference price effects over a time horizon. Examples of such strategies are presented for a retailer selling a national brand of peanut butter. We obtain insights into how promotion prices, timing, and profits are affected by changes in costs, interest rates, consumers' reactions to reference price effects, and error in estimates used in the model. The retailer's optimal reaction to a trade deal is also examined. This strategy involves a phase of increased promotion activity sandwiched between phases of decreased activity. We explain these results using the effects described in the single-period model.
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In this paper, we study both the local and global convergence of various iterative methods for solving the variational inequality and the nonlinear complementarity problems. Included among such methods are the Newton and several successive overrelaxation algorithms. For the most part, the study is concerned with the family of linear approximation methods. These are iterative methods in which a sequence of vectors is generated by solving certain linearized subproblems. Convergence to a solution of the given variational or complementarity problem is established by using three different yet related approaches. The paper also studies a special class of variational inequality problems arising from such applications as computing traffic and economic spatial equilibria. Finally, several convergence results are obtained for some nonlinear approximation methods.
Chapter
This paper focuses on joint dynamic pricing and demand learning in an oligopolistic market. Each firm seeks to learn the price-demand relationship for itself and its competitors, and to set optimal prices, taking into account its competitors’ likely moves. We follow a closed-loop approach to capture the transient aspect of the problem, that is, pricing decisions are updated dynamically over time, using the data acquired thus far. We formulate the problem faced at each time period by each firm as a Mathematical Program with Equilibrium Constraints (MPEC). We utilize variational inequalities to capture the game-theoretic aspect of the problem. We present computational results that provide insights on the model and illustrate the pricing policies this model gives rise to.
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In this paper we introduce and study a general iterative scheme for the numerical solution of finite dimensional variational inequalities. This iterative scheme not only contains, as special cases the projection, linear approximation and relaxation methods but also induces new algorithms. Then, we show that under appropriate assumptions the proposed iterative scheme converges by establishing contraction estimates involving a sequence of norms in En induced by symmetric positive definite matrices Gm. Thus, in contrast to the above mentioned methods, this technique allows the possibility of adjusting the norm at each step of the algorithm. This flexibility will generally yield convergence under weaker assumptions.
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Fluid dynamics models provide a powerful deterministic technique to approximate stochasticity in a variety of application areas. In this paper, we study two classes of fluid models, investigate their relationship as well as some of their applications. This analysis allows us to provide analytical models of travel times as they arise in dynamically evolving environments, such as transportation networks as well as supply chains. In particular, using the laws of hydrodynamic theory, we first propose and examine a general second-order fluid model. We consider a first-order approximation of this model and show how it is helpful in analyzing the dynamic traffic equilibrium problem. Furthermore, we present an alternate class of fluid models that are traditionally used in the context of dynamic traffic assignment. By interpreting travel times as price/inventory–sojourn-time relationships, we are also able to connect this approach with a tractable fluid model in the context of dynamic pricing and inventory management.
