Sales Policies for a Virtual Assistant
SSales Policies for a Virtual Assistant
Wenjia Ba, Haim Mendelson and Mingxi Zhu ∗ Graduate School of Business, Stanford University, Stanford, CA 94305
August 2020
Abstract
We study the implications of selling through a voice-based virtual assistant. The seller has aset of products available and the virtual assistant dynamically decides which product to offerin each sequential interaction and at what price. The virtual assistant may maximize theseller’s profits; it may be altruistic, maximizing total surplus; or it may serve as a consumeragent maximizing the consumer surplus. The consumer is impatient and rational, seeking tomaximize her expected utility given the information available to her. The virtual assistantselects products based on the consumer’s request and other information available to it (e.g.,consumer profile information) and presents them sequentially. Once a product is presented andpriced, the consumer evaluates it and decides whether to make a purchase. The consumer’svaluation of each product comprises a pre-evaluation value, which is common knowledge to theconsumer and the virtual assistant, and a post-evaluation component which is private to theconsumer. We solve for the equilibria and develop efficient algorithms for implementing thesolution. In the special case where the private information is exponentially distributed, theprofit-maximizing total surplus is distributed equally between the consumer and the seller, andthe profit-maximizing ranking also maximizes the consumer surplus. We examine the effects ofinformation asymmetry on the outcomes and study how incentive misalignment depends on thedistribution of private valuations. We find that monotone rankings are optimal in the cases ofa highly patient or impatient consumer and provide a good approximation for other levels ofpatience. The relationship between products’ expected valuations and prices depends on theconsumer’s patience level and is monotone increasing (decreasing) when the consumer is highlyimpatient (patient). Also, the seller’s share of total surplus decreases in the amount of privateinformation. We compare the virtual assistant to a traditional web-based interface, wheremultiple products are presented simultaneously on each page. We find that within a page, thehigher-value products are priced lower than the lower-value product when the private valuationsare exponentially distributed. This is because increasing one product’s valuation increases thematching probability of the other products on the same page, which in turn increases their pricesof other products. Finally, the web-based interface generally achieves higher profits for the sellerthan a virtual assistant due to the greater commitment power inherent in its presentation.
Electronic communication and commerce are in the early stages of a transition from the weband app eras, which were based on a screen- and keyboard-based user interface, to the use ofnatural language, primarily in the form of voice interfaces that are natural, fast and convenient for ∗ To contact the authors, email wenjiaba, haim, [email protected]. a r X i v : . [ ec on . T H ] S e p onsumers. With speech recognition reaching the point where natural language can be interpretedcorrectly more than 95% of the time, most of today’s operating systems come with voice interactivityacross multiple devices ranging from phones and laptops to smart speakers and TVs. Amazon’s Echofamily, Google Home and Alibaba’s Tmall Genie are examples of virtual assistant (VA) devices,and Amazon’s Alexa, Google’s Assistant, Microsoft’s Cortana and Apple’s Siri are examples ofvoice-based user interfaces. The U.S. household penetration of VA devices was 32% in 2019 and itis expected to increase to 51% by 2022 (Tiwari et al. (2019)). Further, these devices are expected tobe integrated into cars, home electronics and consumer electronics devices that can be connected toplatforms such as Amazon’s Alexa. These new devices and interfaces offer a new way for consumersto access online information and increasingly, to engage in commerce. While the use of virtualassistants in electronic commerce is still in its infancy, it promises to become an important gatewayto the marketplaces of the future, with more than sixty percent of marketers being optimistic aboutvoice commerce compared to 11% who are pessimistic (Voicebot.ai (2019b)).JuniperResearch (2020) estimated that there were more than 4 billion digital voice assistantsworldwide in 2000, a number it expected to increase to 8.4 billion by 2024 due to the introductionof voice assistants in new devices including wearables, smart home, and TV devices. Voicebot.ai’sSmart Speaker Consumer Adoption reports found that 15% of U.S. smart speaker owners madepurchases by voice on a monthly basis in 2018, up from 13.6% in 2017 (Voicebot.ai (2019a), p. 17).Moreover, 20% of consumers already shopped using smart speakers, and 44% would make new oradditional purchases if the interface was supported by more brands in 2019 (Adobe (2019)).The voice interface offers speed and convenience but it also changes the nature of the interactionbetween the consumer and the marketplace. When using a virtual assistant, instead of searchingand browsing a list of products within a page, the consumer uses natural language to specify whatshe is looking for and the virtual assistant presents products one by one through voice, based on theconsumer’s requirements and other information. If the consumer rejects the first offer, the virtualassistant presents the next product, and so on. Thus, unlike traditional online channels where theconsumer is largely in control of the search process, the sequential nature of voice response givesthe virtual assistant a greater degree of control.In this paper we study the implications of these new interfaces for product sales. We considera seller with a set of products available for sale. The VA decides which product to offer in eachsequential interaction and at what price to maximize the seller’s expected profit, the total sur-plus or the consumer surplus. The consumer is rational, seeking to maximize her expected utilitygiven the information available to her. The consumer has a limited attention span. If the con-sumer’s search time exceeds her attention span, she leaves the market without making a purchase.When the consumer is active, the VA selects products and presents them to the consumer one byone. Once a product is presented to the consumer, she takes some time to evaluate it and decidewhether to make a purchase. The consumer’s valuation of each product comprises two parts: apre-evaluation component which is common knowledge to the consumer and the VA, and a post-evaluation component which is private to the consumer. Based on a comparison of value and price,the consumer decides whether to buy the current product or request another product, taking intoaccount upcoming product opportunities as well as her limited purchase window.We first solve for the equilibria obtained when the consumer maximizes her expected utility andthe VA maximizes the seller’s expected profit. We find efficient algorithms for solving the problemand compare them to the optimal results. We examine the effects of information asymmetrieson the outcomes—pricing, ranking and surplus allocation. We consider the effects of incentivemisalignment and find the equilibrium results for an altruistic VA (which maximizes the totalsurplus) and a consumer agent (which maximizes the consumer surplus). We study the effectsof the distribution of private valuations and then compare the virtual assistant to a traditional2eb-based interface, where multiple products are presented simultaneously on each page. We findthat the platform’s surplus share is higher for the web-based interface. We also obtain closed-formresults for the case where the private valuations are exponentially-distributed. Although the use of VAs in shopping is still an emerging phenomenon, the academic literature (e.g.,Kumar et al. (2016), Rzepka et al. (2020)) agrees with practitioners on its disruptive potential andexpected growth. This literature is largely empirical or descriptive, focusing on the adoption,benefits and implications of VAs. Kumar et al. (2016) provide a taxonomy of intelligent agenttechnologies as well as a framework for studying their adoption. Nasirian et al. (2017) study thedrivers of VA adoption by consumers, identifying interaction quality as a key driver of trust which(together with personal innovativeness) influences the intention to use a VA. Sun et al. (2019)provide a comprehensive analysis of data from Alibaba’s VA, TMall Genie, showing that the VAincreases shopper engagement and purchase levels. They distinguish between the effect of VAs onpurchase quantity, which is pronounced for high-income, younger, and more active consumers, andthe effect on spending amount, which is larger for low-income, younger, and less active consumers.Several papers discuss both the benefits and limitations of VAs. Rzepka et al. (2020) examinethe benefits and costs that users expect and obtain from voice commerce through semi-structuredinterviews with Amazon Alexa users. They identify convenience, efficiency and enjoyment as keyperceived benefits and limited transparency, lack of trust and technical immaturity as perceivedshortcomings. Jones (2018) uses a case study approach to illustrate these tradeoffs and theirimplications for consumers and marketers. She points out the challenges associated with consumers’loss of control and privacy. Kraus et al. (2019) study factors that influence consumer satisfaction invoice commerce vs. e-commerce. They find that convenience is a key driver of satisfaction for bothe-commerce and voice commerce, with a larger impact on the latter. They also bring a cognitiveinformation processing perspective (Davern et al. (2012)) to the analysis, with the lower richnessof voice commerce requiring a larger cognitive effort than for e-commerce. Mari (2019) and Mariet al. (2020) use interviews with brand owners and AI experts, coupled with an analysis of thefunctional characteristics of VAs, to study the implications of the technology for consumers andbrands. They identify the importance of ranking algorithms, finding that “the virtual assistantmay reduce consumers’ visibility of product alternatives and increase brand polarization. In thiscontext, product ranking algorithms on virtual assistants assume an even more critical role thanin other consumer applications” (p.8), as the sequential nature of choice limits visibility, productalternatives and consumer choice. Similar limitations are also discussed in Rzepka et al. (2020) andJones (2018). One of our objectives in this paper is to study theoretically the implications of theranking effect in light of the limitations of the technology.The limitations of buying through a VA are due to two factors. First, choices are made se-quentially, which significantly limits consumers’ ability to recall earlier options and make directcomparisons with them. Second, the voice interface is more limited than visual interfaces, whichincreases consumers’ cognitive effort and again limits choice. Both drivers have an adverse effecton consumers as they increase cognitive load. Basu and Savani (2019) and the papers they sur-vey compare the effects of simultaneous and sequential presentation on consumer behavior. Thisliterature shows that in general, sequential presentation results in inferior decisions compared tosimultaneous presentation. Viewing options sequentially makes it difficult to properly compare thecurrent choice to previous ones, whereas simultaneous presentation allows comprehensive compar-isons. Kraus et al. (2019) review the literature on the adverse effects of voice commerce due tothe cognitive limitations of the auditory interface. Munz and Morwitz (2019) show through six3xperiments that information presented by voice is more difficult to process than the same infor-mation presented visually. As a result of these cognitive difficulties, consumers tend not to recallthe earlier options they have reviewed. Indeed, research in psychology and consumer behavior findsin multiple settings that consumers sampling products sequentially are often likely to choose thelast product presented to them (e.g., de Bruin and Keren (2003), Bullard et al. (2017)). Biswaset al. (2014) argue that when consumers sample sensory-rich products, they may choose the firstor the last product sampled depending on the degree of similarity across sensory cues. We expectthat in the case of a VA, where the consumer decides when to stop searching, she is even morelikely to select the last product considered when making a purchase.The existing VA literature is largely empirical. As researchers (e.g., Mari (2019), Mari et al.