Rainer Schlosser

Optimal advertising and pricing in a class of general new-product adoption models

In [21], Sethi et al. introduced a particular new-product adoption model. They determine optimal advertising and pricing policies of an associated deterministic infinite horizon discounted control problem. Their analysis is based on the fact that the corresponding Hamilton–Jacobi–Bellman (HJB) equation is an ordinary non-linear differential equation which has an analytical solution. In this paper, generalizations of their model are considered. We take arbitrary adoption and saturation effects into account, and solve finite and infinite horizon discounted variations of associated control problems. If the horizon is finite, the HJB-equation is a 1st order non-linear partial differential equation with specific boundary conditions. For a fairly general class of models we show that these partial differential equations have analytical solutions. Explicit formulas of the value function and the optimal policies are derived. The controlled Bass model with isoelastic demand is a special example of the class of controlled adoption models to be examined and will be analyzed in some detail.

Oligopoly Pricing and Advertising in Isoelastic Adoption Models

This paper deals with deterministic dynamic pricing and advertising differential games which are stylized models of special durable-good oligopoly markets. We analyze infinite horizon models with constant price and advertising elasticities of demand in the cases of symmetric and asymmetric firms. In particular, we consider general saturation/adoption effects. These effects are modeled as transformations of the sum of the cumulative sales of all competing firms. We specify a necessary and sufficient condition such that a unique Markovian Nash equilibrium for such games exist. For two classes of models we derive solution formulas of the optimal policies and of the value functions, and we show how to compute the evolution of the cumulative sales of each firm. The analysis of these games reveals that the existence of the Nash equilibrium relies on the possibility to separate a component, which is specific for each firm, from a [market] component, which is the same for all firms. The common factor is a function of the decreasing untapped market size. The individual factor of each firm reflects its individual market power and has an impact on equilibrium prices; each such coefficient depends on the price elasticities, unit costs, arrival rates, and discount factors of all competing companies. Formulas for these coefficients reveal how equilibrium prices depend on the number of competing firms, and how the entry or exit of a firm affects the price structure of the oligopoly.

Dynamic Pricing and Advertising of Perishable Products with Inventory Holding Costs

We examine a special class of dynamic pricing and advertising models for the sale of perishable goods, including marginal unit costs and inventory holding costs. The time horizon is assumed to be finite and we allow several model parameters to be dependent on time. For the stochastic version of the model, we derive closed-form expressions of the value function as well as of the optimal pricing and advertising policy in feedback form. Moreover, we show that for small unit shares, the model converges to a deterministic version of the problem, whose explicit solution is characterized by an overage and an underage case. We quantify the close relationship between the open-loop solution of the deterministic model and the expected evolution of optimally controlled stochastic sales processes. For both models, we derive sensitivity results. We find that in the case of positive holding costs, on average, optimal prices increase in time and advertising rates decrease. Furthermore, we analytically verify the excellent quality of optimal feedback policies of deterministic models applied in stochastic models.

An Interactive Platform to Simulate Dynamic Pricing Competition on Online Marketplaces

E-commerce marketplaces are highly dynamic with constant competition. While this competition is challenging for many merchants, it also provides plenty of opportunities, e.g., by allowing them to automatically adjust prices in order to react to changing market situations. For practitioners however, testing automated pricing strategies is time-consuming and potentially hazardously when done in production. Researchers, on the other side, struggle to study how pricing strategies interact under heavy competition. As a consequence, we built an open continuous time framework to simulate dynamic pricing competition called Price Wars. The microservice-based architecture provides a scalable platform for large competitions with dozens of merchants and a large random stream of consumers. Our platform stores each event in a distributed log. This allows to provide different performance measures enabling users to compare profit and revenue of various repricing strategies in real-time. For researchers, price trajectories are shown which ease evaluating mutual price reactions of competing strategies. Furthermore, merchants can access historical marketplace data and apply machine learning. By providing a set of customizable, artificial merchants, users can easily simulate both simple rule-based strategies as well as sophisticated data-driven strategies using demand learning to optimize their pricing strategies.

