Thank you for your encouraging review and insightful questions.

Simplifying assumptions such as linear classifiers and linear costs

Our choice to focus on a simple setup stems from several considerations. Indeed, one consideration is tractability (of both pricing and learning problems). Another consideration is simplicity as a guiding principle: as a first step to exploring market-inducing classification, we believe our choices are reasonable. Note also that almost all works on strategic classification focus on linear models and simple costs (e.g., linear, 2-norm). Since our formalism layers on an additional market mechanism, preserving this structure seemed useful. Finally, we hope you agree that despite our simplifying assumptions – the phenomena that arise from the market mechanism are sufficiently interesting, and even surprising, to merit simplifying assumptions. That said – we certainly agree that more elaborate settings are worth pursuing in future work.

No analysis or comparison of the surrogate loss function or why it is worth including

This is a great suggestion! We will gladly add a comparison of the 0-1 loss vs. our m-hinge loss for settings where computing the 0-1 loss is tractable. In fact, we already have such results for some settings; for example, in Fig. 4 (which shows 0-1 accuracy), the hinge loss can be shown to replace “flat” regions with linear slopes – as can be expected. This suggests that the hinge serves as a useful proxy when slopes push the model towards “lower” flat regions. We can add this to the figure (it was actually removed due to clutter), and include further analysis on other examples in the Appendix.

In the market setting, most individuals are able to attain the positive classification

We think we understand your concern, but this claim is not entirely precise. Looking at the results in Fig. 2, it is true that most points end up being able to cross. Fig. 3 suggests that this effect reduces when budgets are more diverse. But both examples consider a single “cluster” of points; once there are more clusters, then it is no longer the case that all or most points will move. The more general phenomena we believe is at play is that clusters move together, i.e., the price setter will tend to be an extreme point of some cluster. This can be seen for example in Fig. 4 (bottom right): note how changing the budget ratio (y-axis) causes the price setter to jump from being the extreme point of one cluster to that of another.
Given your input, we think it will be valuable to add further empirical investigation of this phenomena. We already have some initial results on this and will gladly add them, either in the Appendix or using the extra page of the final version.

How well [does] the model describe real-world strategic classification contexts?

This is a good question which we feel summarizes well many of the previous comments. Real world market dynamics are very likely more complex than our model posits, especially if the market forms in response to a classifier. We believe our model, albeit its simplicity, is still able to capture (at least coarsely) certain effects of transitioning from fixed costs to those of an induced market. But of course this is speculation, and a definitive answer requires much further investigation and research efforts.
In regards to production capacity and costs, these are certainly a natural next step to consider. One reason we focused on no constraints or costs is that this significantly reduced the number of free parameters required to specify the setup. Another subtler point is that it isn’t immediate how constraints and costs operate when transitioning from a finite sample (as in training) to expected outcomes (on which we aspire to evaluate). Having no constraints and costs makes it possible to have a single well-defined market mechanism that captures both and supports the notion of generalization.

I didn’t understand the connection between Theorem 2 and the fact that almost all points cross

Thm. 2 states that, for the considered distributions, (i) there is a unique price setter, and (ii) it lies beyond the peak of uf(u). Intuitively, since this point is typically more extreme than the peak of f(u) itself, the price setter will be positioned at an extreme quantile. We will make this clearer in the next revision.

Response:

Thank you for your positive review! We were happy to hear that you found our paper exciting and novel. If you have any further questions we would be glad to discuss.

Would've preferred to validate on multiple datasets

We are happy to report that we have extended our experimental section to include an additional dataset. For details please see our response to Rev. 8pCZ.

Simplifying assumptions such as linearity in cost modeling etc.

We agree that supporting more complex costs and models would have been a nice addition. However, as a first step, we think that it is reasonable to focus on a linear construction – especially given that most works on strategic classification consider also linear classifiers and simple cost functions (e.g., linear, 2-norm, or squared).

Thank you for your review and comments. We were glad to hear you see our paper as making a clear conceptual contribution – this was indeed our primary aim and focus.

Your review mentions that you believe our results can be strengthened, in particular by considering (i) alternative proxy losses and (ii) more datasets. In regards to (ii), and as per your suggestion, we have extended our empirical results to include an additional dataset based on Folkstable. Results here confirm most of our previous findings and provide additional insights. Regarding (i), we address this and all other points below.

The proposed method underperforms compared to a standard strategic method

This is true, but only for small budget scales ($\le 8$). Note the original data has scale $2^{10}$ (star marker), for which our approach clearly outperforms the baselines and by a large margin. Note also that the x-axis is in logarithmic scale: our method is better in the range $[16,1024]$.
In terms of results, we consider this phenomena an interesting finding. In contrast to our approach (MASC) which anticipates prices that adapt to the learned classifier (at equilibrium), the standard approach (strat) assumes fixed prices. Our interpretation of the results is that when the distribution of budgets approaches uniform, price adaptation becomes either mild or inconsequential. In this regime, the “price” we pay to enable differentiability turns out to be larger than the benefits of accounting for price equilibration.

Perhaps another loss surrogate can perform better?

This is certainly possible, and we would love for future work to develop better solutions. Note however that designing proxy losses for strategic learning tasks can be quite challenging. Even for standard strategic classification, fundamental concepts such as margins can break completely (see Levanon & Rosenfeld (2022)). The only existing approaches that we are aware of and that apply to our setting are the s-hinge (which we build on) and Hardt et al. (2016), which underlies our baseline. For our market setting, even the connection to the s-hinge is not straightforward, as it is intended for the 2-norm cost – not linear, and the generic extension of the s-hinge to other costs is generally intractable.

