The Limits of Optimal Pricing in the Dark

1 Introduction . Pricing is a basic question in microeconomics [ 24 ] as well as a ubiquitous problem in today ’ s digital markets [ 16 ] . In its textbook style setup , there are two agents : a buyer ( he ) and a seller ( she ) . The seller produces d types of divisible goods for sale and has production cost c ( x ) for producing bundle x ∈ Rd+ of the d goods . Using a standard linear pricing scheme with unit price pi for goods i ∈ [ d ] , the seller charges the buyer x · p when he purchases bundle x ∈ Rd+ under price vector p ∈ Rd+ . Naturally , the buyer ’ s optimal bundle for purchase depends on his value function v ( x ) about the products . We assume that , given price vector p , the rational buyer will always pick the optimal bundle that maximizes his quasilinear utility [ v ( x ) − x · p ] . The key question of interest is how the seller can compute the revenue-maximizing optimal prices assuming rational buyer purchase behaviors . ∗This work is supported by a Google Faculty Research Award and an NSF grant CCF-2132506 . 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . The above basic problem can be easily solved if the seller has access to the buyer ’ s value function v ( x ) . However , in practice , this value function is usually private and unknown to the seller . To address this challenge , there has been a rich line of recent research which looks to design the optimal pricing scheme against an unknown buyer . Some of these work have adopted learning-based approaches that aim to compute the profit maximizing price by interacting with the buyer and gleaning information about the buyer ’ s utility function [ 3 , 33 , 32 ] . These are motivated by the wide spread of e-commerce today where the seller can repeatedly interact with the same buyer or buyers from the same population with similar preferences . For example , in online advertising , ad exchange platforms learn to price advertisers from past behaviors ; in online retailing , retailers learn to price customers from their past purchase history ; and in crowdsourcing , platforms learn to reward workers ’ efforts . Another line of works look to design dynamic pricing mechanisms that dynamically adjusts the price based on the observed past buyer purchase behaviors with the objective of maximizing the aggregated total revenue [ 4 , 5 , 26 , 27 , 38 ] . A key challenge of pricing against such unknown buyers , regardless through learning or dynamic pricing , is that the buyer may manipulate his responses in order to mislead the seller ’ s algorithm and induce more favorable outcomes for themselves . This is a fundamental issue when learning from strategic data sources , and is particularly relevant in this pricing setup due to the strategic nature of the problem . In this case , the buyer controls the information content to the seller , therefore he naturally has incentives to utilize this advantage to gain more payoff by strategically misleading the seller ’ s algorithms . Indeed , as observed in previous studies , online advertisers may strategically respond to an ad exchange platform ’ s prices in order to induce a lower future price [ 22 ] ; Consumers strategically time their purchases in order to obtain lower prices at online retailing platforms [ 23 ] . To understand the limits of the seller ’ s optimal pricing against such a strategic and possibly manipulative buyer , we put forward a natural model which augments the above basic pricing problem with only one additional step — i.e. , we assume that the game starts with the buyer committing to a different value functions , coined the imitative value function , after which the seller will compute the optimal pricing scheme against this imitative buyer value function . The buyer ’ s commitment to an imitative value function captures a simple yet powerful buyer manipulation strategy that can be used against any seller learning algorithms — that is , the buyer simply behaves consistently according to this imitative value function during the entire interaction process . Intuitively , the motivation of such commitment assumption comes directly from the fact that the seller has no information about the buyer and has to optimize pricing “ in the dark ” . Consequently , if the buyer consistently behaves according to some different imitative value function , the seller is not able to distinguish this buyer from another buyer who truly has the value function ( see more justifications of the commitment assumption in Section 3 ) . Moreover , such an imitation strategy is also easy to execute in practice by the buyer regardless whether the seller is learning from him or is adopting dynamic pricing schemes . In fact , the buyer could even just report his imitative value function to the seller directly at the beginning of any interaction . In this situation , no learning or dynamic pricing will be needed as the buyer will indeed always behave according to the imitative value function . Therefore , all the seller can do is to directly apply the optimal pricing scheme for the buyer ’ s imitative value function . We remark that such imitation-based manipulation strategy has attracted much interest in recent works , with similar motivations as us — i.e. , trying to understand how to manipulate learning algorithm or conversely , how to design strategy-aware learning algorithms to mitigate such manipulation . However , most of these works have focused on the general Stackelberg game model [ 19 , 10 ] as well as the Stackelberg security games [ 18 , 30 , 29 ] . Our optimal pricing problem is also a Stackelberg model and thus a natural fit for the study . The crucial difference between our work and previous studies is that both agents in our model have continuous utility functions whereas all these previous works [ 19 , 18 , 30 , 10 ] have discrete agent utility functions . Therefore , our model leads to a functional analysis and optimization problem , which is more involved . Fortunately , we show that the optimal functional solution can still be characterized by leveraging the structure of the pricing problem . 