Your task now is to draft a high-quality review for a top-tier Machine Learning (ML) conference for a submission:\n\nTitle:\n```\nModeling content creator incentives on algorithm-curated platforms\n```\n\nKeywords:\n```\nNash equilibria,producer incentives,attention monetizing platforms,recommenders,differentiable games,exposure game\n```\n\nAbstract:\n```\nContent creators compete for user attention. Their reach crucially depends on algorithmic choices made by developers on online platforms. To maximize exposure, many creators adapt strategically, as evidenced by examples like the sprawling search engine optimization industry. This begets competition for the finite user attention pool. We formalize these dynamics in what we call an exposure game, a model of incentives induced by modern algorithms including factorization and (deep) two-tower architectures. We prove that seemingly innocuous algorithmic choices\u2014e.g., non-negative vs. unconstrained factorization\u2014significantly affect the existence and character of (Nash) equilibria in exposure games. We proffer use of creator behavior models like ours for an (ex-ante) pre-deployment audit. Such an audit can identify misalignment between desirable and incentivized content, and thus complement post-hoc measures like content filtering and moderation. To this end, we propose tools for numerically finding equilibria in exposure games, and illustrate results of an audit on the MovieLens and LastFM datasets. Among else, we find that the strategically produced content exhibits strong dependence between algorithmic exploration and content diversity, and between model expressivity and bias towards gender-based user and creator groups.\n```\n\nFigures/Tables Captions:\n```\nFigure 1: Exposure game. Items s i \u2208 S d\u22121 placed to maximize exposure to consumers c \u223c P c .\nFigure 2: YouTube revenue streams incentivizing exposure maximization (\u00d8rmen & Gregersen, 2022).\nFigure 3: A) A game with no PNE (see the proof of Theorem 2). A PNE would exist if the strategy space was convex, and utility quasi-concave (Fan, 1952). B) and C) demonstrate lack of quasi- concavity even if we allow \u2225s i \u2225 \u2264 1: B) n \u2212 1 producers at midpoint, s 1 along slice \u03bbc 1 + (1 \u2212 \u03bb)c 2 (dashed line); C) Change in utility along the slice in B) demonstrates lack of quasi-concavity. D) A non-negative game with very different PNE depending on \u03c4 . E) PNE with \"protective positioning.\" of proposition 4 due to Simon (1987). The producer utilities u i are not differentiable in the hardmax case though, which means we cannot use gradient information to find NE as in the softmax case. The only procedure we know for finding NE in hardmax games requires solving the hitting set problem which is NP-complete (Dasgupta et al., 2008). See Appendix B for further discussion.\nFigure 4: Clustering of strategic producers depends on the exploration level \u03c4 . A cluster is a set of points whose Euclidean distances from one another are less than 10 \u22125 \u221a d. As Theorem 3 predicts, large \u03c4 (e.g., more exploration) leads to higher concentration, i.e., creating content which appeals to more users. Left: MovieLens. Right: LastFM. See Section 3.2 for more discussion.\nFigure 5: Targeting of incentivized content by gender on MovieLens. Left: Difference between median c\u2208G {max i\u2208[n]ri (c)} for men and women (group G), withr i (c) the normalized rating (cosine similarity between c and the strategic s i ). Positive values imply bias towards men (higher median). Note the higher bias when d = 50 (more expressive algorithm); especially NMF incentivizes more biased content relative to the pre-adaptation baseline 'b'. Right: Difference in proportions of s i with best (normalized) rating by women/men. Positive values imply bias towards men (more items best-rated by men). Bias again more pronounced at d = 50. See Section 3.2 for more discussion.\nFigure 6: Incentivized content and creator gender on LastFM. Quantifying relative difficulty of strategic adaptation for female and male content creators, Uses baseline creator embeddings (and associated gender), and their cosine distance from strategic embeddings. Left: Difference between fractions of male and female creators in increasingly large neighborhood of each strategic item. Values above zero imply bias towards male producers. Higher embedding dimension (model expressivity) again results in larger bias. The bias also seems to be larger for higher \u03c4 and for the PMF rating model. Right: Difference between median cosine distance to female and male creators within increasingly large neighborhood of each strategic item. Values above zero imply bias towards male producers. Higher bias is again associated with higher embedding dimension and the PMF rating model, but the impact of temperature \u03c4 is less pronounced. See Section 3.2 for more discussion.