(2020)) have noted, “providing structure and guidance to researchers and marketers in order tofurther explore this emerging stream of research (VA) is fundamental” (p.8). In this paper, we aimto narrow the gap between theory and practice by modeling the VA buying process, determininghow a VA should price and rank products to maximize alternative objective functions given rationalconsumer choice, studying the implications of the resulting equilibria, and comparing the outcomesobtained through a VA to those obtained from a web interface.Analytically, our model shares some aspects (in particular, consumer impatience) with Izhutovand Mendelson (2018), who consider a two-sided marketplace for services such as tutoring and derivepricing algorithms to maximize social surplus or seller profits. In Izhutov and Mendelson (2018),the seller can only control prices (there is no ranking problem) and consumer behavior is exogenous.In contrast, our model studies a VA selling physical products and controlling both the prices andthe order of presentation to the consumer. Our optimization problem is also analytically related tothe retail assortment planning problem in the Operations Management literature. This literaturemay be classified into three overlapping groups: (i) traditional assortment planning analysis withno ranking effect, which is more foundational in nature; (ii) ranking through an auction or similarmeans, where an intermediary platform designs a mechanism that ranks offers based on suppliers’bids or similar information signals; and (iii) more complex (mostly two-stage) models of assortmentranking.The traditional assortment planning literature (group (i) ) derives algorithms to compute theoptimal assortment for a retail operation. The problem may be addressed in either static or dynamicsetting. K¨ok et al. (2008) provide an extensive review of the static assortment planning literature.Mahajan and Van Ryzin (2001) and Honhon et al. (2010) study the optimal assortment whenconsumers can only choose among the products that are still in stock. Golrezaei et al. (2014) andBernstein et al. (2015) consider the problem of dynamically customizing assortment offerings basedon the preferences of each consumer and the remaining product inventories. Motivated by fastfashion operations, Caro et al. (2014) study how to release products from a fixed set into storesover multiple periods, taking into account the decay in product attractiveness once presented in thestore. Davis et al. (2015) study the assortment planning problem for a seller that sequentially addsproducts to its assortment over time, thereby monotonically increasing consumers’ considerationsets. In this setting, the profit margin is exogenously given and the consumer is myopic. Davis et al.(2015) derive an approximation algorithm that achieves at least (1 − /e ) of the optimal revenue.Saban and Weintraub (2019) consider the mechanism design problem for a procurement agencythat selects suppliers or product assortments made available to consumers. Suppliers may compete for the market (entry/assortment competition) and in the market (price competition within theassortment). They characterize the optimal buying mechanisms, showing how restricting the entryof close substitutes may increase price competition and consumer surplus.Ferreira and Goh (2019) introduce information effects to the assortment planning literature,considering the impact of concealing products which are in the full product catalog from consumers4n an attempt to induce them to buy additional products later. They compare the case where theseller presents the entire assortment for the entire selling season (similar to a traditional departmentstore or a web-based catalog) to the case where the seller intentionally introduces products one at atime (fast-fashion being a canonical example). The difference between the seller’s expected profitsin the latter vs. the former case is the value of concealment. When consumers are non-anticipating(i.e., make decisions a period at a time ignoring the future), the retailer successfully induces themto buy more products, which increases its profit. When consumers are strategic and anticipate theretailer’s future actions, the value of concealment is ambiguous. While most of the paper focuses onthe value of concealment for a given ranking and assortment, (Ferreira and Goh (2019), Corollary1 and 2) also show that under a special set of circumstances, the seller will induce the consumerto buy more by concealing the more valuable products so she will buy them later. This happenswhen the consumer is non-anticipating and either (i) the prices (assumed exogenous throughoutthe paper) are the same and the valuation distributions follow a stochastic dominance relationship,or (ii) the valuations have the same distribution which has an increasing failure rate. Ferreira andGoh (2019) thus provide a nice model of information disclosure as a tool sellers can manipulate totheir advantage.As discussed above, the product ranking decision is particularly important when selling througha VA (cf. Mari (2019), Mari et al. (2020)). This problem may be related to the vast literaturestudying how a search engine should rank paid ads. These models are typically three-sided, involv-ing consumers, advertisers and the search engine, and they focus on a tradeoff between relevanceand revenue (see, e.g, Agrawal et al. (2020) for a review). l’Ecuyer et al. (2017) bring this tradeoffto the assortment planning domain considering multiple sellers selling one item each through aplatform. l’Ecuyer et al. (2017) summarize the behavior of consumers using a click-through ratefunction that represents the purchase probability of each item as a function of its position andintrinsic characteristics. The platform maximizes expected revenue by calculating for each item ascore that balances its revenue and relevance and ranking them by decreasing scores. Chu et al.(2020) study a three-sided market involving a consumer and multiple suppliers with a platformin-between. The platform’s objective is to maximize a weighted average of the suppliers’ surplus,consumer surplus, and platform revenue. They study a position auction (in the spirit of search en-gine position auctions) to sell the first k available slots to n suppliers. They rank suppliers based ona surplus-ordered ranking (SOR) mechanism that considers suppliers’ contributions to the objectivefunction (realized for the first k and expected for the remaining suppliers), and show numericallythat this mechanism is near-optimal. From the consumers’ point of view, products are ex-antehomogeneous, which allows Chu et al. (2020) to show that, in a simpler setup with no auction, theplatform’s optimal ranking is based on realized SOR with full information and expected SOR withno information.Ranking decisions may be made by the consumer, by the seller, or by some combination of thetwo. The classic framework for ranking by the consumer is due to Weitzman (1979), who proposesan elegant solution to a consumer choice model he calls “Pandora’s Box”: a consumer faces N closedboxes, where box i has a prize whose (unknown) value comes from a known distribution, and thecost to open box i is c i . The consumer has to decide in what order to open and inspect the boxes (tolearn the realized value of the prize inside), and when to stop opening and take home the best prize,allowing perfect recall. The solution ranks boxes by a reservation price index, with the consumerstopping when the maximum sampled reward exceeds the reservation price of every remaining closedbox. A few papers in the assortment planning literature use the idea of a two-stage consumer choiceprocess, where the consumer first forms a consideration set in the spirit of Weitzman (1979), and inthe second stage selects the product within that consideration set. This is an effective model for aweb-based seller but not for a VA, where the order of inspections is determined by the VA, not by5he consumer. Wang and Sahin (2018) study assortment planning and pricing where the consumerfollows a two-stage model with homogeneous search costs. The consumer includes products in herconsideration set by comparing their incremental net utility to the search cost. In the second stage,the consumer uses the Multinomial Logistic (MNL) model (assuming Gumbel random valuations)to select from the consideration set. Wang and Sahin (2018) develop an algorithm to calculate theoptimal assortment and prices. They find that without a search cost, all products have the sameprice—a consequence of the MNL assumption. Derakhshan et al. (2018) develop a two-stage modelwith heterogeneous search costs which are increasing in the item’s position. Prices are exogenousand consumers follow a process similar in spirit to Weitzman (1979) to form a consideration set.The final choice is then made following an MNL choice model similar to Wang and Sahin (2018).They show that the problem is NP hard and propose a polynomial-time solution algorithm. Theyfind that even though ranking products in descending order of intrinsic utilities is suboptimal, itachieves a multiplicative approximation factor of 1/2 and an additive factor of 0.17.The web-interface model developed by Gallego et al. (2018) also adopts the two-stage consumerchoice approach: the consumer first forms a consideration set (all products in the first k pages)according to her type (which determines k ) and then follows the MNL model to select a productwithin the consideration set. The seller knows the distribution of consumer types and maximizesits expected revenue over rankings and prices. Gallego et al. (2018) recommend a descendingranking in value gaps (net utilities when products are sold at their unit wholesale costs). Theyshow that the price markups are the same for all products on the same page, and are increasingin the page index. Aouad et al. (2015) propose a two-stage model outside the Weitzman (1979)setup. Instead, different consumer types have exogenous consideration sets and product preferences,and the platform determines the overall assortment from which consumers select their preferredproduct. Aouad et al. (2015) formulate the problem as one of maximizing expected revenue over agraph and derive a recursive algorithm for solving it.To our knowledge, our model is the first that combines optimal ranking and pricing decisions bya seller under strategic consumer choice in the setting of a VA. Because of our different objectives,both our model structure and our results are significantly different from those in the assortmentoptimization literature. Relative to the traditional assortment planning literature, these differencesare obvious, as our model is designed to jointly determine the optimal ranking and pricing forthe particular setting of a VA. This is why ranking and presentation are at the core of our modelas opposed to most traditional assortment planning models (group (i) in our classification of theliterature). In contrast to the myopic consumer assumed in most traditional models, we modela rational, forward-looking consumer. As a result, the temporal dynamics are the equilibriumoutcome of the interactions between the seller and the consumer, unlike traditional (group (i) )assortment planning models where the dynamics are driven by the seller’s inventory level. Theseand other differences are to be expected as this literature was not designed to address the issueswe focus on in this paper.Our model is closer in structure to those of l’Ecuyer et al. (2017), Chu et al. (2020), Wangand Sahin (2018), Derakhshan et al. (2018), Gallego et al. (2018), Aouad et al. (2015) and Ferreiraand Goh (2019), which we reviewed above. The key differences between these models and ours aresummarized below.1. Pricing.
Most (but not all) of the above papers focus on assortment planning per se andtherefore take product prices as given (Ferreira and Goh (2019), l’Ecuyer et al. (2017), Chuet al. (2020) Derakhshan et al. (2018), Aouad et al. (2015)), or use the MNL assumption(Wang and Sahin (2018) Gallego et al. (2018)). As is well-known, the MNL assumptiongreatly simplifies the price optimization problem, however it is restrictive and is designed to6ead to the same price or margin for all products (Anderson et al. (1992)). Indeed, Gallegoet al. (2018) find the same price margin within a same page and Wang and Sahin (2018) findthe same price for all except possibly one product. In our setting, the seller has two leversat hand, pricing and ranking, and our model allows it to take full advantage of both, whichenables us to derive key insights on their interaction and impact.2.
Consumer choice and ranking model.
The literature covers multiple consumer choiceand ranking models, all of which significantly differ from ours. Wang and Sahin (2018) andAouad et al. (2015) differ from Derakhshan et al. (2018) in that the former focus on two-stageassortment optimization without an actual position (i.e., ranking) effect whereas the latterconsiders position-dependent consumer search costs. In all three papers, the consumer hasthe freedom to choose which product to review—an appropriate structure for a store or aweb-based seller, but not for a VA, where the VA, rather than the consumer, makes thischoice. In Gallego et al. (2018)’s two-stage model, the consumer views all the products in thefirst few pages without deciding when to stop, as the consumer type determines the numberof pages (and products) she views. Similar to Derakhshan et al. (2018) and Wang and Sahin(2018) and unlike our setting, the final choice is made by the consumer based on the MNLchoice model. l’Ecuyer et al. (2017) and Chu et al. (2020) model a platform in the spiritof the search engine literature, with the consumer choice problem not being influenced bythe platform’s strategy, or being modeled indirectly. Chu et al. (2020)’s three-sided modelinvolves a platform, suppliers and consumers. The consumer’s decisions need not depend onthe platform’s ranking strategy as all products are ex-ante homogeneous from the consumer’spoint of view. l’Ecuyer et al. (2017) model the consumer choice problem indirectly through aclick-through-rate function that combines the effects of position and product characteristics.Unlike the foregoing papers, Ferreira and Goh (2019) consider information strategies designedto increase the number of units bought by a consumer over a selling season. For the most part,they consider the implications of a given ranking and assortment on the value of concealment.Two of their Corollaries show show that under a special set of circumstances, a particular rank-ing (ascending order in product valuations) is optimal. The consumer choice model leadingto this result is entirely different from ours (an infinitely patient non-anticipating consumerbuying multiple units) as are the assumptions (same prices or same valuation distributions)and results, differences which are driven by their different objective (to study the value ofconcealment).3.
