Marginal Estimation + Price Optimization for Multi-Product Pricing Problems

Transcription

1 Marginal Estimation + Price Optimization for Multi-Product Pricing Problems Zhenzhen Yan Cong Cheng Karthik Nataraan Chung-Piaw Teo February 7, 2018 Abstract In this paper, we develop a data-driven approach for the multi-product pricing problem by exploiting properties of the representative consumer model in discrete choice. We relate a special case of the representative consumer model to a semparametric choice model called the Marginal Distribution Model) and show that the multi-product pricing problem in this case is convex when the marginal probability density functions are log-concave. In the special case with exponential marginal distributions, this recreates the convexity results of multi-product pricing under the Multinomial Logit model, while generalizing the result to other marginal distributions such as normal, logistic, extreme value or Laplace distributions. Using this approach, we establish a set of closed-form relationships between prices and market shares for the products. While it is difficult to calibrate the shape of a general regularization function for the representative consumer model from data, we show that using a separable regularization function provides good estimates on the structure of the choice model and guides the algorithm in the search for optimal prices. In this way, we develop second order conic and linear programs to estimate the shape of the separable regularization function in the representative consumer model. Mixed integer linear programming models are used to find the optimal prices when side constraints are present and linear programs when side constraints are absent. This partially addresses the problem of model misspecification for pricing problems, since we do not explicitly assume the structural form in the consumer s utility model. Extensive tests National University of Singapore, Singapore. a @u.nus.edu Northeastern University, China. chc 5588@163.com Engineering System and Design, Singapore University of Technology and Design, Singapore. karthik nataraan@sutd.edu.sg National University of Singapore, Singapore. bizteocp@nus.edu.sg 1

2 using both simulated data and industry data demonstrates clearly the benefits of this Marginal Estimation + Price Optimization approach. 1 Introduction Pricing, as a strategy to shape the demand in a market, is commonly used in industry and has been extensively studied both by academicians and practitioners see Talluri and van Ryzin 40)). The primary goal is to try and understand how demand is influenced by prices and to then optimize prices to maximize the profit. The growth of the Internet has led to a drastic increase in the capability of companies to make changes in prices to learn the demand model while doing price optimization. Einav et al. 15) analyze the targeted pricing and auction design variations with ebay sales data and find that of the 100 million listings on a given day, more than half will reappear as a separate listing, often with modified sales parameters such as prices). In fact, with the boom in e-commerce activities, the practice of changing prices strategically to learn more about the customer s behavior has been adopted by many companies. Online retailers are turning to data to help them compete, and they have strategic price ranges that they play between, says Meghan Heffernan, a spokeswoman for Savings.com. The availability of data from pricing experiments presents companies with an opportunity to increase profits if they can use it effectively. Hence, research on revenue maximization using data from pricing experiments has exploded in recent years. In the multi-product setting, the difficulty in the pricing problem lies in capturing the impact of the prices on the cross-elasticity of demand, especially if the customer can only purchase one out of a set of products offered. Ferreira et al. 26) discuss the operational challenges faced by the online fashion retailer Rue La La which offers price discounts. They use regression trees and machine learning techniques to estimate the demand model with the use of integer programming to do price optimization. In another recent work, Bertsimas and Vayanos 10) study the exploration-exploitation tradeoff using dynamic pricing where the demand and price relationship is modeled through uncertainty sets that encode the seller s beliefs about the demand curve parameters. Using mixed integer conic optimization, they develop an adaptive dynamic pricing mechanism to do price optimization. The demand and price relationship in their work is modeled through a linear function with additive error terms where the error terms are assumed to lie in a bounded uncertainty set. Fisher et al. 18) discuss a competition-based dynamic pricing model for the Chinese online retailer Yihaodian where pricing experiments are used to estimate price sensitivities and a best-response pricing algorithm is used to choose prices. In their work, a Nested Logit NL) model is used to capture the customer s choice behavior. 2

3 While this stream of research showcases the power of the choice model estimation and optimization framework in pricing products using data from experiments, a natural concern is the accuracy of the estimation model in capturing the customer s purchase behavior, and the computational tractability of the pricing optimization model arising from the estimation technique. The critical step is to find a proper model to characterize the market demand, based on the available sales data. Our experience in working with data from a large automobile manufacturer is that the market demand model is often a black box to companies which predicts the sales for each product through a complex, time consuming market simulator using a pricing proposal with all other features fixed) as the input to the black box. The market share of each product is then a reflection of the expected percentage of the total sales in the market that is earned by a particular product. There are essentially two approaches used to address this problem, reflecting the trade-off between flexibility and tractability in modeling the market shares as a function of the prices which we discuss below: A flexible model refers to a model which can mimic the mechanism of the black box well for different sets of input and output, by learning from data. Machine learning tools have become popular recently due to its modeling flexibility, compared to the traditional econometric methods. For instance, neural networks are well known for their capability to fit a black box system, given a large enough amount of training samples. However, the relationships between the input prices) and ouput sales/market share) described by the neural networks are complicated essentially a composition of several nonlinear functions), and lack theoretical ustification. Furthermore, the pricing optimization based on the market response function described by neural networks appears to be intractable. There is an alternative stream of research that focuses on identifying tractable instances of the pricing problem using discrete choice models. Most of these results are applicable to specific choice models, such as the Multinomial Logit MNL) model see Song and Xue 39), Dong et al. 14)) and its generalization to attraction models see Keller et al. 27)), the NL model with identical price-sensitivity parameters for products within a nest see Li and Huh 29)) and under some restrictions to non identical price-sensitivity parameters see Gallego and Wang 21)), the d-level NL model see Huh and Li 25), Li et al. 28)), the Paired Combinatorial Logit see Li and Webster 30)), the Generalized Extreme Value GEV) model see Zhang et al. 42)) and the exponomial choice model see Alptekinoǧlu and Semple 4)). Model misspecification is however a general concern in these models. For instance, in the estimation of NL model, the products in the different nests and the levels of the nesting structure 3

