Policy Design for Two-sided Platforms with Participation Dynamics

We would like to thank the reviewer for their time and efforts on the review. We respond to the key comments and questions below.

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(factual discussion)

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W3 [polarized equilibrium]

We clarify the reasoning behind the paragraph below Proposition 2 step by step. First, consider the case where $f$ and $\bar{\lambda}$ are monotonically increasing concave functions. This means that the upper bound of the first order derivative, $C_1$ and $C_2$ are monotonically decreasing to the viewer/provider populations.

Then, let us assess if a polarized equilibrium can be allowed (i.e., satisfies Ineq. (8)) given some specific policies. When the equilibrium population is polarized, this means that some population diminishes, and thus $C_1$ and $C_2$ can be large. Therefore, RHS of Ineq. (8) becomes small (e.g., near-zero). While the exposure-fair policy may exclude this point due to the violation of Ineq. (8), the exposure-concentrated policy can allow this point (Ineq. (8) is satisfied) by letting the LHS of Ineq. (8) to be (near-)zero or smaller than RHS.

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Theoretical guarantees of look-ahead policy.

Please also refer to the rebuttals for Reviewer E67u.

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Baselines in experiments.

Please refer to the rebuttals for Reviewer 54YD.

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(contributions)

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W1 [why our results are not trivial]. While our results may appear straightforward, we respectfully disagree that they are trivial and would like to clarify why they are novel.

Theorem 1, a notable insight of our result is the guaranteed existence of pure NE in a two-sided market setting. This contrasts with findings in relevant prior works, such as [1], [2], and [3], which study one-sided markets involving only strategic content creators and demonstrate that pure NEs may fail to exist or only exist under restrictive conditions. Our result highlights a meaningful and somewhat surprising message: when the passive resource (e.g., user attention) in previous models becomes an active participant (as in our two-sided market), the game structure always ensures the existence of pure NE. We recognize this as an important conceptual contribution and will highlight it more clearly in the revised version.

Proposition 2, we would like to emphasize two non-trivial messages. First, it is well-established (e.g., [4], [5]) that the existence of pure NE does NOT automatically imply convergence under multi-agent gradient dynamics; in fact, such dynamics may get stuck in local NEs [5] or even non-Nash stationary points [4]. In contrast, our model establishes that gradient dynamics provably converge under certain conditions, which is an important theoretical contribution. Second, the sufficient conditions we identify for convergence—namely, weak population effects and more equitable exposure allocation—are not only intuitive but also offer actionable insights for policy design. We briefly discussed these points in the paragraph following Eq. (8) and will make them more explicit in the revision.

Theorem 3, we agree that fully characterizing the optimality of myopic-greedy policies is technically challenging. However, our synthetic experiments in Section 6 do well support our theoretical insight in Section 4. In fact, the main purpose of Section 4 is to conceptually illustrate the scenarios in which myopic-greedy policies perform well versus when they fall short. As a result, Section 4 serves as a motivating example that leads naturally into our exploration of optimal policy design in subsequent sections.

[1] Modeling content creator incentives on algorithm-curated platforms.

[2] Supplyside equilibria in recommender systems.

[3] How Bad is Top-$K$ Recommendation under Competing Content Creators?

[4] On finding local nash equilibria (and only local nash equilibria) in zero-sum games.

[5] User welfare optimization in recommender systems with competing content creators.

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Additional clarifications about our contributions.

We would like to kindly argue that these contributions are actually acknowledged by reviewers KANq and 54YD as “this paper studied an interesting problem ... The theoretical and numerical results both appear sound and provide interesting insights.” “the model is interesting, as are the results.”

If the proposed method appears to be straightforward, it means that our method is well-motivated by the theoretical analysis provided in advance in the paper (especially Theorem 2). Moreover, as we have discussed about Theorem 1 and Proposition 2, non-uniqueness and stability of NEs do NOT actually hold in many other similar settings. Providing theoretical evidence for a seemingly reasonable hypothesis, highlighting the overlooked problem, and giving a reasonable explanation is crucial for scientific research. We would appreciate it if the reviewer could acknowledge our contribution regarding this point.