Recall.
All the above papers assume that consumers have perfect recall. Whereas thisassumption is reasonable for a traditional web-interface (at the page level), as discussedabove, choice under a sequential presentation (especially with a voice interface, which furtherincreases cognitive load) is better characterized by no recall. It will be interesting to examinethe effects of imperfect recall, which covers both assumptions as special cases, in future work.4.
Comparison of VA to web interface.
An important issue studied in our paper is thecomparison between selling through a web interface and selling through a VA. Naturally, thepapers cited above were not designed to provide such a comparison.The differences between the objectives and modeling choices of our paper vis-a-vis the foregoingpapers also drive key differences in the results:1.
Optimal pricing . Most of the above papers take prices as given, Wang and Sahin (2018)and Gallego et al. (2018) being the exceptions. The MNL consumer choice model used in7oth papers naturally gives rise to pricing results that are significantly different from ours.Wang and Sahin (2018) find that either all products are priced the same, similar to the MNLliterature, or all but one product are priced the same (the lowest value product may be priceddifferently so it will enter the consideration set without affecting the prices of higher-valuedproducts). Gallego et al. (2018) find that with a web interface, all products within each pagehave the same price margin, a result which is mainly driven by the MNL assumption. Incontrast, we find—for our web interface model—the counter-intuitive result that within apage, prices are typically decreasing in product valuations (or margins), which means thathigher-valued products tend to have a lower price. Higher-valued products are priced lower toincrease the matching probability of the entire page, thus allowing lower-valued products toset a higher price. This insight does not apply to Gallego et al. (2018)’s model, where a givenconsumer can simultaneously inspect all products on the pages she has access to. Gallegoet al. (2018) also find that across web pages, the price margin is increasing in the page index.In contrast, we find that the reverse may also happen depending on problem parameters suchas the amount of private information and the consumer patience level.2.
Optimal ranking . Among the papers where the seller solves a ranking problem, one suggestsan approximation algorithm with performance guarantees (Derakhshan et al. (2018)), andtwo characterize the optimal results (Chu et al. (2020), l’Ecuyer et al. (2017)). Derakhshanet al. (2018) develop an approximate ranking algorithm and show that the intuitive ranking(descending in product utility), though not optimal, has a performance guarantee. In Chuet al. (2020), the optimal ranking is descending in the product’s contribution to the platform’s(weighted) objective function. In l’Ecuyer et al. (2017), the optimal ranking is decreasing inproduct scores that summarize each product’s contribution to expected revenue. In all threepapers, products are ranked by a decreasing score that quantifies their contribution to theobjective function. Due to the flexibility of our model, we find the optimal ranking may be anarbitrary permutation of the available products depending on the consumer patience level andother problem parameters. Further, under certain conditions the optimal ranking is actually increasing in product valuations. Wang and Sahin (2018) and Aouad et al. (2015) considerthe case where the seller provides an assortment to the consumer, and the consumer ranks theproducts to form a consideration set and then selects the best item within the considerationset. The optimal ranking in their model is descending in the consumer utility (Wang andSahin (2018)) or in the consumer’s preference list (Aouad et al. (2018)). In our paper, theoptimal ranking for a profit-maximizing seller is highly-dependent on the consumer’s patiencelevel. In particular, the optimal ranking is descending in valuations for a highly-impatientconsumer and ascending in valuations for a sufficiently patient consumer. This is intertwinedwith our model’s pricing decisions: an ascending order, for example, allows the seller toinduce the consumer to stay and pay a higher price for lower-ranked products. This strategy,however, can work only when the consumer is sufficiently patient. Even for a consumer agentthat maximizes consumer surplus, the optimal ranking is still ascending in valuations whenthe consumer is patient as the consumer benefits from viewing more products.3.
Comparison of alternative objective functions . Most of the above papers (Ferreira andGoh (2019), l’Ecuyer et al. (2017), Gallego et al. (2018), Aouad et al. (2015), Ferreira andGoh (2019)) consider a profit- (or revenue-)maximizing seller or platform, Chu et al. (2020)and Derakhshan et al. (2018) being the exceptions. Chu et al. (2020)’s objective function is aweighted average of the platform revenue, supplier surplus and consumer surplus. They findthat the optimal ranking for a profit-maximizing platform is descending in product prices;8or maximizing supplier surplus, it is descending in the seller valuations; and for maximizingconsumer surplus, it is descending in the consumer’s utility. Derakhshan et al. (2018) alsodevelop an algorithm for maximizing consumer welfare. Both papers provide algorithms toapproximate the optimal ranking and, unlike our paper, do not focus on structural differencesin the results obtained under alternative objective functions. In our paper, we study thedifferences in surplus allocations and strategies for VAs that maximize seller profits, overallsurplus (an altruistic VA) or consumer surplus (a consumer agent). Our specification allowsus to study how the surplus allocation depends on the distribution of the private signalsand on the consumer’s patience level. Interestingly, when the private valuation distributionis exponential, a profit-maximizing seller always extracts half the total surplus. We alsofind that the seller makes zero profit when the VA is altruistic or acts as a consumer agent.Again, unlike Chu et al. (2020) and Derakhshan et al. (2018), we can also study the effecton product prices. We find that an altruistic seller (maximizing total surplus) follows exactlythe same ranking and pricing strategy as a consumer agent. The strategies of a a seller-profit-maximizing VA reflect its market power and its ability to manipulate the presentationto its advantage. Nevertheless, for both highly patient and highly impatient consumers andexponentially-distributed private valuations, the optimal rankings under all three objectivefunctions are the same.
The plan of the paper is as follows. Following this Introduction, Section 2 presents our model. Thepricing problem for a given ranking is solved in Section 3. Section 4 addresses the optimal ranking.Section 5 discusses the implications of using a VA, Section 6 compares the VA to a traditionalweb-interface and Section7 offers our concluding remarks.
We consider a virtual assistant serving a seller that sequentially offers products to consumers andprices them dynamically. The seller (through the VA) selects from N products i = 1 , , · · · , N thatare available for sale, presenting and pricing them one at a time. When the seller sells product i , it incurs a cost c i which includes payments to the vendor, logistics and shipping (if applicable)costs, etc., and it pockets the difference between the price it charges, p i , and its cost c i . Anarriving consumer specifies what she is looking for and the VA then presents the available productssequentially at its discretion. When a product i is presented, the seller prices it dynamically at p i based on the information available to it. The consumer evaluates the product and decides whetherto buy it or move on to the next product (in which case product i becomes unavailable). Eachproduct presentation and evaluation lasts an exponential amount of time with rate τ p and expectedvalue 1 /τ p . The consumer’s overall search lasts for an exponential amount of time with rate τ c andexpected value 1 /τ c . The consumer leaves the system following this search time or after she exhaustsall N available products without making a purchase. All of the evaluation and search times areindependent. We denote by ρ = τ p τ p + τ c the consumer’s patience parameter—the probability that fora given product, the consumer’s evaluation is completed before she leaves the system (so she canbuy it if she chooses to). The seller uses the VA to rank and dynamically price each product so asto maximize its expected profit. With this objective function, the VA is subservient to the seller.We also consider other objectives (maximizing total surplus and maximizing consumer surplus) in9ection 3.2. As consumers arrive sequentially, we can perform the analysis for one consumer at atime.When a consumer arrives, she provides information about the product she is looking for (e.g.,in the form of a search query). This information, along with consumer profile information, isconverted by the seller to an estimated base valuation v i for each product i . Once the consumerhas evaluated a product presented to her, she values the product at v i + (cid:15) i , where (cid:15) i is the privateinformation unobservable by the seller. When product i is priced at p i , the consumer’s net utilityfrom purchasing it is thus u i = v i − p i + (cid:15) i . We assume the (cid:15) i are i.i.d. random variables sampledfrom a common distribution F ( · ) with positive support.The consumer aims to maximize her expected net utility. We show later that in equilibrium, theconsumer will buy product i if and only if its net utility u i is above a threshold level that balancesthe net utility of product i against the opportunity to buy a product yielding a higher net utility inthe future. We call this form of consumer policy “threshold policy”. For simplicity, we assume nodiscounting (as is well known, a discount factor may be incorporated in τ c ). If product i were soldat cost, its value to the consumer would be v i + E ( (cid:15) ) − c i . We call v i + E ( (cid:15) ) − c i the value margin of product i ( i = 1 , , . . . , N ) and assume for simplicity that the ( v i − c i ) are non-positive and (cid:15) ispositively-supported.In summary, the seller decides on a product ranking and dynamic pricing policy to maximizeits objective function knowing all v i , c i ( i = 1 , , · · · , N ) and which products have been rejected bythe consumer. The consumer dynamically decides on whether to accept or reject each offer, andthe process continues until the consumer departs. Our notation is summarized in appendix A. Each party (the seller and the consumer) hold beliefs about the other’s strategy, and each optimizesits own objective function taking the other’s strategy as given. In equilibrium, these beliefs areconsistent. The seller’s strategy is defined by two n -dimensional vectors:(i) a permutation σ = ( σ , · · · , σ N ) specifying the ranking of products for presentation by theseller, and(ii) a price vector p = ( p σ , · · · , p σ N ) specifying the price assigned by the seller to each product.The consumer’s strategy is defined as a mapping from the history up to time t , H t , to { accept, reject } .In equilibrium,(i) given the price p and ranking σ , the consumer’s acceptance strategy at each time t is optimal;and(ii) given the consumer’s strategy, the rank-price combination ( σ , p ) maximizes the seller’s ob-jective function.Next, we show that in equilibrium, the consumer’s optimal strategy must have the threshold pol-icy structure. Assume, without loss of generality, that for a given seller strategy ( σ , p ), σ =(1 , , · · · , N ) by renumbering the products. Let V ci ( (cid:15) i ) be the expected consumer utility fromround i until the consumer departs (possibly purchasing a product) when she observes a privatevalue of (cid:15) i . Let V ci = E V ci ( (cid:15) i ) where the expectation is taken over (cid:15) i . Then, if the consumer decidesto buy in round i , she’ll get v i – p i + (cid:15) i . Otherwise, the consumer waits for the next round andreceives an expected utility of ρV ci +1 . It follows that V ci ( (cid:15) i ) = max { v i – p i + (cid:15) i , ρV ci +1 } . i if and only if u i = v i – p i + (cid:15) i ≥ ρV ci +1 . Thus, the consumerwill choose a threshold policy: u i ≥ δ i := ρV ci +1 , where δ i is the consumer’s optimal threshold inround i . We have: Proposition 2.1 (Consumer threshold policy).