4 need to be pre-determined, and often involves a fair amount of ingenuity and understanding of the consumer market to arrive at an appropriate model. A maximum likelihood method is then typically used to estimate the parameters of the choice model and prices are set using the proposed choice model. A natural concern is what happens if the model is misspecified? We use next a small simulated example to illustrate the challenge of model misspecification for this class of pricing problems. Example 1. We generate a set of sales data with the underlying choice model being a NL model with 4 branches and 5 products in each branch. For a given set of parameters in the NL model, we uniformly generate a set of prices and calculate its corresponding market shares using the assumed NL choice model the details of the parameters are provided in Section 5.1.2). We estimate this data set using MNL choice model and optimize the prices based on the estimated MNL model. The true profit generated by the obtained pricing solution can be evaluated by calculating the choice probability using the assumed NL choice model. In contrast, we can also get the true optimal profit by solving the pricing with NL model, which is a convex optimization model as shown by Li and Huh 29). We change the parameters in the NL model and repeat the procedure above to get a set of profits generated by the MNL based approach and the corresponding true optimal profits. We plot the kernel density functions of the two sets of profits in Figure 1. When the price sample size is 200 points, there is a huge gap between the profits obtained from the MNL based approach and the true optimal profits. More importantly, this gap can not be eliminated by simply increasing the sample size to even 2000 points. In other words, there exists a systematic error if the underlying choice model is misspecified. To resolve the problem of model misspecification, more sophisticated choice models such as the random coefficient logit has been proposed in the marketing and economics literature that relaxes the assumption that all consumers are identical and allows for consumer heterogeneity. Estimation of the random coefficient logit model however becomes much more complicated. Berry et al. 9) develop a two-step estimation method to estimate the random coefficient logit model from the aggregate sales data set. Solving the pricing problem with the random coefficient logit model is however an outstanding open challenge that still needs to be resolved. The complexity of the problem is further exacerbated by the fact that pricing decisions in practice are often constrained by business strategies and operational concerns. For instance, in a study of the pricing problem in an bilevel retailing example, Harsha et al. 23) found that the following are common business rules that affect the pricing decisions: a) Volume constraint - prices set must ensure that the sales target for certain products are met; b) Price monotonicity constraint - prices in certain channels must be set at a discount of the prices offered in other 4

5 Figure 1: Kernel density function estimate of the profits under MNL versus the true optimal profit channels, for market positioning purposes; and c) Price bounds - new prices must be confined to within certain range, as drastic changes in price levels may turn away customers. Harsha et al. 23) use attraction demand models to characterize customers demand and propose a general pricing model to handle these side constraints in the pricing problem. Mixed integer linear programs are proposed to solve these pricing problems. In this paper, we restrict our attention to a recently proposed class of choice models which uses the properties of the marginal distributions of random utilities and identify conditions under which the pricing problem is computationally tractable. We show how this modeling perspective turns the estimation of choice model to that of estimation of marginal distribution functions, and develop a semiparametric method that partially addresses model misspecification and consumer heterogeneity considerations. Based on the estimation result, we propose a mixed integer programming based pricing model, which is able to incorporate different side constraints on the prices due to business strategies and reduces to a linear program in the absence of side constraints. The rest of the paper is organized as follows: In Section 2 we provide a literature review of discrete choice models with a focus on pricing and estimation. In Section 3, we shows the connection between a special instance of the representative consumer model used in choice modeling, and the Marginal Distribution Model MDM) introduced in Nataraan et al. 38). We also provide a robust interpretation of the model in this section. We use this connection to 5

6 obtain a general closed-form relationship between the prices and market shares, and show that the pricing problem becomes convex and polynomial time solvable when the marginal density functions are log-concave. This nicely extends some of the currently known results for the single product pricing problem to the multi-product pricing problem. In Section 4, we develop the Marginal Estimation + Price Optimization approach for pricing with aggregate sales data. We discuss computational experiments to demonstrate the performance of the proposed approach in Section 5 using both simulated data and industry data from an automobile manufacturer and a fast food company. 2 Literature Review There is now a significant amount of literature on the multi-product pricing problem using discrete choice models of customer demand. In this section, we review some of the key results that are relevant to the current paper. 2.1 Pricing with Choice Models Myerson 34) provides a complete description of how a seller should choose a price to maximize the expected profit when selling a single item to buyers whose valuations the seller is uncertain about. Assuming that w is the cost of the product and F p) is the seller s assessment of the probability that a buyer has a valuation less than or equal to a price p, the optimal mechanism is to sell the item at a fixed take-it-or-leave-it price p where: p = arg maxp w)1 F p)). 1) p 0 Alternatively, this problem can be reformulated in the choice probability market share) variable x = 1 F p) where the optimal market share x is the solution to: x = arg max x [0,1] F 1 1 x) w)x, 2) and the optimal price is p = F 1 1 x ). Log-concavity of the complementary distribution function 1 F ) implies quasi-concavity of the profit function in the price variable in 1) and concavity of the profit function in the market share variable in 2) which makes this single dimension optimization problem straightforward to solve see Bagnoli and Bergstrom 6)). Examples of such distributions include the exponential, uniform, normal, logistic, extreme value, Laplace distributions among others. Generalizing this result to multiple products is however 6