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While every firm in a supply chain bears supply risk (the cost of insufficient supply), some firms may, even with wholesale price contracts, completely avoid inventory risk (the cost of unsold inventory). With a push contract there is a single wholesale price and the retailer, by ordering his entire supply before the selling season, bears all of the supply chain’s inventory risk. A pull contract also has a single wholesale price, but the supplier bears the supply chain’s inventory risk because only the supplier holds inventory while the retailer replenishes as needed during the season. (Examples include Vendor Managed Inventory with consignment and drop shipping.) An advance-purchase discount has two wholesale prices: a discounted price for inventory purchased before the season, and a regular price for replenishments during the selling season. Advance-purchase discounts allow for intermediate allocations of inventory risk: The retailer bears the risk on inventory ordered before the season while the supplier bears the risk on any production in excess of that amount. This research studies how the allocation of inventory risk (via these three types of wholesale price contracts) impacts supply chain efficiency (the ratio of the supply chain’s profit to its maximum profit). It is found that the efficiency of a single wholesale price contract is considerably higher than previously thought as long as firms consider both push and pull contracts. In other words, the literature has exaggerated the value of implementing coordinating contracts (i.e., contracts that achieve 100 % efficiency, such as buy-backs or revenue sharing) because coordinating
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We study competition in a supply chain where multiple manufacturers compete in quantities to supply a set of products to multiple risk-averse retailers who compete in quantities to satisfy the uncertain consumer demand. For the symmetric supply chain, we give closed-form expressions for the unique equilibrium. We find that, provided there is a sufficiently large number of manufacturers and retailers, the supply chain efficiency (the ratio of the aggregate utility in the decentralized and centralized chains) can be raised to 1 by inducing the right degree of retailer differentiation. Also, risk aversion results in triple marginalization: retailers require a strictly positive margin to distribute even when they are perfectly competitive, because otherwise they are unwilling to undertake the risk associated with the uncertainty in demand. For the asymmetric supply chain, we show how numerical optimization can be used to compute the equilibria, and we find that the supply chain efficiency may drop sharply with the asymmetry of either manufacturers or retailers. We also find that the introduction of asymmetric product assortment reduces the degree of competition among retailers and thus has an effect similar to that of reducing the number of retailers. We show that, unlike in the symmetric chain, the asymmetric chain efficiency depends on product differentiation and risk aversion because of the interaction between these features and the asymmetry of manufacturers and retailers.
Article
We consider a two-echelon distribution system in which a supplier distributes a product toN competing retailers. The demand rate of each retailer depends on all of the retailers' prices, or alternatively, the price each retailer can charge for its product depends on the sales volumes targeted by all of the retailers. The supplier replenishes his inventory through orders (purchases, production runs) from an outside source with ample supply. From there, the goods are transferred to the retailers. Carrying costs are incurred for all inventories, while all supplier orders and transfers to the retailers incur fixed and variable costs. We first characterize the solution to the centralized system in which all retailer prices, sales quantities and the complete chain-wide replenishment strategy are determined by a single decision maker, e.g., the supplier. We then proceed with the decentralized system. Here, the supplier chooses a wholesale pricing scheme; the retailers respond to this scheme by each choosing all of his policy variables. We distinguish systematically between the case of Bertrand and Cournot competition. In the former, each retailer independently chooses his retail price as well as a replenishment strategy; in the latter, each of the retailers selects a sales target, again in combination with a replenishment strategy. Finally, the supplier responds to the retailers' choices by implementing his own cost-minimizing replenishment strategy. We construct a perfect coordination mechanism. In the case of Cournot competition, the mechanism applies a discount from a basic wholesale price, based on thesum of three discount components, which are a function of (1) annual sales volume, (2) order quantity, and (3) order frequency, respectively.
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We provide sharp lower and upper bounds on the ratio of decentralized to centralized profits when multiproduct firms offering differentiated products engage in price competition. The bounds depend on the demand sensitivity matrix but are independent of marginal costs.
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The paper provides a comprehensive review of the recent development of revenue management in different industries. We discuss research on different revenue management strategies including pricing, auctions, capacity control, overbooking and forecasting. Related issues such as economic concerns, customer perception, competition and consolidation, implementation, performance evaluation, and common techniques and approaches used for solving revenue management problems are also discussed. Finally, we give our suggestion on some important areas that warrant further research.
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A convenient technique used for solving several types of economic problems is to cast them as optimization problems having the line integral of the (inverse) demand and/or supply functions in the objective. But if these functions are non-integrable (having a nonsymmetric Jacobian matrix) then alternative techniques must be used. This paper surveys these and proposes an additional solution technique, based on making the given functions integrable by making a small number of substitutions and iteratively adjusting a parameter associated with each substitution until a solution is found. The technique is particularly applicable when certain neoclassical assumptions are satisfied.
Advantages and drawbacks of variational in-equality formulations. Giannessi, F., A. Maugeri eds, Variational Inequalities and Network Equilibrium Problems
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