Stochastic dynamic pricing and advertising in isoelastic oligopoly models

In this paper, we analyze stochastic dynamic pricing and advertising differential games in special oligopoly markets with constant price and advertising elasticity. We consider the sale of perishable as well as durable goods and include adoption effects in the demand. Based on a unique stochastic feedback Nash equilibrium, we derive closed-form solution formulas of the value functions and the optimal feedback policies of all competing firms. Efficient simulation techniques are used to evaluate optimally controlled sales processes over time. This way, the evolution of optimal controls as well as the firms’ profit distributions are analyzed. Moreover, we are able to compare feedback solutions of the stochastic model with its deterministic counterpart. We show that the market power of the competing firms is exactly the same as in the deterministic version of the model. Further, we discover two fundamental effects that determine the relation between both models. First, the volatility in demand results in a decline of expected profits compared to the deterministic model. Second, we find that saturation effects in demand have an opposite character. We show that the second effect can be strong enough to either exactly balance or even overcompensate the first one. As a result we are able to identify cases in which feedback solutions of the deterministic model provide useful approximations of solutions of the stochastic model.

A Stochastic Dynamic Pricing and Advertising Model Under Risk Aversion

This article analyzes a dynamic pricing and advertising model for the sale of perishable products under constant absolute risk aversion. We consider a time-dependent version of Gallego and van Ryzin’s dynamic pricing model with exponential demand and include isoelastic advertising effects as well as marginal unit costs. We derive closed-form expressions of the optimal risk-averse pricing and advertising policies of the value function and of the certainty equivalent. The formulas provide insight into the (complex) interplay between risk-sensitive pricing and advertising decisions. Moreover, to evaluate the optimally controlled sales process over time we propose efficient simulation techniques. These are used to analyze the characteristics of different degrees of risk aversion, particularly the concentration of the profit distribution and the impact on the expected evolution of price and advertising rates.

Stochastic Dynamic Multi-Product Pricing with Dynamic Advertising and Adoption Effects

We analyze stochastic dynamic multi-product pricing models for durable goods and consider a single advertising channel to promote all types of products. We include general adoption effects, unit costs as well as inventory holding costs. In case of isoelastic, exponential and linear demand, we derive solution formulas for the expected profit, the optimal feedback prices for all types of products, and the optimal advertising rate. In order to evaluate optimally controlled sales processes over time, we use efficient simulation techniques. Moreover, for the case of exponential demand, we demonstrate how to include risk aversion in the model.

Data-Driven Inventory Management and Dynamic Pricing Competition on Online Marketplaces

