Producers Equilibria and Dynamics in Engagement-Driven Recommender Systems
Authors: Krishna Acharya
Varun Vangala
Jingyan Wang
Juba Ziani
Venue: Accepted by TMLR ()

Abstract: Online platforms such as YouTube, Instagram heavily rely on recommender systems to decide what content to present to users. Producers, in turn, often create content that is likely to be recommended to users and have users engage with it. To do so, producers try to align their content with the preferences of their targeted user base. In this work, we explore the equilibrium behavior of producers who are interested in maximizing user engagement. We study two variants of the content-serving rule for the platform's recommender system, and provide a structural characterization of producer behavior at equilibrium: namely, each producer chooses to focus on a single embedded feature. We further show that specialization, defined as different producers optimizing for distinct types of content, naturally emerges from the competition among producers trying to maximize user engagement. We provide a heuristic for computing equilibria of our engagement game, and evaluate it experimentally. We highlight i) the performance and convergence of our heuristic, ii) the degree of producer specialization, and iii) the impact of the content-serving rule on producer and user utilities at equilibrium and provide guidance on how to set the content-serving rule.

Published in Transactions on Machine Learning Research (02/2025)
Producers Equilibria and Dynamics in Engagement-Driven Recommender Systems
Krishna Acharya
krishna. krishna.ocharya wwwaraaaatttchhhh.....oooooooo edu
Georgia Institute of Technology
Varun Vangala Georgia Instilute of Technology
wwongola3@gatecc.com edu
Jingyan Wang
jingyonw@ttic.edu
Toyota Technological Institute at Chicago
Juba Ziani
jriani3@gatech. edu
Georgia Institute of Technology
Reviewed on OpenReview: https: // openreview.net. net/ forum? id= id= EWT4GajGDS
Abstract
Online platforms such as YouTube, Instagram heavily rely on recommender systems decide what content present to users. Producers, turn, often create content that is likely to be recommended to users and have users engage with it. To do so, producers try to align their content with the preferences of their targeted user base. Tn this work, we explore the equilibrium behavior of producers who are interested in maximizing user engagement. We study two variants of the content-serving rule for the platform's 's recommender system, and provide a structural characterization of producer behavior at equilibrium: namely, each producer chooses to focus on single embodded feature. We further show that specialization, defined as different producers optimizing for distinct types of content, naturally emerges from the competition among producers trying to maximize user engagement. We provide heuristic for computing equilibria of our engagement game, and evaluate it experimentally, We highlight i) the performance and convergence of our hcuristic, ii) the degree of producer specialization, and iii) the impact of the content-serving rule on producer and user utilities at equilibrium and provide guidance on how to set the content-serving rule !
Introduction
Recommender systems have transformed our interactions with online platforms like Instagram, Spotify, and
YouTube (Stigler Committee, 2019 Qian and Jain 2022 These systems curate content to enhance user
experiences, lostering user relention and engagement. (Gooroow 2021 Instagram 2023). This often translates into increased revenue for the platform. Due to their importance to today's digital industry, there has been a and improving existing recommendation algorithms, c.g., (Koren)
large body of work aiming at developing
et al. 2009 Li el al.] 2010: Lü et al. 2012 Wang and Zhang 2012:
Luo el al. 2014 Covington el al. 2016
He et al. 2017 Yi et
al. 2019
In 2023, the creator cconomy, driven largely by these systems, was valued at a staggcring $250 billion PPrclli
2023 Content producers have adapted to this landscape to act stralegically (Milli et al. 2023 Merrill and Mack 2019). They often compete against each other and tailor their content to maximize exposure (or how many users does a producer reach) or engagement (or how much users engage with a given
Oremus 2021
Code available htt.::///gttaaacccrrrrracccaaaes

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producer's content). This competition be modelled game, where understanding the dynamics offers insights into content creation incentives and phenomena like content specialization. However, comprehending these dynamics, both theoretically and empirically, presents challenges when considering the complex nature of user preferences and content strategics in high-dimensional spaces. Finding Nash equilibria in general a computationally difficult process, especially as the number of players and the action space increase. In the case of recommender systems, user preferences and producer contents are represented by high-dimensional vectors, and the set of possible strategy profiles for the producers rapidly grows intractably large as the size of the game increases.
In this work, we aim to provide new insights into producer competition and the equilibria of recommender systems. We take a departure from much of the related literature that aims at understanding equilibria and dynamics when producers try to maximize exposure (i.e., how many users see their content). We instead focus on producers interested in maximizing engagement, metric that encompassos not just exposure but. also how much users interact with content.