7 much more complicated. Consider the multi-product pricing problem where the set of products offered by the seller is denoted as N = {1,..., N} and {0} is the outside option. Let the price of product be p, the cost of product be w where without loss of generality, the outside option parameters are assumed to be w 0 = p 0 = 0. The seller s expected profit maximization problem is formulated as: max p 0,p 0=0 =1 p w )x p), 3) where p = p 0, p 1,..., p n ) is the price vector and x p) is the market share of product given the price vector p. In the simplest MNL choice model, the choice probabilities are given as: x p) = 1 + e v αp, = 1,..., N, e v k αp k k=1 4) where v is the deterministic utility of product excluding the price), α is the price-sensitivity parameter where without loss of generality, we assume v 0 = p 0 = 0 for the outside option. In model 4), the price-sensitivity parameter for each product is assumed to be identical. Then with probability x 0 = 1/1 + k N ev k αp k ) the customer does not buy a product from the set. Hanson and Martin 22) showed that the multi-product profit function is unfortunately not a quasi-concave function of the price variables for the MNL model. Though a non-convex optimization problem, Akçay et al. 2) showed that the profit function is unimodal in the prices and demonstrated the efficiency of using the first order conditions in determining the optimal prices. An alternate approach to tackle this problem was developed by Song and Xue 39) and Dong et al. 14) who showed that pricing problem is a convex optimization problem for MNL in terms of the market share variables and reformulated the problem by optimizing over these variables rather than the prices. The pricing problem for MNL in terms of the choice probability variables is formulated as: max x s.t. p x) w )x =1 x = 1, =0 x 0, = 0, 1..., N, 5) 7

8 where the price as a function of choice probabilities is given as: p x) = 1 α v ln x ) + ln x 0 )), = 1,..., N, 6) It is straightforward to verify that the obective function in 5) is concave in the choice probability variables when the inverse choice probability equation is given as 6) for the MNL model. The optimal prices for this model are known to have the equal markup property when the price sensitivities are identical. Building on this technique, Li and Huh 29) studied the multi-product pricing problem under the NL model. In the NL model, the set of N products is divided into K nests where N k denotes the number of products that belong to nest k. Assuming that τ k [0, 1] is a parameter describing the dissimilarity among products in nest k, v k is the deterministic utility of product in nest k excluding the price), p k is the price of product in nest k and α k is the price-sensitivity parameter for the products in nest k, the customer s probability of choosing product in a nest k is given as: Nk ) τk 1 e v k α k p k e v lk α k p lk x k p) = K 1 + l=1 Nt t=1 l=1 ) τt, = 1,..., N k, k = 1,..., K. e v lt α tp lt Li and Huh 29) showed that the multi-product pricing problem is a convex optimization problem in the market share variables in this case too. Gallego and Wang 21) showed that for productdifferentiated price sensitivities, the convexity result for the NL model fails to hold and identified specific conditions on the price sensitivities and nest parameters for which the optimization problem remains convex in the choice probability variables. Generalization of the pricing problem to a multi-level NL model has been recently studied in Li et al. 28) who developed an iterative algorithm to find a stationary point of the revenue function. The expected revenue is however not concave in the product prices for this case. Huh and Li 25) generalized these results to multi-stage attraction models and identified specific conditions under which the optimal prices can be found efficiently and characterize the optimal markup for each product. Li and Webster 30) studied the class of PCL models which allows for more general correlation structures as compared to the NL model and identify conditions under which an unique optimal price solution is computable. Zhang et al. 42) showed the pricing problem with the GEV model can also be efficiently solved based on an explicit formula for the optimal markup in terms of the Lambert-W function. Alptekinoǧlu and Semple 4) developed a 8

9 discrete choice model which they referred to as the exponomial choice model where the pricing problem was shown to be a convex optimization problem. As these results indicate in comparison to the single product pricing problem for which convexity results are known for a large class of distributions, there are fewer convexity results known for the multi-product pricing problem and this is often obtained by careful analysis on a case by case basis. In this paper, we showcase an approach that can nicely exploit the generality of results from the single product case to the multi-product case. 2.2 Estimation of Choice Models We review some of the key estimation techniques for discrete choice models in this section, starting with the MNL model. The most popular method used in the parametric estimation of choice models is the maximum likelihood method. The deterministic term in the utility specification captures part of the customer s i utility of product and is represented as v i = β z i, where z i is the vector of product s attribute levels excluding price) for customer i and β represents the customer s partworth. Under the assumption of homogeneous partworth across all the customers with z i0 = 0, p 0 = 0, the maximum log-likelihood estimation of the parameters with the MNL choice model is given as: max β,α i=1 =0 M y i ln 1 + e β z i αp k=1 e β z ik αp k, 7) where M = {1,..., M} is the set of customers and y i = 1 if customer i selects product and 0 otherwise. The obective function in 7) is concave in the β and α variables and the estimation problem is efficiently solvable see McFadden 35)). One of the main restrictions of the MNL model is that it requires the error terms to be independent and identical. However, in practice, this assumption can be easily violated. The red bus-blue bus paradox is a classic example to illustrate this. To address this drawback, models such as the NL model have been proposed. Daganzo and Kusnic 13) and Mishra et al. 37) showed the maximum log-likelihood estimation of the NL model in the β and α variables is solvable as a convex optimization problem when the parameters τ k are in [0, 1] and fixed. The estimation problem is however non-convex when the τ k parameters also need to be estimated. There has also been a significant interest in incorporating heterogeneity at the customer level in discrete choice models. One such model is the random coefficient logit choice model 9

