Dynamic Service Fee Pricing under Strategic Behavior: Actions as Instruments and Phase Transition

Thank you for your comments and encouragement. The followings are our responses to your questions.

Re Weaknesses: Thank you for your suggestion. In the camera-ready version, we will include some core proofs on the additional page of the main body.

Re Questions: $P_{St}$ represents the payment received by the seller, reflecting the supply curve, while $P_{Dt}$ represents the price paid by the buyer, reflecting the demand curve. Since the platform charges a service fee $a_t$, there is a gap between these two prices at equilibrium. In other words, at the equilibrium point where we use $Q_t^e$ to denote equilibrium quantity, we have $P_{Dt}(Q_t^e) = P_{St}(Q_t^e) + a_t$. However, buyers might not purchase according to their true demand, i.e., their willingness to pay. For example, a buyer may be able to afford Uber's price but choose to wait for some time until the price drops. Consequently, Uber may perceive the buyer's purchasing power as lower and reduce prices in future personalized pricing, thereby giving the buyer higher long-term utility. As a result, we use $P_{Dt}'$ to represent the buyer's apparent demand, based on which they choose the quantity to purchase, and it might be different from $P_{Dt}$ reflecting strategic behaviors. Therefore, we finally have $P_{Dt}'(Q_t^e) = P_{St}(Q_t^e) + a_t$.

Re Limitations: Thank you for your comment. In Appendix B, we discuss how to generalize the 100% civilian-run seller assumption to $\alpha$ civilian-run. We chose $\alpha=100$% to concisely express our main contributions, namely (1) how to use endogenous actions as IVs, (2) how to deal with buyers' strategic behavior, and (3) how to explore the unknown environment through adaptive randomness injection.

Thank you again for your comments. We hope you find the response satisfactory. Please let us know if you have further questions.

Thank you for your detailed remarks and questions. The followings are our responses to your questions.

Re Weakness 1 and Question 1: One application scenario could be that some online platforms, e.g., Amazon, sells a large number of products daily, and takes service fees between sellers and buyers. At the same time, we notice that on ride-hailing platforms like Uber, prices in the app frequently fluctuate due to frequent changes in supply. While the booking fee doesn't change too often, promotions can change quickly and be highly personalized. Therefore, our model can also be applied to dynamic promotion pushes. This is another application scenario for our model.

Re Weakness 2: We agree that this is an important and challenging future direction. We would like to note that although our results on $\gamma$ may not be tight, we have improved upon previous results in the literature. Please refer to Lines 245-252. Compared to $O(1/(1-\gamma)^2)$ regret in another similar pricing problem [37], we improved it to $O(1/(1-\gamma))$ in Thm 3.2. Additionally, our algorithm does not require prior knowledge about $\gamma$.

Re Question 2: The revelation principle [63] tells us that for any mechanism that maximizes revenue, there exists an incentive-compatible (IC) mechanism that can achieve the same revenue. IC means that buyers can purchase according to their true demand, and the mechanism's incentives will guarantee them higher utility. Therefore, we study how to motivate and approximate within the IC mechanism framework, narrowing down the policy space and making it easier to find an approximately optimal mechanism. This not only simplifies the analysis but also makes it easier to apply in real-world scenarios (buyers don't need to struggle with searching for a good strategy).

Re Question 3: $a_t$ is not i.i.d. because it depends on the supply from $1$ to $t$ and the demand from $1$ to $t-1$. Therefore, $a_m$ and $a_n$ are correlated, since they, for example, both depend on the supply at $t=1$. However, since the platform announces $a_t$ before observing the equilibrium at time $t$, $a_t$ and $P_{Dt}$ are conditionally independent given $P_{S1}, ..., P_{St}, P_{D1}, ..., P_{D(t-1)}$. Therefore, we model it as a martingale for parameter estimation. Thank you for your comment; we will include this discussion in the next version.

Re Question 4: In our experiments, we chose $T_0 = 10\log_2 T$. We also used $T_0$ data points to empirically estimate $\sigma_S^2$. If the estimated value is greater than $10/\sqrt{T}$, we consider it to belong to $\mathbb{H}_1$; otherwise, we consider it to belong to $\mathbb{H}_0$. The parameter 10 does not affect the order of the regret bound in terms of $T$, and the concrete value is usually determined empirically, such as through cross-validation with past data. We hope this helps with your understanding.

Re Question 5: Thank you for your suggestion. We increased the number of trajectories to 100 and observed that the bandwidth significantly decreased. Please refer to Figure 1 in the newly added pdf. Additionally, we tested the regret under different $\sigma_S^2$ values: 0.5, 1, 1.5, 2. Please refer to Figures 2-4 in the newly added pdf. We found that the larger the $\sigma_S^2$, the smaller the regret, which confirms our theoretical results.

Thank you again for your comments. We hope you find the response satisfactory. Please let us know if you have further questions.

Thank you for your kind remarks and questions. The followings are our responses to your questions.

Re Weakness 1: Thank you for your suggestion. If there is a suitable opportunity, we will consider conducting experiments on real-world data.