\n```\n\nMain Content:\n```\n\nSection Title: INTRODUCTION\n  INTRODUCTION In 2018, Jonah Peretti (CEO, Buzzfeed) raised alarm when a Facebook main feed update started boosting junk and divisive content ( Hagey & Horwitz, 2021 ). In Poland, the same update caused an uptick in negative political messaging ( Hagey & Horwitz, 2021 ). Tailoring content to algorithms is not unique to social media. For example, some search engine optimization (SEO) professionals specialize on managing impacts of Google Search updates ( Marentis, 2014 ;  Dennis, 2016 ;  Shahzad et al., 2020 ;  Patil et al., 2021 ;  Goodwin, 2021 ). While motivations for adapting content range from economic to socio-political, they often translate into the same operative goal: exposure maximization. We study how algorithms affect exposure- maximizing content creators. We propose a novel incentive-based behavior model called an exposure game, where producers compete for a finite user attention pool by crafting content ranked highly by a given algorithm (Section 1.1). When producers act strategically, a steady state-Nash equilibrium (NE)-may be reached, with no one able to unilaterally improve their exposure (utility). The content produced in a NE can thus be interpreted as what the algorithm implicitly incentivizes. We focus on algorithms which model user preferences as an inner product of d-dimensional user and item embeddings, and rank items by the estimated preference. Section 2 presents theoretical results on the NE induced by these algorithms. We identify cases where algorithmic changes seemingly unconnected to producer incentives-e.g., switching from non-negative to unconstrained embeddings-determine whether there are zero, one, or multiple NE. The character of NE is also affected by the level of algorithmic exploration. Perhaps counter-intuitively, we show that high levels of exploration incentivize broadly appealing content, whereas low levels lead to specialization. In Section 3, we explore how creator behavior models can facilitate a pre-deployment audit. Such an audit could be particularly useful for assessing the producer impact of algorithmic changes, which is hard to measure by A/B testing for two important reasons: (1) producers cannot be easily randomized to distinct treatment groups, and (2) there is often a delay between deployment and content adaptation. Our hope is that this new style of auditing will enable detection of misalignment between the induced and desired incentives, and thus flag issues to either immediately address, or monitor in content filtering and moderation. For demonstration, we execute a pre-deployment audit on the MovieLens and LastFM datasets using the exposure game behavior model, and matrix factorization based recommenders. We find a strong dependence between algorithmic choices like embedding dimension and level of exploration, and properties of the incetivized content such as diversity (confirming our theory), and targeting of gender-based user and creator groups.\n\nSection Title: SETTING AND THE EXPOSURE GAME INCENTIVE MODEL\n  SETTING AND THE EXPOSURE GAME INCENTIVE MODEL We assume there is a fixed recommender system trained on past data, and a fixed population of users (consumers). Together, these induce a demand distribution P c which represents typical traffic on the platform over a predefined period of time. Content is created by n \u2208 N producers who try to maximize their expected exposure (utility). Denoting consumers by c \u223c P c , an item created by the i th producer by s i (strategy), s := (s i ) i\u2208[n] , and s \\i := (s j ) j\u0338 =i , we define (expected) exposure as the proportion of the \"user attention pool\" captured by the i th producer u i (s) = u i (s i , s \\i ) := E c\u223cPc [1{c is exposed to s i }] \u22c6 = E c\u223cPc [p i (c)] , (1) with p i (c) \u2265 0 the probability that the algorithm exposes c to s i rather than any s \\i . As common in game theory, we can extend from deterministic single item strategies to stochastic multi-item strategies s i \u223c P i for some distribution P i . This extension is discussed in more detail in Section 2. The assumption that E[1{c is exposed to s i }] \u22c6 = E[p i (c)] does not explicitly model interactions not mediated by the algorithm (e.g., YouTube videos linked to by an