10 see McFadden and Train 36)) where estimation is typically carried out with simulation based optimization methods. However the estimation problem for the mixed logit model for customer level data is non-convex. Berry et al. 9) consider aggregate sales data set and propose a two-step estimation method to address both partworth heterogeneity and price endogeneity. Under the homogeneous population assumption, the market share equals the individual choice probability and hence the estimation of the choice probability directly applies to the market share. In the case that the customers are heterogeneous in preference weights, the market share is the average of customers choice probability. In the first step, they propose to invert the market share function by solving a system of equations using iterative methods. In the second step, they estimate the distribution of the random coefficients using General Method of MomentsGMM) and use instrumental variables to address the price endogeneity issue. Another popular approach to estimation is to combine maximum likelihood estimation model with regularization technique adapted from machine learning to penalize deviations of customer specific partworths β i from the population average partworth β. The reader is referred to Evgeniou et al. 16) for a MNL model that incorporates heterogeneity at the partworth level using such an approach. There has also been growing interest in semparametric and nonparametric estimation methods. Building on the pioneering work of the maximum score estimator of Manski 32), Fox 19) showed that the semiparametric maximum-score estimator of choice models is consistent when using data only on subsets of choices. Farias et al. 17) proposed a nonparametric approach to estimate the choice model from a set of automobile sales transaction data. In their approach, the choice model is viewed as distributions over product s rankings. Compared to parametric methods, which rely on the correctness of the assumed underlying choice model, semiparametric and nonparametric models make much fewer assumptions. Such approaches have however been primarily limited to the context of descriptive analytics rather than prescriptive analytics such as pricing problems. The paper most closely related to our work is the class of choice models proposed by Nataraan et al. 38) who develop a Marginal Distribution Model MDM) where only the marginal distributions are specified but the oint distribution is not. By focusing on the oint distribution that maximizes customer s expected utility, Mishra et al. 37) showed that MDM is able to recover a large number of widely used choice probabilities such as the MNL model and NL model by appropriately chosen marginal distributions. Mishra et al. 37) developed a scale heterogeneity version of this model that allows for customers to have different perception variances of the outside option by allowing for non-identical marginal distributions. Furthermore, they proposed a constrained maximum-likelihood estimation method for estimating the parameters 10

11 of MDM with disaggregate data. Computational tests on simulated and real data on preferences of safety features in automobiles in Mishra et al. 37) illustrate that the model is suitable for capturing both product and consumer level heterogeneity. Ahipasaoglu et al. 1) extended this model to compute traffic equilibrium flows in transportation networks and illustrated the modeling flexibility that capturing the marginal distributions provides in this context. We build on this model in the paper and apply it to solve multi-product pricing problems. 3 Multi-Product Pricing with MDM The multi-product pricing problem can be viewed as a bilevel optimization problem where in the outer step, the retailer sets the price for each product while in the inner step, the customers observe the prices and then make purchasing decisions. We use market shares as the metric to model the market response to price changes. A flexible model of the market shares is expected to fit the prices and sales data well in many cases but results in a more complicated estimation and pricing optimization procedure. Model misspecification in such a case might arise from a wrong guess of the underlying choice model generating the sales data. Our key contribution in this paper is to provide a model which can achieve modeling flexibility and computational tractability simultaneously. The proposed model is built on a representative consumer who captures the stochastic choice behavior. Let v i denotes a consumer i s deterministic valuation for product and p denotes the price of product. Then v i αp represents consumer i s surplus on product where α > 0 is a homogeneous price sensitivity parameter across products. We assume that x i denotes consumer i s randomization strategy which is the probability of consumer i choosing product. The classical random utility framework uses a random noise ɛ i to model choice behaviour, assuming that customer i will choose the product with the largest utility where the utilities are given by: U i := v i αp + ɛ i. 8) The representative consumer model uses however a convex regularization term Cx) to serve as a reward to the consumer s randomization strategy x as follows: max x N v i αp )x i Cx) 9) =0 11

