Modeling content creator incentives on algorithm-curated platforms
Authors: Jiri Hron
Karl Krauth
Michael Jordan
Niki Kilbertus
Sarah Dean
Venue: ICLR 2023 notable top 5% (2023)
Keywords: Nash equilibria
producer incentives
attention monetizing platforms
recommenders
differentiable games
exposure game

Abstract: Content 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—e.g., non-negative vs. unconstrained factorization—significantly 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.

Published as a conference paper at ICLR 2023
MODELING CONTENT CREATOR INCENTIVES ON ALGORITHM-CURATED PLATFORMS
Jiri Hron" Karl Krauth** " Michael I. Jordan 2, Niki Kilbertus Sarah Dean
University of Cambridge, UC Berkeley, TU Munich & Helmholtz Munich, *Connell University
ABSTRACT
Content 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, 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 exposure game, a model of incentives induced by algorithms, including modern factorization and (deep) two-tower architectures. We prove that seemingly innocuous algorithmic choices-ee e.g.. non-negative vs. unconstrained factorization fictorization-siiifiiiunlll affect the existence and character of (Nash) equilibria in exposure games. We proffer use of creator behavior models, like exposure games, 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 audit the MovieLens and LastFM datasets. Among clse, we find that the strategically produced content exhibits strong dependence between algorithmic exploration and content diversity, and between model expressivity and bias towards gender-bases user and creator groups.
INTRODUCTION
In 2018, Jonah Peretti (CEO, Buzzfeed) raised alarm when 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.
rI producers
fixed demand distribution
c Pe
$2
8n
utility of 8q
Ee- (f 8 )
Figure 1: Exposure game. Items 8; E S placed maximize exposure to consumers c Po-
We study how algorithms affect exposure- maximizing content creators. We propose a novel incentive-basee 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)-ma be reached, with no one able unilaterally improve their exposure (utility). The content produced in NE can thus be interpreted as what the algorithm implicitly incentivizes.
We focus on algorithms which model user preferences and item embeddings, and rank items by the estimated preference. Section 2 presents theoretical results 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 zero, one, multiple NE. The character of NE is also
inner product of d-dimensional user

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affected by the level of algorithmic exploration. Perhaps counter-intuitivve 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 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 al 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- basser user and creator groups.
1.1 SETTING AND THE EXPOSURE GAME INCENTIVE MODEL
We assume there is fixed recommender system trained on past data, and a fixed population of users (consumers). Together, these induce a demand distribution Pe which represents typical traffic on the platform over a predefined period of time. Content is created by n maximize their expected exposure (utility). Denoting consumers by c Pe, an item created by the 7th producer by 8; (strategy), 8 == (s)))e(m) and 81, == (s;))zi, we define (expected) exposure as the proportion of the "user attention pool" captured by the 1th producer
producers who try to
(s(s)= uz 8.) = Ee [1 is exposed to s;}] E
(1)
with (c) 2 0 the probability that the algorithm exposes to rather than any sy As common in game theory, we can extend from deterministic single item strategies stochastic multi-item strategies 8; ~ P, for some distribution P; This extension is discussed in more detail in Section 2.
The assumption that E[ {c is exposed to H = E [c)] does not explicitly model interactions not mediated by the algorithm (e.g.. YouTube videos linked 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-definet 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 (c.g., Li et al., 2010; Chen et al., 2019; Ben-Porat et al., 2020; Curmei et al., 2021), alternative interaction models are important future research direction.
Unlike previous work (Section 1.2), we focus on the popular class of factorization- -baed algorithms. These models rank items by score estimated by the inner product of user and item embeddings c, 8 E Rd. The larger this score, the higher the probability of exposure, which we model as
pe(c) = exp(T (c, s;))
EV= expep-ppppppp (c)) = softmax (I-- 8y 81))
where is temperature parameter which controls the spread of exposure probabilities over the top scoring items. When 7 = 0 (i.e., hardmax), these probabilities correspond to top- I recommendation or absolute first-position bias. Taking 7 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-basse systems.
We are now ready to formalize exposure games, an incentive-based model of creator behavior.
An exposure game consists of an embedding dimension d E N, a demand distribution P. E P(R)), and n E N producers, each of whom chooses a strategy 8; E Sd-1 {v E Rd: |||||| = 1}. to maximize their utility ux(x) = P.c. [P((c) with pt (c(c) in Equation (2) for given 2 0.
Definition 1.
We restrict items s; to the unit sphere S g A norm constraint is necessary as otherwise exposure could be maximized by inflating ||s + oo, which is not observed in practice.' We distinguish
Possiily due the often finite rating scale, use of gradient clipping, and various forms of regularization.
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non-negative games where all embeddings lie in the positive orthant; this includes algorithms ranging from TF-IDF, bag-of-words. to non-negan 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 exposure game where the support of Pa restricted to the positive orthant, iee, Paffc E I : Cj 2 0, Vj E [d]) =
We assume all producers are rational, omniscient, and fully control placement of in Sd These assumptions are standard both machine learning and economics literature, including in the related facility location games (see Section 1.2). They often provide 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 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- -ontext-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.
Box 1: How our assumptions map onto YouTube (YT) as an illustrative example. On YT. strategy 8, an embedding of video, with creators able produce multiple videos (mixed strategy 8; P.). Rational behavior: YT creators receive income proportional to their view numbers (Figure 2), which motivates exposure maximization. Most creators do not significant income, but the majority of traffic is driven by only a few popular and high-earning creators (Cheng al., 2008). This motivates focus on these few producers and their strategic behavior. Complete information and full-control. YT creators cannot directly manipulate the embeddings of their videos st, 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.
sponsored content
payment
donation services, uag. Fatroof
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YouTube creator income
YouTube Premium
affillate marketing. Ispannared lines)
memberships
streams
webshops
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Figure 2: YouTube revenue streams
incentivizing exposure maximization (Ørmen & Gregersen, 2022).
1.2 RELATED WORK
Most relevant setup are works 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 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-Base 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 predefined finite item catalog. This excludes the popular factorization-bart -hased algorithms trantthmm---nniingggg from standard matrix factorization (Koren et al., 2009) to (deep) two-tower architectures (Huang et al., 2013; Yi et al., 2019)-hosse 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 [o, 1] scalars, which is equivalent to the special case of two-dimensional non-negative exposure games. Continuous embedding spaces 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 al. (2021) study design of recommender systems which optimize for both user and producer utility.
recently studied in (Mladenov et al., 2020; Zhan al., 2021),
Concurrently but independently, Jagadeesan et al. (2022) study a model equivalent to hardmax non- negative exposure games, except the ||>:|| constraint is replaced by production cost, yielding
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u,(s) Epp[cc]] - || for some norm || and 3 2 (higher norm interpreted 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 3 - oo 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 al., 2021). 1). The social impact and potential disparate effects of strategic adaptation have been analyzed in (Milli et al., 2019; Hu al., 2019; Liu al., 2020). Most relevant for us is 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 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 scores-(e, 8)) 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 known algorithm. Thus, 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).
