Paper: Pricing with Contextual Elasticity and Heteroscedastic Valuation (ICLR-2024)

Reviewer comment (Theory):
The motivation for this contextual price elasticity seems unclear.

Author response:
As in the generalized linear demand model $S(\alpha p + x_t^\top\beta)$, the real-world motivations of assuming the price elasticity $\alpha=x_t^{\top}\eta^*$ as contextual are mainly from the fact that different products have different price elasticities [Anderson et al. 1997], given how crucial they are for our daily life and work.

The assumption that $\alpha=x_t^\top\eta^*>0$, i.e. a negative elasticity, is necessary for the monotonicity of a demand-price function, given that the link function $S(\cdot)=1-F(\cdot)$ is non-increasing. Stated as *the law of demand* [Gale, 1955] [Hildenbrand, 1983], that the quantity purchased varies inversely with price, is derived from the law of diminishing marginal utilities and has been widely accepted and used [Marshall, 2009].

The assumption that $\exists C_{\beta}>0$ as a positive constant such that $x_t^{\top}\eta>C_{\beta}>0$ comes from the boundedness of the noise variance. Recall that an equivalent valuation model for our generalized linear demand is $y_t = \frac1{x_t^\top\eta^*}\cdot(x_t^\top\theta^*+N_t)$ where $N_t$ is an iid noise. Therefore, we know that $Var(y_t) = \frac1{(x_t^\top\eta^*)^2}Var(N_t)$. Therefore, it is necessary to assume the existence of $C_{\beta}$ in order to achieve a finite variance bound of $y_t$.

We will include our explanation above and add more details in the updated version.

Reviewer comment (Theory):
Certain assumptions, such as $x^\top \eta$ having a positive lower bound, lack a real-world explanation.

Author response:
The assumption that $\alpha=x_t^\top\eta^*>0$, i.e. a negative elasticity, is necessary for the monotonicity of a demand-price function, given that the link function $S(\cdot)=1-F(\cdot)$ is non-increasing. Stated as *the law of demand* [Gale, 1955] [Hildenbrand, 1983], that the quantity purchased varies inversely with price, is derived from the law of diminishing marginal utilities and has been widely accepted and used [Marshall, 2009].

The assumption that $\exists C_{\beta}>0$ as a positive constant such that $x_t^{\top}\eta>C_{\beta}>0$ comes from the boundedness of the noise variance. Recall that an equivalent valuation model for our generalized linear demand is $y_t = \frac1{x_t^\top\eta^*}\cdot(x_t^\top\theta^*+N_t)$ where $N_t$ is an iid noise. Therefore, we know that $Var(y_t) = \frac1{(x_t^\top\eta^*)^2}Var(N_t)$. Therefore, it is necessary to assume the existence of $C_{\beta}$ in order to achieve a finite variance bound of $y_t$.

We will include our explanation above and add more details in the updated version.

Reviewer comment (Experiments):
Lack of applying this framework to real-data studies.

Author response:
We are actually motivated by real-world scenarios to consider a heteroscedastic setting where the price elasticity is feature-based. However, it is unfortunate that we are unable to have real-world evaluations of our algorithm, which requires either massive investments or confidential commercial-use data. On the one hand, a pricing algorithm cannot be fairly evaluated until being put into market. This is because customers’ valuations demand curves are never revealed precisely, causing a totally unknown demand on any not-proposed prices. Therefore, if an algorithm does not propose the historical price, then we cannot know how the demand would have been if this algorithm was applied. On the other hand, although big online retail companies could build up real-world demand simulators that are trained on historical sales records, we still have no access to either the simulator or the data as they are confidential. Besides, our contributions are mainly in theoretical aspects, and a real-world demand simulator is itself a challenging research topic that is out of the scope of this work.

Reviewer comment (Theory):
About the linear assumption on $x_t^\top \eta$, can this be generalized to some non-linear function of $x_t$? Also, when $x_t$ is stochastic, can the assumption of $x_t^\top \eta>0$ be relaxed to $E[x_t^\top \eta]>0$, where $E[\cdot]$ is the expectation over $x$?

Author response:
As for the generalization of our linear price elasticity model (i.e. $\alpha=x_t^\top\eta^*$), we believe that our algorithm design and analysis are still applicable within a slight generalization from Euclidean space to *known* kernel spaces. As for the generalization from $x_t^\top\eta^*>0$ to $\mathbb{E}[x_t^\top\eta^*]>0$ for stochastic setting, we do not think it is necessary given *the law of demand*, i.e. the necessity of assuming $x_t^\top>0$ listed above as our response to your first question.

Reviewer comment (Experiments):
Can the authors provide a real-world (or semi-real) data study on evaluating the performance of algorithms in real-life situations?