For any given ( σ , p ) , the consumer’s bestresponse is a threshold policy δ . An equilibrium is thus defined by three n -dimensional vectors ( δ , σ , p ) such that: δ maximizes E [ V c ( δ , σ , p )] , and (1)( σ , p ) maximizes E [ V p ( δ , σ , p )] , (2)where V p ( V c ) is the expected seller (consumer) surplus. We first solve the profit-maximizing pricing problem for a given ranking σ , where the i.i.d. privatevaluations (cid:15) i follow a general distribution F . We then consider the special case where they areexponentially distributed. In section 3.2, we examine what happens under alternative objectivefunctions. For a given ranking σ (assuming, without loss of generality, σ = (1 , , · · · , N )), the equilibriumprices and the consumer’s optimal thresholds can be calculated by backward induction, yieldingthe following proposition. Proposition 3.1 (Profit-maximization for a given ranking with (cid:15) ∼ F ( · ) ). Given σ =(1 , , · · · , N ) , the equilibrium ( δ , p ) satisfies the following recursion:(a) In stage N ,The consumer’s threshold is δ N = 0 .The equilibrium price at stage N is given by p ∗ N = arg max p P N ( p − c N ) , and the equilibrium probability of purchase is given by P N = 1 − F ( p ∗ N − v N ) . Under the optimal price, the seller’s expected profit in period N is given by V pN = P N ( p ∗ N − c N ) , and the consumer’s expected surplus in period N is given by V cN = P N E ( (cid:15) N + v N − p ∗ N | (cid:15) N + v N − p ∗ N ≥ δ N ) . (b) In stage i = N − , · · · , , he consumer’s threshold is δ i = ρV ci +1 .The equilibrium price in stage i is given by p ∗ i = arg max p P i ( p − c i ) + (1 − P i ) ρV pi +1 , and the probability of purchase is given by P i = 1 − F ( p ∗ i − v i + ρV ci +1 ) . Under the optimal price, the seller’s expected profit in period i is given by V pi = P i ( p ∗ i − c i ) + (1 − P i ) ρV pi +1 , and the consumer’s expected surplus in period i is given by V ci = P i E ( (cid:15) i + v i − p ∗ i | (cid:15) i + v i − p ∗ i ≥ δ i ) + (1 − P i ) ρV ci +1 . In the special case where the private valuations are exponentially distributed, Proposition 3.1simplifies to:
Corollary 3.2 (Profit-maximization: Equilibrium for a given ranking with (cid:15) ∼ exp( α ) ). Given σ = (1 , , · · · , N ) , the equilibrium ( δ , p ) satisfies the following recursion:(a) In stage N ,The consumer’s threshold is δ N = 0 , the equilibrium price is p ∗ N = c N + 1 α , and the probability of purchase is given by P N = exp( α ( v N − c N ) − . Under the optimal price, the seller’s expected profit and the consumer’s expected surplusare both given by V pN = V cN = 1 α exp( α ( v N − c N ) −
1) = 1 α P N (b) At stage i = N − , . . . , ,The consumer’s threshold is δ i = ρV ci +1 ,The optimal price is p ∗ i = c i + 1 α + ρV pi +1 , and the probability of purchase is P i = exp( α ( v i − c i ) − · exp( − αρ (2 V pi +1 )) Under the optimal price, the seller’s expected profit and the consumer’s expected surplusat stage i is V pi = V ci = 1 α N X k = i ρ k − i P k .
12y Proposition 3.1, the optimal price in the last stage is the “monopoly” price that maximizes theseller’s expected profit in the single-product case. This is because in the last stage, the seller iseffectively making a final ”take-it-or-leave-it” offer to the consumer with no continuation option.In the exponential case (Corollary 3.2), the monopoly price in the last stage is independent ofthe last product’s valuation, and the price of the product offered in each stage i is independentof its own valuation v i – it depends only on the valuations of products i + 1 , . . . N through thecontinuation value ρV pi +1 . The optimal price has a special structure – it equals the monopoly priceplus the future continuation value. This implies that products viewed in earlier stages receive ahigher price markup to induce the consumer to view more products which, in turn, increases theexpected private valuation of the product purchased and with it, the seller’s expected profit.Another interesting consequence of Corollary 3.2 is that in the exponential case, for any givenranking, the seller’s expected profit is equal to the expected consumer surplus. This implies thatthe ranking that maximizes the seller’s profits also maximizes the consumer surplus: Corollary 3.3
When (cid:15) is exponentially distributed, the profit-maximizing pricing solution equatesthe seller’s expected profit to the consumer surplus for any given ranking. Thus, the profit-maximizingranking also maximizes the consumer surplus.
In the next subsection, we solve the optimal pricing problem for a VA with different objectivefunctions: (i) a consumer agent that maximizes the expected consumer surplus, and (ii) an altruisticVA that maximizes the expected total surplus.
The results of Section 3.1 can be directly generalized to VAs with different objective functions. TheVA may be altruistic, maximizing the total surplus V s = V c + V p , or it may serve as a consumeragent, maximizing the consumer surplus V c . As we show below, in both cases the seller extractsno surplus and each product is priced at cost: Proposition 3.4 (Altruistic/Consumer agent VA: Equilibrium for a given ranking with (cid:15) ∼ F ( · ) ): Given σ = (1 , , · · · , N ) , the equilibrium ( δ , p ) satisfy the following recursion:(a) In stage N ,The consumer’s threshold is δ N = 0 .The equilibrium optimal price is p ∗ N = c N and the equilibrium probability of purchase is P N = 1 − F ( c N − v N ) . Under the optimal price, the seller’s expected profit is V pN = 0 , and the consumer’s expected surplus is V cN = P N E ( (cid:15) N + v N − c N | (cid:15) N + v N − c N ≥ δ N ) . (b) In stage i = N − , · · · , , he consumer’s threshold is δ i = ρV ci +1 .The equilibrium optimal price is p ∗ i = c i and the equilibrium probability of purchase is P i = 1 − F ( c i − v i + ρV ci +1 ) . Under the optimal price, the seller’s expected profit is V pi = 0 , and the consumer’s expected surplus is V ci = P i E ( (cid:15) i + v i − c i | (cid:15) i + v i − c i ≥ δ i ) + (1 − P i ) ρV ci +1 . By Proposition 3.4, both the altruistic VA and the consumer agent follow a very simple pricingrule, i.e., p i = c i . In addition, for any ranking, the two objective function values (total surplus andconsumer surplus, respectively) are the same. It follows that the optimal rankings under the twoobjective functions are also the same. Corollary 3.5 (Altruistic/Consumer agent VA: Equilibrium and surplus allocation).
In equilibrium, an altruistic VA and a consumer agent share the same optimal ranking and set allprices p i at cost c i . In expectation, the consumer extracts the entire surplus. In summary, the percentage of total surplus extracted by the profit-maximizing seller depends onthe distribution of (cid:15) and is 50% for the exponential distribution (we’ll examine other distributionsin Section 5). Under both altruistic and consumer agent VA, the seller’s expected profit is zero.In addition, the equilibrium results (ranking, pricing, and consumer thresholds) are the same for aconsumer agent and an altruistic VA, independent of the distribution of (cid:15) . ku
Using Proposition 3.1, we can solve the optimal ranking problem by complete enumeration, i.e., bycomparing the objective function values across all possible rankings. Finding the optimal ranking isthen the combinatorial problem of searching among all N ! permutations, which becomes intractablefor large N . In this Section, we provide two algorithms for efficiently computing or approximatingthe optimal ranking. We propose an efficient algorithm which produces the optimal ranking based on checking local pair-wise optimality. The algorithm, which we call the “Greedy Pairwise Switch” or “GPS” Algorithm,operates as follows (we describe the algorithm for a profit-maximizing seller; the adjustments forother objective functions are straightforward). Let σ = ( σ , · · · , σ N ) be the current ranking and R ( σ ) be the set of all pairwise-switched rankings starting from σ . For example, if N = 3 and σ = (1 , , R ( σ ) = { (2 , , , (1 , , , (3 , , } . A pairwise-switched ranking is obtainedby switching two products in the current ranking σ and keeping the ranks of all other productsunchanged. Formally, R ( σ ) = { σ = ( σ , · · · , σ i − , σ j , σ i +1 , · · · , σ j − , σ i , σ j +1 , · · · , σ N ) , i < j } . σ , the algorithm checks whether there is a local profit improvement for all σ ∈ R ( σ ).If there is no local improvement (i.e., V p ( σ ) ≥ V p ( σ ) for all σ ∈ R ( σ )), the algorithm returns σ as the optimal ranking. If there is a local improvement, the algorithm updates the current rankingto the local switch that achieves the largest improvement. The algorithm terminates in finite timesince we have a finite number of possible permutations and it never cycles since each switch providesa strictly positive improvement.Figure 1(a) shows that the number of iterations required for the GPS algorithm increases almostlinearly in N and Figure 1(b) shows the average iteration ratio between complete enumeration andthe GPS algorithm. Clearly, GPS is much more efficient than complete enumeration, and in all thecases we examined in a large number of experiments, it achieved the optimal profit. (a) Complexity of the GPS algorithm. (b)
Complexity ratio for complete enumeration rel-ative to the GPS algorithm.
Figure 1:
Complexity comparison between complete enumeration and the GPS algorithm. Shown is theaverage number of iterations as a function of the number of products, N , for each algorithm. The complexityof the GPS algorithm is linear in N (left figure), a significant improvement over complete enumeration (rightfigure) that grows as ( N − When the survival probability ρ approaches 0 (extremely impatient) or 1 (extremely patient con-sumer), we obtain the following limits. Proposition 4.1 (Limiting case analysis)
Assume the private valuations are exponentially distributed. Then, for each of the three objectivefunctions (profit-maximizing, altruistic or consumer agent VA):(a) There exists a ρ > such that for all ρ ∈ [0 , ρ ) , the descending ranking in value margins ( v i + E ( (cid:15) ) − c i ) is optimal. Further, for a profit-maximizing seller, the optimal prices are givenby the monopoly prices p i = c i + α , i = 1 , , ..., N .(b) There exists a ρ < such that for all ρ ∈ ( ρ , , the ascending ranking in value margins ( v i + E ( (cid:15) ) − c i ) is optimal. These limiting results suggest an approximating algorithm which we call the “double-rank approx-imation.” 15 lgorithm. 4.1 (Double-rank approximation).