2 EQUILIBRIA IN EXPOSURE GAMES
This section presents theoretical results on incentives in exposure games. We focus on the impact of the recommenderinffomaaiio 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 (see Equation (2)). We thus distinguish the softmax (r > 0) and the hardmax (T 0) case.
In competitive settings, a key question is whether there are equilibria in which players are satisfied with their strategies, 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 NE (5d-13)" where no player can increase their utility by unilaterally deviating from sN E 5d-1, 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 choose randomized (mixed) strategies P. E P(S Rather than selecting single piece of content, a creator following mixed strategy samples s; ~ P,. Alternative interpretation that producers create multiple items, splitting their time/budget proportionally to the P:-probabilities.
In later sections, we explore the weaker solution concepts of e-NE, local NE (LNE), and their combination c-LNE. An e-NE is approximate NE where no producer unilaterally increase their utility by more than 6 (NE are "O-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.
2.1 PURE AND MIXED NASH FQUILIBRIA
We begin by characterizing the existence of pure and mixed NE in general exposure games.
Theorem 1. Every exposure game has least one mixed Nash equilibrium.
A key property of softmax games is that the utilities u; continuous in s. This, and the compactness of the strategy space Sd guarantees existence of MNE (Glicksberg, 1952, section 2). In the hardmax case (7 = 0), can show that MNE are guaranteed to exist through a direct application
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Al non existence
Siice position
Utility change along sice
DJ Nan neg example
example
0.00
-002
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0.
Figure 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 ||s S 1: B) n - producers at midpoint, $1 along slice λcι + (1 - A)e2 (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 т. E) PNE with "protective positioning."
of proposition 4 due to Simon (1987). The producer utilities u, are not differentiable the hardmax case though, which means cannot use gradient information to find NE 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.
We now turn to existence of pure NE, which is the setting where creators strategically design 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 (T 0) softmax (T 0) exposure games.
Figure 3A illustrates the non-existence result. The counter-example holds even for n 2 players and planar (d = 2) strategies. reader familiar with classic PNE results may ask if PNE would appear we relaxed the 5d-1 strategy space to the convex B {v: |||||| < 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 -Brat et al., 2019a; Procaccia & Tennenholtz, 2013). The next proposition lists several special cases in which we understand existence and character of PNE.
Proposition 1.
PNE always exists in n = = non-negative hardmax, games, but may not without
non-negativity or when d > 2. For n 2 non-negative softmax games with è == 1 (1 Bic F O, the only possible PNE is 81 = 82 = č with č == é/||ē|| (independently of d), but a PNE may not exist. When n 2, non-negative sofimax games can have a PNE other than $1
S
Figure 3D illustrates 4-player non-negative exposure game. Depending on the temperature, we observe either the collapsed 8; = ē (large T), or what we term "protective positioning" (small r). In Figure 3D, players place their strategies between 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 these centers, but again offset towards the most contested consumers.
2.2 C-NASH EQUILIBRIA
While existence of NE is not guaranteed, the situation changes when we adopt the weaker solution concept of c-NE, in which no producer can unilaterally increase their utility by more than c.
The existence and character of such equilibria strongly depends on the temperature т. When oc, exposure is equally likely p:(c) = _ for all and e regardless of the adopted strategies. Thus, every strategy profile is an NE. Considering a sequence of increasing (Fi) we can therefore argue that the limit of any convergent sequence of NE indexed by is NE at 7 = co. Interestingly, Theorem 3 shows that a sufficiently large but finite T > 0 is sufficient for existence of e-PP)NE. The result is constructive, showing that the c-NNE is parallel to the average consumer embedding.
Theorem 3. For any > O and P. E P(Rd) with compact support and Ejc] + 0, > 0 s.s. 81 _... Sn is an e-PNE for all 2 To. Moreover, for all To, the smallest e, for which e is an PNEE satisfies 6 < &. If e < [|e]], then the set of beeteer-resposes to ē
:u v (() = {o E 1 (v. ........,, ... 2 ui 2
(3)
is subset of (c) {v: ||| c|| < 8) with 5 2t/ 2+/(1|@] c). and 8 0 as co.
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This result shows that all e-improvements concentrate near the consumer average c-PNE as T co.
Additionally, the "consumer symmetry" ||ق|| = (1- - ) [ determines how quickly 5 + 0. When consumers are spread approximately symmetrically w.r.. the origin, the degenerate equilibrium appears only for large v. However, smaller r are sufficient for more directionally concentrated P.- A high number of producers also slows the concentration the appeal of u. u., with n. We conclude with a corollary based on our development so far.
decreases
Corollary 1. There a fixed Eo and demand distribution Pe deiich-d on the chosen i-inuuce zero, one, multiple, or infinitely many e-NE for all e S 60-
Corollary 1 underscores the sensitivity of exposure games the temperature parameter т, with uniformly homogeneous content at one end (high r), and potentially persistent oscillation behavior in competition when no NE exist (low 7). A higher т > can be a result of algorithmic exploration (Chen et al., 2019; Cesa- Desa-Bianchi et al., 2017; Lattimore & Szepesvári, 2020), which is provably necessary for optimal performance in static environments (Lattimore & Szepesvári, 2020). In contrast, our results show that in environments with strategic actors, exploration may incentivize content which 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 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 (T high), and encouraged to mercilessly seek and protect their own niche when the algorithm performs little exploration (7 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.
2.3 LOCAL NASH EQUILIBRIA
In local NE, each s; 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.
Proposition 2. Every T > 0 exposure game with E(c] * 0 has a critical point at 81 =... = 8n = C.
As have seen, 81 .... Sn may be equilibrium (Proposition 1). To distinguish LNE from mere critical points, we use the Riemannian second derivative test, treating 5d-1 as a Riemannian submanifold of Rd as usual. For background, see (Boumal, 2022, sections 3 & 5).
Definition (Boumal, 2022, lemma 5.41). A point s in strategy space satisfies the second derivative s; un (s(s)) is zero, and (2) the Riemannian Hessian test test vi (1) the Riemannian gradient (T (188 V
(I (18 [V2, [s)] (1- s;ss) (st, V., (s))(())(I -1;,,)),
strictly negative definite in the subspace perpendicular to st.
This condition is sufficient but not necessary for critical point to be an LNE. LNE which satisfy Definition 3 are termed differentiable NE (Ratliff et al., 2016; Balduzzi et al., 2018). The distinction is similar to that between the flat minimum of 24 at zero the more well-behaved z??