12 where the optimization is over the unit simplex defined as: N = x RN+1 + x i = 1. 10) Such a representative consumer model has been studied in Anderson et al. 3) and is known to recreate the MNL choice probabilities when Cx) = i x i log x i is the entropy function. Hofbauer and Sandholm 24) showed that for any random utility model when the error terms have a strictly positive density function, there exists a strictly convex regularization function Cx), such that the solution to the representative cosumer model provides the choice probability in the random utility model. However it is also known that the representative consumer model is not equivalent to the random utility model. For example, there is no random utility model which is equivalent to the representative consumer model when Cx) = i log x i see Proposition 2.2 in Hofbauer and Sandholm 24)). Fudenberg et al. 20) recently studied a special case of the representative consumer model under the assumption that the regularization term is separable as follows: max x N =0 =0 v i αp )x i C x i ), 11) where each C ) is a convex function and showed that in this case, the choice probabilities satisfy a weaker form of the IIA property. However they do not discuss applications of their =0 model to pricing problem which is our focus in this paper. In fact, for each differentiable convex function C ) : [0, 1] R, there exists a valid cumulative probability distribution F ) such that C x i ) = 1 1 x i F 1 t)dt 1. In the following analysis, we focus on this particular representation of the separable convex function, turning our attention from a general convex function to a probability distribution function. This representation of the regularization function is motivated from the Marginal Distribution Model MDM) proposed by Nataraan et al. 38). Their model is built on characterizing the choice probabilities for the extremal distribution in the set of all oint distributions with the given marginal distributions of the error terms that maximizes expected consumer utility see Nataraan et al. 38) and Theorem 1 in Mishra et al. 37)). Specifically, we solve the following convex 1 To see it, we can let F t) = 1 C 1 t), where C ) denotes the derivative of function C ) and C 1 ) denotes the inverse of function C ). The convexity of C ) ensures the monotonicity of its first derivative, i.e. C 1 x i) is nondecreasing in x i. Additionally as C ) is a function mapping from [0, 1] to R, the inverse of the derivative function maps to [0, 1], indicating F ) is a valid distribution function. 12

13 optimization problem to compute the choice probabilities: max x N v i αp )x i + =0 1 F 1 1 x i t)dt ), 12) Nataraan et al. 38) have shown that for a fixed pricing solution p, MDM can recover the whole generalized extreme value GEV) family by properly defining the marginal distribution functions. Mishra et al. 37) showed that all the choice probabilities in the relative interior of a simplex can be recreated by MDM under appropriate assumptions on the marginal distributions. We note that there is an alternate interpretation of the choice probabilities in the MDM model from a robust optimization perspective. To this end, we show that a modeler who assumes that the utilities of the product lies in an uncertainty set and uses the worst-case realizations to estimate the choice probabilities can recreate the MDM choice probabilities under an appropriately choice of the uncertainty set. We consider a convex uncertainty set that is defined from the marginal distributions of the error terms as follows: Uδ) = ɛ E F [ ɛ i ɛ i ] + δ. 13) =0 In this set, if δ 2 δ 1, then clearly Uδ 1 ) Uδ 2 ). Consider a robust optimization problem of the following form: max x N min ɛ U,δ 0 =0 v i + ɛ i + δ)x i, 14) where the modeler estimates the choice probabilities assuming that the error terms for each product and the budget is chosen by nature so as to put larger weights on the products chosen with lower probability. For larger values of δ, the worst-case utility will be smaller while for smaller values of δ, the worst-case utility will be larger. We incorporate the budget decision variable δ in to the obective function so as to capture this tradeoff and to ensure that the solution will not be too conservative. As we shall see next in the next theorem, the proposed robust choice model 14) is exactly equivalent to the convex optimization formulation of MDM. Theorem 1. The optimal x variables in the solution to the robust optimization problem 14) for the uncertainty set U in 13) is exactly the choice probabilities in the convex optimization formulation of MDM in 12). The proof of Theorem 1 is a direct application of convex optimality conditions and is provided in Appendix A. The result is closely related to the recent work of Fudenberg, Iiima and Strzalecki 20) who develop an additive perturbed nonlinear utility model which is related to 13

14 MDM. Furthermore, they show that the choice corresponds to an ambiguity averse preference of a modeler who is uncertain about the true utility and uses a regularization term in making the choices. In Theorem 1, we explicitly construct an uncertainty set and show that choice probabilities in MDM can be viewed as the solution to a robust optimization problem, thus providing an alternative perspective on discrete choice models, beyond random utility theory. 3.1 Pricing Model We now consider the pricing model under MDM. Clearly, 12) is a convex optimization problem. We assume the error term is independent of the price of the products, i.e. F ), = 0, 1,..., N is not a function of p. The optimality condition of 12) then yields p = v i + F 1 1 x i ) F x i0 ), = 1, 2,..., N. α This provides a wide class of closed-form relationship between prices and choice probabilities. With a slight abuse of notation, we use F ) to denote the marginal distribution function of the generalized random error term v i + ɛ i, and assume that the deterministic valuation v i = 0 in our model. In this case the relationship between price and market share can be written as p = F 1 1 x ) F x 0), = 1, 2,..., N. α and the choice probability vector is the solution to the convex optimization problem: x = arg max x N αp x + =0 1 F 1 1 x t)dt ), 15) We plug the closed-form solution between prices and market shares to the seller s pricing model in 5) to obtain an equivalent reformulation of the multi-product pricing problem with the market share decision variables x as follows: max x s.t. w x + 1 α =1 x = 1, =0 =1 x 0, = 0, 1,..., N. x F 1 1 x ) 1 α 1 x 0)F0 1 1 x 0 ) 16) This brings us to the following result in Theorem 2, which provide conditions that guarantee that the pricing problem 16) is tractable see Appendix A). 14

