Originality: To the best of my knowledge the model of general user retention dynamics and corresponding statements evidencing negative feedback loops are novel contributions to the literature in sequential fairness works. The contributions of the paper would be clearer if citations were provided for methods and models introduced in earlier works (for example, I suggest adding citations for the fairness criteria in lines 149-158, for user departure models in 197-208, and for the statement in lines 173-174, if applicable). Since the full related work is deferred to the appendix, I see no need to cite [2, 3, 7, 10, 15, 16] without distinction between them. More context on what these works do and how they relate to your work is useful for readers to contextualize your contributions; please expand on the discussion of these papers. Quality: The simple and unifying model of sequential decision making presented is very valuable in my opinion. The assumptions necessary for the technical results (lemmas and theorems) are stated clearly. The assumptions regarding monotonicity and group separation seem fairly restrictive, but are technically illuminating if not practical for real-data regimes. The statement in lines 173-174 regrading strictly increasing function \phi_{C,t} either needs to be backed up mathematically with reference to a proof, or with a citation. Clarity: I could not find forward links to any of the appendix proofs in the main text. Please provide references to the appendix where applicable so that the interested reader can find them easily. Additionally, since the paper is very notation heavy, it would be nice to provide a link to the notation table at the front of the appendix. The figures are highly informative, and add to the exposition and understanding, but are unfortunately not conducive to the page size (figures are too small to be read without significant zooming). Specifically: - Figures 1, 2, and 3 are too small to be read in a printed version. Please either make the figure sizes bigger or make sure that the quality of the images is high-resolution and the text within figures is big enough that it can be read on a printed - The color scale for figures 4 and 5 needs to be a sequential color scale. As it currently is, the plots are not intelligible in a black and white print out since the orange, green, and magenta all correspond to the same grey-scale. I'm not sure that Table 1 is the most effective way to show the group representation disparity changes, as the expressions themselves are quite hard to parse. Perhaps there is a more interpretable way to summarize this set of results, for example by showing properties of the resulting functions, rather than the functions themselves, in the table. Parentheses in Table 1 should be bigger for the fractions, i.e. use \left( and \right) Overall the prose is well written in the main text. I suggest you revise the appendix for grammar, particularly in the mathematically heavy sections (some sentences did not parse). Purely to aid in your revisions, here's a list of specific lines at which I found typos/grammatical mistakes/etc.: 18, 39, 118 (missing space), 167-168, 176, 367, 764. Significance: While the assumptions are not necessarily indicative of real-data regimes, they are theoretically illuminating, and clearly stated. I believe that this is a good starting point for future work which could extend the results presented here to characterize regimes in which one could hope to say, incentivize decisions that in the long term equalize representation rates. The definition and analysis of simple models here is a valuable building block for the community. However, the presentation of these results is slightly hindered by the sheer amount of notation and technical results without major discussion. Figures are not readable at regular scale, and little to no intuition for proofs is given in the main text (while no forward links to the appendix are provided for proofs). Lastly, little is said with regards to the feasibility and applicability of the proposed model in different real-world contexts. The paper, and I believe it's ultimate impact, would be strengthened by a discussion of what the proposed framework adds to scholarship on sequential decisions in the fairness literature, beyond the results presented here. Therefore, in addition to points raised above, I request the authors to address the following more open-ended questions in their response: - When aiming to promote stable group representations via the proportion of each group in the total population, what ensures that the absolute sizes of both groups are not driven to be very small (\beta_a, \beta_b), even if the ratios become more even? Is this a scenario that is considered bad in your framework? - Regarding the experimental results in Section 4, is the case of EqLos special, or do you believe that other dynamics models for group retention will correspond to the 'right' notion of fairness being an already defined fairness constraint?

Update: I have read the other reviewers' comments and the author feedback. I update my score to 5. A few more comments: 1. Since the authors did not provide the result (for multi-dim) that they claimed, I am not able to comment on the veracity or significance of such a result. In general, I believe that one can find some setting where MC holds. The question is whether these conditions are reasonable. 2. Using the adult dataset: I'm not sure what the relevant application is here. Why would there be any dynamics at all in this setting? This is more of a proof of concept. I don't think my concerns about the lack of pertinent application domains were fully addressed by this experiment. 4. Hashimoto et al.'s Corollary 1 gave general conditions for when the "fair" fixed point is not stable, which directly implies that the dynamics will converge to some fixed point where there is representation disparity. In other words, this already implies the authors' results (e.g. Theorem 3). Therefore the main technical contribution of this work seems to be to show that "static fairness criteria" does not always help with representation disparity. ==== Originality: User retention model is mostly taken from previous work, i.e. Hashimoto et al 2018. One add is considering "User departure driven by intra-group disparity" but this is given quite little justification apart from lines 204-205, which does not explain why this is a model researchers and practitioners should care about. The idea that representation disparity worsens over time is also introduced in Hashimoto et al 2018 with a more general model (i.e. not just one dimensional features) so the conclusion of this paper is somewhat unsurprising. The authors also considered applying fairness constraints at each step and worked out the population ratios in this case (Theorem 2), but this result appears to be trivial under their setting and assumptions. It's perhaps unsurprising that applying fairness criteria without knowing the dynamics will not prevent representation disparity from getting worse in general. That being said, showing this in the one-dim setting is a contribution and a natural enough extension of previous work, and the proofs look like a fair amount of work even though the arguments are elementary. One concern is that there was no meaningful comparison made between different fairness criteria and the absence of fairness criteria, since there's only a blanket negative result. I'm not sure if this is an interesting takeaway at all, as you can always build a worst case that is not close to anything in reality. Quality: The notation is cumbersome (e.g. g_k,t^i, f_b,t^1(x)), even though most of the results rely on rather simple ideas, e.g. follows from the monotonicity condition, and does not seem to warrant such heavy notation. Clarity: In the 3.1 binary classification problem, what is y? Its distribution was not given. Again, the heavy notation did not help with clarity. Significance: In terms of significance, a weakness of the paper is that all the conclusions are proved in the one dimensional setting, and it's not clear what their analogues are in the multidimensional setting. The authors claimed (falsely, I think) that "the main conclusion holds for multi-dimensional scenarios, when it's not at all obvious that is the multi-dim analogue for Assumption 1, even (and not to mention Theorem 2). Assumption 1 is very specific to the one dimensional setting. The toy one-dim setting seems very far from any application where one might care about user retention, so it does not make a very strong point. The experiments section seemed lacking. The synthetic experiment in the one-dim setting did not connect the paper's results meaningfully with any relevant application. One question is in what application do we suspect that there is widening user disparity over time? Is there any real data that can be used to illustrate this?

This paper provides a framework to study group representation dynamic over time, provides theoretical analysis from the perspective of departure dynamic and in-group representation change. The paper is well-written and most importantly, clearly delivers several key insights in this sequential setting. 1. The paper shows that group representation disparity can easily exacerbate under a monotonicity condition, which means the retention rate goes higher when a group's representation increases. While this observation is not new, the authors further show that the exacerbation cannot be solved under commonly fairness criterion because the group feature distribution still can be affected, which is nontrivial but often ignored in fairness research that focus on static supervised learning tasks. 2. Another key contribution of this paper is to highlight the importance of aligning fairness criteria with what drives user retention. Fig 5 in Section 4 shows the distribution of the converged group proportion under combinations of arriving rates. 3. The authors propose a method of finding proper fairness criteria through a constrained optimization step. They show the ideal stable fixed point in a heatmap, but real experiment results would be more effective, especially in the case when perfect fairness is not feasible.