external website). This may be a reasonable approximation for infinite feed platforms (e.g., Twitter, Facebook, TikTok) where most consumers scroll through items in the algorithm-defined order, and search engines (e.g., Google, Bing) where first-page bias is well documented ( Craswell et al., 2008 ). While similar assumptions are common in the literature (e.g.,  Li et al., 2010 ;  Chen et al., 2019 ;  Ben-Porat et al., 2020 ;  Curmei et al., 2021 ), alternative interaction models are an important future research direction. Unlike previous work (Section 1.2), we focus on the popular class of factorization-based algorithms. These models rank items by a score estimated by the inner product of user and item embeddings c, s i \u2208 R d . The larger this score, the higher the probability of exposure, which we model as where \u03c4 \u2265 0 is a temperature parameter which controls the spread of exposure probabilities over the top scoring items. When \u03c4 = 0 (i.e., hardmax), these probabilities correspond to top- 1 recommendation or absolute first-position bias. Taking \u03c4 > 0 models the effects of ranked position, injected randomness for exploration, and can partially adjust for user randomness and other factors which make top-ranked items receive more but not all of the traffic. While an approximation in some settings, Equation (2) has been directly used, e.g., by YouTube ( Chen et al., 2019 ). We emphasize that we make no assumption on how the embeddings are obtained. Our conclusions thus apply equally to classical matrix factorization and deep learning-based systems. We are now ready to formalize exposure games, an incentive-based model of creator behavior. Definition 1. An exposure game consists of an embedding dimension d \u2208 N, a demand distribution P c \u2208 P(R d ), and n \u2208 N producers, each of whom chooses a strategy s i \u2208 S d\u22121 = {v \u2208 R d : \u2225v\u2225 = 1}, to maximize their utility u i (s) = E c\u223cPc [p i (c)] with p i (c) as in Equation (2) for a given \u03c4 \u2265 0. We restrict items s i to the unit sphere S d\u22121 . A norm constraint is necessary as otherwise exposure could be maximized by inflating \u2225s i \u2225 \u2192 \u221e, which is not observed in practice. 1 We distinguish non-negative games where all embeddings lie in the positive orthant; this includes algorithms ranging from TF-IDF, bag-of-words, to non-negative matrix factorization ( Lee & Seung, 1999 ), topic models ( Blei et al., 2003 ), and constrained neural networks ( Ayinde & Zurada, 2017 ). Definition 2. A non-negative exposure game is an exposure game where the support of P c is restricted to the positive orthant, i.e., P c ({c \u2208 R d : c j \u2265 0 , \u2200j \u2208 [d]}) = 1. We assume all producers are rational, omniscient, and fully control placement of s i in S d\u22121 . These assumptions are standard in both machine learning and economics literature, including in the related facility location games (see Section 1.2). They often provide a good first order approximation, and an important basis for studying the subtleties of real-world behavior. Full control is perhaps the least realistic, since producers can modify content features, but they often do not know how these changes affect the content embedding. This assumption has a significant advantage though: it abstracts away an explicit model of producer actions (cf. the variety of SEO techniques). Appropriateness of rationality and complete information are then context-dependent; they may be respectively reasonable in environments where strong profit motives or user profiling tools are common. However, investigating alternatives to each of the above assumptions is an important direction of future work.\n\nSection Title: Box 1: How our assumptions map onto YouTube (YT) as an illustrative example\n  Box 1: How our assumptions map onto YouTube (YT) as an illustrative example\n\nSection Title: Rational behavior\n  Rational behavior YT creators receive income proportional to their view numbers ( Figure 2 ), which motivates exposure maximization. Most creators do not earn significant income, but the majority of traffic is driven by only a few popular and high-earning creators ( Cheng et al., 2008 ). This motivates focus on these few producers and their strategic behavior.\n\nSection Title: Complete information and full-control.