15 Theorem 2. Assume that the following two conditions hold: C1. xf 1 1 x) is a concave function in x for each = 1,..., N. C2. xf 1 0 x) is a convex function in x. Then, the pricing problem 16) is a convex optimization problem in the market share x variables and the optimal prices are computable in polynomial time. Moreover, if the optimal solution is x, then the optimal pricing strategy is p = F 1 1 x ) F x 0), = 1, 2,..., N. 17) α In fact, Condition 1 and 2 are known to be fairly mild conditions as we see in Proposition 1 and Corollary 1. Proposition 1. Let F x) for = 0,..., N be the marginal distributions. Then, i) Function xf 1 1 x) for = 1,..., N is concave if and only if function 1 1 F x) = 1,..., N is convex. ii) Function xf 1 0 x) is convex if and only if function 1 F 0x) is convex. Let F x) = 1 F x) for = 1,..., N. Then, we have the following result. Corollary 1. Conditions C1 and C2 hold if the marginal distributions satisfy the following conditions: i) The tail distribution F ) for = 1,..., N is log-concave; ii) The distribution F 0 ) is log-concave. Both the conditions i) and ii) are satisfied when the probability density function f ) for = 0,..., N is log-concave. Since log-concavity is satisfied by many common probability distributions see Bagnoli and Bergstrom 6)), such as the normal, logistic, exponential, extreme value and Laplace distributions, this result identifies a large class of problems for which the pricing problem is now tractable. We show in Appendix B that by defining appropriate marginal functions, the method is able to recover the pricing result for MNL. By considering the optimality condition of the pricing model 16), we can further characterize the optimal markup of each product. The optimality condition implies for w + 1 α F 1 1 α F 1 1 x ) + 1 α x 1 F 1 F 0 1 x 0) 1 α 1 x 0) 1 F 0 1 F 1 x )) + λ = x 0 )) + λ = 0 18) Combining this with 17), we get the following characterization of the markup. 15

16 Proposition 2. The optimal markup of each product is characterized by the following: p w = 1 α x x 0 F 1 F 1 x )) α F 0 1 F0 1 x 0 where x is the vector of choice probabilities from MDM. )), 19) This brings us to the following corollary which recovers the equal markup property of the MNL Model. Corollary 2. If the utility of each product shares the same hazard rate 1 FU) F U), then the optimal markup of each product p w is the same. For exponential marginal distributions, the hazard rates are the same implying that the optimal markup is the same. markup will not be the same under this choice model. For more general marginal distributions, however the optimal 4 Data-Driven Price Optimization with MDM A key part in optimizing the prices from data is to calibrate the consumer s choice behavior. To mitigate the model misspecification issue, ideally we would try to estimate the convex regularization term, which would lead to the underlying representative consumer model. However, the regularization function is multi-dimensional, making it hard to estimate and furthermore there is no guarantee that the corresponding pricing problem is tractable. To achieve a balance between the estimation accuracy and price optimization tractability, we turn our estimation of the general convex function to that of a separable convex function. As shown in the previous section, this approximation can represent a family of common choice models used in practice and at the same time ensures the tractability of the price optimization problem. We denote this method as Marginal Estimation + Price Optimization. In this section, we will show how to estimate the marginal distribution function from aggregate sales data and then solve the pricing problem by solving a linear program LP) and/or mixed integer linear program MILP). We assume that the data is given as follows: There are M periods of sales data, which provides the prices and market shares of N products and the market share of the outside option. We denote the data as {p t, x t } t=1,...,m,=0...,n where without loss of generality we assume that p t0 = 0 for all t. 16

17 4.1 Estimating Marginals The main idea of the estimation technique is to use the closed-form solution of the price in 17) in terms of the market shares to fit the sales data {p t, x t } t=1,...,m,=0...,n. The goal is to estimate a set of valid marginal distribution functions such that the deviation of the fitted prices from the observed prices is minimized as follows: α>0,f 1 min ):increasing M =1 t=1 ) F x t ) F x t0 ) p t, 20) α α This estimation model is inspired by one of the most commonly used nonparametric estimation method in shape constrained estimation. This method is favored due to several reasons. Firstlu, it is free of tuning parameters and requires fewer assumptions, which provides it with model identifying power see Matzkin 33)). Secondly, incorporating the shape restrictions provides desirable sample properties of the estimator see Beresteanu 7)) and achieves a high prediction accuracy under small sample sizes. Interested readers are referred to Chen 12) and Xu 41) for a detailed introduction of shape constrained estimation. In this paper, we restrict ourselves to one special shape constrained estimation model additive isotonic regression, which was first proposed by Bacchetti 5). The estimation model considered in their work is as follows: min c,f l :monotonic M Yi c f 1 Xi 1 ) f L Xi L ) ) 2 i=1 21) where X 1 i, X2 i,..., XL i, Y i) is ith observed data for i = 1,..., M. Mammen and Yu 31) analyzed rates of convergence of the estimator of this model and demonstrated finite sample properties of the estimator through simulation experiments. It is clear that 20) is a straightforward application of 21), with two additive functions F 1 1 x i ) and F0 1 1 x i0 ) for each = 1,..., N. To solve the optimization problem, we modify 20) by defining y x) := xf 1 1 x)/α for = 1,..., N, y 0 x) := 1 x)f x)/α. Denote the estimated function value of y ) at point x t as y t for t = 1,..., M, = 0,..., N. Then F 1 1 x t )/α = y t /x t and F x t0 )/α = y t0 /1 x t0 ). We add in the monotonicity shape constraints according to the natural monotonicity structure of F ). Then the estimation model can be written as follows: min y M M =1 t=1 yt x t y ) 2 t0 p t, 22) 1 x t0 17