Our main goal is to understand the extent and conditions under which content specialization occurs defined different producers choosing to create distinct types of content instead of all producers producing the same homogeneous content.
Summary of contributions
Our main contributions are as follows:
In Section 2 we formally introduce our model. We assume producers aim maximize user engagement instead of user exposure, where the latter is typical in works characterizing producer equilibria in recommender systems (Hron et al..
Jagadeosan et al. 2022).. We rely on the softmax rule used in previous work (Hron et al. 2022 Chen et al. 2019) for showing content to users, but also
2022:
introduce new linear-proportiona content-serving rule as a baseline for comparison.
In Section we provide main theoretical structural characterization result. We mathematically prove that at equilibrium, producers prefer producing content that targets single (embedded) feature at a time, rather than investing across several fcatures.
In Section 4 we characterize the pure Nash equilibrium with the linear content-serving rule for a simplified setting in which we assume all users are "single-minded"; i.e. are only interested in a single type of content. The closed-form equilibrium we derive exhibits specialization, defined when different producers split themselves across different types of content, rather than all producing the same, homogeneous content. While this setting is simplified, we note in the experimental results of Section 5 that the insights it provides do carry on to the general softmax content-serving rule and to more general user preferences.
In Section 5 we present experiments synthetic data and three real-world datasets Moviclens- Amazon Music, and Rent the Runway (Harper and Konstan, 2015 Ni et al. 2019 Misra et al. 2018). First, we introduce a computationally efficient heuristic based on best-response dynamics (Algorithm 1 for computing pure Nash equilibria of our engagement game. We observe that this heuristic almost always converges to a pure Nash equilibrium within relatively few steps. Second, our experiments further characterize producer specialization, showing that it occurs even under more complex settings than those in Section 4 We then study the effect of the temperature parameter in the softmax content-serving rule both producer specialization and utility. Specifically, we demonstrate that the degree of producer specialization is monotonie with respect to temperature: higher temperatures (represcnting greater exploration in the content shown to users) incentivize more homogeneous content production, while lower temperatures (favoring the most relevant content for each user) encourages specialization, with producers focusing on distinct types content. Lastly, we show that both producer and user utilitics decrease monotonically with increasing temperature. A low softmax temperature yiclds the highest utility for both producers and users, emphasizing the benefits of selecting lower temperatures in the softmax content-serving rule.
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1.1 Related work
The study of strategic behavior and incentives among producers in recommender systems has seen much interest recently. These interactions can broadly be classified into two types of games: exposure-based, where producers are rewarded for maximizing the reach of their content, and engagement-based where producer rewards depend not only on reach but also on how well the recommended content aligns with user preferences.
Exposure games The seminal works of Raiter et al. (0117) |: Basat et al. 2017]; Ben-Porat and Tennenholtz (2018); Ben-Porat al. (2019b a 2020) introduce game- models of competition among producers aiming to maximize exposure. While these early studies constrain content creators to finite strategy spaces by requiring them to select from a pro-specified, finite catalog, more recent works-suse as Jagadcesan et al.
(2022) Hron et al. (2022), which are most closely related to ours have relaxed this assumplion by focusing creators with continuous strategy sets.
Engagement games The works of Yao et al. 2023a; Immorlica et al. 2024 Huttenlocher et al. 2024 model producer rewards based on engagement. Yao et al. (2023a] study social welfare in scenarios where content creators compete without the modiation
contral recommender, and focus on bounding the Price of Anarchy Immorlica al. (2024) examine engagement games from a different perspective, exploring the trade-off between producing high-quality content and gaming the recommender system by creating low-quality, "clickbait" content. Huttenlocher al. (2024) model the problem two-sided marketplace with departing users and creators and show that maximizing total engagement in such a setting is NP-hard.
Mechanism design Works Exposure and Engagement games focus characterizing producer competi- tion and equilibrium under fixed content-serving rule and producer reward. In contrast, Yao et al. (2023b) 2024) adopt a mechanism design perspective, designing rewards and serving rules incentivizing equilibria that have high social welfare. Hu et al. (2023) explore a similar sctting but model the platform as a lincar contextual bandit.
User dynamics Our work studies adaptive producers who change their content vectors maximize user engagement. The user preferences are static, consistent with the literature on Exposure and Engagement games. Another line of work by (Dean and Morgenstern 2022: Kalimeris et al. 2021 studies shifting user preferences based the content recommended to them. Lin et al. (2024) adopt this user preference shift model but also model evolving producers, providing conditions for user polarization, though their producer evolution is not game-theoretic.