3 PRE-DEPLOYMENT AUDIT OF STRATEGIC CREATOR INCENTIVES
Beyond regularly retraining on new data, online platforms continuously roll out algorithm updates. While A/B testing can detect changes in user metrics, like satisfaction or churn, prior to the full-scale deployment (Tang et al., 2010; Hohnhold et al., 2015; Xu et al., 2015; Gordon et al., 2019), assessing the impact on content producers is comparatively harder due to the longer delay between the roll-out and corresponding content adaptation. Furthermore, since producers cannot be easily assigned to distinct treatment groups without limiting their content to only a subset of consumers, modern A/B testing methods must eschew making causal statements about producer impact (Nandy et al., 2021: Ha- Thuc et al., 2020; Huszár et al., 2022). Undesirable results including promulgation of junk and abusive content then have be addressed via post-hoc measures like content filtration and moderation.

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clusters # producen
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Figure 4: Clustering of strategic producers depends on the exploration level т. A cluster is a set of points whose Euclidean distances from one another are less than 10 10-5vd. As Theorem predicts, large (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.
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-basel 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 useful illustration of the insights the proposed audit can offer to platform developers.
3.1 SETUP
We use the MovieLens 1etennss1000 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), matplot ib (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, {cy [cmm C Rd, were used to construct the demand distribution P. = 1 E3 Oey, and evaluate the recommendation probabilities (c). Details in Appendix C.I.
The only algorithm for finding NE 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; Benaim & Hirsch, 1997), we are not aware of any technique applicable to n-player exposure games. We therefore focus on pure e-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, 8, see Appendix C.2 for comparison with gradient descent) combined with reparametrization s; = 0./1|0, for each producer, where we iteratively update 0r, 02 + a a VV... (4 ( for shared step size 0, and
V Vo,u:(5) [ (T
(iss) Elps(c)(1 ())()))))()))))))))) = T (T 818s (.)))
(4)
Equation (4) shows the update direction is parallel to the Riemannian gradient of us(s) w.r.t. 8; E S (Section 2.3). We also experimented with the related Riemannian gradient ascent optimizer (Boumal, 2022), but abandoned it after (predictably) observing little qualitative difference. We note that the local updates themselves define better-response dynamics linked iterative minor content changes; investigation of their relation to real-world producer behavior is an interesting future direction.
We investigate the sensitivity of the incentivized content to the: (i) rating model E {PMF, NMF), (ii) embedding dimension d E (3, 50), and (iii) temperature log E (-2,-11, 0). We further vary the number of producers n {10, 100) to examine scenarios with different producer to consumer ratios (user count is fixed to the full 943 for Movielens, and 13,698 for LastFM). The above values were selected
preliminary sweep as representative of the effects presented below. For every setting, we used five random seeds for initialization of the recommender (affects P.), and for each ran the gradient ascent algorithm 10x to identify possible e-LNE. We applied early stopping when e-change in parameters between iterations dipped below 10 8 vd; the number of iterations was set to 50K so convergence was achieved for every run. We only report runs where the second-order Riemannian test from Section 2.3 did not rule out an e-LNE, Additional results, including those where the Riemannian test was conclusive, are in Appendix C.3.
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dit
0.05
10
A % nearest
0015
pmf
à 0.06
b.001011
1.0 b0010.
01 temperature (T)
temperature (1)
Difference between mediancee [m]] }:(c) for men and women (group G), with F:(c) the normalized rating (cosine similarity between e and the strategic s,). 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 adeapatttiionn baseline 'b'. Right: Difference in proportions of s; 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.
Figure 5: Targeting of incentivized content by gender on MovicLens. Left:
3.2 RESULTS
Emergence of clusters with growing т. Theorem 3 shows that producers concentrate around E [lcl/|] for sufficiently high т. Figure 4 corroborates the result on both MovieLens and LastFM, with the concentration happening already at r = =1 regardless of the embedding dimension d and producer count n. We also see that lower r can lead to "local clustering" where only few producers converge onto the same strategy. We hypothesize that the simultaneous local updates of the consumers create "attractor zones" where close-by producers collapse onto each other; they will remain collapsed henceforth due to equality of their gradients (by symmetry). Theorem 3 does tell us collapse is to be expected for high т. and it is possible that a local version of the result with more than one clusters is true for intermediate values of This highlights how crucial the algorithmic choice of r is for the induced incentives within our model.
Targeting of incentivized content by gender. The MovieLens dataset contains binarized user gender information. In Figure 5, we examine targeting of incentivized content on women and men. To do so, we employ aggregate statisties of predicted ratings. While predicted ratings may differ from actual user preferences, they do determine recommendations and thus user experience. To help disentangle effect of exposure maximization, we also include statistics based on the original item locations (labeled by 'b'), ), i.e., the content created before producers adapt to the recommender. Since the baseline embeddings need not satisfy the unit norm constraint (see Definition 1), we measure normalized ratings [c) ;= (c.s.)/ [s to facilitate comparison. The normalization also alleviates the known issue of varying interpretation of ranking scales between users (Lynch Jr et al., 1991).
In Figure (left), medianegmen (max; F; (c() median (max; f:())) measures if the incentivised content is predicted to appeal women/men; Figure 5 (right) shows the fraction of creators incentivised to target women/men: Σi= 1 (argmax, (c) E men] 1 (argmax, F. (c) E women]. Positive values signify content crafted for male audience (users are 71% male). Higher embedding dimension results in more bias, presumably due to the larger model expressivity, and thus enables more fine-grained targeting. NMF consistently incentivizes more biased content.
Association between incentivized content and creator gender. Platform developers may want to know if some creators are being disadvantaged (Chokshi, 2017: Farokhmanesh, 2018; Rodriguez, 2022). While solutions were proposed in the static case (e.g, Beutel et al., 2019; Wang et al., 2021), understanding if the algorithm (de)incentivizes content by particular creator groups may limit future harm. In Figure 6, measure the difference between the proportion of (left) and the median distance to (right) baseline creator embeddings (learned by the recommender before strategic adaptation), within increasingly large neighborhoods of each strategic s,. Since the baseline embeddings need not be unit norm, we use the cosine distance to define the neighborhoods.
Starting with the proportion (left), higher embedding dimension (more flexible model) incentivizes content more typical of male artists. This may be related to the higher prevalence of men in LastFM, combined with training by average loss minimization. The gender imbalance also explains why the proportion (left) stabilizes at positive value, whereas the median distance (right) reverts to zero, as the number of considered neighbors grows. The bias is also related to the choice of rating model, where especially PMF at high temperatures results in significant advantage for male artists.
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dim
reighbus of gender
1.00
med. medurw d55.
1.0
0.4
0.50
0.4
60c
-052
neighbors
neighbors
nnéghhors
neighbors
Figure 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 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 T less pronounced. See Section 3.2 for more discussion.
4 DISCUSSION
From social media and streaming to Google Search, many of interact with recommender and information retrieval systems every day. While the core algorithms have been developed and analyzed years ago, the socio-economic context in which they operate received comparatively little attention in the academic literature. We make two main contributions: (a) we define exposure games, an incentive- based model of content creators' interactions with real-world algorithms including the popular matrix factorization and two-tower systems, and (b) we formulate a pre-deployment audit which employs a model of creator behavior identify misalignment between incentivized and desirable content.