18 where M is defined by a set of shape monotonicity) constraints: M = y y t x t y t x t for all t, t ) such that x t x t for = 1,..., N y t0 1 x t0 y t 0 1 x t 0 for all t, t ) such that x t0 x t 0. In addition, to calibrate the marginal distribution functions with limited experiment data, we can add additional structure on the functional form. It has been shown in Corollary 1 that log-concavity of the density function implies conditions C1 and C2 which make the pricing problem computationally tractable and is satisfied by many common probability distributions. Hence we can also incorporate the convexity conditions from this in our estimation model. In summary, we add the following four constraints in our estimation model. i) xf 1 1 x) for = 1,..., N is concave in the x variable, ii) 1 x)f x) is convex in the x variable, iii) F 1 1 x) for = 1,..., N is a monotone decreasing function in the x variable, iv) F x) is a monotone decreasing function in the x variable. We enforce these monotonicity and convexity conditions in our model as constraints. To do this, we sort for each product, the market shares in the ascending order. We denote the corresponding indices of the sorted data set for a given as s = s 1,..., s M ). Then the constraints i)-iv) can be added to the estimation model 22) as linear constraints. Formally, the shape constraints we consider in our estimation problem are defined as S = y 1 η t )y s t+1 + η ty s t 1 y s t, t = 2,..., M 1, = 1,..., N 1 η t0 )y s 0 t η t0 y s 0 t 1 0 y s 0 t 0, t = 2,..., M 1 y s t x s t y s 0 t 0 1 x s 0 t 0 y s t 1 x s t 1, t = 2,..., M, = 1,..., N y s 0 t 1 0, t = 2,..., M. 1 x s 0 t ) where η t = x s t+1 x s )/x t s t+1 x s t 1). Then the estimation problem can be modeled as a second order conic program SOCP), which can be efficiently solved as follows: min y,φ s.t. M t=1 =1 yt x t y S φ t y ) 2 t0 p t φ t, t = 1,..., M, = 1,..., N, 1 x t0 24) 18

19 It is also possible to use other estimators by minimizing the L 1 norm instead of L 2 norm, which turns the estimation problem to a linear program as follows: min y,φ s.t. M t=1 =1 y t φ t y t0 p t φ t, t = 1,..., M, = 1,..., N, x t 1 x t0 yt y ) t0 p t φ t, t = 1,..., M, = 1,..., N, x t 1 x t0 y S 25) The novelty of the estimation method under this choice model is that we estimate the marginal distribution functions rather than oint distributions by taking advantage of closed-form relationship between prices and market shares. The fitting of the marginal distribution functions from the aggregate sales data leads us to an appropriate choice model for the pricing problem, which partially overcomes the model misspecification issue that might arise by assuming a fixed structural form of the choice model as we show in the numerical experiments. 4.2 Optimizing Prices With a set of fitted values y t, we now consider the problem of optimizing prices. Using the definition of y ) from the previous section, the obective function in the pricing optimization model 16) is essentially given by the summation of all the y ) functions. Therefore, it is natural to consider a piecewise linear approximation of the obective function. Define piecewise linear function value P F x; s; f) as follows: P F x; s; f) = M y = λ t f t t=1 λ, z) : λ 1 z 1, λ t z t 1 + z t, t = 2,..., M 1 λ M z M 1, z t {0, 1}, t = 1,..., M 1, M λ t = 1, λ t 0, t = 1,..., M, t=1 M 1 M z t = 1, x = λ t s t, t=1 t=1, where the first argument x denotes the independent variable at which we need to compute the function value. The second argument s is the ordered input market share data and the third argument f indicates the corresponding function values. The intuition behind the constraints is as follows: For a point x, y) on the piecewise linear curve, the y value is the same convex combination of the f t values as the x value is a convex combination of the s t values. Specifically, if x = λ t s t + λ t+1 s t+1, with λ t + λ t+1 = 1, λ t, λ t+1 0, then y = λ t f t + λ t+1 f t+1. The binary 19

20 variable z is introduced to indicate the interval in which the point is located in. Thus, P F x; s; f) is a singleton, whose value denotes the piecewise approximation of the y value at x. Define / as the element wise division between two vectors. Then the price optimization problem can be formulated in the following manner: Π := max x,δ,fi w x + δ δ 0 =1 =1 s.t. δ P F x ; x s ; y s ), = 1,..., N, δ 0 P F x 0 ; x s 0; y s 0) F I P F x ; x s ; y s /x s ), = 1,..., N, F I 0 P F x 0 ; x s 0; y s 0 /1 x s 0)) F I F I 0 Ω, = 1,..., N, x = 1 =0 x x s M, = 0,..., N, x x s 1, = 0,..., N, x 0, 26) where the first constraint provides the piecewise linear approximation of the function x F 1 1 x )/α at x, the second constraint provides the approximation of 1 x 0 )F 1 1 x 0 )/α at x 0. Similarly, function inverse value F I, = 0,..., N provides the approximation of F 1 1 x )/α at x. Hence the optimal price p can be represented as F I F I 0. We encapsulate all the price constraints in Ω, = 1,..., N. For example, this may include bound constraints on the price p, e.g. u p ū. Finally, we limit x to lie within the range of the data since we have no additional information beyond the range unless we make some additional assumptions. When the set Ω is described through linear and integer constraints, this problem is solvable as a mixed integer linear program. Figure 2 provides a small example to illustrate the intuition of the piecewise linear approximation method with N = 2, M = 10). The estimation model provides a pointwise estimation of x F 1 1 x ) α, = 1,..., N and 1 x 0)F x 0 ) α at each sample point. For any market share x, we can use a convex combination of the function values x F 1 1 x ) α resp. 1 x0)f x 0 ) ) α 20