The seller compares the objective functionvalues for only the descending ranking and the ascending ranking in value margins ( v i + E ( (cid:15) ) − c i ) ,and selected the one that achieves the higher value. The double-rank approximation is obviously computationally efficient since it only requires 2 N iterations. But how close are its results to the optimal results? We evaluate the efficiency of thealgorithm through the ratio of expected profits under the double-rank algorithm to the optimalobjective function value. Multiple numerical experiments show that in spite of the simplicity of thedouble-rank approximation, it is surprisingly efficient.Our numerical experiments use the Gamma distribution, which is widely used in applications(Appendix B summarizes key features of the Gamma distribution and lists the parameters ofour numerical experiments). We present here the results for three private valuation distributions:exponential with unit mean (Figure 2(a)), Gamma (2 , .
5) (Figure 2(b)), and Gamma (5 ,
10) (Figure2(c)). As seen in the figures, the descending order remains optimal even for moderate values of ρ and the ascending order kicks in only at ρ = 0 .
99 for the Gamma (5 , .
1) and Exp(1) cases, andat ρ = 0 . ,
10) (where the private valuations have a highervariance ab ). The results indicate that with lower information asymmetry, the descending orderin value margins is more likely to be optimal. Throughout, the double-rank algorithm achieves atleast 95% of the optimal profit. Similar results are obtained for the other objective functions. (a) Gamma(1 ,
1) = Exp(1) (b)
Gamma(2 , . (c) Gamma(5 , Figure 2:
Efficiency of the double-rank approximation for Gamma-distributed private valuations. Thefigure shows the ratio of the expected profit achieved by the double-rank approximation to the optimalexpected profit, and how it depends on the consumer patience level ρ . Shown are results for (i) Exp(1), (ii)
Gamma (2,0.5), and (iii)
Gamma (5,10).
In this Section, we discuss some of the implications of the foregoing results. We have consideredthree different objective functions for the VA: a consumer agent, an altruistic VA and a (seller)profit-maximizing VA. By proposition 3.4, under the first two objective functions (consumer agentor altruistic VA), p i = c i for all i and the consumer extracts the entire surplus. When the VAmaximizes the seller’s profit, the optimal prices and surplus allocation depend on the consumer pa-tience parameter and on the distribution of private information. When the private information isexponentially distributed, the surplus is equally split between the consumer and the seller, indepen-dent of the other problem parameters. The results are more complex when the private informationfollows a general distribution and we illustrate them using numerical examples.16e consider N = 6 products with expected valuations (0 , / , / , / , / , /
6) and zero costs( c i = 0) (see Appendix B for a full listing of the parameters). Fixing these parameters, we firstexamine the structure of the optimal solution (pricing and ranking) when the private valuationsare exponentially distributed (Subsection 5.1). We then consider the structure of the solution whenthe private valuations follow the more general gamma distribution (Subsection 5.2) and we finallyconsider how information asymmetry affects the pricing, ranking and surplus allocation (Subsection5.3). We first consider the problem with (cid:15) ∼ exp(1) for different levels of the consumer patience parameter ρ (Figure 3). Figure 3:
Comparison of virtual assistant prices when (cid:15) is exponentially distributed, (cid:15) ∼ exp(1), for ρ = 0 . , . , . ρ = 1 (blue line). The horizontal axis shows the six product valuations.The vertical axis shows the corresponding prices. The presentation rank is either descending (brown lines)or ascending (blue line) in the expected valuations. When ρ is 0.9 or less (brown lines), the optimal ranking is descending in the expected valuations,and the seller ranks the most valuable product (with expected valuation 5 /
6) first, similar to thelimiting results in Proposition 4.1(a). When ρ = 1 (blue line), the optimal ranking is ascending inthe expected valuations, and the seller ranks the most valuable product last, as shown in Proposition4.1 (b).One might expect higher-valued products to command higher prices. In this case, however,Corollary 3.2 shows that for a given ranking, the optimal price of each product i is independent ofits own expected valuation, and it depends primarily on its rank which determines ρV pi +1 through theexpected valuations of products i + 1 , i + 2 , . . . N . The pricing decision balances two considerations:on the one hand, an impatient consumer is unlikely to buy the products presented late, suggestinga higher margin for products presented earlier, which are more likely to be bought by the consumer.On the other hand, the different products effectively compete with one another, with less candidatesremaining to be viewed, the seller has greater market power so it can extract higher margins(at the extreme, when only one product is left, Corollary 3.2 shows that the seller charges themonopoly price). The optimal balance depends on the consumer’s impatience parameter ρ : Whenthe consumer is relatively impatient, the seller prefers to charge higher margins on the productspresented earlier since there is only a small probability that she will buy products presented later.17hen the consumer is patient, there is a high probability that the consumer will wait and the sellercharges higher margins on the products presented later. Accordingly, Figure 3 shows that when ρ is 0.9 or less (brown lines), the margins decrease as the product rank increases, whereas for ρ = 1(blue line), the margins increase along with the product rank.Figure 3 also shows that for any given ranking σ , each product’s optimal price increases in theconsumer patience parameter ρ . The Proposition below generalizes this finding. Proposition 5.1 (Monotonicity in Optimal Prices)
When the private valuations (cid:15) are exponentially distributed, under any given ranking, the optimalprices increase in the consumer patience parameter ρ . Proposition 5.1 follows from the fact that under any given ranking, product i ’s optimal price isthe sum of the monopoly price α + c i and the markup ρV σ i +1 , which in turn is increasing in ρ .Intuitively, a more patient consumer receives a higher surplus, which allows the seller to extractmore of that surplus through higher prices. When the private valuations are not exponentially distributed, the optimal solution becomes morecomplex. By Proposition 4.1, at the limits as ρ goes to zero or 1, the optimal rankings are monotonein the products’ value margins. In the example below, we show that, for the Gamma distribution, (i) monotone rankings remain optimal for a wide range of (although not all) ρ values; and (ii) inthe narrow range where the optimal ranking is not monotone, it changes quickly from descendingto ascending through multiple switches.We consider (cid:15) ∼ Gamma(2 , .
5) and vary ρ from zero to 1 (Table 1 and Figure 4). As shownin the Table, the descending ranking is optimal for all ρ ∈ [0 , . ρ ∈ (0 . , ρ ∈ (0 . , . i
16 13 12 23 56 (0 . , . . , . . , . . , . . , . . , . . , . . , . . , . . , . . , . . , . Table 1:
Optimal rankings for different consumer patience levels ρ when (cid:15) ∼ Gamma(2 , . ρ values in (0,1]. For example, the optimal ranking permutation is(6 , , , , ,
2) when ρ ∈ (0 . , . igure 4: Optimal prices for different values of consumer patience parameter ρ when (cid:15) ∼ Gamma(2, 0.5).
Shown are the optimal prices for the six products in each range of ρ values in [0 . ,
1] (the behavior in (0 , . . , . Similar to the exponential case, the consumer patience parameter ρ influences the optimal pricesthrough the markup and the optimal ranking. As we observe in Figure 4, in each interval wherethe optimal ranking remains constant, the optimal price is increasing in ρ for each product. In this Subsection, we study the effects of information asymmetry on the allocation of surplus.Whereas the seller knows only the v i s, the consumer can also learn the realization of the (cid:15) i s. Thisgives the consumer an informational advantage that, intuitively, should increase with the varianceof (cid:15) . To study how this information asymmetry affects our results, we consider Gamma-distributedprivate valuations with shape parameter a and scale parameter b , fixing the mean at E ( (cid:15) ) = ab = 1and changing the shape parameter a . With a constant mean, increasing the shape parameter a isequivalent to reducing the variance (Var( (cid:15) ) = ab = a ) of the consumer’s private information.Figure 5 shows that as the shape parameter a increases, the seller’s surplus share increasesas expected. When a = 1, the distribution of (cid:15) is exponential and the seller extracts 50% of thesurplus. The limit as a → ∞ (not shown) is deterministic with no private information. In thiscase, the seller extracts all the surplus and consumer gets 0 . Figure 5:
Seller’s surplus share for a profit-maximizing VA. The private information (cid:15) ∼ Gamma ( a, b ) , ab =1. As a increases, the variance a decreases. Traditional web-based sellers such as Amazon, eBay and Taobao dominate today’s electronic com-merce market. What would be the effect of the predicted shift to sales through virtual assistantson pricing, seller profits and consumer surplus? To answer this question, we model sales through aweb interface and compare the results to those we obtained for a virtual assistant. We focus on a(seller) profit-maximizing VA.
Consider a web interface that enables the presentation of k products per page. In our model, theconsumer specifies what she is looking for and the seller presents the available products a page ( k products) at a time. The consumer then examines the entire page and decides whether to choose aproduct within that page, or to proceed to the next page. The process continues until the consumerexits.As in our virtual assistant model, the consumer is impatient and stays on for an exponentialamount of time with mean τ c . We assume that with the web interface, the time to evaluate a pageis exponentially distributed with rate κτ p /k , where κ is an acceleration factor specifying how fastthe consumer views one page (1 ≤ κ ≤ k ). When κ = k , the expected time to view an entire pageis the same as that for evaluating one product with the virtual assistant, whereas when κ = 1,viewing a page takes k times as long as evaluating a single product with the virtual assistant. Wedenote by u l,i = v l,i + (cid:15) l,i − p l,i the realized utility when the consumer buys the i th product on page l , and by v l,i the valuation of the i th product on page l .With this specification, the virtual assistant corresponds to the special case of a web interfacewith k = κ = 1. We compare the behavior of prices and the allocation of surplus between the webinterface and the virtual assistant. 20 .2 Profit-maximization For a given ranking, the equilibrium prices and the consumer’s optimal thresholds under the webinterface are given by Proposition 6.1, whose straightforward proof is omitted.
