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Author: Chenhao Du
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Languages : en
Pages : 0
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We consider a seller's problem of determining revenue-maximizing prices for an assortment of products that exhibit network effects. Customers make purchase decisions according to a multinomial logit (MNL) customer choice model, modified -- to incorporate network effects -- so that the utility each individual customer gains from purchasing a particular product depends on the market's total consumption of that product. In the setting of homogeneous products, we show that if the network effect is comparatively weak, then the optimal pricing decision of the seller is to set identical prices for all products. However, if the network effect is strong, then the optimal pricing decision is to set the price of one product low, and to set the prices of all other products to a single high value. This boosts the sales of the single low-price product in comparison to the sales of all other products. These results can be compared to the optimal pricing policy for the classical MNL model (without network effects) in which it is optimal to set identical prices and obtain identical sales quantities for homogeneous products. We obtain comparative statics results that describe how optimal prices and sales levels vary with a parameter that determines the strength of the network effects. We extend our analysis of settings with heterogeneous products as well as to settings with inter-product network effects, and we describe efficient computational methods.
Author: Chenhao Du
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Book Description
We consider a seller's problem of determining revenue-maximizing prices for an assortment of products that exhibit network effects. Customers make purchase decisions according to a multinomial logit (MNL) customer choice model, modified -- to incorporate network effects -- so that the utility each individual customer gains from purchasing a particular product depends on the market's total consumption of that product. In the setting of homogeneous products, we show that if the network effect is comparatively weak, then the optimal pricing decision of the seller is to set identical prices for all products. However, if the network effect is strong, then the optimal pricing decision is to set the price of one product low, and to set the prices of all other products to a single high value. This boosts the sales of the single low-price product in comparison to the sales of all other products. These results can be compared to the optimal pricing policy for the classical MNL model (without network effects) in which it is optimal to set identical prices and obtain identical sales quantities for homogeneous products. We obtain comparative statics results that describe how optimal prices and sales levels vary with a parameter that determines the strength of the network effects. We extend our analysis of settings with heterogeneous products as well as to settings with inter-product network effects, and we describe efficient computational methods.
Author: Zhenyu Cui
Publisher:
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Languages : en
Pages : 25
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We obtain exact semi-explicit solutions to the optimal pricing problem for the multinomial logit (MNL) consumer choice model with network effects, through a novel conditioning argument and the use of the Lambert W function. Then we manage to characterize the exact optimal solution in the presence of negative, positive and weak or positive and strong network effects, and also in the setting of heterogeneous market parameters. Furthermore, we provide a semi-explicit expression of equilibrium sales quantities in terms of equilibrium prices, based on which we establish sufficient conditions for the uniqueness of market equilibria. In addition, we provide an exact semi-explicit expression for the steady-state choice probability, based on which we discuss the uniqueness of the steady-state choice probability.
Author: Qing Feng
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Languages : en
Pages : 0
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In this paper, we study a dynamic multinomial logit choice model with network effect (we call it the "dynamic model'' in short). In our model, a continuum of customers stay in a market for an infinite horizon. We adopt a random utility model to capture customer's valuation of each product in which the expected utility of each product is determined by its intrinsic value, its price and the market share at the current time period. In each time period, for a customer, if the highest utility among products is higher than the utility of the no-purchase option, then this customer will choose the product with the highest current utility to purchase and leave the market. Otherwise this customer will not purchase any product and will stay in the market in the next time period. This process continues until it reaches a steady state. We call the market shares under the steady state the choice probabilities under the dynamic model.We study the properties of the dynamic model, particularly its choice probabilities, and compare the choice probabilities under the dynamic model with those under a previously proposed static MNL model with network effect in the literature (we call it the "static model" in short). We find that under proper dominance conditions, the dominant product will have a higher choice probability than the other products. We also find that under mild conditions, the choice probabilities under the dynamic model tend to be more balanced than those of the static model, and the total choice probability of all products will be smaller than that under the static model. Then we study the operational decision problems under the dynamic model, including the optimal pricing problem and the assortment optimization problem. For the optimal pricing problem, we propose an approximation scheme of the final market shares, and establish an efficient algorithm for the optimal pricing problem with logarithmic utility function. We also propose a gradient descent method that works for general optimal pricing problems under the dynamic model. For the assortment optimization problem, we propose a new class of assortments called k-proximity assortments, and investigate the optimality of these assortments both theoretically and numerically.