Author response:
We are actually motivated by real-world scenarios to consider a heteroscedastic setting where the price elasticity is feature-based. However, it is unfortunate that we are unable to have real-world evaluations of our algorithm, which requires either massive investments or confidential commercial-use data. On the one hand, a pricing algorithm cannot be fairly evaluated until being put into market. This is because customers’ valuations demand curves are never revealed precisely, causing a totally unknown demand on any not-proposed prices. Therefore, if an algorithm does not propose the historical price, then we cannot know how the demand would have been if this algorithm was applied. On the other hand, although big online retail companies could build up real-world demand simulators that are trained on historical sales records, we still have no access to either the simulator or the data as they are confidential. Besides, our contributions are mainly in theoretical aspects, and a real-world demand simulator is itself a challenging research topic that is out of the scope of this work.

Reviewer comment (Presentation):
In terms of the presentation of simulation results, could the authors present log-log plots and compare them with the $1/2 \log T$ curve? Since it would be hard to see the regret order if they are not presented in this way.

Author response:
It’s a good catch that a log-log plot would better show the regret rate. In fact, our plots are indeed presented in log-log diagrams, and therefore a slope-$\alpha$ line indicates an $O(T^{\alpha})$ regret.To show the regret rate, we also plotted the linear asymptote of each regret curve. The curve of $\frac12\log T$ on a logT-diagram as you suggested should be a straight line with slope $0.5$ adding some constant. This idea is brilliant, but it is not necessary since we already have such asymptotes.

Reviewer comment (Novelty):
In my opinion, Ban and Keskin (2021) should be given more credits. As far as I know, Ban and Keskin (2021) is the first to consider the heterogenous price elasticities which are formulated to be linear with context. At least when introducing the formulation, I think the paper should be cited and discussed more.

Author response:
1,  We have attributed our model to Ban and Keskin (2021) who introduced the generalized linear demand model with heterogenous price elasticity (coefficient) for the first time, which well-motivates a feature-based heteroscedasticity and how it affects the price-demand relationship.  We also reduce a broadly adopted linear-valuation model to this generalized linear-demand model and show that they are equivalent.

Our demand model is generally similar to theirs, but we generalize the demand model from two perspectives: (1) We assume the incoming feature sequence $[x_t]_{t=1}^T$ is adversarial while they assume iid series. (2) We assume a Boolean-censored demand while they assume an add-on iid noise on the demand. Those generalizations escalate the problem hardness substantially. As a result, their ILQX algorithm, which highly depends on the concentration of the empirical Fisher Information, shall not achieve $O(d\sqrt{T}\log T)$ in an adversarial (instead of iid) environment as we assume.

Compared with our previous submission, we have added more in-depth discussion about Ban and Keskin (2021) in the current literature review. Please kindly see Sec 2.

Reviewer comment (Theory):
I understand that a known link function is a good starting point and a common practice. One direction that I think might further improve the paper is to consider (or at least discuss about) an unknown link function. The reason why I mention this point is that Fan et al. (2021) studies a problem with unknown noise distribution. According to equivalence of the two formulation, it seems that it is not undoable to consider a version without knowing the link function.

Author response:
2, Thanks for your suggestions! Unfortunately, our algorithm is unable to be generalized to the online contextual pricing problem with linear valuation and unknown noise distribution that has been studied by Fan et al. (2023). Indeed, the problem becomes substantially harder when the noise distribution is unknown to the agent. Existing works usually adopt bandits or bandit-like algorithms to tackle that problem. For example, Fan et al. (2022) approaches it with a combination of exploration-first and kernel method (or equivalently, local polynomial), Luo et al. (2023) uses a UCB-styled algorithm, and Xu et al. (2022) adopts a discrete EXP-4 algorithm. However, none of them close the regret gap even under the *homoscedastic elasticity* environment as they assumed, and the known lower bound is at least $\Omega(T^{\frac23})$, or $\Omega(T^{\frac{m+1}{2m+1}})$ for smooth ones (Wang et al. 2021). On the other hand, we study a parametric model, and it is not quite suitable for a bandit algorithm to achieve optimality in regret. In a nutshell, these two problems (known vs unknown noise distributions), although seem similar to each other, are indeed substantially different.

Compared with our previous submission, we have included a detailed discussion about this unknown link function setting. Please kindly refer to Appendix C.

References:

Fan, Jianqing, Yongyi Guo, and Mengxin Yu. "Policy optimization using semiparametric models for dynamic pricing." Journal of the American Statistical Association (2022): 1-29.

Luo, Yiyun, Will Wei Sun, and Yufeng Liu. "Distribution-free contextual dynamic pricing." Mathematics of Operations Research (2023).

