Online Pricing with Offline Data: Phase Transition and Inverse Square Law

This paper investigates the impact of pre-existing offline data on online learning, in the context of dynamic pricing. We study a single-product dynamic pricing problem over a selling horizon of $T$ periods. The demand in each period is determined by the price of the product according to a linear demand model with unknown parameters. We assume that before the start of the selling horizon, the seller already has some pre-existing offline data. The offline data set contains $n$ samples, each of which is an input-output pair consisting of a historical price and an associated demand observation. The seller wants to utilize both the pre-existing offline data and the sequential online data to minimize the regret of the online learning process. We characterize the joint effect of the size, location and dispersion of the offline data on the optimal regret of the online learning process. Specifically, the size, location and dispersion of the offline data are measured by the number of historical samples $n$, the distance between the average historical price and the optimal price $\delta$, and the standard deviation of the historical prices $\sigma$, respectively. We show that the optimal regret is $\widetilde \Theta\left(\sqrt{T}\wedge \frac{T}{(n\wedge T)\delta^2+n\sigma^2}\right)$, and design a learning algorithm based on the "optimism in the face of uncertainty" principle, whose regret is optimal up to a logarithmic factor. Our results reveal surprising transformations of the optimal regret rate with respect to the size of the offline data, which we refer to as phase transitions. In addition, our results demonstrate that the location and dispersion of the offline data also have an intrinsic effect on the optimal regret, and we quantify this effect via the inverse-square law.Comment: Forthcoming in Management Scienc

Online pricing for multi-type of Items

LNCS v. 7285 entitled: Frontiers in algorithmics and algorithmic aspects in information and management: joint international conference, FAW-AAIM 2012 ... proceedingsIn this paper, we study the problem of online pricing for bundles of items. Given a seller with k types of items, m of each, a sequence of users {u 1, u 2, ...} arrives one by one. Each user is single-minded, i.e., each user is interested only in a particular bundle of items. The seller must set the price and assign some amount of bundles to each user upon his/her arrival. Bundles can be sold fractionally. Each u i has his/her value function v i (·) such that v i (x) is the highest unit price u i is willing to pay for x bundles. The objective is to maximize the revenue of the seller by setting the price and amount of bundles for each user. In this paper, we first show that the lower bound of the competitive ratio for this problem is Ω(logh + logk), where h is the highest unit price to be paid among all users. We then give a deterministic online algorithm, Pricing, whose competitive ratio is O (√k·log h log k). When k = 1 the lower and upper bounds asymptotically match the optimal result O(logh). © 2012 Springer-Verlag.postprin

Online Pricing Incentive to Sample Fresh Information

Today mobile users such as drivers are invited by content providers (e.g., Tripadvisor) to sample fresh information of diverse paths to control the age of information (AoI). However, selfish drivers prefer to travel through the shortest path instead of the others with extra costs in time and gas. To motivate drivers to route and sample diverse paths, this paper is the first to propose online pricing for a provider to economically reward drivers for diverse routing and control the actual AoI dynamics over time and spatial path domains. This online pricing optimization problem should be solved without knowing drivers' costs and even arrivals, and is intractable due to the curse of dimensionality in both time and space. If there is only one non-shortest path, we leverage the Markov decision process (MDP) techniques to analyze the problem. Accordingly, we design a linear-time algorithm for returning optimal online pricing, where a higher pricing reward is needed for a larger AoI. If there are a number of non-shortest paths, we prove that pricing one path at a time is optimal, yet it is not optimal to choose the path with the largest current AoI. Then we propose a new backward-clustered computation method and develop an approximation algorithm to alternate different paths to price over time. Perhaps surprisingly, our analysis of approximation ratio suggests that our algorithm's performance approaches closer to optimum given more paths.Comment: 14 pages, 13 figure

Did the Euro Foster Online Price Competition? Evidence from an International Price Comparison Site

We study the impact of the Euro on prices charged by online retailers within the EU. Our data spans the period before and after the Euro was introduced, covers a variety of products, and includes countries inside and outside of the Eurozone. After controlling for cost, demand, and market structure effects, we show that the pure Euro changeover effect is to raise average prices in the Eurozone by 3% and average minimum prices by 7%. Finally, we develop a model of online pricing in the context of currency unions, and show that these price patterns are broadly consistent with clearinghouse models.Price competition, internet

A Review of The Effect of Pricing Strategies on The Purchase of Consumer Goods

This study examined the effect of pricing strategies on the purchase of consumer goods. Also examined in this research is the effect of internet (online presence) on informed purchase decision. The research intended to answer questions on the extent to which competitor's price affects purchase of products, how customers perceive the value-based pricing concept of firms and the extent to which online pricing inform customer purchase decision. This paper being descriptive and historical relied heavily on secondary sources of information. The research utilized a descriptive and historical method and relied heavily and solely on secondary instruments as sources of data. Findings from the data obtained indicate that consumers have a perception of value reflected in prices of firms’ products. It also shows that competitors price affect the purchase of firm products and that online pricing informs and affects purchase decision. This study contributes to knowledge in series of issues associated with pricing strategies and purchase decision process. This research recommends that as much as firms should focus on communicating value to customers through prices, firms should also be on the watch for competitor’s prices and examine how much it affects purchase of their products

The importance of attractive prices in pricing dynamics

Nominal rigidities have an important role in macro models used for the analysis of monetary policy. Re-cently, attractive prices (also known as price points) have often been referred to as one important potential explanation of nominal rigidities. An increased interest on attractive prices as an explanation for price ri-gidities rests on online pricing, in the context of which it has been shown that prices are rigid also on the internet, where physical costs are not important. Our empirical analyses using micro data on consumer prices in Finland indicate that a specific form of attractive prices – 9-ending prices – have a considerable effect on pricing dynamics. The results of the study show that changes to prices with 9 endings are more often decreases than are changes to prices with other endings. Price changes to 9-ending prices are also of smaller size than are changes to other endings.rigidity; price endings; attractive prices; 9-prices

A Study of Pricing Evolution in the Online Toy Market

We examine the pricing trends in the online toy markets based on a unique set of panel data collected across three years’ span. The analysis was made through panel data regression models with error components and serial correlation, allowing comparisons of prices and price dispersions between the two types of online retailers as well as examinations of dynamics of prices and price dispersions. Our results indicate that both online branch of multichannel retailers (OBMCRS) and dotcoms charge similar prices on average, and over time their prices move in tandem. Although the OBMCR retailers charge significantly different prices, the dotcoms do charge similar prices. Moreover, both retailer types demonstrate different magnitudes of price dispersion that move at different rates over time. Although the price dispersion of OBMCRS is higher than that of the dotcoms at the beginning, the gap narrows over time.e-commerce, online pricing strategies, online toy market, price dispersion, pricing trends