We would like to thank the reviewer for valuable feedback and the acknowledgment of the contributions. We respond to the key comments and questions below.

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Could some related works that model the population departure be included as baselines?

Thank you for the great point. Unfortunately, the related works, which model the population departure, cannot be directly applied to our setting. There are two reasons: (1) The existing papers assume that the system dynamics are linear, but we consider the non-linear dynamics. Therefore, their linear programming approach is not solvable in our setting. (2) The existing papers do not model the population increase, and even if the dynamics are linear, it is not trivial how to apply the existing work taking the population increase into account (because we do not have a binary indicator of the population). Our paper proposes the first reasonable baseline for optimizing long-term outcomes under flexible and non-linear dynamics of the population.

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That being said, I think the paper should have a more in-depth discussion on the related works which appear to be missing from the main body as of now.

Thank you for the thoughtful suggestion, and we appreciate your point. We will have an additional page in the camera-ready version, and we plan to move the discussion of related work that is currently in the Appendix to the main text upon publication.

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The authors proposed to use the softmax policy as the approximation of $\pi_t^1$; would this impact the performance of the policy?

This is a great point. By using the softmax policy as the approximation of $\pi_t^1$, we expect some underestimation of the objective function ($R(\cdot)$ in Eq. (10)). However, as long as the maximizer of this objective function ($\pi$) is the same (which is often the case), the performance of the policy should not change by a large amount.

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It appears that one needs to know the viewer satisfaction between each viewer/provider pair in order to solve the optimization problem for the look-ahead policy. What if the platform lacks such knowledge in a realistic setting?

This is also a great question. We consider the scenario where we work on sub-group level, and thus the (expected) satisfaction is usually accessible as a subgroup level, and this is also a standard assumption in existing work [1]. However, when this information is unavailable, we need some additional estimation process of viewer satisfaction using past interaction data. We may then need additional consideration of exploration-exploitation tradeoffs, which should be an interesting future direction.

[1] Optimizing Long-term Social Welfare in Recommender Systems: A Constrained Matching Approach.

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I also think the authors should offer more discussions how to best select $\beta$ in real-world applications and whether any theoretical results can be obtained.

Could you establish any guarantees on short-term outcomes under the fully look-ahead policy?

These are indeed important points. While we cannot provide guarantee on the short-term outcomes, it is possible to validate the short-term outcome before deploying the look-ahead policy. Throughout the paper, we consider the situation where the immediate outcome is accessible for both baseline and the proposed methods (following existing work). Thus, once we identify $\pi_t^{(d)}$ using Eq. (10), we can compare the short-term (i.e., myopic) outcome of the proposed method and myopic policies. If the platform aims to guarantee some certain short-term outcome, the platform can adjust the value of $\beta$ to satisfy such constraints.

We would like to thank the reviewer for their time and effort on the review. We respond to the key comments and questions below.

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(factual discussion)

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there is a lack of theoretical results in support of their proposed look-ahead policy

Thank you for the valuable feedback. We would like to kindly argue that providing a theoretical guarantee to the look-ahead policy is challenging due to two reasons: (1) interactive dynamics between viewers and providers, and (2) non-linearity and non-convexity of the systems.

First, one of the related works, [1] considers the departure of viewers and providers in two-sided platforms. While [1] considers a more restrictive setting than ours in modeling linear dynamics and not modeling the increase in viewer and provider participation, identifying the optimal policy is proven to be an NP-hard problem. Second, guaranteeing sub-optimality in non-linear systems is in general considered a hard problem, and without assuming convexity, we often cannot get a theoretical guarantee [2].

The most related theoretical analysis to our method is [3]. This paper studies minimizing loss function for predictions when the population gradually reacts to the model in a single-sided platform. This paper provides some theoretical guarantees when replacing the reference point to be the (estimated) fixed point and under the convexity assumption.

