__ Summary and Contributions__: The authors consider the model of content generation game in a recommender system. The particular model was introduced by Ben Basat et al., JAIR 2017 and has previously been studied in a series of papers by Ben Porat and authors published in NeurIPS, EC and AAAI.
The model studied is as follows: It's a game between authors. Each author chooses a topic to write about. They are shown to a viewer randomly if they are one of the highest quality and derive a value equal a conversion factor that depends on the author and the topic.
The authors extend previous work by showing that in this game better response dynamics converge under more general conversion structure that satisfies simple monotonicity properties (content of higher quality yields has higher conversion rate). They show that better response dynamics can still take exponential time to converge and provide an algorithm for directly computing a pure Nash equilibrium in these games.

__ Strengths__: Soundness of the claims:
The authors justify all their claims with complete proofs. The proofs are involved but very thorough. I was able to follow all the details that I tried to understand.
Significance and novelty: The proofs provided by the authors are non-trivial. They require novel ideas and constructions.
Relevance to the NeurIPS community: The model studied by the authors has previously been published in JMLR, AAAI, NeurIPS, EC. It also seems to be of practical relevance as it captures the natural competition between content creators on platforms like Medium.

__ Weaknesses__: Significance and novelty: Considering this work by itself, I wonder why pure Nash equilibria, better response dynamics are the main concepts to study in this model. Would no-regret learning and correlated/coarse-correlated equilibrium be a more natural model. Isn't this better thought of as a repeated game between content creators? At the least, wouldn't authors choose a mixed strategy where they write about a variety of topics and see how they fair compared to their competition? Perhaps this is covered by previous work but might be useful to repeat, to justify further effort on pure Nash equilibria and better response dynamics.
I was also curious about the focus on better response dynamics instead of best response dynamics. In particular since the authors' last result is about computing a pure Nash equilibrium, can the pure Nash equilibrium be reached using best response dynamics? My apologies if there is a theorem that says that consider best response and better response in equivalent.
Post-rebuttal: I am still not fully convinced why better response dynamics are the right dynamics to consider. I am curious to know if any best response dynamics is guaranteed to converge in a polynomial number of steps. As for mixed vs pure strategies - the authors are free to choose any topic that gives them best utility so it is still conceivable that they could randomized between a few different topics.

__ Correctness__: Based on the proofs I read in detail they seem correct. I didn't read all the proofs completely.

__ Clarity__: The paper is well written. Between the main paper and the appendix, the authors have include all the details. I think it's great that the authors provided the example for the better response dynamics not converging it gives a better flavor of the construction instead of providing the full details. (which are included in the appendix). They also include a handy table with all the notation to make it easier to follow the proof.

__ Relation to Prior Work__: Authors discuss relationship to prior work when relevant. Perhaps authors should include more details about the results from the closely related work [4, 5, 6, 7] - that could provide further motivation for why the results considered here fit into the broader research agenda.

__ Reproducibility__: Yes

__ Additional Feedback__: * The authors index the topics according to D_i in one place and according to a different metric in another. I hope that doesn't break anything. Could the earlier one be dropped completely?
* In Theorem 2, note that if T is a constant then (T-2/P + 1)^P is not exponential in P. I think T has to be poly(P).
* In Algo1 change step 10 to create a *new* node x associated with topic k^*
* Minor typos:

__ Summary and Contributions__: The paper studies the following game that captures the decisions content providers must make in choosing the topics they cover: Each player (content provider) selects from a finite set of topics. If player j writes on topic k, the quality of this content is an exogenously determined parameter q_{jk}. Readers interested in each topic choose the highest quality content provided (with ties broken uniformly at random), and the content providers are rewarded based on the fraction of readers they get, multiplied by a conversion rate that depends on the content provider and the topic.
The main results of the paper are the following theoretical results: this game always has a pure-strategy Nash equilibrium, the better-response dynamics always converges but might take exponential time, and there is an efficient algorithm to compute a Nash equilibrium.
The game is essentially something in between a congestion game with player-specific payoff functions and a stable marriage game. Given all the previous work on similar settings (in particular the AAAI'19 paper), the results are not surprising, although actually working out all the technical details is non-trivial. It's worth noting that (if I'm not mistaken) when there is no ties in quality parameters, the game is essentially a stable marriage game, and a Nash equilibrium can be found by an author-proposing algorithm. Therefore, all the complications of Algorithm 1 has to do with ties among quality scores.
In terms of motivation, the paper falls under the category of theory papers loosely motivated by a practical scenario. In particular, the model is really about how authors choose a topic to write about, and there's no real connection to the recommender systems, since the model assumes a trivial model of recommendation (the recommender system that always picks an item with the highest quality, for known quality scores).

__ Strengths__: Nice theoretical results. The algorithm for computing a NE is non-trivial and interesting.

__ Weaknesses__: The marginal contribution over the previous work (e.g., AAAI'19 paper) is not that substantial.
Problem is not very well motivated.

__ Correctness__: I haven't checked the proofs (that are in the appendix), but the results seem reasonable.

__ Clarity__: Yes.

__ Relation to Prior Work__: Acceptable, but I expected more discussion of the connection to congestion games and the deferred acceptance algorithm for stable marriage.

__ Reproducibility__: Yes

__ Additional Feedback__:

__ Summary and Contributions__: This paper formalizes a game theoretical scenario between content provides (players) of the recommendation system. With the solution concept of pure Nash equilibrium, the decentralized algorithm, better-response dynamics, is shown to converge in exponential time, while a centralized algorithm is designed to find a PNE efficiently. The convergence result indicates the existence of PNE is such a game, which is important and the centralized algorithm is mainly leveraging the perfect matching to find the corresponding PNE.