Proposition 6.1 (Web-interface equilibrium for a given ranking with (cid:15) ∼ F ( · ) ) Given σ = (1 , , · · · , N ) , an equilibrium ( δ , p ) must satisfy the following recursion:(a) In stage n := d Nk e ,The consumer’s threshold is δ n = 0 .For any price vector p , the probability of purchasing product i in stage n is given by P n,i = P ( u n,i ≥ δ n , u n,i ≥ u n,j for all j = i, j is on page n ) , where u n,i = v n,i + (cid:15) n,i − p i .The seller’s expected profit in stage n is given by V pn = max p ∈ R k X i P n,i ( p n,i − c n,i ) . Let p n be the maximizer of the above equation, then the consumer’s expected surplus instage n is given by V cn = X i P n,i E ( (cid:15) n,i + v n,i − p n,i | u n,i ≥ δ n , u n,i ≥ u n,j , for all j = i, j is on page n ) . (b) In stage l = n − , · · · , , The consumer’s threshold is δ l = ρV cl +1 .For any price vector p , the probability of purchasing product i in stage n is given by P l,i = P ( u l,i ≥ δ l , u l,i ≥ u l,j , for all j = i, j is on page l ) , where u l,i = v l,i + (cid:15) l,i − p i . The seller’s expected profit in stage l is given by V pl = max p ∈ R k X i P l,i ( p l,i − c l,i ) + ρ (1 − X i P l,i ) V pl +1 . Let p l be the maximizer of the above equation, then the consumer’s expected surplus instage l is given by V cl = X i P l,i E ( (cid:15) l,i + v l,i − c l,i | u l,i ≥ δ l , u l,i ≥ u l,j , for all j = i, j is on page l )+(1 − X i P l,i ) ρV cl +1 . Proposition 6.1 holds regardless of whether N is divisible by k . It enables us to compute the optimalranking following the methodology of Section 5.We next illustrate the effects of the user interface on price behavior using numerical examples.As before, we first consider a seller selling 6 products with expected valuations 0 , / , / , / , / /
6, where the private valuations are exponentially distributed with unit mean, (cid:15) ∼ exp(1).21e compare the virtual assistant and the web interface with k = 2 , (a) ρ = 0 . (b) ρ = 0 . (c) ρ = 1 Figure 6:
Price comparison between virtual assistant and web interface with k = 2 , κ = 1. We present the optimal prices under the optimal ranking for ρ = τ p τ c + τ p = 0 . . P.i means that the product is presented on page i . Some page labels arenot shown for nearly-overlapping data points. Across pages , the price pattern we observe is similar to the one we obtained for the virtualassistant. For low and moderate values of ρ , the virtual assistant ranked products in descending order of product valuations and the prices were increasing in product valuations. Similarly, the web-interface places the products with higher valuations on earlier pages (Figure 6(a),(b)), and theseproducts are priced higher. Although under the web-interface, the there is no elegant decompositionof price into a monopoly price and a continuation markup, the intuition carries over: when theconsumer is willing to accept a product presented early, the seller infers that she has a favorableprivate valuation, which the seller exploits to extract a higher profit. For large ρ , the optimalranking and pricing rules for the virtual assistant were reversed: the optimal order was ascending ,and prices were descending , in product valuations. We observe a similar pattern under the web-interface: the order is descending in the product valuations and the optimal prices are higher forthe earlier pages (Figure 6(c)). 22owever, the prices within a page are monotone decreasing in the product valuations in all threefigures. We explain in detail why this happens in Appendix D for the case of two products on asingle page. In the Appendix, we prove that for a web interface with two products on the same page,the optimal prices reverse the order of product valuations. This happens because the equilibriumprice of p is not a function of v , but is an increasing function of the other product’s price margin v − p . Similarly, the equilibrium price of p is not a function of v , but is an increasing function ofthe other product’s price margin v − p . An increase in v does not have a direct effect on product1’s equilibrium price, but it increases p through the increase in the price margin of product 1(equation (6) in Appendix D). However, increasing on p further decreases the value-price marginof product 2, hence it further decreases the equilibrium price of product 1. And we have the pricesare in reverse order of valuations. To compare the surplus allocation under the two regimes, we consider again the Gamma-distributedprivate valuations studied in Section 5 with shape parameter a , scale parameter b , and constantmean ab = 1. We increase the shape parameter a from 0 . a andthereby the amount of private information. Figure 7 shows the seller’s surplus share (the ratio V p V p + V c of seller profit to total surplus) for N = 6 products for different values of the impatienceparameter ρ , number of products on a page k and acceleration factor κ . (a) ρ = 0 . , k = 2 . (b) ρ = 0 . , k = 3 . (c) ρ = 0 . , k = 6 . (d) ρ = 1 , k = 2 . (e) ρ = 1 , k = 3 . (f) ρ = 1 , k = 6 . Figure 7: seller surplus share as a function of the Gamma shape parameter a for fixed mean ab = 1 and N = 6 products for different values of ρ and k and for κ = 1 or k . As a increases, the variance of the privatevaluations decreases. We first consider how the problem parameters (the Gamma shape parameter a , which deter-mines the amount of private information; the acceleration factor κ , which affects the consumer’s23peed of evaluation; and the number of products presented per page k ) affect the surplus allocationunder the web-interface.Figures 7 show that the seller surplus share is decreasing in the shape parameter a . This isconsistent with our prior intuition and results for the virtual assistant case (Figure 5 in Section 5):as a increases, the amount of private information decreases, which allows the seller to extract moreof the total surplus.Second, as the acceleration factor κ increases, the consumer can explore more products, whichbenefits both the consumer and the seller. As a result, the levels of both the consumer surplusand the seller profit increase. Figures 7 show that this acceleration has only a small effect on thesurplus ratio .Finally, looking across Figures 7 (with the same a ), we observe that the seller’s surplus shareincreases with k . This effect is best interpreted in conjunction with a comparison of the web-interface and the virtual assistant. A key driver of this comparison is the seller’s power of pricecommitment under the web-interface. A well-known fundamental result in game theory is thatbeing able to commit to a strategy before other players move is generally beneficial (cf. Courtyand Hao (2000)).When comparing the web-interface to the virtual assistant, the former enables the seller tocommit to a full page of prices while the virtual assistant dynamically prices a product at a time.Thus, the web interface has higher commitment power, which is an increasing function of thenumber of products per page k . Thus, we expect (i) the web-interface to result in a higher surplusshare for the seller, and (ii) the surplus share to increase with k , which is what we observe in Figure7. As for (ii) , an increase in k also allows the consumer to view more products, but as we haveseen earlier, the effect of this factor on the surplus share is small.However, the viewing rate is always higher in a web interface (web interface: τ p κ , virtualassistant τ p ), which enables the consumer to view more products in a web interface. As we discussedbefore, it favors both consumer and the seller, thus has little influence on the surplus allocation .Overall, the effect of commitment power dominates the effect of viewing rate and as we observefrom figure 7, the surplus allocation is always higher in a web-interface. With an increasing number of consumers choosing to purchase products through virtual assistants,this emerging channel is expected to become an important gateway to commerce. It is importantto understand how the special features of virtual assistants (in particular, the sequential nature ofproduct presentation) affect the market outcomes.In this paper, we developed a model of a forward-looking consumer who strategically makessequential purchase decisions after submitting a request to a virtual assistant which makes rankingand pricing decisions. The virtual assistant may operate on behalf of a profit-maximizing seller, itmay be altruistic, or it may act as a consumer agent. We find the optimal prices under a generalprivate valuation distribution and derive them in closed-form when the distribution is exponential.We find that in the exponential case, a profit-maximizing seller extracts the same surplus as theconsumer. As a result, the profit-maximizing ranking also maximizes the consumer surplus. Foran altruistic seller or a consumer agent, we find that pricing at cost is optimal and the consumerextracts the entire surplus. We develop algorithms for optimally ranking products and find thatthe simple descending or ascending rankings are optimal when consumers are highly patient orimpatient. We propose the double-rank approximation algorithm which is shown to capture atleast 95% of the surplus in numerical experiments.24n July 2020, the European Commission launched an inquiry into the market for consumerproducts and services linked to the Internet of Things with a focus on voice assistants. Citing the“incredible potential” of these devices, Commission Executive Vice-President Margrethe Vestagerfocused on the “risk that some of these players could become gatekeepers of the Internet of Things,with the power to make or break other companies. And these gatekeepers might use that power toharm competition, to the detriment of consumers... whether that’s for a new set of batteries foryour remote control or for your evening takeaway. In either case, the result can be less choice forusers, less opportunity for others to compete, and less innovation” (Vestager (2020)).Analyzing the effects of selling through a VA compared to selling through a traditional webinterface calls for an explicit analysis of the two equilibria. We perform such an analysis where theseller may sell through a VA or through a web interface, where products are presented on multipleweb pages with each page showing multiple products simultaneously. One might expect that whenthe VA maximizes the seller’s profits, it will exploit its gatekeeper control over product presentationto extract a larger share of the surplus. We find that the opposite is true: the seller’s equilibriumsurplus share is in fact larger with a web interface, where it can credibly commit to fixed prices oneach page. We also find that when the consumer’s private valuations are exponentially distributed,the optimal prices within a page are decreasing in product valuations.There are several interesting extensions to this work. First, it will be useful to extend ouranalysis to the case where the consumer has imperfect recall. Second, it is interesting to study athree-sided platform model where strategic suppliers and consumers are mediated through a VA.The VA might first decide on the price of each position. Observing these prices, suppliers mightthen decide which positions to acquire and bring in their products with associated prices. Finally,consumers might use the platform to make buying decisions.25 eferences