Note that many existing papers empirically demonstrate the performance of the algorithm when some assumption (e.g., convexity) is violated. Our paper also provides empirical evidence that leveraging the reference point (instead of the fixed point) performs reasonably well (i.e., at least competitive or better than both myopic and uniform baselines) across multiple configurations. We will clarify these points and provide additional discussion about the theoretical connection to existing work in the revision.

[1] Matching of Users and Creators in Two-Sided Markets with Departures.

[2] Performative Prediction in a Stateful World.

[3] How to Learn when Data Gradually Reacts to Your Model.

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What additional insight do we get from multiple simulation experiments?

This is to show that the look-ahead policy performs adaptively well in the case where the myopic succeeds and fails. As we can see in the experiment result, both uniform and myopic can be a good choice in one of the scenarios, however, fail catastrophically in the opposite scenario. In contrast, because the proposed approach can care about both myopic outcome (i.e., viewer satisfaction) and investment for provider population growth (i.e., exposure) by design, ours works adaptively well across two opposite situations. Our real-data experiment also provides an example of how an actual population effects can arise in real data. Reviewer 54YD acknowledges our contribution on this point as “it's good that the authors evaluate their method on both synthetic and real-world data.”

(contributions)

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that platforms should attend to producer welfare as well as consumer welfare, is reasonably intuitive and the paper fails to explore more implications of their model beyond this basic point

Thank you for sharing your concerns, and we would like to kindly emphasize that, even if the outcome seems reasonably intuitive, highlighting the overlooked problem is crucial for scientific research.

Regarding this point, we share a similar contribution to existing papers: e.g., [4], which points out the importance of guaranteeing provider exposure under the modeling of viewer/provider departure. We generalize the discussion in the setting where the viewer/provider both increases and decreases their population (i.e., a more complex situation), and demonstrate that caring about producer welfare is indeed beneficial.

Our paper provides the theoretical implications and evidence for the platform designers that caring for provider welfare is indeed crucial for the long-term success of the platform, not only for the ethical perspectives. Reviewer 54YD also acknowledges this point “I believe this work contributes to the broad literature on two-sided platform. It touches on important subjects including balancing the interests of strategic agents and balancing the tradeoff between short-run and long-run welfare.”

[4] Optimizing Long-term Social Welfare in Recommender Systems: A Constrained Matching Approach.

We would like to thank the reviewer for valuable feedback and the acknowledgment of the contributions. We respond to the key comments and questions below.

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I am a bit wary about some of their claims, though. For example, they state, after Theorem 3 and Proposition 3, (overstatement)

Thank you for the thoughtful feedback. We appreciate your point, and we plan to rephrase the pointed-out sentence in the revision as follows.

• (Current) “.. the myopic-greedy policy is optimal only under highly restrictive ..".
• (Revision) “.. the myopic-greedy policy is guaranteed optimal only under highly restrictive ..”

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It would be good to have some explanation as to why the greedy policy worked so well in the real-world experiments. In fact, the greedy policy seems to have the highest (or almost the highest) welfare throughout all the timesteps.

Thank you for the great point. Throughout the experiments (including the ones not presented in the paper), we found that there are tradeoffs in concentrating and distributing exposures. Which is better often depends on the problem instance.

For example, suppose the special case where viewers do not change their population ($\eta_k = 0, \forall k \in [K]$) and the total viewer population is fixed to 100. Then, consider a scenario with 100 provider groups. In this situation, distributing exposure among different subgroups can result in expected exposure of 1 for each provider group. In such cases, concentrating the exposure to one provider group can be a better strategy than distributing allocation for the total population growth. This is why myopic policy performs well in some scenarios, and our argument is that the proposed method can work adaptively well (i.e., at least better or competitive than both myopic and uniform) regardless if the myopic policy succeeds or falls short.

Note that the reason for having a small gap between the myopic policy and the proposal in the real-data experiment is that our method involves an optimization process (e.g., estimating dynamics and optimizing policy) and thus has small modeling errors. We hope this answer could resolve your question.