21 Figure 2: Illustration of piecewise linear approximation of profit function at two adacent sample points containing x to approximate x F 1 1 x ) α resp. 1 x 0)F x 0) ), α hence the profit value at x can be represented in a linear form N δ δ 0. The binary variable z is introduced to indicate which interval x 1 F 1 x locates in. Similarly, ) α is approximated by the same convex combination of the value F 1 1 x ) α at the two two adacent sample points containing x, which is denoted as F I in 26). Then F I F I 0 provides the approximated price p, = 1,..., N at the given market share x. Finally, it is worth observing that if Ω only includes nonnegativity constraints, the optimization model can be further simplified as a linear program by taking advantage of the structure of the marginal distribution functions. Notice that for a concave function, a linear approximation of the function value at point x can be calculated as min t y s t +y s t+1, y s t )/x s t+1, x s t )x x s t ) and for a convex function, the corresponding approximated value at point x 0 is obtained from max t y s 0 t 0+y s 0 t +1,0 y s 0 t 0)x s 0 t+1,0 x s 0 t 0)x 0 x s 0 t 0). Therefore, the optimization problem =1 21

22 can be modeled as the following linear programming problem: Π := max x,δ s.t. w x + δ δ 0 =1 =1 y δ y s t + s t+1, y s t x s t+1, x x x s s t t ), t = 1,..., M 1, = 1,..., N, δ 0 y s 0 t 0 + y s 0 t +1,0 y s 0 t 0 x 0 x x s 0 t+1,0 x s 0 s 0 t 0), t = 1,..., M 1, t 0 x = 1 =0 x x s M, = 0,..., N, x x s 1, = 0,..., N, x 0. 27) 4.3 Uniqueness of the Optimal Prices and Profit Functions Note that in both 24) and 25), there are MN + 1) variables y t to be estimated including the outside option, but there are only M N constraints from the optimality conditions. There could hence be multiple solutions that can satisfy these set of conditions, leading to issues in model identification. This is because all the marginal distributions F ) s can be scaled and shifted by a constant, without affecting the market share attained by each product. We show next that all the optimal solutions to this problem will generate the same optimal market shares and optimal prices when the samples are suitably generated and the sample size goes to infinity. This provides a theoretical ustification on the uniqueness of the optimal prices and profit under this approach. To see this, we denote the optimal solution in the estimation model which leads to the true underlying marginal distribution function as yt, = 1,..., N; t = 1,..., M. Notice that every y satisfying p t = yt x t yt0 1 x t0 is a feasible solution to the estimation model. We can without loss of generality write the solutions as y t = y t + δ t for = 0, 1,..., N and t = 1,..., M, where δt x t = δt0 1 x t0, for any = 1,..., N and t = 1,..., M. Consider an arbitrary market share x and assume x is located between x s t, and x s t +1,, for = 0, 1,..., N, and 22

23 x = λ x s t, + 1 λ )x s t +1, Then the price of product is given as: p = λ = λ y s t, x s t, y s t, y + 1 λ ) s t +1, x s + 1 λ ) y t +1, s t +1, x x s t, s t +1, λ δ s x t, s t +1,+1 λ )δ s t +1,x s, t x s t, x s t +1, λ y s 0 t0,0 0 1 x s 0 t0,0 λ y s 0 t, x s 0 t0,0 ) + 1 λ 0) y s 0 t 0 +1,0 1 x s 0 t0 +1,0 y s 0 t 0 +1,0 ) + 1 λ 0) 1 x + s 0 t0 +1,0 λ ) 0 δ s 0 t,0 1 x s 0 0 t 0 +1,0)+1 λ 0 )δ s 0 t 0 +1,01 x s 0,0) t 0 1 x s 0 t0,0 )1 x s 0 t 0 +1,0) 28) The last two terms indicate the price deviation from the true price under x. Let δt x t = δt0 1 x t0 := t, for any = 1,..., N. The price deviation can be rewritten as δ s λ t,0 x s t, 1 x x s s t,0 t +1,+1 λ ) x s t +1, x x s t, s t +1, λ ) 0 δ s 0 t,0 1 x s 0 0 t 0 +1,0)+1 λ 0 )δ s 0 t 0 +1,01 x s 0 t,0 ) 0 1 x s 0 t0,0 )1 x s 0 t 0 +1,0) = λ δs t,0 1 x s t,0 ) + 1 λ ) δs t +1,0 1 x s = λ s t + 1 λ ) s t +1 t +1,0 ) δ s t +1,0 1 x s t +1,0 x s t, λ 0 δ s 0 t,0 1 x s 0 0 t 0 +1,0)+1 λ 0 )δ s 0 t 0 +1,01 x s 0 t,0 ) 0 ) λ 0 s 0 t0 + 1 λ 0) s 0 t x s 0 t0,0 )1 x s 0 t 0 +1,0) ) We now claim that under the following assumption, the price deviation in 29) goes to 0. 29) Assumption *) 1. The pricing experiments are generated such that for each market share outcome x, there is another experiment with market share x such that max =0,1,...,N x x ɛ. 2. The inverse of marginal distribution function has a bounded first derivative within the range of data: max x [min i max x i,max i 1 F ) 1 x) D for some D > 0. x i] The assumptions hold when the data sample is carefully generated and the sample size is sufficiently large with strictly positive market shares for each experiment. We first show in Lemma 1 that under these assumptions, the deviations of y in different experiments are bounded. Lemma 1. Under Assumption *), i t Dɛ for any i t. Then we are ready to claim 23