Learning in reverse causal strategic environments with ramifications on two sided markets

We thank the reviewer for the helpful comments, we address particular concerns below.

Soundness concern: Repeated risk minimization (RRM) plays a central role in the theoretical analysis (eq. (2.2) ,eq. (3.1)), and convergence is claimed to be due to “repeated risk minimization, which is known to converge to performatively stable policies (Perdomo et al., 2020)” \ldots I was unable to find a discussion of these assumptions and their applicability in the paper, and therefore it is not clear why RRM is guaranteed to converge in this context.

In response to this soundness concern we have made the following changes to the submission:

1. We have re-written the discussion of performative prediction in the Coate-Loury model (section 3.1) in order to clarify that our results DO NOT depend on the convergence of RRM. Each theorem is a comparison of the impacts of the two types of solutions in performative prediction (stable and optimal points) on labor markets, and for such a comparison to be valid we only need that both solution types exist, which is always guaranteed. We also point out that the conditions provided in (Perdomo et al. (2020)) are sufficient, but not necessary, and our empirical results demonstrate that in practice the convergence of RRM/reactive firms is not an issue in a wide range of markets.

2. A more thorough discussion on the market conditions needed to guarantee RRM convergence is added to the appendix (see appendix D). The primary benefit of imposing such conditions on the market is that all ``reactive" employers are guaranteed to eventually stabilize. This is not needed for our results but is still nice conceptually. Given this, we have added two new theorems (now theorems 3.2 and 3.4) which provide results in markets which are compatible with these conditions (particularly we show our main take aways are similar in low wage markets).

Additional related results in Perdomo et al.: In the paragraph below the statement of Theorem 3.1, it is claimed that “Theorem 3.1 gives conditions for there to be an appreciable gap. This complements prior results (for example, in Perdomo et al. (2020)) that provide conditions under which the gap is small.”. In contrast, Theorem 4.3 in Perdomo et al. 2020 predicts that the gap between the PO and RRM policies is expected to be small. What is the relation between the gaps presented in this paper and Theorem 4.3 in Perdomo et al.? If some required Theorem 4.3 are not met, which ones? And how does it relate to the RRM convergence guarantees discussed in the question above?

None of the conditions in (Perdomo et al. (2020)) are broken, but the ``$\epsilon$-sensitivity condition" is only satisfied for a large value of $\epsilon$ when the $w$ is large. Thus the upper bound on the gap between PO and PS policies from (Perdomo et al.(2020)) is also large under the conditions of our theorems 3.1 and 3.3. Appendix D now clarifies the importance that wage plays in convergence of RRM.

Learning setting: At what stage data is available to the employer, and how do they learn from it? How do the main results extend to scenarios where predictors are learned from finite datasets?

In a stochastic (or finite data) setting the order of learning at time $t$ is as follows:

1. The firm deploys decision $\theta_t$

2. Firm recieves data (and corresponding loss/reward) drawn from $\mathcal{D}(\theta_{t})$. The firm may use this to either deploy a reactive decision or in some algorithm that converges towards an optimal policy (the discussion on this in appendix A assumes a finite data set)

Although we work in the population setting, we expect similar results in the finite-sample setting in which the firm has to learn its predictor from data. Under standard assumptions that guarantee generalization, the firm's learned predictor will be close to its optimal predictor in the population setting that we study.

Do similar results hold for the content creation scenario described in Example 2.2? What would be required in order to apply the results in other scenarios?

This is an intersting line of follow up research, the content creation game is quite different, as the ``content creators" actually compete with one other for user attention. This means the response of the creators are participants in a game and their response will be some type of equilibria of the game. This type of agent response is very different from the agent best response that we study.

Small question about notations: What is the difference between $w\int_X 1_{f(x)=1}d\Phi(x|1)$ and $w\int_{[0,1]} 1_{f(x)=1}d\Phi(x|1)$ in Example 2.1?

This is a typo, $X$ should be [0,1] in this case.

Code is not provided, making it hard to validate and reproduce the results of Section 4, which rely on numerical evaluation.

We have now uploaded our code.

Dear reviewer L1Mv,

Thank you for taking time to review our paper and provide valuable feedback. Hopefully you have had time to read our response. Please let us know if you have further questions or comments, we would be happy to address them. If you found our responses and updated submission satisfactory, please consider updating your score.

We thank the reviewer for the encouraging words on our work, and alert them that in response to another reviewer a second version of the submission has been posted.