Our main theoretical contributions focus on the properties of Nash equilibria in exposure games. We found that seemingly innocuous algorithmic choices like temperature т, embedding dimension d. or non-negativity constraint on embeddings can have serious impact on the induced incentives. For example, high 7 incentivizes uniform broadly appealing content, whereas low 7 motivates targeting smaller consumer groups. Since higher is often linked to exploration, which is necessary for optimal performance in static settings (e.g.. Lattimore & Szepesvári, 2020), this result highlights the importance of considering the socio-economic context in algorithm development.
Our producer model has several limitations from assuming rationality, complete information, and full control, to taking the skill set of each producer to be the same, their utility to be linear in total exposure, and ignoring algorithmic diversification of recommendations. We also consider the attention pool as fixed and finite, neglecting the problematic reality of the modern attention economy, where online platforms constantly struggle to increase their user numbers and daily usage (Covington et al., 2016; Williams, 2018; Bhargava & Velasquez, 2021). Our theoretical understanding is incomplete as, e.g., our understanding of the influence of constraining embeddings to be non-negative is limited to the two-dimensional case. The empirical evaluation of our behavior model is hindered by the lack of academic access to the almost exclusively privately owned platforms (Greene et al., 2022).
Due to their sizable influence on individuals, societies, and economy (Milano et al., 2020), information and recommender systems are of critical importance from an ethical and societal perspective. While we hope that better understanding of the incentives these algorithms create will mitigate their negative social consequences, this also entails risks. Perhaps the most important is the possibility of employing an optimizer such as the one in Section to game real-world algorithm. This is especially relevant to the current debate about transparency (e.g.. Sonboli et al., 2021; Rieder & Hofmann, 2020; Sinha & Swearingen, 2002), and the proposal to (partially) open-source the Twitter code base (Knight, 2022). Due to the aforementioned limitations, we also caution against treating the predictions of our srrentive-based behavior model as definitive, especially given the significant complexity of many
algorithms and the environments in which they operate.
Going forward, we want to deepen our understanding of exposure games, and make pre-deplymment audits a practical addition to the algorithm auditing toolbox. We hope this research enriches the debate about online platforms by a useful perspective for thinking about harms, (over)ampliffcatton, and design of algorithms with regard to the relevant incentives of the involved actors

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A PROOFS
List of abbreviations:
s.t. such that
a.s. almost surely
sh.s. = left-hand side
r.h.s. = right-hand side
w.l.o.g. without loss of generality
A.1 STAND-ALONE STATEMENTS
Theorem 2. PNE need not exist in either the hardmax (T 0) or softmax (T > 0) exposure games.
Proof Theorem 2. (I) Hardmax: Take n = d == 2 and P - Σ3= Sc, where the angle between any сj # C₁ is 2r Assume (#1, $2) is PNE. W.L.o.g $2 on the geodesic connecting C2 and C3 which has higher dot product with both C2 and 0g than s1- Hence (s) > 2/3 by the assumption that is PNE. The same argument implies ua (s) 2 2/3. This a contradiction since 2; (s))) = by definition of the exposure utility.
argmax, (s1. Then there is
3
(II) Softmax: Let n = 2, and P. = (28 + 8Gz) where é = [1. TT and C2 = T By Proposition 1, we know that the only possible PNE is 81 82
E(c] [2, 1]/3, where both
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players enjoy (s(s) u2(s) 2. Let $ (2 + cc1)/l ceill for some 0. As w1 (8 3 by continuity. Hence Fro S.t.L 81 82 is not PNE for all 70-
Theorem 3. For any > 0 and Pa E P(R") with compact support and Elc] + 0, 3.o > 0 s.f.
81 .... = Sn is an e-PNE for all 2 To. Moreover, for all To, the smallest E, for which is an e,EE satisfies I +. al e < ||ë||, then the set of better-responses to غ
(c) = (o E Sd : (v. ,,,,, un e, cc.,,,....
(3)
is subset of (C) = fo: || a|| s 6) with = 2 - c). and - 0 7
Proof of Theorem 3. We w.l.o.g. focus on the defection strategies for By the mean-value theorem
= ui (31, 2) - 4n (c. , - (0' 81 -),
where 01 V u1 (1 .. a) for some s on the line connecting 81 and č. While the rigorous
argument below relies on few technicalities, the main idea is simple: as r + co, T gf + é =
E[c] uniformly over 8i 5 (Lemma 1), and thus A ~ (č, 81 - [1] 0.
Lemma 1. 1im sup SuPs,ES6-1 ||T (/-l-1---- -cl +
Proof of Lemma Since sP) compact by assumption, and r T (c) p (c)) all we need is (i(c) (c()) [1 ( pointwise in (dominated convergence), and uniform over of Bd {u: ||l||| < 1} (mean-value theorem yields si on the line connecting 81 with e). As
exp(r oxp(r-1(e.5)) (0,5))) (i(c) exp( (c,,,)))( exxxp( (c,e))
monotonic in T 7 (c, s1), and sup,, (c, s1) = ||||| < oo by compactness, (1 will converge to [1 1) uniformly over 3d by continuity of the exponential function at zero.
For any given > 0, Lemma 1 can be combined with
T A < (2, 91 - -) + şi - ||||| -cl,
where (2, s1 - d) < for all s1 e sdl1 by è/||ē| obtain A for sufficiently large т. In particular, Lemma yields such that TT 91 - é|l - e/2, which ensures
|| s |||| - c < 2/|T of 2| 6,
for all r 2 To. Hence č is at least an ş PNE for all r 2 To (w.l.o.g. To 2 1).
The above can be used obtain bound on ||s| a|l for 81 E (c)). Using orthogonality
A = ((I- )g' 81 - é) + (2, -
<- - 1ll []] -6([[-=6111) اill (c, - gi) || - -a]]
by the triangle inequality, and (z, $1 c; - (2(2, .1) - 2) - - all by $1, S Sd-1, The terms in the square bracket the r.h.s. can be bounded using the Pythagoras' theorem
||T oi - a|1 [|(I 06 (T 91 2 + ||c0" (r 0012
[|(I - σ8 ThTT) )r 1/122 (2, + 91) ||||
where we used (I - हc ) 0 and ||ब|| = 1. Because || gi - all < e, the same is true for (the square
roots of) both terms on the r.h.s. above. By a simple algebraic manipulation of these inequalities
r 7 < v (1|=l -
(5)
The r.h.s. is positive only when 0 2e/((a) - <). Since in Equation (5) is only used as an upper bound || -cll, and Lemma tells this norm converges zero, > 0 as - [
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Proposition 1. A PNE always exists in d 2 non-negative hardmax games, but may not without
non-negativity or when >> 2. For n = 2 non-negative softmax games with é = 1 1 n. Eic + O, the only possible PNE is $1 82 č with č := é/||@|| (independently of d), but a PNE may not exist. When n > 2, non-negative softmax games can have a PNE other than 81 = ...