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Econometrica: Journal of the EconometricSociety ppendices A Notation • i : Product index; i = 1 , , . . . , N , where N is the number of products.• τ c : a consumer request is live for an exponential duration with rate τ c .• τ p : the presentation and evaluation time of each offer is exponentially distributed with rate τ p .• v i + (cid:15) i : valuation of product i , where• v i : observable part of the valuation of product i ;• (cid:15) i : unobservable (random) part of the valuation of product i , (cid:15) i ∼ F .• α : when (cid:15) i are exponentially distributed, E ( (cid:15) i ) = 1 /α .• σ : product ranking permutation, σ = ( σ , . . . , σ n ).• ρ : probability that the consumer completes a product evaluation.• p i : consumer price of product i .• c i : platform cost to acquire and ship product i when it’s sold to the consumer.• δ i : consumer threshold valuation. The consumer buys product i when u i = v i − p i + (cid:15) i ≥ δ i .• P i = P ( (cid:15) σ i ≥ p σ i + δ σ i − v σ i ) : probability that the consumer buys the i th product offered. B Parameters in Numerical Examples
The numerical examples in sections 4–6 consider the following default setting:• N = 6 products with valuation vector v = (0 , , . , , ) and costs c i = 0( i = 1 , , . . . , (cid:15) ∼ Gamma ( a, b ).Gamma ( a, b ) is the Gamma distribution with shape parameter a and scale parameter b . Thisdistribution is widely used in applications; we summarize here some of its salient properties:• Mean E ( (cid:15) ) = ab ;• Variance V ar ( (cid:15) ) = ab ;• For a fixed mean, increasing the shape parameter a reduces the variance ab = E ( (cid:15) ) /a .• (cid:15) is exponential when a = 1 and has the uniform distribution as its limit as a → ∞ .28 Proofs
Proof of Proposition 3.1
The proof is by backward induction. (a)
Stage N : In stage N , the consumer will purchase if and only if (cid:15) N + v N − p N ≥ δ N = 0 , and the probability of purchase at any price p is 1 − F ( p − v N ). The seller’s expected profitis then given by (1 − F ( p − v N )) · ( p − c N ); denote the maximizer by p ∗ N .The maximizer exists because both at p N = c N and as p N → ∞ , the expected profit goes tozero. The optimal expected profit is positive since p N = c N results in zero profit, and thederivative at c N (given by (1 − F ( c N − v N )) is positive (since ( v i − c i ) are non-positive and F ( · ) is positively supported).The equilibrium probability of purchase in stage N is clearly P N = 1 − F ( p ∗ N − v N ) , and the seller’s expected profit is thus V pN = P N ( p ∗ N − c N ) . The consumer’s expected surplus under the optimal price is equal to the consumer’s expectedvaluation times the purchase probability conditional on the valuation exceeding the stage- N threshold, hence V cN = P N E ( (cid:15) N + v N − p ∗ N | (cid:15) N + v N − p ∗ N ≥ δ N ) . (b) We next use backward induction. In stage i = N − , . . . ,
1, the consumer will purchase if andonly if her current-stage realized valuation is greater than her expected future return, i.e., if (cid:15) i + v i − p i ≥ ρV ci +1 , which gives us a threshold-policy structure with δ i = ρV ci +1 .It follows that the probability of purchase given any price p is 1 − F ( p − v i + ρV ci +1 ). Theseller’s expected profit is the sum of its expected gain when the consumer accepts the currentoffer and its expected gain when the consumer continues to the next stage, given by:(1 − F ( p − v i + ρV ci +1 ))( p − c i ) + F ( p − v i + ρV ci +1 ) ρV pi +1 . A similar argument to the one for i = N shows that a finite maximizer always exists, theequilibrium probability of purchase in stage i is P i = 1 − F ( p ∗ i − v i + ρV ci +1 ), the seller’sexpected profit in stage i is V pi = P i ( p ∗ i − c i ) + (1 − P i ) ρV pi , and the expected consumer surplus is V ci = P i E ( (cid:15) i + v i − p ∗ i | (cid:15) i + v i − p ∗ i ≥ δ i )) + (1 − P i ) ρV ci +1 . roof of Corollary 3.2 The results follow from Proposition 3.1 with (cid:15) ∼ exp( α ). (a) In stage N , δ N = 0 and since (cid:15) ∼ exp( α ), the seller’s expected profit is given by (1 − F ( p − v N ))( p − c N ) with optimal price p ∗ N = c N + 1 − F ( p N − v N ) f ( p N − v N ) = c N + 1 α . Plugging p ∗ N into the expression for P N , we have P N = exp( − α ( c N + 1 α − v N )) = exp( α ( v N − c N ) − V pN = P N ( p ∗ N − c N ) = 1 α exp( α ( v N − c N ) −
1) = 1 α P N . Now, plugging p ∗ N and δ N into the expression for V cN and using the memeoryless property ofthe exponential distribution, we get V cN = P N E ( (cid:15) N + v N − p ∗ N | (cid:15) N + v N − p ∗ N ≥ δ N )= 1 α exp( α ( v N − c N ) −
1) = 1 α P N . (b) For stages i = N − , . . . ,
1, Proposition 3.1 implies δ i = ρV ci +1 with optimal price p ∗ i = arg max p (1 − F ( p − v i + ρV ci +1 ))( p − c i ) + F ( p − v i + ρV ci +1 ) ρV pi +1 = c i + 1 − F ( δ i ) f ( δ i ) + ρV p i +1 = c i + 1 α + ρV pi +1 . The equilibrium probability of purchase in stage i is given by P i = 1 − F ( p ∗ i − v i + ρV c i + 1) = exp( α ( v i − c i ) −
1) exp( − αρ ( V ci +1 + V pi +1 )) , the expected seller profit is V pi = P i ( p ∗ i − c i ) + (1 − P i ) ρV pi +1 = 1 α P i + ρV pi +1 = 1 α N X k = i ρ k − i P k , and the expected consumer surplus is V ci = P i E ( (cid:15) i + v i − p ∗ i | (cid:15) i + v i − p ∗ i ≥ δ i )) + (1 − P i ) ρV ci +1 = 1 α P i + ρV ci +1 = 1 α N X k = i ρ k − i P k . roof of Proposition 3.4 We prove proposition 3.4 by backward induction. (a)
In stage N , the optimal threshold is δ N = 0 and the total surplus is V sN = E ( (cid:15) N + v N − c N | (cid:15) N + v N − p N ) P ( (cid:15) N + v N − p N ≥ δ N )= Z ∞ δ N + p N − v N ( x + v N − c N ) f ( x )d x. Now, ∂V sN ∂p N = − ( δ N + p N − c N ) f ( δ N + p N − v N ) = − ( p N − c N ) f ( p N − v N ) . Since F ( · ) has positive support and v N ≤ c N , V sN is constant in ( −∞ , v N ), strictly increasingin ( v N , c N ], and strictly decreasing in ( c N , ∞ ) . Thus, the optimal price is p N = c N , V sN = V cN and V pN = 0 . (b) In stage i = N − , . . . , δ i = ρV ci +1 = ρV si +1 , the expected total surplus is V si = E ( (cid:15) i + v i − c i | (cid:15) i + v i − p i ≥ δ i ) P ( (cid:15) i + v i − p i ≥ δ i ) + (1 − P ( (cid:15) i + v i − p i ≥ δ i )) ρV si +1 = Z ∞ δ i + p i − v i ( x + v i − c i ) f ( x )d x + (1 − Z ∞ δ i + p i − v i f ( x )) ρV si +1 = Z ∞ δ i + p i − v i ( x + v i − c i − ρV si +1 ) f ( x )d x + ρV si +1 , and ∂V si ∂p i = − ( δ i − ρV si +1 p i − c i ) f ( δ i + p i − v i ) = − ( p i − c i ) f ( δ i + p i − v i ) . Again, V si is constant in ( −∞ , v i − δ i ), strictly increasing in ( v i − δ i , c i ], and strictly decreasingin ( c i , ∞ ) . Thus, the optimal price is p i = c i and the inductive hypothesis holds.Finally, since V ip = 0 for all i , p i = c i maximizes both the expected consumer surplus and thetotal surplus. Proof of Proposition 4.1 (a) We show that there exists a ρ > ρ ∈ [0 , ρ ), the descending ranking (invalue margins) is optimal. We first prove the result for a profit-maximizing seller and then for analtruistic seller or a consumer agent.Let the ranking permutation be σ . For a profit-maximizing VA, following Corollary 3.2, theseller’s expected profit in the first stage is given by V p = 1 α N X k =1 ρ k − P k . ρ →
0, as P k are bounded for all k , the the expected first-stage profit is given by V p = 1 α P + o ( ρ )by Corollary 3.2. Let M σ ( i ) = exp( α ( v σ ( i ) − c σ ( i ) ). Then, P ( ρ ) = exp( α ( v σ (1) − c σ (1) ) −
1) exp( − αρ (2 V p )) = M σ ( i ) exp( −
1) exp( − αρ (2 V p )) . Taking ρ → − αρ (2 V p )), we have P ( ρ ) = M σ ( i ) exp( − − αV p ρ + o ( ρ ) . Since V p is bounded, V p = M σ ( i ) exp( −
1) + o ( ρ ) . Since M σ ( i ) is increasing in v σ ( i ) − c σ ( i ) , the VA maximizes V p by placing the product with thehighest value-margin first.It is of course possible that the consumer will progress to the second stage (due to a small enoughrealization of (cid:15) ). By Corollary 3.2, the corresponding expected profit is V p = 1 α N X k =2 ρ k − P k . Similar to our analysis of stage 1, V p = M σ (2) exp( −
1) + o ( ρ ) , and the optimal σ (2) is given by the remaining product with the highest value-margin, namely theone with the second-highest value margin. By induction, the optimal ranking as ρ → p ∗ i = c i + 1 α + ρV pi +1 = c i + 1 α + o ( ρ ) , since V pi +1 are bounded.We next consider the case of a consumer agent VA. Let the ranking permutation be σ . ByCorollary 3.5, the expected first-stage surplus is given by V c = P E ( (cid:15) + v σ (1) − c σ (1) | (cid:15) + v σ (1) − c σ (1) ≥ ρV c ) + (1 − P ) ρV c . As (1 − P ) V c is bounded from above, taking ρ → V c = P E ( (cid:15) + v σ (1) − c σ (1) | (cid:15) + v σ (1) − c σ (1) ≥ ρV c ) + o ( ρ )= (1 − F ( − ( v σ (1) − c σ (1) ))) E ( (cid:15) + v σ (1) − c σ (1) | (cid:15) + v σ (1) − c σ (1) ≥
0) + o ( ρ ) . For t = v i − c i , ∂∂t (1 − F ( − t )) Z ∞− t ( (cid:15) + t ) f ( (cid:15) ) d(cid:15) = (1 − F ( − t )) Z ∞− t f ( (cid:15) ) d(cid:15) + f ( − t ) Z ∞− t ( (cid:15) + t ) f ( (cid:15) ) d(cid:15) ≥ , V c is increasing in the value margin of the first product presented. To maximize V c , the VAshould thus place the item with the highest value-margin first.Proceeding to stage 2, from Corollary 3.5 the expected consumer surplus is given by V c = P E ( (cid:15) + v σ (2) − c σ (2) | (cid:15) + v σ (2) − c σ (2) ≥ ρV c ) + (1 − P ) ρV c . As before, as ρ → V c is dominated by the expected consumer surplus at stage 2: V c = (1 − F ( − ( v σ (2) − c σ (2) ))) E ( (cid:15) + v σ (2) − c σ (2) | (cid:15) + v σ (2) − c σ (2) ≥
0) + o ( ρ ) . Following the foregoing analysis, since V c is increasing in the value-margin of the second product,the σ (2) that maximizes V c is given by the product with second-highest value margin. By induc-tion, the optimal ranking as ρ → Claim: If σ = ( σ (1) , · · · , σ ( N )) is the optimal ranking, the sub-permutation ¯ σ i := ( σ ( i ) , σ ( i +1) , · · · , σ ( N )) is the optimal ranking for the subproblem from stage i through N , i.e., V pi (¯ σ i ) =max σ V pi ( σ ). Proof:
Let M i = exp( α ( v i − c i )) and rewrite V pi as a function of M σ ( i ) and V pi +1 : V pi = f ( M σ ( i ) , V pi +1 ). Now, V pi = f ( M σ ( i ) , V pi +1 ) = P i ( p σ ( i ) − c σ ( i ) ) + (1 − P i ) ρV pi +1 (3)= P i ( 1 α + ρV pi +1 ) + (1 − P i ) ρV pi +1 = 1 α exp( α ( v σ ( i ) − c σ ( i ) ) −
1) exp( − αρV pi +1 ) + ρV pi +1 = 1 αe exp( − αρV pi +1 ) M σ ( i ) + ρV pi +1 . We have that ∂f∂V = ρ (1 − e e − αV M ) >
0, which implies that V pi is increasing in V pi +1 .Now, when we fix the order of the first i − ∂V p ∂V pi = ∂V p ∂V p ∂V p ∂V p · · · ∂V pi − ∂V pi >
0. Therefore,if ¯ σ i does not maximize V pi , there exists another ¯ σ i such that V pi ( ¯ σ i ) > V pi (¯ σ i ). But then V p (( σ (1) , · · · , σ ( i − , ¯ σ i )) > V p (( σ (1) , · · · , σ ( i − , ¯ σ i )) = V p ( σ ) , thereby contradicting the optimality of σ. We next show by backward induction that for ρ = 1 and an optimal ranking σ , each sub-permutation¯ σ i satisfies the monotonicity condition v σ ( i ) − c σ ( i ) ≤ v σ ( i +1) − c σ ( i +1) ≤ · · · ≤ v σ ( N ) − c σ ( N ) , orequivalently M σ ( i ) ≤ M σ ( i +1) ≤ · · · ≤ M σ ( N ) for all i .• Baseline case:
The statement is trivially true for the baseline case when i = N .• Induction:
Assume the statement holds for for k = i + 1 , · · · , N . We prove it also holds for k = i, namely M σ ( i ) ≤ M σ ( i +1) . 33ssume by contradiction that M σ ( i ) > M σ ( i +1) . Then, we switch products σ ( i ) and σ ( i + 1)to obtain the permutation σ : σ = ( σ (1) , σ (2) , · · · , σ ( i + 1) , σ ( i ) , σ ( i + 2) , · · · , σ ( N )) . We will show that the sub-permutation ¯ σ i := { σ ( i ) , σ ( i + 1) , · · · , σ ( N ) } doesn’t maximize V pi , contradicting the inductive hypothesis.Since σ and σ only differ in the i − th and ( i + 1)-th positions, ¯ σ i +2 = ¯ σ i +2 and we de-note V pi +2 (¯ σ i +2 ) = V pi +2 ( ¯ σ i +2 ) by ¯ V .