1.1 Our Results and Implications . Given the effectiveness and easy applicability of the buyer ’ s imitative strategy described above , this paper studies what the optimal buyer imitative value function is and how it would affect buyer ’ s surplus and the seller ’ s revenue . Our main result provides a full characterization about the optimal buyer imitative value function . We show that the optimal buyer imitative value function u∗ features a specific bundle of products x∗ that is most desirable to the buyer . Interestingly , it turns out that u∗ is a Leontief-type piece-wise linear concave value function [ 2 ] such that the buyer would only proportionally value a fraction of the desired bundle x∗ and nothing else . Moreover , we also characterize the seller revenue and buyer surplus at equilibrium as well as necessary and sufficient conditions under which the seller obtains strictly positive revenue . The optimal buyer imitative value function turns out to depend crucially on the seller ’ s production cost function . When the cost function is concave , we show that the optimal buyer imitative value function u∗ will “ squeeze ” the seller revenue essentially to 0 . The fortunate news for the seller , however , is that finding out the u∗ turns out to be an NP-hard task for the buyer . In fact , we prove that it is NP-hard to find the u∗ that can guarantee a polynomial faction of the optimal buyer surplus . For the widely adopted convex production cost , we show that the equilibrium seller revenue is the Bregman divergence between production 0 and the x∗ bundle mentioned above . This illustrates an interesting message that convex production costs are “ better ” at handling buyer ’ s strategic manipulations . Note that production costs are indeed more often believed to be convex since economic models typically assume that marginal costs increase as quantity goes up [ 35 , 34 ] . All our characterizations so far focus on the standard linear pricing scheme . Our last result examines the possibility of using a more general class of pricing schemes to address the buyer ’ s strategic manipulation . Surprisingly , we show that for the strictly more general class of concave pricing functions ( i.e. , the seller is allowed to use any concave function as a pricing function ) , the equilibrium will remain the same as described above . Therefore , the more general pricing schemes do not necessarily help to address buyer ’ s manipulation behavior . In fact , we show that there exist examples where strictly broader class of pricing functions leads to strictly worse seller revenue . This is because more general pricing schemes may “ overfit ” buyer ’ s incentives , which renders it easier to manipulate . This illustrates an interesting phenomenon in learning from strategic data sources and shares similar spirit to overfitting in standard machine learning tasks . 1.2 Additional Related Works . Due to space limit , here we only briefly discuss the most related works while refer readers to Appendix A.1 for more detailed discussions and comparisons . Closely related to ours is a recent study by Tang and Zeng [ 36 ] . They study the bidders ’ problem of committing to “ imitate ” a fake value distribution in auctions and acting consistently as if the bidder ’ s value were from the fake distribution . This is similar in spirit to our buyer ’ s commitment to an imitative value function . However , there is significant difference betweens our setting and that of [ 36 ] , which leads to very different conclusions as well . Specifically , the seller in [ 36 ] auctions a single indivisible item with no production costs whereas our seller sells multiple divisible items with production costs . Another recent work [ 28 ] also studies buyer ’ s strategic manipulation against seller ’ s pricing algorithms , but in a single-item multi-buyer situation . Moreover , they assume a fixed seller learning strategy ( thus not adaptive to buyer ’ s strategy ) with separated exploration-exploitation phases motivated by [ 4 ] . Another very relevant literature is learning the optimal prices or optimizing aggregated total revenue by repeatedly interacting with a single buyer [ 4 , 5 , 26 , 27 , 38 ] . These works all focus on designing learning algorithms that can learn from strategically buyer responses . Our work complements this literature by studying the limits of what learning algorithms can achieve . Moreover , the setups of these previous works are also different from us – they either assume buyer values are drawn from distributions [ 4 , 5 , 27 ] or the seller sells a single indivisible good with discrete agent utilities . Thus , their results are not comparable to us . There have also been studies on learning the optimal prices from truthful revealed preferences , i.e. , assuming the buyer will honestly best respond to seller prices [ 8 , 39 , 6 , 42 , 3 , 33 , 32 ] . Our works try to understand if the assumption of truthful revealed preferences does not hold and if the buyer will strategically respond to the seller ’ s learning , what learning outcome could be expected when the buyer simply imitates a different value function that is optimally chosen . From this perspective , these works serve as a key motivation for the present paper . More generally , our work subscribes to the general line of research on learning from strategic data sources . Most works in this space has focused on classification [ 11 , 20 , 43 , 17 , 25 , 21 , 12 ] , regression problems [ 31 , 15 , 13 ] and distinguishing distributions [ 40 , 41 ] . Our work however focuses on learning the optimal pricing scheme .

Summary: This paper consider a problem of buying/selling multiple divisible items with a posted price mechanism between one buyer and one seller, in a noiseless setting. It consider the problem that through repeated trades, buyer and seller learn from each other, leading them to adopt strategies that maximizes their revenue over time, breaking classical incentive compatibility properties. More precisely, this paper considers that  * given a mechanism (seller) and a value function (buyer), the buyer plays a one-shot best response. * given a value function (buyer), the seller chooses the optimal mechanism (knowing the buyer will play best response) * given this previous knowledge, the buyer chooses a *fake* value function to announce to the seller to optimize his objective.  This paper describes the properties of the solutions of this tri-level optimization problem and characterize the complexity of solving it.