C.
Proof of Proposition 1. (I) Hardmax: For existence when n = 2, let Og be the angle of c from (w.l.o.g.) [1. 0J. and let A C C' denote the set of angles such that for every On E A, PPO,. I 0m) 2 and P(O. 2 0m.) 2 2. with P implied by the underlying Pe. Then any (21, EAX A A PNE.
For non-existence when d > 2, consider d = 3 and P. - 3 deg where Cj are the three canonical basis vectors, Assume = (31, $2) is a PNE. Disregards of $1 location, there will be point 82 on the great circle connecting the two most distant points from 81 (break ties arbitrarily) which is closer to both of the two. Hence uz(s) 2 2/3 by the assumption that is a PNE. The same argument implies in (s) 2 2/3. This is a contradiction since >. us(s) = by definition.
For non-existence without non-negativity in d see the hardmax part of the Theorem proof.
(II) Softmax: In the n = 2 case, a necessary condition for (51,82) 82) to be PNE is that the Riemannian gradients of the utility, (I - 5,8; )9; with y; = (s), are zero. Since V., (s) = -- (1) pc(c))cl, 9i belongs the first orthant by the definition of a non-negative game, and is not zero (for r >> 0, all probabilities lie in (0, 1). and c not a.s. zero since we assumed E(c] * 0). Hence the Riemannian gradients can only be zero if 8; X 9i. and in particular 8; = st/l[g:||> ||2 because this is the direction which makes dot products with all vectors in the first orthant positive.
Crucially, 91 = 92 in 2-player games due to the symmetry of pl (c) - (e)) = pi (c)p2((() = (1 (e)). Therefore at PNE, 81 82 in which case pc(c) 2 for all c. Thus gi(s) x
Elc, implying $1 = 82 = ē is the only possible PNE. To show it may not be a PNE, consider P.= 22ee, + 6c,) for arbitrary non-zero C1 # 02 in the first orthant. Then
2eg + c2 with ui (1 c) (2 c)- = 1/2. Fixing 81 = / ///llc ||2 and taking 7 + O, we get in (81, 6) + 2/3, which
means there exists a > 0 for which 81 / /|/cal ||2 is strict improvement over 81 when 82 C.
For the >>> 2 case, we focus on two-dimensional = game with P. [ + 6eg)) ) with a = OT and and C₂ [0, 1] (the two canonical basis vectors). In particular, investigate existence of NE of the form 81 = 82 and $3 = 84. Since d = 2, the strategies are restricted to 5 which means will further restrict our attention to the symmetric case 0 02 = 0 and 03- = 04 = 0 for some 0 E [0, f = K. This allows us define
use polar coordinates parameterize s; = ((())) = [cos(B,),, sin)))) We
Q = (i )
and look for values of 8 E K where (w.l.o.g.) f(0) = Su, (s) 8s 8= (91 Q=1) is equal zero.
Note that in the definition of f. all s; and Ji vary with @ according to the relationship s; = (00,) O. However, [(0) is only the derivative of un (s) w.r.t. 0,, ignoring the dependence of 82, 83 and 84 on O. This definition of f means that only the roots of can possibly be NE. The next lemma will help us locate these roots.
with 0, 0₂ 0 and 03- 04 =
Lemma 2. For a sufficiently small r > 0, f: 0 + (g1. Q81) is strictly convex on K.
Proof of Lemma 2. It is sufficient to prove that f" on K. For this, observe f((0) ||Qs 1l (91,81) where Hi == v (s), and f*(0)= ||Qsile (Qs1,3Hiss + g1) 2
1Q0sl12,, H. - 3| H1 - llar||2, where by construction
(c)) - -
(m(e)ii 1 - (m))
H1
(1 (=-p
Ve, H1
2
1
(1 - 6pt (cx) - -m
(1 ppiteett-pieeippcelll-----
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gracient
9.55
0.24
64-
66-
64-
-000
0.2
12
-306
6A
0.25
0.0 0.2
0.30
0.25
375
2.00 325
350
0.00 0.5 3.50
6.15
300
1.5
150
32
Figure 7: n > 2 softmax case from the proof of Proposition 1. Left: Symmetric PNE location (here
for T = 1). Middle left: Plot of f(0) = (91 (s), Qsa) with 81 = 52 (00) and 83 = 54 = qp( 2 0). Right: Plot of utility and its gradient for all possible defection strategies 81 (p(8)) with $2, 83, 84 kept put in the positions defined by 8 from the left plots. Vertical line shows π (right end K).
11 ~ ~ |Hill ~ ~ and || ~T implying that for low enough, the Hence [|
positive term ||@ H, dominates (using that all expressions share the term pı(c) - pi (c)). and thus after dividing and observing -> gives pi (c() close to either one zero, we get that all the terms scale as pi (1) pi (c))/T* for the appropriate E (1,2,3))
Lemma implies there are at most two NE (f is strictly convex, and f(0) = 0 is a necessary condition). At 0 7. 82 83 84 by definition, which know is critical point, so f(1) = 0. Since 91 & Elpr - (1)p))) A for any T > 0 (c cannot be a.s. 0 by E[c] * and the noggtiigt assumption), f(0) = (91 Qs1) > 0 (s1 = (0)) = = = [1,0]T) The other possible root of thus could only be in the interior (0, () of K. For small enough T, moving from
= - towards e = [1. OJ will increase utility, implying f( < 0. Hence there exists T 0 and 87 (0, } s.t. f(0;)
by the mean value theorem.
So far we have established that corresponding small т. By symmetry, it is sufficient to check if there a defection strategy for s1. Any defection to 0, E (0,, _ -00 0*] will result in (c() < for both = C1, e2, and thus worse utility. Defection to (2 Ar 2 will not yield utility greater than defection to [0, 87) since
= (0;)),
5 07) is local NE for the
$3 $1 pp( 07). so it is sufficient to focus on 0 [0, 60. Here
Ui = (g), Qsi)
(e2) pı (c2)) cos(O:) (1())( Pa (c1)) 2 (c) - pı (ci)) (c(o)))))B))))) (01) (in)
since Pi(ci) grows quicker than pi (c2) decays. By construction, 0r < T. and we know cos(O) s0n(O) > 0 for 0 E F). In other words, the utility of 81 is strictly increasing on 0, 0, 0;), i.e., none of the corresponding 81 (0(0) is an improvement. Hence 81 = 82 ( $3 = $4 = _ 0:) is NE. (The construction illustrated in Figure 7.)