\n  Complete information and full-control. YT creators cannot directly manipulate the embeddings of their videos s i , or observe the user embeddings. However, popular creators have a myriad of analytic tools at their hand, with information about views, demographics (e.g., gender, age, region), acquisition channels, drivers of engagement, competition and more. They can also observe and adopt behaviors of other creators. Taking the strong monetary incentives into account, motivated creators will actively optimize their exposure using trial-and-error, making complete information and full-control an imperfect yet not unreasonable model of their behavior.\n\nSection Title: RELATED WORK\n  RELATED WORK Most relevant to our setup are works on the incentives of exposure-maximizing creators induced by recommender and retrieval systems ( Ben-Porat et al., 2020 ;  Raifer et al., 2017 ;  Ben-Basat et al., 2017 ;  Ben-Porat & Tennenholtz, 2018 ;  Ben-Porat et al., 2019b ;a). Interesting aspects of these works which we omit include (i) repeated interactions ( Ben-Porat et al., 2020 ;  Raifer et al., 2017 ;  Ben-Porat et al., 2019b ), (ii) user welfare ( Ben-Porat et al., 2020 ;  Ben-Basat et al., 2017 ;  Ben-Porat & Tennenholtz, 2018 ;  Ben-Porat et al., 2019a ), and (iii) incomplete information ( Raifer et al., 2017 ). The most important distinction of our approach is that the above works constrain creators to a predefined finite item catalog. This excludes the popular factorization-based algorithms-ranging from standard matrix factorization ( Koren et al., 2009 ) to (deep) two-tower architectures ( Huang et al., 2013 ;  Yi et al., 2019 )-whose continuous embedding space translates into an infinite number of possible items. The only exception is ( Ben-Porat et al., 2019a ) where items are represented by [0, 1] scalars, which is equivalent to the special case of two-dimensional non-negative exposure games. Continuous embedding spaces were recently studied in ( Mladenov et al., 2020 ;  Zhan et al., 2021 ), but neither studies producer incentives or competition.  Mladenov et al. (2020)  consider producers who decide whether to stay or leave the platform if their exposure is too low.  Zhan et al. (2021)  study design of recommender systems which optimize for both user and producer utility. Concurrently but independently,  Jagadeesan et al. (2022)  study a model equivalent to hardmax non- negative exposure games, except the \u2225s i \u2225 = 1 constraint is replaced by a production cost, yielding u i (s) = E[p i (c)] \u2212 \u2225s i \u2225 \u03b2 for some norm \u2225 \u00b7 \u2225 and \u03b2 \u2265 1 (higher norm interpreted as higher quality). The authors investigate how the cost function influences the economic phenomena exhibited by NE, from formation of \"genres\" (multiple directions with non-zero probability), to the possibility of realizing positive profits (utility). In contrast, we investigate how NE depend on algorithmic and environmental factors (non-negativity, exploration, dependence of exposure on ranking), and propose an algorithmic audit which leverages the creator model. While taking \u03b2 \u2192 \u221e in the Jagadeesan et al.'s cost recovers our unit norm constraint, understanding the NE behavior at the limit remains a subject of future work (e.g., pure NE exist only in our setup). Our works are thus largely complementary. Literature on adaptive behavior in the presence of a prediction algorithm is also relevant ( Hardt et al., 2016 ;  Kleinberg & Raghavan, 2020 ;  Perdomo et al., 2020 ;  Jagadeesan et al., 2021 ). The social impact and potential disparate effects of strategic adaptation have been analyzed in ( Milli et al., 2019 ;  Hu et al., 2019 ;  Liu et al., 2020 ). Most relevant for us is a recent paper by  Liu et al. (2022)  which studies strategic adaptation in the context of finite resources (e.g., number of accepted college applicants). Unlike us, the authors assume a single score for each competitor, who can pay cost to improve it. A principal then designs a reward function which allocates the finite resource based on the scores, and the authors study how different choices affect various notions of welfare. The preliminary results on multi- dimensional scores (appendix B) assume the scores and individual improvements are independent, whereas our scores-\u27e8c, s i \u27e9 for each c-imply complex dependence and trade-offs. Finally, our proposed methods for auditing recommender and information retrieval systems belong to a rapidly growing algorithm auditing toolbox. We focus on understanding producer incentives caused by a known algorithm. Thus, we complement prior work that aims to audit these systems based upon: the degree of consumer control ( Curmei et al., 2021 ), fairness ( Do et al., 2021 ), compliance with regulations ( Cen & Shah, 2021 ), and dynamical behavior in simulations ( Krauth et al., 2020 ;  Lucherini et al., 2021 ) or deployed systems  Haroon et al. (2022) .