Xu, Jianyu, and Yu-Xiang Wang. "Towards agnostic feature-based dynamic pricing: Linear policies vs linear valuation with unknown noise." International Conference on Artificial Intelligence and Statistics. PMLR, 2022.

Wang, Yining, Boxiao Chen, and David Simchi-Levi. "Multimodal dynamic pricing." Management Science 67.10 (2021): 6136-6152.

Reviewer comment (Theory):
Although the proposed demand model extends existing models by considering the feature-dependent price elasticity, the proposed model and online algorithm still rely on linear forms of elasticity and valuation. Remember ICLR is a deep learning conference. A potentially more suitable treatment may be substituting the linear functions with a neural tangent kernel and then devising online algorithms correspondingly.

Author response:
> W1. "ICLR is a deep learning conference" "replace linear with NTK"

While ICLR starts as a deep-learning focused conference, I believe it is now widely regarded as exchangeable with ICML and NeurIPS.  The topic list in the 2024 Call for Papers seems very inclusive and we believe our paper is within scope. The AC and SAC should confirm this?

As for replacing our model with a kernelized counterpart, it is potentially doable with techniques from Calandriello, Lazaric  & Valko, ICML'2017 on "Second-order kernel online convex optimization with adaptive sketching".    In short, we might be able to replace the dimension d in our regret bound with an effective dimension d_eff.  However, we emphasize that this extension would be complementary to our current work. The main new techniques that allowed us to solve the linear case would still be essential for the kernel case.  We will see if it is feasible to add this in the appendix (if it ends up being a straightforward extension).

Reviewer comment (Theory):
As the authors mention in Ethic issues, personalized pricing may have fairness issues. Therefore, it is essential to discuss how to deal with the cases when we add some fairness regularization terms or fairness constraints to the optimization problem.

Author response:
> W3. "fairness in personalized pricing", "fairness inducing regularization"

> W4. "the objective is purely the interest of the platform" "how does personalized pricing affect buyer well-being" 

We agree these are important problems. However, our problem setting is *not* personalized pricing!  The context $x_t$ describes only the features of the product being sold and the same price is applied to any buyer.  This is clear in the motivating example for the whole line of work on "feature-based dynamic pricing" or "contextual dynamic pricing" started in Cohen et al. 2020, and Javanmard & Nazerzadeh, 2019. 

We agree that our algorithm can be used for personalized pricing if $x_t$  includes features from user profiling --- that is why we are discussing this point in the first place in the paper, i.e., to warn against potential unethical applications. Also, this is not specific to our work but rather an issue of the entire body of research on contextual pricing. More fundamentally, every new technology has a potential risk of being abused. In our humble opinion, this should not be the reason why we stop research that pushes the scientific frontier. Specific to contextual pricing, our algorithm can certainly help improve market efficiency.


References:

Adel Javanmard and Hamid Nazerzadeh. Dynamic pricing in high-dimensions. The Journal of Machine Learning Research, 20(1):315–363, 2019.

Cohen, M. C., Elmachtoub, A. N., & Lei, X. (2022). Price discrimination with fairness constraints. Management Science, 68(12), 8536-8552.

Reviewer comment (Evaluation):
Still about personalized pricing. As the objective is purely the interest of the platform, I would like to see discussions or experimental results on how the personalized pricing algorithm affects customer well-being metrics such as consumer surplus.

Author response:
> W3. "fairness in personalized pricing", "fairness inducing regularization"

> W4. "the objective is purely the interest of the platform" "how does personalized pricing affect buyer well-being" 

We agree these are important problems. However, our problem setting is *not* personalized pricing!  The context $x_t$ describes only the features of the product being sold and the same price is applied to any buyer.  This is clear in the motivating example for the whole line of work on "feature-based dynamic pricing" or "contextual dynamic pricing" started in Cohen et al. 2020, and Javanmard & Nazerzadeh, 2019. 

We agree that our algorithm can be used for personalized pricing if $x_t$  includes features from user profiling --- that is why we are discussing this point in the first place in the paper, i.e., to warn against potential unethical applications. Also, this is not specific to our work but rather an issue of the entire body of research on contextual pricing. More fundamentally, every new technology has a potential risk of being abused. In our humble opinion, this should not be the reason why we stop research that pushes the scientific frontier. Specific to contextual pricing, our algorithm can certainly help improve market efficiency.


References:

Adel Javanmard and Hamid Nazerzadeh. Dynamic pricing in high-dimensions. The Journal of Machine Learning Research, 20(1):315–363, 2019.

Cohen, M. C., Elmachtoub, A. N., & Lei, X. (2022). Price discrimination with fairness constraints. Management Science, 68(12), 8536-8552.