Proposition 2. Every T > 0 exposure game with E(c] + 0 has a critical point at $1 = Sn
Proof of Proposition 2. When 81 =.... = 8n = C. the gradient from Equation (4) is the same for all producers, and it is proportional to (T ec. )č. This is equal by 6/1A/
A.2 INLINE STATEMENTS
Lemma 3. The distribution from the part (I) of the proof of Theorem 2 = 2, P-= Σ3 dey with 0j # Ck
25 apart-admits admits a mixed NE P1 = P2 = Pa at 7 =
Proof of Lemma 3. By symmetry, u(Po, P.) 1. Vi. Since for any s1 E supp(P.) (2,
Eg,s, [n1 (81, 92) [1] (s1, + (11, c2) + un (5), -111 1 +2 [1 + + =
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all we need is to show that (ute][11(11) [1] s1]= 2 for any $1 & supp(P.). W.l.ogg. assume $1 lies on the geodesic connecting 61 and C3 (i.e., on the are opposite of c2). Such an $1 is closer to ej and Og than 02 (u, (31, C2) f)), but is further from ej
and c2 (resp. Og and 02) than C3 (resp. ej). Hence
Be ($1, s2) $1] um (51, c1) un u1 02) un ($1, ca)] 3
=
Since
1 2 $1 has no incentive to move any of its mass away from supp (Po).
B HARDMAX GAMES
In this section we present two different algorithms for finding mixed Nash equilibria in two-player hardmax games. We note that the set of allowable mixed strategies must be restricted in some way since certain distributions with support on the unit-sphere 5d-1 require infinite storage. Hence, our first algorithm finds mixed NE for a discretized strategy space, while our second algorithm considers settings where P. is discrete and finds mixed NE with support over finite number of pure strategies in the original non- con-discretized space, assuming such a mixed NE exists.
We caution that both of these algorithms can only find mixed NE for small exposure games due to their poor scaling properties. We list them here highlight the difficulty of solving hardmax games when compared to the softmax setting and to serve as inspiration for future research into more efficient algorithms.
B. DISCRETIZED GAMES
We first consider the setting where both players may only choose mixed strategies with support over a finite subset A {s(1), s(2)) ., s(m)) Sd-1 of pure strategies. This setting includes embeddings that are represented using floating point numbers although A will be very large. In this case the mixed strategy of the players can be expressed as an m-dimensional probability vector x; with Tij = Ps (s()). Since there are a finite set of pure strategies a mixed NE is guaranteed exist (Nash Jr, 1950). Furthermore since this is two-player constant-sum game we can find a mixed NE by solving the following linear program (Dorfman, 1951)
maximize
a
subject to Ux 2 a1
1 r =1
x; 2 0, == I,...,m. m,
where Ui u1 (())) The strategies where Tı π2 correspond to mixed NE. While such a problem simple to formulate and solve, the number of possible strategies grows rapidly with d for most discretization schemes. For example, we might create uniform grid of k points over each spherical coordinate, in which case we will have m Ad-1 pure strategies to consider.
B.2 FINITE SUPPORT
Next, we consider the setting where both players choose mixed strategies with support over at most m pure strategies, and the support of P. is over I points, supp
c...........00 ..). Unlike in the discretized case, the players may choose any pure strategy that lies on s We begin by outlining method that, given a mixed strategy P. finds all pure strategies 1 that dominate it:
En, ($1, $2) U2(81, $2)] < 0 for all 82 E D. By symmetry, we w.l.o.g. assume Player provides the mixed strategy. We will then use this method as subroutine to find mixed NE.
Lemma 4. P2) a mixed NE if and only if EN [n, ($1, s)] 2 and Exgary [uz(s, 82) 2 for all pure strategies s 5d
Proof of Lemma 4. Assume (P1, P2) is a mixed NE, then by definition (fs,,s))~((ApPP)1 [u $2) 2
for all mixed strategies P E P(gd-1), since each pure strategy is also a mixed strategy it follows that (P,. (81, s)) 2 for all 5d-1, Similarly for Player 2.
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Now assume have two mixed strategies (P1, such that Esi~A wn) (s1 2 2/2 and
Pa (1, 82)) 2 for all E Sd-1, Given mixed strategy Pe P( it follows that
((P 82)] Engap [ (01, 82)]
Enj-P, (2),s)))
sa
1 dP((2) = 2"
ga-1 2
Similarly for Player 2.
Lemma allows to only consider pure strategies when checking if strategies are mixed NE.
Now given mixed strategy P with finite support PP) {s((1, ,!?)...., } we can find every subset of 5d-1 that does not satisfy the condition Lemma 4. By noting that any arbitrary strategy can be either closer, farther, or at the same distance from consumer as a given (0); we see that each (() partitions 5d-1 into 3' disjoint partitions based upon the distance the strategies
g(m)
to each consumer Cs. That is, x = {x (i) x(6) ..... x.,x x(()
(8)) j. with x satisfying
2 if (0(0, cz) > {8, cx)
jk
1 if (s(0, ct) = (8,ck)
0 if (0(0, cx) Cx).
for all pure strategies E x where ja is the k-th digit in the ternary representation of j. By considering all m partitions created by the strategies in supp(P),, we can further partition the space into 3 disjoint partitions y {Y₁, Y2..., Yzm) with Y, = N x!!) where iz is the j-th digit of the 3' aay representation of
For every Y y we have E,,~ [u (51, $)] = Ee,- Thus, find the set of all pure strategies D that dominate P by iterating over V, testing single point in each partition, and taking unions:
(81, s')] for all s, E Y by construction.
{ = Y VEY:EY = (E 8)] <
11 D= UY Yez
follows from Lemma 4 that P) is mixed NE if and only D is empty.
Finally, we outline method to find mixed NE. We first note that for every positive integer m, every pure strategy s E 1 defines a feasible set F, of all mixed strategies with support over at most m pure strategies that not dominated by s, that is:
1 2
F.
T
T.U (s
where и is an m-dimenniionll probability vector. follows from Lemma that if P is mixed strategy with support over at most m points then (P, P)) is mixed NE if and only if P NNEsa- F,. We can frame finding such strategy P as an optimization problem
minimize PCP
P
subject to P #0, E
where the set of all mixed strategies with support over at most m pure strategics. An optimal solution with more than element in P indicates that there does not exist mixed strategy with support over m points or fewer, whereas if PP = then (P, P) is a mixed strategy where the singleton element in P.
This is an instance of the implicit hitting set problem. Hence, we can use the algorithm proposed in Section 2.1 of Chandrasekaran al. (2011) solve the above optimization problem. Their
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algorithm assumes oracle that, given proposed subset P P will return subset F, that is not hit P F. = or will certify as a feasible solution to the above optimization problem. We can easily achieve this by finding all dominating pure NE using our proposed method above for each P E P and taking the intersection of the resulting sets. If the intersection is empty then P is a feasible solution, otherwise every element in the intersection represents subset F, that has not been hit by P.