\n\nSection Title: EQUILIBRIA IN EXPOSURE GAMES\n  EQUILIBRIA IN EXPOSURE GAMES This section presents theoretical results on incentives in exposure games. We focus on the impact of the recommender/information retrieval model on the competitive equilibria. Throughout, we find that one of the most important factors determining existence and character of equilibria is the temperature \u03c4 (see Equation (2)). We thus distinguish the softmax (\u03c4 > 0) and the hardmax (\u03c4 = 0) case. In competitive settings, a key question is whether there are equilibria in which players are satisfied with their strategies, as otherwise there may be never-ending oscillation in search for better outcomes. We thus consider several solution concepts (i.e., definitions of equilibria) related to NE. A pure NE (PNE) is a point in strategy space s NE \u2208 (S d\u22121 ) n where no player i can increase their utility by unilaterally deviating from s NE i \u2208 S d\u22121 . In other words, no content producer can increase their exposure by modifying their content. Mixed NE (MNE) refer to the setting where players are allowed to choose randomized (mixed) strategies P i \u2208 P(S d\u22121 ). Rather than selecting a single piece of content, a creator following a mixed strategy samples s i \u223c P i . Alternative interpretation is that producers create multiple items, splitting their time/budget proportionally to the P i -probabilities. In later sections, we explore the weaker solution concepts of \u03f5-NE, local NE (LNE), and their combination \u03f5-LNE. An \u03f5-NE is an approximate NE where no producer can unilaterally increase their utility by more than \u03f5 (NE are \"0-NE\"). LNE are analogous to local optima: points where no player benefits from small deviations from their strategy. The approximate and local perspectives are relevant when deploying local search algorithms to find NE numerically (Section 3). Exposure games are symmetric, meaning that any permutation of strategies forming an equilibrium produces another equilibrium. Our statements on the existence and uniqueness of equilibria hold up to player permutation. All proofs for the results in this section are presented in the appendix.\n\nSection Title: PURE AND MIXED NASH EQUILIBRIA\n  PURE AND MIXED NASH EQUILIBRIA We begin by characterizing the existence of pure and mixed NE in general exposure games. Theorem 1. Every exposure game has at least one mixed Nash equilibrium. A key property of softmax games is that the utilities u i are continuous in s. This, and the compactness of the strategy space S d\u22121 , guarantees existence of MNE ( Glicksberg, 1952 , section 2). In the hardmax case (\u03c4 = 0), we can show that MNE are guaranteed to exist through a direct application We now turn to existence of pure NE, which is the setting where creators strategically design a single piece of content. Unlike MNE, PNE are not guaranteed to exist even in the softmax case. Theorem 2. PNE need not exist in either the hardmax (\u03c4 = 0) or softmax (\u03c4 > 0) exposure games. Figure 3A illustrates the non-existence result. The counter-example holds even for n = 2 players and planar (d = 2) strategies. A reader familiar with classic PNE results may ask if PNE would appear if we relaxed the S d\u22121 strategy space to the convex B d = {v : \u2225v\u2225 \u2264 1} ( Glicksberg, 1952 ;  Debreu, 1952 ;  Fan, 1952 ). This is not true as the exposure utility is not quasi-concave (Figure 3B&C). We now move to non-negative exposure games (Definition 2). For n = d = 2, non-negative hardmax exposure games are equivalent to Hotelling games ( Hotelling, 1929 ), and more generally to facility location games on a line ( Ben-Porat et al., 2019a ;  Procaccia & Tennenholtz, 2013 ). The next proposition lists several special cases in which we understand existence and character of PNE. Proposition 1. A PNE always exists in n = d = 2 non-negative hardmax games, but may not without non-negativity or when d > 2. For n = 2 non-negative softmax games with\u0109 := 1 n (1 \u2212 1 n ) E[c] \u0338 = 0, the only possible PNE is s 1 = s 2 =c withc :=\u0109/\u2225\u0109\u2225 (independently of d), but a PNE may not exist. When n > 2, non-negative softmax games can have a PNE other than s 1 = \u00b7 \u00b7 \u00b7 = s n =c. Figure 3D illustrates a 4-player non-negative exposure game. Depending on the temperature, we observe either the collapsed s i =c (large \u03c4 ), or what we term \"protective positioning\" (small \u03c4 ). In Figure 3D, players place their strategies between a consumer and the next closest producer. Figure 3E illustrates protective positioning for a higher number of consumers and n = 3. Here, consumers are roughly clustered around three centers (blue dots). The producer strategies are close to these centers, but again offset towards the most contested consumers.