Online markets are characterized by competition and limited demand information. In E-commerce, firms compete against each other using data-driven dynamic pricing and ordering strategies. Successfully managing both inventory levels as well as offer prices is a highly challenging task as (i) demand is uncertain, (ii) competitors strategically interact, and (iii) optimized pricing and ordering decisions are mutually dependent. Currently, retailers lack the possibility to test and evaluate their algorithms appropriately before releasing them into the real world. To study joint dynamic ordering and pricing competition on online marketplaces, we built an interactive simulation platform. To be both flexible and scalable, the platform has a microservice-based architecture and allows handling dozens of competing merchants and streams of consumers with configurable characteristics. Further, we deployed and compared different pricing and ordering strategies, from simple rule-based ones to sophisticated datadriven strategies which are based on state-of-the-art demand learning techniques and efficient dynamic optimization models. 1 Joint Ordering and Pricing Competition E-commerce has become highly dynamic and competitive. Merchants are able to automatically adjust prices to react to changing market situations within milliseconds, [Greenwald and Kephart, 1999]. Similarly, they can flexibly reorder items taking into account (i) estimated demand, (ii) inventory holding costs, and (iii) delivery times. To derive effective pricing and ordering decisions over time is challenging considering that numbers of combinations are enormous, demand is uncertain, and markets are steadily changing (cf. [Tsai and Hung, 2009], [Adida and Perakis, 2010], [Chen and Chen, 2015]). Moreover, pricing and ordering strategies mutually affect each other [Rajan et al., 1992]. Simulating the performance of automated ordering and pricing strategies is important as testing is potentially hazardous when done in production. To the best of our knowledge, however, flexible simulation platforms that allow evaluating specific strategies under competition do not exist. Existing platforms, e.g., [Kephart et al., 2000], [DiMicco et al., 2003], are limited in their capabilities: Simulations run on a single machine, offer a limited set of consumer behaviors, simulate solely short sales horizons, and pricing updates or orders are restricted to predefined discrete points in time. We built a continuous time framework (mimicking production marketplaces such as Amazon or eBay [Boissier et al., 2017; PriceWars, 2018]) to simulate dynamic pricing and ordering under competition. Our setup supports customers with heterogeneous buying behaviors. The competitors’ offers include multiple dimensions such as price and quality. The platform allows an arbitrary number of merchants to compete simultaneously. Each merchant can run his preferred repricing and ordering strategy to adjust prices on the marketplace and to order products, respectively. Due to the strategic interaction of competing merchants’ price reactions market situations steadily change. Simulating streams of customer decisions allows generating realized sales events and the firms’ sales revenues. The firms’ inventory levels, their holding costs as well as their ordering costs are determined by their ordering strategies. Finally, the user can easily study the complex interplay of ordering and repricing strategies and, most importantly, their performance (e.g., short and long-term profits). Our system supports self-adapting learning strategies. The platform logs each interaction such as orders, price updates, stock-outs, new offers, sales, etc. This historical data – which is defined as partially observable (sales are private knowledge) – is requested and numerically analyzed by data-driven merchants. Various state-of-the-art machine learning approaches can be applied to quantify how demand (i.e., sales probabilities) is affected by a merchant’s pricing decisions. Further, merchants are able to deploy optimization models [Transchel and Minner, 2009; Schlosser and Boissier, 2017; Yabe et al., 2017]. which are calibrated by estimated sales probabilities to compute optimized data-driven pricing and ordering strategies. In this context, it can be even tried to learn about competitors’ strategies in order to take anticipated price reactions into account. Finally, our framework also allows controlling and measuring the influence of (i) the customers’ buying behavior, (ii) price adjustment frequencies, as well as (iii) the exit or entry of competitors on a strategy’s performance. In addition, different demand learning techniques and optimization approaches can be compared regarding their accuracy and efficiency. Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence (IJCAI-18)

Dynamic Pricing under Competition on Online Marketplaces: A Data-Driven Approach

Most online markets are characterized by competitive settings and limited demand information. Due to the complexity of such markets, efficient pricing strategies are hard to derive. We analyze stochastic dynamic pricing models in competitive markets with multiple offer dimensions, such as price, quality, and rating. In a first step, we use a simulated test market to study how sales probabilities are affected by specific customer behaviors and the strategic interaction of price reaction strategies. Further, we show how different state-of-the-art learning techniques can be used to estimate sales probabilities from partially observable market data. In a second step, we use a dynamic programming model to compute an effective pricing strategy which circumvents the curse of dimensionality. We demonstrate that the strategy is applicable even if the number of competitors is large and their strategies are unknown. We show that our heuristic can be tuned to smoothly balance profitability and speed of sales. Further, our approach is currently applied by a large seller on Amazon for the sale of used books. Sales results show that our data-driven strategy outperforms the rule-based strategy of an experienced seller by a profit increase of more than 20%.

Dealing with the Dimensionality Curse in Dynamic Pricing Competition: Using Frequent Repricing to Compensate Imperfect Market Anticipations

Most sales applications are characterized by competition and limited demand information. For successful pricing strategies, frequent price adjustments as well as anticipation of market dynamics are crucial. Both effects are challenging as competitive markets are complex and computations of optimized pricing adjustments can be time-consuming. We analyze stochastic dynamic pricing models under oligopoly competition for the sale of perishable goods. To circumvent the curse of dimensionality, we propose a heuristic approach to efficiently compute price adjustments. To demonstrate our strategys applicability even if the number of competitors is large and their strategies are unknown, we consider different competitive settings in which competitors frequently and strategically adjust their prices. For all settings, we verify that our heuristic strategy yields promising results. We compare the performance of our heuristic against upper bounds, which are obtained by optimal strategies that take advantage of perfect price anticipations. We find that price adjustment frequencies can have a larger impact on expected profits than price anticipations. Finally, our approach has been applied on Amazon for the sale of used books. We have used a sellers historical market data to calibrate our model. Sales results show that our data-driven strategy outperforms the rule-based strategy of an experienced seller by a profit increase of more than 20%.