C
EXPERIMENTS
C.I SETUP
The LastEM dataset was preprocessed by Shakespeare et al. (2020). Original larger scale sweep was executed with {10, 25, 100, 500, 1500), d E {3. 10, 50, 100}, stepsize in {10 10 10010-1 }. and r E (10-2, 10-1 .0.25, 0.5, 1.0). We only used 2 random seeds for the recommender, and random seeds for our LNE-finding algorithm (i.e., 6 runs in total per configuration). For the reported the number of steps was upper bounded by 50,000 (all runs have successfully converged a fixed point as mentioned). While our code contains an option to scale_ __rr________________ by_ tempera at ure (see the config. py file in the provided code), which multiplies the stepsize by before its use, we did not use this option in the experiments.
results, stepsize sweep was restricted to {10-2 2,002110111 ]; 10-1];
The second-orde Riemannian test (Definition 3) is implemented in mani fold. py. Defining the tangent space projection IT; := (T 8,8; ). we consider a candidate strategy profile s E (S"-1)" as violating the second order test if any of the Riemannian gradients II,V a, (s(s) had (2-morm higher than 10-5, 5. vd. or the Riemannian Hessian I1,[V?, u, (s)][I, (8,, v., (s)) had strictly positive eigenvalue (no tolerance used here).
The final MovieLens and LastFM experiments were on 72 AWS machines, each with 4 CPU cores, for 5 hours. Including preliminary and failed runs, we used over 50K CPU hours.
C.2 OPTIMIZER
The gradient ascent optimization technique (Singh et al., 2000; Balduzzi et al., 2018) we employ is very similar to standard gradient descent algorithm from machine learning literature. Here we provide a short description of the similaritics and differences between the two.
The optimizer the gradient of the utility (s) w.r.t. 0g, E [ri]. as described around Equation (4) (recall 8; = (//1e.1)). or,+1 is obtained using 0j.1 for all j # i, i.e., the locations of the other producers from the last step. All n optimizers execute these steps at the same time, iterating until all of them converge. See optimisationn py, particularly the optax minimisatii satiiiiio on method, for more details.
simultaneously runs independent gradient descent optimizers, each following
C.3 ADDITIONAL PLOTS
Appendix C.3.1 contains plots where the second-order test confirmed and LNE. Appendix C.3.2 then offers comparison to a third ranking algorithm: standard matrix factorization (MF; Koren et al., 2009), i.e.. PMF with additional bias terms. The bias terms effect interpretation of T values, and we also ignore them when running the LiEdfinding algorithm. This makes the comparison with PMF and NMF difficult, which is why excluded MF from the main text. Results in Appendix C.3.2 again contain runs where the second-order test did not rule out LNE
C.3.1 LNE CONFIRMED BY THE SECOND-ORDER TEST
As mentioned, the plots shown in the main body of the paper are for runs where the second-order Riemannian test did not rule that the found pure strategy profile is a LNE. Here show exactly the same plots with only the runs where the test confirmed a LNF. The difference is that here we exclude the runs where the Riemannian Hessian had least one zero eigenvalue associated with direction perpendicular to 9;, for at least one i E [r)]. As you see below, this had little effect on the LastFM results, but has noglneglyyy reduced the number of admitted runs for MovieLens.
C.3.2 MATRIX FACTORIZATION (PMF WITH BLASES) RESULTS
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dim
dim 30
dim 3
dim 50
1.0
chusars # poodussss
130
96
004
9.1
0.01 semperatuue (F)
tetuperature (r)
Figure 8: A counterpart to Figure 4 with runs where LNE test was inconclusive excluded.
din
dim 80
1003
10
1o0k 1005 5,60
0015
0.03
-10
Ierppmmtune
Imngerattrr
Figure 9; A counterpart to Figure 5 with runs where LNE test was inconclusive excluded.
diz
dan 90
dan
din 50
4.% gesite
-078
Teiyhbore
tetgshons
tetyhbore
Figure 10: counterpart to Figure with runs where LNE test was inconclusive excluded.
clustess * prod/ccms
dim 3
dim 50
+ dustess + prodocars
1.0
dim
nf
nert
00001 1000n01 10amun3 Ca
81 1001003 1000003 Lb
temperature (r)
Figure 11: A counterpart to Figure 4 with added MF results.
dim.
dim
8.0
211
-13
391 31 13 000 61 1E a1 20
Iemperddore (7)
tamprraton (F)
Figure 12: A counterpart to Figure 5 with added MF results. Baselines omitted to reduce clutter.
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2 2
Figure 13: A counterpart to Figure 6 with added MF results.
22


Reviews:
[{"review_id": "j8fdQ0wRr3", "reviewer": "Reviewer_49Ro", "summary": "", "questions": "", "limitations": "", "rating": 10, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper study how technical changes in recommendation algorithms - giving exposure of a group of producers' products to a group of users - can inventive strategic content creators. They model such context through an exposure game with strategic rational omniscient agents fully capable of changing their strategies. They demonstrate both theoretically and experimentally the impact of a parameter of their model on the Nash equilibrium to which the content creators can converge. The authors propose to use this game as an audit tool to determine the incentives of strategic creators before modifying a recommendation algorithm in production.", "summary_of_the_review": "This paper is very pleasing to read. Although some proofs are highly challenging, this paper seems novel, gives many highly relevant references for the topic and the goal of the paper will be of great use as a tool to evaluate incentives towards content creators in recommendation models, which has many social implications. The quality of this paper excuses for the few unclear moments. I would be more than glad to raise my score once my concerns are raised.\n\nEDIT: The authors addresses my concerns on their work, I increased my score in response.", "strength_and_weaknesses": "### Strengths\n\n- The proofs of this paper are very (sometimes too) elegant.\n- The model proposed would be highly beneficial when used to assess recommendation systems and potential bias that could emerge prior to deploying them to production.\n- The paper is well written and, except for the proofs and a few other points, easy to follow.\n- As the author claim: their work “apply equally to classical matrix factorization and deep learning-based systems.” Which is appreciable, given how deep learning-based systems are still underresearched.\n- The authors acknowledge the limitations of their model and do not present it as a miracle solution.\n\n### Weaknesses\n\n- Page 2: You define $S^{d-1}$ after using it. Please define it before/within Definition 1.\n- Figure 3, plot B & C: if 1 producer is at midpoint, shouldn't the utility reach $\\frac{1}{n}$ at $\\lambda = 0.5$? Or did you not normalize the vector $s_1$?\n- Page 6: You mention “Riemannian second derivative test” Can you please be more specific? The reference is long, dense, and do not mention this term.\n- Page 17, Figure 7: It seems to me like there is an issue with the two central plots as $f(\\theta) = \\nabla_{\\theta_1} u_1 (s)$ but the plot do not seem to show this relationship. $f(\\frac{\\pi}{4}) = 0$ but $u_1(s)$ appears decreasing at that value of $\\theta = \\frac{\\pi}{4}$.\n- Page 18: It seems to me like the partitions $\\mathcal{X}^{(i)}$ and $\\mathcal{Y}$ are unnecessarily complex, I didn't fully get the grasp on them. Isn't there a simpler way to deal with it?\n\nNot really weaknesses:\n\n- Page 3: \"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.\" Or is it? This trial-and-error is done in an always-changing environment (many hours of video uploaded every second on youtube), so is it that informative?\n- Page 6: “Exploration effects are typically thought of as having negative impact on user experience through immediate reduction in quality of service as a result of suboptimal recommendations.” Being trapped in an \"algorithmic bubble\" with always the same type of recommendation is critisized and not appreciated.\n- Proposition 1: isn't there a typo? \"For $n=2$ *non-negative* softmax *games*\". Shouldn't it be $n \\geq 2$?