\n\nSection Title: \u03f5-NASH EQUILIBRIA\n  \u03f5-NASH EQUILIBRIA While existence of NE is not guaranteed, the situation changes when we adopt the weaker solution concept of \u03f5-NE, in which no producer can unilaterally increase their utility by more than \u03f5. The existence and character of such equilibria strongly depends on the temperature \u03c4 . When \u03c4 = \u221e, exposure is equally likely p i (c) = 1 n for all i and c regardless of the adopted strategies. Thus, every strategy profile is an NE. Considering a sequence of increasing (\u03c4 i ) i\u22651 , we can therefore argue that the limit of any convergent sequence of NE indexed by \u03c4 is a NE at \u03c4 = \u221e. Interestingly, Theorem 3 shows that a sufficiently large but finite \u03c4 > 0 is sufficient for existence of \u03f5-(P)NE. The result is constructive, showing that the \u03f5-PNE is parallel to the average consumer embedding. Corollary 1 underscores the sensitivity of exposure games to the temperature parameter \u03c4 , with uniformly homogeneous content at one end (high \u03c4 ), and potentially persistent oscillation behavior in competition when no NE exist (low \u03c4 ). A higher \u03c4 > 0 can be a result of algorithmic exploration ( Chen et al., 2019 ;  Cesa-Bianchi et al., 2017 ;  Lattimore & Szepesv\u00e1ri, 2020 ), which is provably necessary for optimal performance in static environments ( Lattimore & Szepesv\u00e1ri, 2020 ). In contrast, our results show that in environments with strategic actors, exploration may incentivize content which is uniform and broadly appealing rather than diverse. This may contradict the intuition that more exploration should lead to greater content diversity due to the higher exposure of niche content. One way to understand this result is the tension between randomization and the ability of niche creators to reach their audience: producers may be discouraged from creating niche content when the algorithm is exploring too much (\u03c4 high), and encouraged to mercilessly seek and protect their own niche when the algorithm performs little exploration (\u03c4 low). When the algorithm captures user preferences well, exploration is typically thought of as having negative impact on user experience through immediate reduction in quality of service as a result of suboptimal recommendations. However, the above results show secondary long-term effects.\n\nSection Title: LOCAL NASH EQUILIBRIA\n  LOCAL NASH EQUILIBRIA In a local NE, each s i is optimal on some of its neighborhood within the embedding space. Sometimes motivated as a form of bounded rationality, LNE can often be found by local search algorithms (e.g.,  Mazumdar et al., 2019 ). Since our motivation in studying exposure games is ultimately better system understanding and audits, we are particularly interested in these algorithmic benefits. Practical first-order algorithms for identifying LNE operate analogously to gradient descent, implying they may terminate in critical points that are not LNE. Unlike NE, critical points always exist. A tool for ex-ante (pre-deployment) assessment of producer impact could thus limit the harm experienced by users, moderators, and other affected parties. We demonstrate how to utilize a creator behavior model for this purpose, using the exposure game as a concrete example. The incorporation of factorization-based algorithms in exposure games allows us to use real-world datasets and rating models. While exposure games have limitations as a behavior model, we believe our experiments provide a useful illustration of the insights the proposed audit can offer to platform developers.