The expected profits under the two rankings are asfollows: V pi +1 (¯ σ i +1 ) = ¯ V + M σ ( i +1) αe e − α ¯ V ,V pi (¯ σ i ) = ¯ V + M σ ( i +1) αe e − α ¯ V + M σ ( i ) αe e − αρ ¯ V − e e − α ¯ V M σ ( i +1) ,V pi +1 ( ¯ σ i +1 ) = ¯ V + M σ ( i ) αe e − α ¯ V , and V pi ( ¯ σ i ) = ¯ V + M σ ( i ) αe e − α ¯ V + M σ ( i +1) αe e − α ¯ V − e e − α ¯ V M σ ( i ) . Showing that V pi (¯ σ i ) < V pi ( ¯ σ i ) is equivalent to showing that the function g ( x ) := 1 x − exp( − e exp( − α ¯ V ) x ) x is increasing in x . Let e exp( − α ¯ V ) =: k <
1, then g ( x ) = − x + exp( − kx ) x . The first derivative of g ( x ) is g ( x ) = 1 − exp( − kx )( kx + 1) x . Since exp( − kx )( kx + 1) < < kx <
1, it follows that g ( x ) > V pi (¯ σ i ) < V pi ( ¯ σ i ).This contradicts the claim that ¯ σ i maximizes V pi .From equation (3) we know that V pi ( σ ) andV pi ( σ ) are continuous in ρ. By continuity, thereexists a ρ such that the monotonicity would hold for all ρ ∈ ( ρ , ρ such that for all ρ ∈ ( ρ , M σ ( i ) < M σ ( i +1) < · · · < M σ ( N ) .We next prove the result for an altruistic VA and a consumer agent. Notice that in the proofof the profit-maximizing case, the key was to show that (i) V pi is monotone in V pi +1 , and (ii) V pi (¯ σ i ) < V pi ( ¯ σ i ) when ρ = 1. To prove (i) , we have for a consumer agent (and an altruisticplatform) with ρ = 1: V pi = f ( M σ ( i ) , V i +1 ) = M σ ( i ) α exp( − αρV pi +1 ) + ρV pi +1 , and∂f∂V pi +1 = ρ − ρM σ ( i ) exp( − αρV pi +1 ) > (i) . Similarly, V pi (¯ σ i ) = ¯ V + M σ ( i ) α exp( − α ¯ V ) + M σ ( i ) α exp( − α ¯ V − M σ ( i +1) exp( − α ¯ V )) ,V pi ( ¯ σ i ) = ¯ V + M σ ( i ) α exp( − α ¯ V ) + M σ ( i +1) α exp( − α ¯ V − M σ ( i ) exp( − α ¯ V )) , andαM σ ( i ) M σ ( i +1) ( V i ( ¯ σ i ) − V i (¯ σ i )) = 1 M σ ( i ) exp( − α ¯ V − M σ ( i ) exp( − α ¯ V )) − M σ ( i ) exp( − α ¯ V ) − M σ ( i +1) exp( − α ¯ V − M σ ( i +1) exp( − α ¯ V )) − M σ ( i +1) exp( − α ¯ V . ) Now, letting k = exp( − α ¯ V ) < g ( x ) = − x + exp( − exp( − α ¯ V ) x ) x is increasing in x . Thus, the monotonicity holds for ρ = 1 as inthe profit-maximizing case and continuity implies that there exist a ρ such that for all ρ ∈ ( ρ , v i − c i ) is optimal. Proof of Proposition 5.1
We prove Proposition 5.1 by backward-induction. We’ll be using Corollary 3.2 that V pi = V ci inthe calculation below.We first prove the monotonicity of the continuation valuations in the customer patience parame-ter ρ . Specifically, we show that under any given ranking σ and in any given stage i ∈ { , . . . , N − } ,the continuation value V pi +1 is monotone increasing in ρ .• Inductive hypothesis: V ci ( V pi ) is (weakly) monotone increasing in ρ , i.e., ∂V ci ∂ρ ( ∂V pi ∂ρ ) ≥ i ∈ { N, N − , . . . , } .• Baseline case:
When i = N , the expected consumer surplus (which equals to the platformprofit by Corollary 3.2) is given by V cN = V pN = 1 α exp( α ( v N − c N ) − , which is constant and is weakly increasing in ρ. • Induction:
In period i , i = N − , · · · ,
1, the expected value is V ci = V pi = 1 α P i + ρV ci +1 = 1 α exp( − α (2 ρV ci +1 + c i − v i + 1 α )) + ρV ci +1 , and ∂V ci ∂ρ = ∂V pi ∂ρ = ( − ρ ) exp( − α (2 ρV ci +1 + c i − v i + 1 α )) ∂V ci +1 ∂ρ + V ci +1 + ρ ∂V ci +1 ∂ρ = ρ ∂V ci +1 ∂ρ [1 − e exp( − α (2 ρV ci +1 + c i − v i ))] + V ci +1 . By the inductive hypothesis, ∂V ci +1 ∂ρ is non-negative. In addition, 2 ρV ci +1 + c i − v i is non-negativebecause c i ≥ v i and 2 ρV ci +1 is non-negative. Therefore, [1 − e exp( − α (2 ρV ci +1 + c i − v i ))] isnon-negative, which completes the proof. 35ombining this result with corollary 3.2, we have, p ∗ i = c i + α + ρV pi +1 , where V pi +1 is increasing in ρ . It follows that each product’s optimal price is an increasing function of the consumer patienceparameter ρ . D Web interface: On the relationship between valuations andprices within a page
In this Appendix we analyze the profit-maximizing pricing of two products within a single page( k = 2) to better understand the surprising inverse relationship between prices and expected valua-tions within a page. As in Section 5, we assume the private valuations are exponentially distributedwith unit mean c = c = 0. The following Proposition shows that the product with the higherexpected valuation is priced lower . Proposition D.1
For a profit-maximizing VA, v > v implies p < p . Proof : The prices p and p are set to maximize the seller’s expected profit P p + P p , P i ( i = 1 , i is given by P i = Z ∞ p i − v i Z x + v i − p i − v j + p j exp( − ( x + y ))d y d x = Z ∞ p i − v i exp( − x )(1 − exp( − ( x + v i − p i − v j + p j ))d x (4)= exp( − ( p i − v i )) −
12 exp( v i + v j − p i − p j ) . The first order conditions simplify to p = 1 + p p − v ) − , and (5) p = 1 + p p − v ) − . (6)Equations (5)-(6) imply that p is increasing in v and decreasing in p , and that 1 < p , p < p + p − p − p − v ) , and (7) p + p − p − p − v ) . (8)Dividing equation (7) by equation (8) and rearranging, we get η ( p ) η ( p ) = exp( − ( v − v )) < , where η ( p ) = p − p ) . Since η ( p ) is an increasing function for p < η ( p ) < η ( p ) implies p < p , whichcompletes the proof.A more intuitive way of reaching this result is by starting with the case of two products withequal valuations, v = v , and increasing v . Obviously, when v = v , p ∗ = p ∗ . Now increase v to v > v . First, notice that this has no direct effect on p : by equation (7), p is not a function of v . Raising v does increase p : by equation (8), p is an increasing function of v . Now, increasing36 leads to a decrease in p : by equation (4), p is a decreasing function of p . We thus movedfrom an initial equilibrium with p ∗ = p ∗ to a new equilibrium ( p , p ) with p < p ∗ and p > p ∗ (see Figure 8). Thus, prices have moved in the opposite direction to that of expected valuations. Figure 8:
Equation (5)-(6) with different parameters v i . The red and the blue lines correspond to equations(5) and (6), respectively, when v = v = 1. The dotted blue line corresponds to equation (6) with v > v ..