\n- Page 15, proof of proposition 1: $\\mathbb{P}(\\theta \\geq \\theta_m) = \\frac{1}{2}$. What is $\\theta$?\n- Page 16, typo: $P_c = \\frac{1}{2}(\\delta_{c_1} + \\delta c_2)$, should be $\\delta_{c_2}$.\n- Page 17, typo: \"$c_3$ then $c_2$\" should have been \"than\".\n- Page 17, typo: \"$c_1$ (resp. $c_3$)\" it seems to me like they should be switched.\n\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "L8LBcPRZgZF", "reviewer": "Reviewer_rzZd", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This is a very cool paper, considering the strategic incentives of multiple content creators when producing content for a YouTube-like recommender system. The article addresses questions of whether the setup of the recommending algorithm has an effect on the type of aggregate content-producer behaviour we would expect to see. This is an important and interesting question, and my first reaction on reading the paper was that I wanted to hear these people present their work and then sit down and discuss it with them in detail! The key claims are that restricting the recommender system to non-negative domains improves the behaviour of the whole system, and that by the recommender system being more explorative it induces content creators to be more conservative (tackling the central point of the content consumer distribution, instead of specialising).\n\nUnfortunately I am writing this review in something of a rush, having only been asked to do it because others dropped out. It is quite possible that I have missed some details in my reading of it.", "summary_of_the_review": "A really exciting piece of research, let down by a bit of over claiming, and some sloppiness in the presentation.", "strength_and_weaknesses": "Strengths:\n- The topic is very interesting, and important.\n- The authors use serious game-theoretical approaches to assess the likely outcome of the strategic considerations by content creators\n- The results are, so far as I can tell on an extremely quick read, more or less correct.\n- The results tell us something interesting - more explorative recommender systems can help to avoid over-targeting by producers, but might reduce the diversity of content that is created.\n\nWeaknesses:\n- There are several problems in the proofs (see below).\n- Insufficient details are given about the experimental results. Section 3.2 was extremely difficult to follow - the plots are very small, with poorly explained axes; which concept of clustering is being used; etc etc.\n- The claims about switching from non-negative to unconstrained feature spaces having a strong effect is only actually present in Proposition 1, in the case of two players and a two-dimensional feature space (since features are constrained to have norm 1, what has actually happened is that the feature space becomes a line instead of being a circle, but with higher dimensions there is less topological distinction between non-negative and unconstrained).\n\nProblems in the maths:\n- The proof of Theorem 2 appears to be incomplete. In the hardmax case, the given proof says that for any s_1 that is close to c_1, there is a better position on the geodesic between c_2 and c_3. This means that switching to that point will get better reward than staying at s_1, in the absence of player 2. However the whole point is that there is a player 2, who needs to be taken into account. The logic would need to go something like: for any s_1, player 2 will select this particular s_2, which means that player 1 is not playing a best response to player 2. It is quite possible that the theorem is true (I do not have the time to check) but the proof is not correct.\n- In the statement of Proposition 1, what is the point of defining \\hat{c}? It is a scaling of E(c), to which the first (and only) thing that is done is a further scaling to get \\bar{c}.\n- In the proof of Proposition 1, I think more care needs to be taken. In the first part, what if there is no such theta_m? For example if there is a theta_m such that P(theta_c >= theta_m) >0.5 and P(theta_c <= theta_m) > 0.5 (i.e. P(theta_c=theta_m) is positive, and it just so happens that this is right in the middle of the distribution). For the non-existence when d>2, same complaint as for the proof of Theorem 2 - you need to complete the argument instead of just showing that s_1 is not a best response to itself. (I’ve just had a thought - are you assuming everything is symmetric, so a pure Nash equilibrium must have all s_i equal? That seems to be an anomalous piece of logic - just because the game is symmetric does not mean that the equilibrium must be symmetric - although is this actually the result that is buried inside the 2nd paragraph of the softmax part of the proof, and does that apply t hardmax too?).\n- I have not checked the proofs of Lemmas 1 and 2.\n- Proposition 2 is not proved\n\n\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}, {"review_id": "8BrU8NRzUm-", "reviewer": "Reviewer_6hAx", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The paper studied the interaction effects between content producers and consumers. The dynamics in the interactions are well formalized as an exposure game about incentives. They provided a comprehensive theoretical analysis of the properties of Nash equilibria in the games and proposed tools for numerically finding equilibria in exposure games. Both theory and empirical experiments verify the dependence between algorithmic exploration and content diversity, and between model expressivity and bias.", "summary_of_the_review": "In general, I think it's a good paper with clear novelty. Some of the descriptions and empirical experiments in the paper may require further explanations.", "strength_and_weaknesses": "Strengths:\n- The paper is well-written and flows very smoothly. Despite the paper's modest space, it provides the required context and necessary explanation in a good job. \n- Most parts of the paper have clear motivations. Many of my questions are well answered in the paper.\n- Formulation and theoretical analysis of the exposure game in the recommender system is insightful. Most papers studied the recommender system in a fixed manner. I agree understanding the interaction effects/dynamics between the producer and customer is important and useful.\n- The analysis of diversity and exploration is interesting. The paper mentioned the game may concentrate on uniform distribution, and exploration may incentivize content that is uniform and broadly appealing rather than diverse. From the recent papers, I also found deeper models can achieve diversity/coverage and accuracy at the same time. These two concepts may not be contradictory. If could further reveal the relationship between these two, it'll be very interesting.\n- The paper identified several factors that can influence the recommendation seriously and provided a pre-deployment audit tool that can benefit the community.\n\nWeaknesses/Questions:\n- In eq. (2), the paper introduced the temperature parameter $\\tau$ to control the spread of exposure probabilities. Many of the follow-up analyses and experiments are built on this. However, most recommender systems didn't include this in their objectives. Could you please explain the rationale or connectivity between these?\n- On page 3, I'm confused about the full control assumption. What's the difference between full contry and partial control and why it can abstract away the explicit model of producer actions? Is it possible to list some concrete examples? \n- For experiments, how are the producers defined? In my opinion, there are only user/item information in these data sets. \n- For the $\\epsilon$-LNE solver in eq. (4), is this the paper originally proposed or adapted from others? I'm not familiar with the NE-based methods. It seems it's also updating the model with the gradient. Could you please illustrate a little more on the common points and difference between this one and the normal gradient descent method used to optimize ML model?", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Decision: Accept: notable-top-5%