Author: Mohan Gopalakrishnan
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Languages : en
Pages : 0
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Motivated by direct interactions with practitioners and real-world data, we study a monopoly firm selling multiple substitute products to customers characterized by their different social network degrees. Under the multinomial logit model framework, we assume that the utility a customer with a larger network degree derives from the seller's products is subject to more impact from her neighbors and describe the customers' choice behavior by a Bayesian Nash game. We show that a unique equilibrium exists as long as these network effects are not too large. Furthermore, we study how the seller should optimally set the prices of the products in this setting. Under the homogeneous product-related parameter assumption, we show that if the seller optimally price-discriminates all customers based on their network degrees, the products' markups are the same for each customer type. Building on this, we characterize the sufficient and necessary condition for the concavity of the pricing problem, and show that when the problem is not concave, we can convert it to a single-dimensional search and solve it efficiently. We provide several further insights about the structure of optimal prices, both theoretically and numerically. Furthermore, we show that we can simultaneously relax the multinomial logit model and homogeneous product-related parameter assumptions and allow customer in- and out-degrees to be arbitrarily distributed whilemaintaining most of our conclusions robust.
Author: Hongmin Li
Publisher:
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Category :
Languages : en
Pages : 32
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In this paper, we study price optimization with price-demand relationships captured by the paired combinatorial logit (PCL) model which overcomes restrictions of the well-studied multinomial logit (MNL) and nested logit (NL) models. The PCL model allows for choice-correlation and, like the NL model, includes the MNL model as a special case. Compared to the NL models, the PCL model does not restrict the sequence of the choice structure and allows for different covariances among all pairs of choices. This additional flexibility in structure enables a more accurate representation of some choice settings and broadens its empirical applications. Hence, it is of both theoretical and practical interests to extend the normative studies on the MNL and NL models to the PCL model and examine the pricing problem under this model. Due to cross-nesting of choice alternatives, the pricing problem under the PCL model poses a greater challenge than the MNL and NL models. However, using the concept of P-matrix, we are able to identify conditions for a unique optimal price solution and develop an efficient and theoretically sound approach for finding the optimal prices. We show that the analysis and solution approach are generalizable to other GEV family models with cross-nested alternatives.
Author: Aydin Alptekinoglu
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Languages : en
Pages : 50
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We investigate the use of a canonical version of a discrete choice model due to Daganzo (1979) in optimal pricing and assortment planning. In contrast to multinomial and nested logit (the prevailing choice models used for optimizing prices and assortments), this model assumes a negatively skewed distribution of consumer utilities, an assumption we motivate by conceptual arguments as well as published work. The choice probabilities in this model can be derived in closed-form as an exponomial (a linear function of exponential terms). The pricing and assortment planning insights we obtain from the Exponomial Choice (EC) model differ from the literature in two important ways. First, the EC model allows variable markups in optimal prices that increase with expected utilities. Second, when prices are exogenous, the optimal assortment may exhibit leapfrogging in prices, i.e., a product can be skipped in favor of a lower-priced one depending on the utility positions of neighboring products. These two plausible pricing and assortment patterns are ruled out by multinomial logit (and by nested logit within each nest). We provide structural results on optimal pricing for monopoly and oligopoly cases, and on the optimal assortments for both exogenous and endogenous prices. We also demonstrate how the EC model can be easily estimated--by establishing that the loglikelihood function is concave in model parameters and detailing an estimation example using real data.
Author: Lei Guan
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Languages : en
Pages : 0
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{Problem Definition:} This paper considers the operational management problems under a newly proposed choice model that captures the effect of focality. The offered assortment is separated into the focal set and the non-focal set under this new model due to the bias of focality, which is identified by the focal sets and an assortment-dependent focal parameter. A prospective consumer is more likely to choose a product from the focal set, while she may still choose one from the non-focal set for a variety of reasons such as previous purchase experience or brand loyalty. This focal multinomial logit model generalizes the famous multinomial logit model and several well-studied consideration-set choice models. In addition, it has the capability to describe and explain a variety of irrational choice behaviors often observed in practice, such as the context effect, halo effect, and choice overload. {Methodology/results:} In this paper, we primarily focus on the threshold focal set and various focal parameter settings, including the constant, cardinality, and linear focal multinomial logit models, as well as a broader model that satisfies certain regularity conditions and subsumes the above models. We analyze the computational complexity and propose polynomial-time exact or approximation algorithms to solve the assortment optimization problems under different focal parameters. We then characterize the optimal strategy for the joint price and assortment optimization problem. Additionally, we develop a mixed integer conic programming reformulation method that converges to a global optimal estimator for the model calibration problem. {Managerial Implications:} We use these methods to conduct numerical experiments on both synthetic and real data sets. The results demonstrate the efficiency of our proposed algorithms, the predictive power, and the increase in revenue for the focal multinomial logit model. Our extensive analysis implies that in practice retailers may take into account the effect of focality in consumer purchase behavior because it could increase the accuracy of demand estimation and therefore improve operational performance.