Re Weakness 2: We are sorry for the confusion and misunderstanding. We discussed in Lines 165-177 why we use a linear model (based on market observations and the robustness of the linear model [15]) and how to go beyond it (using GMM and RKHS). We assumed a linear model to avoid heavy notations while expressing our main points: (1) how to use endogenous actions as IVs, (2) how to deal with buyers' strategic behavior, and (3) how to explore the unknown environment through adaptive randomness injection.

Re Weakness 3 and Question 3: Recall that our goal is to maximize the total revenue. The revelation principle [63] tells us that there exists a DSIC mechanism that can achieve the optimal revenue. Therefore, we only need to consider the policy class satisfying DSIC. If we approach the optimal DSIC mechanism, then we can approach any optimal mechanism no matter it is DSIC or not. In other words, we identify there exists an optimal point in a smaller policy space and then approximate it within this smaller space. This not only simplifies the analysis but also makes it easier to apply in real-world scenarios (buyers don't have to struggle with searching for a good strategy).

Re Question 1: Thanks for your question. One application scenario could be that some online platforms, e.g., Amazon, sells a large number of products daily, and takes service fees between sellers and buyers. At the same time, we notice that on ride-hailing platforms like Uber, prices in the app frequently fluctuate due to frequent changes in supply. While the booking fee doesn't change too often, promotions (which can be regarded as a reduction on the service fee) can change quickly and be highly personalized. Therefore, our model can also be applied to dynamic promotion pushes. This is another application scenario for our model.

Re Question 2: We demonstrate the relationship between regret and the noise level in the supply curve in Figure 5. We also add another group of experiments (see pdf) showing their negative correlation. Additionally, a higher noise level in demand leads to greater regret, as it results in larger estimation errors for the parameters. Please refer to Line 688 for this impact on $\hat\beta_1$. [56] informs us that model misspecification causes additional unavoidable errors, while [15] explains that linear models are not so affected by misspecification. This is one of the reasons why we assume linearity.

Thank you again for your comments. We hope you find the response satisfactory. Please let us know if you have further questions.

Thank you for your remarks. The followings are our responses to your questions.

Re Weakness 1: $P_{St}$ is the amount received by the seller at time $t$, and $P_{Dt}$ is the amount paid by the buyer. The gap between them is the service fee, which is $a_t$, and $Q$ is the transaction quantity. Then, $P_{St}(Q)$ and $P_{Dt}(Q)$ represent the supply curve and demand curve respectively.

Re Weakness 2: We are sorry for the confusion and misunderstanding. In Lines 165-177, we introduced the motivation and extension of Assumption 1. Our results hold for general subGaussian noise, and we use Gaussian noise for ease of illustration. We assume linear models because they perform well locally and have strong robustness.

Re Weakness 3: We use the service fee, i.e., action, as an instrumental variable, which is emphasized in the title. Since they are neither i.i.d. nor external, we used a martingale-based approach. Intuitively, at time $t$, $a_t$ only depends on the demand from times 1 to $t-1$; so given those, $a_t$ is independent of the demand at time $t$. We proved that in this case, the service fee can be used as an approximately valid IV.

Re Weakness 4: The challenge in our problem lies in how to add randomness. We carefully choose the amount of data used to make $a_t$ an approximately valid IV. Additionally, we add noise with carefully chosen magnitude to balance between exploration and exploitation.

Re Weakness 5: For Assumption 2.1, we can use the generalized method of moments (GMM), which is standard in econometrics, to handle non-linearity. For Assumption 3.2, we discuss other possible choices in Appendix B. Please refer to the discussion in lines 154-177. We use these assumptions to highlight our technical innovations while avoiding heavy notations, including (1) how to use endogenous actions as IVs, (2) how to deal with buyers' strategic behavior, and (3) how to explore the unknown environment through adaptive randomness injection.

Re Weakness 6: $T_0$ is a constant multiple of $\log T$. The constant does not affect the order of $T$ in the regret bound, and the concrete value is usually determined empirically, such as through cross-validation with past data. In our experimental design, as referenced in Line 1118, we chose $T_0 = 10 \log_2 T$.

Re Weakness 7: Although the idea of hypothesis testing is straightforward, our innovation lies in recognizing the phase transition and finding the optimal algorithm design through injecting randomness adaptively. This optimal algorithm is further achieved through hypothesis testing.

Re Weakness 8: Thanks for your suggestion. From the experimental results, when $\sigma_S^2=1$, we find that the regrets when $T=20000, 40000, 60000, 80000, 100000$ ($\log T$ = 9.90,10.60,11.00,11.29,11.51) are 220, 230, 234, 236, and 237, respectively. These points are even slightly sublinear. So, the actual performance is even better than the theoretical bound. When $\sigma_S^2=0$, $\log(\text{Regret})$ = 6.43, 6.75, 6.95, 7.10, 7.17 when $\log T$ = 9.90,10.60,11.00,11.29,11.51. The estimated slope by OLS is 0.47, testifying our square root $T$ regret. We will include this in the next version.

Thank you again for your comments. We hope you find the response satisfactory. Please let us know if you have further questions.