\n\nSection Title: SETUP\n  SETUP We use the MovieLens-100K and LastFM-360K datasets ( Harper & Konstan, 2015 ;  Bertin- Mahieux et al., 2011 ;  Shakespeare et al., 2020 ), implement our code in Python ( van Rossum & Drake, 2009 ) and rely on numpy ( Harris et al., 2020 ), scikit-surprise ( Hug, 2020 ), pandas (pandas development team, 2020), matplotlib ( Hunter, 2007 ), jupyter ( Kluyver et al., 2016 ), reclab ( Krauth et al., 2020 ), and JAX ( Bradbury et al., 2018 ) packages to fit probabilistic (PMF;  Mnih & Salakhutdinov, 2007 ) and non-negative (NMF;  Lee & Seung, 1999 ) matrix factorization. The models are trained to predict the user ratings (centered in the PMF case). To select regularization and learning rate, we performed a two-fold 90/10 split cross-validation separately on each dataset. The tuned recommenders were then fit on the full dataset, and the resulting user embeddings, {c j } j\u2208[m] \u2282 R d , were used to construct the demand distribution P c = 1 m j \u03b4 cj , and evaluate the recommendation probabilities p i (c). Details in Appendix C.1. The only algorithm for finding NE in hardmax exposure games we know has exponential worst-case complexity. We thus focus on the softmax case. While search for general mixed NE is possible in special cases ( Fudenberg & Kreps, 1993 ;  Kaniovski & Young, 1995 ;  Bena\u00efm & Hirsch, 1997 ), we are not aware of any technique applicable to n-player exposure games. We therefore focus on pure \u03f5-LNE (Section 2.3), where each producer creates a single new item. We employ simple gradient ascent ( Singh et al., 2000 ;  Balduzzi et al., 2018 , see Appendix C.2 for comparison with gradient descent) combined with reparametrization s i = \u03b8 i /\u2225\u03b8 i \u2225 for each producer, where we iteratively update \u03b8 i,t = \u03b8 i,t\u22121 + \u03b1\u2207 \u03b8i,t\u22121 u i (s i,t\u22121 , s \\i,t\u22121 ) for shared step size \u03b1 > 0, and Equation (4) shows the update direction is parallel to the Riemannian gradient of u i (s) w.r.t. s i \u2208 S d\u22121 (Section 2.3). We also\n```\n\n======\nYour task:\nCompose a high-quality peer review of an ML paper submitted to a top-tier ML conference on OpenReview.\n\nThe review form is shown below:\n\n1. Summary and contributions: Briefly summarize the paper and its contributions\n\n2. Strengths: Describe the strengths of the work. Typical criteria include: soundness of the claims (theoretical grounding, empirical evaluation), significance and novelty of the contribution, and relevance to the community.\n\n3. Weaknesses: Explain the limitations of this work along the same axes as above.\n\n4. Correctness: Are the claims and method correct? Is the empirical methodology correct?\n\n5. Clarity: Is the paper well written?\n\n6. Relation to prior work: Is it clearly discussed how this work differs from previous contributions?\n\n7. Reproducibility: Are there enough details to reproduce the major results of this work?\n\n8. Additional feedback, comments, suggestions for improvement and questions for the authors\n\n9.  Overall score:\nYou should NOT assume that you were assigned a representative sample of submissions, nor should you adjust your scores to match the overall conference acceptance rates. The \u201cOverall Score\u201d for each submission should reflect your assessment of the submission\u2019s contributions.\n10): Top 5% of accepted papers. Truly groundbreaking work.\n9): Top 15% of accepted NeurIPS papers. An excellent submission; a strong accept.\n8): Top 50% of accepted NeurIPS papers. A very good submission; a clear accept.\n7): A good submission; accept.nI vote for accepting this submission, although I would not be upset if it were rejected.\n6): Marginally above the acceptance threshold. I tend to vote for accepting this submission, but rejecting it would not be that bad.\n5): Marginally below the acceptance threshold. I tend to vote for rejecting this submission, but accepting it would not be that bad.\n4): An okay submission, but not good enough; a reject. I vote for rejecting this submission, although I would not be upset if it were accepted.\n3): A clear reject. I vote and argue for rejecting this submission.\n2): I'm surprised this work was submitted to NeurIPS; a strong reject.\n1): Trivial or wrong or already known.\n\n10.  Confidence score:\n5): You are absolutely certain about your assessment. You are very familiar with the related work.\n4): You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.\n3): You are fairly confident in your assessment.  It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.\n2): You are willing to defend your assessment, but it is quite likely that you did not understand central parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.\n1): Your assessment is an educated guess. The submission is not in your area or the submission was difficult to understand. Math/other details were not carefully checked.\n\n11. Have the authors adequately addressed the broader impact of their work, including potential negative ethical and societal implications of their work?\n\n12. Does the submission raise potential ethical concerns? This includes methods, applications, or data that create or reinforce unfair bias or that have a primary purpose of harm or injury. If so, please explain briefly.\n\n