Experiments on other datasets

We are happy to report that our experimental section has been extended to include another dataset based on Folktables (thanks for this suggestion!). The target variable is employment status, and budgets derive from income. Results show overall similar trends to Adult, but with some distinctions. All methods improve as inequality increases. Our method outperforms all others – here at all scales $\ge 4$. The % of crosses, welfare, and social burden behave similarly to Adult. Interestingly, and in contrast to Adult, here strat underperforms across all scales, even when compared to naive. We will add these results to the final version.

Can the weights, in general, be both positive and/or negative?

Although possible, our approach does not explicitly constrain weights to be positive. This is in line with the results of Hardt et al. (2016) for non-adaptive linear costs. We do however expect them to be such for the learned classifier (otherwise, users would be paying to decrease feature values).

Vs. Chen et al. (2024):

This recent paper is similar to ours in that it generalizes strategic classification to support dependencies across user response through the cost function. The main difference is that in their setup, dependencies are encoded explicitly as externalities in the cost function, which is fixed and predetermined. This means that, as in standard strategic classification setup, learning requires knowledge of the particular cost function. In contrast, our work models dependencies as forming indirectly through the market mechanism; the cost function itself is adaptive, and learning only requires knowledge of how the market operates. Another distinction is that externalities depend on the classifier only indirectly: explicitly, they depend on how other points move. Market costs depend on demand rather than on actions; given prices, actions become independent. These distinctions make it hard to draw direct connections between the results and methods of our work and theirs.

Minors

• Eq. (4): $\delta$ is defined as the amount purchased. This implicitly depends on b, which constrains how much can be purchased.
• Demand set: Eq. (5) is the definition. Demand set in general is a common construct.
• a vs. $\tau$: Thanks – we will correct this.
• $\Delta$ in Eq. (6): This is true! Thank you for noticing. The fix is to add $x+$ before the argmax.
• Sellers have foresight: See e.g. “Noncooperative Collusion under Imperfect Price Information” (Green & Porter, 1984).

Thank you again for your feedback. We hope our response and improvements meet your expectations, and kindly ask you to consider increasing your score.

Thank you for your careful reading and important comments. Overall, it seems that your concerns are: (i) clarity of exposition and captions, (ii) discussion of the empirical findings in Sec. 6, (iii) the use of a single dataset, and (iv) contribution.
For (i) and (ii), we believe these are easily fixable, especially given the extra page for the final version. For (iii), we have added another dataset, please see our response to Rev. 8pCZ. For (iv), we believe our response below will help establish the contrary by clarifying the role of budgets. In this case, we kindly hope you would be willing to reconsider your evaluation.

Empirical results analysis is insufficient

Thank you for pointing this out. In hindsight we agree that a more thorough discussion of the results in Sec. 6 would be beneficial – we will gladly expand the discussion here. As you note, there are many potential insights to discuss within the existing results.
Please note however that our empirical analysis spans both Sec. 5 (synthetic data) and Sec. 6 (real data). We hope you agree that our discussion in Sec. 5 is adequately comprehensive (as other reviewers note), and that together they provide a useful picture.

Exposition

As per your suggestion, we will add to the introduction a succinct summary of the paper’s contributions as bullet points. We will also make the distinction from standard strategic classification clear and explicit.
On a more general note: Our paper lies at the intersection of two disciplines. Presenting our setup and motivation in a way that is equally readable to both audiences is not an easy task. We have given this much thought, but it is possible that the current version leans more towards one than the other (given that other reviewers were happy with clarity). If there are any particular clarity issues you feel are present, please let us know and we will improve them.

Figures/plots

Thank you for this input. We will gladly make use of the extra page to clarify and further elaborate.

If the performance is dependent on budgets, isn't the classifier really classifying based on the budget?

Generally, no. Budgets are certainly important, and can play a significant role in determining learning outcomes, but are not the only component that matters. If this were the case, then learning itself would become degenerate. Our results indicate that this is not what happens. The relation between budgets and labels affects accuracy through the market mechanism – but the market itself depends on the classifier. Since the classifier h(x) does not take budgets as input, the relations between features x and labels y is what determines the space of possible markets. Learning can come to rely on features for which budget disparities are dominant (if exist), but it is equally plausible that learning will come to avoid such markets. We have included in the appendix a simple 2D example which illustrates this behavior. In the construction, b correlates with y in the direction of x1, but the optimal classifier uses only x2, under which the market is stagnant.

Last sentences in intro:

Please note that these should be read in the context of the preceding sentence – not as a general independent claim.

How does this algorithm perform when budgets are not correlated with the outcome?

This is a good question. In general, standard (linear) classifiers work well when the labels correlate with distances from the decision boundary. In strategic market settings, budgets can improve (or degrade) this correlation–which is why they can be helpful. Thus, what matters is the relation between budgets and labels along the direction of the classifier (or more precisely, the distances on its negative side). This depends on the classifier and is not an a-priori property of the data. This is also why learning should make use of features (see our point above).
If budgets and labels are completely independent, then the choice of classifier should have little effect on the market. But the more likely scenario is that in some directions features will be more informative, and in others, budgets. An optimal solution is likely one that exploits the combination of these two different sources of information through how they form a market.

Do we want to build such classifiers? Or [is this] an accurate description of reality?

The question of what we should “want” is of course multifaceted and depends on context. What we believe our results convey is that: (a) if we learn naively (in a way that does not account for the market), then market forces will obscure our performance, whereas (b) if we learn in a way that accounts for the market, then this can improve performance, but possibly by exploiting financial inequalities.

Minors:

line 363: See Hardt et al. (2016).
line 296: Yes, we will correct this.
line 303: bmin and bmax have been swapped. Fig 5: We will add arrows for a subset of points.