Author: Guillermo Gallego
Publisher: Springer
ISBN: 1493996061
Category : Business & Economics
Languages : en
Pages : 336
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“There is no strategic investment that has a higher return than investing in good pricing, and the text by Gallego and Topaloghu provides the best technical treatment of pricing strategy and tactics available.” Preston McAfee, the J. Stanley Johnson Professor, California Institute of Technology and Chief Economist and Corp VP, Microsoft. “The book by Gallego and Topaloglu provides a fresh, up-to-date and in depth treatment of revenue management and pricing. It fills an important gap as it covers not only traditional revenue management topics also new and important topics such as revenue management under customer choice as well as pricing under competition and online learning. The book can be used for different audiences that range from advanced undergraduate students to masters and PhD students. It provides an in-depth treatment covering recent state of the art topics in an interesting and innovative way. I highly recommend it." Professor Georgia Perakis, the William F. Pounds Professor of Operations Research and Operations Management at the Sloan School of Management, Massachusetts Institute of Technology, Cambridge, Massachusetts. “This book is an important and timely addition to the pricing analytics literature by two authors who have made major contributions to the field. It covers traditional revenue management as well as assortment optimization and dynamic pricing. The comprehensive treatment of choice models in each application is particularly welcome. It is mathematically rigorous but accessible to students at the advanced undergraduate or graduate levels with a rich set of exercises at the end of each chapter. This book is highly recommended for Masters or PhD level courses on the topic and is a necessity for researchers with an interest in the field.” Robert L. Phillips, Director of Pricing Research at Amazon “At last, a serious and comprehensive treatment of modern revenue management and assortment optimization integrated with choice modeling. In this book, Gallego and Topaloglu provide the underlying model derivations together with a wide range of applications and examples; all of these facets will better equip students for handling real-world problems. For mathematically inclined researchers and practitioners, it will doubtless prove to be thought-provoking and an invaluable reference.” Richard Ratliff, Research Scientist at Sabre “This book, written by two of the leading researchers in the area, brings together in one place most of the recent research on revenue management and pricing analytics. New industries (ride sharing, cloud computing, restaurants) and new developments in the airline and hotel industries make this book very timely and relevant, and will serve as a critical reference for researchers.” Professor Kalyan Talluri, the Munjal Chair in Global Business and Operations, Imperial College, London, UK.
Author: Ruxian Wang
Publisher:
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Languages : en
Pages : 0
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This paper incorporates heterogeneous threshold effects into the classical multinomial logit (MNL) model, and studies the associated operations problems such as estimation and assortment optimization. The derived model is referred to as the threshold multinomial logit (TMNL) model and incorporates the recently proposed threshold Luce (T-Luce) model as a limiting case. Under the TMNL model, consumers first form their (heterogeneous) consideration set: If an alternative with significantly low utility is dominated by another one, it will not be included in the consideration set. The TMNL model can alleviate the restricted substitution patterns of MNL due to the independence of irrelevant alternatives (IIA) property, and therefore can model more flexible choice behavior. We develop a maximum likelihood based estimation to calibrate the proposed threshold model and further establish its statistical properties such as consistency and asymptotic normality under mild conditions. An efficient EM algorithm is also developed to handle the scenario with incomplete sales data. Our extensive numerical studies on synthetic and real datasets show that the new model can improve the goodness of fit and prediction accuracy of consumer choice behavior. In addition, we characterize the optimal strategies and provide efficient solutions for the associated assortment optimization problems under the TMNL model. Our theoretical and empirical results suggest that the threshold effects should be taken into account in firms' decision making such as demand estimation and operations management, and ignoring these effects could lead to sub-optimal solutions or even substantial losses for firms.
Author: Stefanus Jasin
Publisher:
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Languages : en
Pages : 0
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In this paper, we study the assortment and price optimization problems under Multinomial Logit (MNL) model with price range effect, where the utility of a product is affected by the relative position of its price with respect to the highest and the lowest prices in the offer set. This model is motivated by the so-called Range Theory popularized in the behavioral economics and psychology literature. It addresses the limitation of a single-point interpretation of reference price, which ignores the impact of all other distributional information. We investigate the pure assortment problem, the pure pricing problem, and the joint assortment and pricing problem under the MNL model with price range effect. For each model, we first identify the structure of the optimal policy, and then we propose tractable algorithms that either output the optimal solution in polynomial time or admit an Fully Polynomial-Time Approximation Scheme (FPTAS).