__ Strengths__: The game formalization in this paper is quite novel and characterize most properties in practical recommendation system. Most proofs, as far as I checked, are correct and well-written.The topic is relevance to the NeurIPS community, as the authors are trying to justify the significance of the recommendation system.

__ Weaknesses__: The choice of better-response dynamics (BRD) is questionable. BRD is not popular algorithm to find NE, neither efficient nor guaranteed to converge in most games. I believe it is mainly used to prove the existence of PNE through non-existence of improvement cycle in BRD, which is well-written although relatively not novel. Meanwhile, BRD should not also be considered as a practical algorithm that may be adopted by each content provider alone, since BRD still requires much information about the conversion matrix and demand function. In other word, the exponential running time of BRD is expected, and somewhat meaningless.

__ Correctness__: I checked most proofs and they are correct.

__ Clarity__: This paper is well written. The presentation is good and the notations, although complicated, are well explained. All the proofs in the Appendix are easy to follow. The main drawback is related to the social welfare part. It seems that the authors want to emphasize the social welfare varies a lot among multiple PNE, and the coordination (centralized algorithm) helps players to obtain the PNE with better social welfare. On one hand, it is not a good way to define social welfare only w.r.t. to consumers but not including content providers' utility. On the other hand, the concept of "Price of Correlation" is a very good way to explain this motivation, but it is only mentioned in Appendix. And there is indeed no discussion on comparing the social welfare under the PNE obtained by decentralized and centralized algorithms.

__ Relation to Prior Work__: It is clearly discussed how this work differs from previous contributions.

__ Reproducibility__: Yes

__ Additional Feedback__: I think it may be better to provide an intuitive proof for Theorem 3, explaining why the output is an equilibrium, instead of the one for Lemma 1.
Some minor issues: Line 146, two players, two topics; Line 256, undefined notation N_G for neighbors.
==========================
Regarding author's response:
Although my two major concerns, the choice of BRD and the analysis of social welfare, are responsed, I think there is no new point there. Basically, these two issues maybe challenging to address, but indeed reduce this paper's significance. Thus, I shall not change my score.

__ Summary and Contributions__: The work studies the game-theoretical dynamics of "ecosystems" such as blogging platforms. On the one hand we have the content producers, that is, bloggers or filmmakers; on the other hand, the content consumers, that is, the final users, who demand different amounts of content on different topics. On top of this there is a recommender system/platform that suggests contents on the basis of their topics and quality. The only actual players are the producers, who try to maximize their profit (for example, the number of readers of their blog). To this end, each producer can adapt the topic of its content, for example by opening a blog on a more profitable subject. But social welfare, measured as the average quality of the blogs, is at stake too. This complex scenario is formalized as a non-cooperative game. The paper gives two main results. First, under natural assumptions, if each producer sequentially makes a move to increase profit, then the game converges to a pure Nash equilibrium, and that this may require a large number of steps. Second, a pure Nash equilibrium for the system can be computed reasonably fast, in polynomial time; this is done in an interesting way by (loosely speaking) computing max-weight perfect matchings.

__ Strengths__: The subject is of interest of the NeurIPS community (although perhaps not exactly central). The work gives a very clear message with two/three main results. The results are nontrivial (the model is complex and not easy to analyse). The fact that the system converges to a pure Nash equilibrium is interesting (it is not obvious that a pure equilibrium strategy exists, unlike for mixed strategies i.e. distributions). The lower bound construction for the convergence time is neat and insightful. The computation of the equilibrium in polynomial time is interesting as well. The work is well presented and pleasant to read. At a higher level, the work sheds light on the interplay between profit maximization (the bloggers' point of view), social welfare maximization (the readers' point of view), and system design (the recommender system's point of view). This is different from the traditional recommender system problem, that is, suggesting relevant content to users, and different techniques are used.
One drawback is that the model is complex, but this is not a fault of the paper. Rather, it is necessary in order to model both the content provider side (bloggers) and the users side (readers) while giving a role to the recommender system/platform.
A second drawback is that the social welfare "disappears" in the paper, unless I am missing something. That is, the two results of the paper are oblivious to the social welfare of the system. They are a function of the content producers' utilities but not of the average quality of the content. The only relationship is in the (natural) assumption that higher quality carries higher profit everything else being equal. I find this a bit weird given that the paper brings as motivation the study of long-term social welfare in these dynamic systems.

__ Weaknesses__: One drawback is that the model is complex, but this is not a fault of the paper. Rather, it is necessary in order to model both the content provider side (bloggers) and the users side (readers) while giving a role to the recommender system/platform.
A second drawback is that the social welfare "disappears" in the paper, unless I am missing something. That is, the two results of the paper are oblivious to the social welfare of the system. They are a function of the content producers' utilities but not of the average quality of the content. The only relationship is in the (natural) assumption that higher quality carries higher profit everything else being equal. I find this a bit weird given that the paper brings as motivation the study of long-term social welfare in these dynamic systems.

__ Correctness__: I could not check the proofs. The proof sketch of the lower bound sounds correct, though.

__ Clarity__: The paper is clear and well polished.

__ Relation to Prior Work__: Yes.

__ Reproducibility__: Yes

__ Additional Feedback__: I have only two specific comments.
- L 146: "three" should be "two"
- Theorem 2 and places where it is mentioned: the bound is not exponential in P unless T is a function of P. If T is fixed than the bound is O(exp(T)) which is constant in P.