A table providing more detailed comparison of our paper to related work along various axes (such as the nature of producer reward, type of equilibrium and dynamics) is available in Appendix A
2
Our model
We consider an engagement game between n producers on an online platform. The producers must decide what type of content to produce in a d-dimensional space to maximize the engagement from users. This space generally not what one may think of as the original feature space, where features are defined, e.g., different genres that producer or may about. Rather, these features are embedded features that are the results of matrix factorization algorithm" whose goal is to learn representative "directions" of the recommendation problem, as in Hron et al. (2022 Jagadeesan et al. (2022). The online platform then uses their recommender system to recommend content to users as a function of how well producer content matches user preferences. More formally:
Producer model We have n producers on the platform. Each producer chooses a content vector 8; from the set = {s: s E Rd. ||>||] < 1). We note that we focus on the f--nrr in order to model the relative
2The ratio between the optimal welfare that at the decentralized cquibrriunnKuuusopppia and Papadimitriou 1909 MF learns latent representations of users and movies which are then used to predict user preferences Farigg. Our results however da depend the specifics of the algorithm used to obtain this embedding.
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amount of weight that each producer has each embedded feature. We let AS be the set of probability distributions over S. Letting s;()) denote the f-th entry of content vector for feature ʃ E [d], one can interpret s(()) as the fraction of producer i's content that targets embedded feature f.
User model We have K users on the platform. Each user & E [K] is described by a preference vector in C:= fc: they altribule to each embedded leature. The more weight on the leature, the more utility the user derives from seeing content that aligns with said feature. We assume that the utility of a user with preferences C₁ who faces content given by ofs..
C E ||c|aa I 1}. We note that C describes user preferences in the form of the amount of weight
Recommender system's content-serving rule The platform uses a recommender system (RS) to decide which producer's content to show to which user. Denote ø = ($1 ( 8a) E S the full profile of producers' production choices. The RS shows producer i's content 8; to user with preferences e with probability Pi(c,, 3) = Pi(ck, 8t, 8-1), where s-, denotes the rest of the producers. We call this probability the content-serving rule. In this paper, we consider two content-scrving rules:
1. The softmar content-serving rule is, in Hron ct al.
2022
Chen et al. 2019
exp
Pe(c, 81,8))
A
(1)
Lj=1 exp
c Si
where denotes the softmax temperature. A low temperature corresponds greedier serving (ie.., only the best fitting producer's content is shown to the user), whereas high temperature corresponds to adding more randomness to the serving ("worse" producers may still have their content shown to the user, albeit with lower probability). The limit - 0 corresponds to a hard maximum, i.e., the producer whose content is best aligned, namcly producer arg maXje[n] c sj. is shown to user c.
The top-k softmar content-serving rule first selects the top-k producers with the highest alignment scores c 8j, where [n]. Among the selected producers, then apply the softmax function with temperature 7:
Tai
c if K.
pr(c, 84, 8-1) A
ESec exp
(2)
otherwise,
where K- e [m] c sy is among the top k values of {c s cT sn When n, the top-k softmax rule reduces to the regular softmax rule defined in and when i: 1, it's defined as the greedy serving rule.
3. The linear-proportional content-serving rule Luuce 1977) choice axiom), where each producer's content s; shown to user eg with a probability directly proportional to 6 84.
if c CTs, > 0, if CT. 0.
palc, 81,88) A
5=
(3)
Note that the linear serving rule is well-defined even when c sy is zero for all producers j. We use this rule as baseline to compare performance to the typical softmax-based rule, and as an alternative rule to derive theoretical insights (our experiments in Section show that theoretical insights for the linear-content serving rule in fact extend to the softmax rule).
The round-robin serving rule serves producers in cyclic order: In the first round, all users are shown producer I's content, in the second round producer 2 and so Formally, the serving probability for
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user in serving round is defined as:
pi &( a [1 if (r-1 mod n)
(4)
otherwise.
Producer utilities One of our contributions to characterize the utility of a producer using engagement, rather than just exposure. In exposure games, the utility of producer simply the probability that this producer's content is shown to user. In contrast, with engagement, the utilities incorporate an additional term that measures how much a user engages with the content once this content is shown to them. Formally, assume that producer who successfully shows content to user k with preferences C2 derives utility 8, Ck from that user. This captures the fact that user whose preferences are better aligned with the producer's s content are more likely to spend more time engaging with that content. Formally, we define the expected engagement utility for producer i as
(s) = (.1 Σ Pi(ck, $) ck cj
(5)
Note that this expected utility is reweighted by the probability of producer showing 8; to user k; as i derives no utility from user ls if said user does not see his content in the first place. The total producer utility Up is then defined
Up
prlck. 5) cfs..
User utilities We similarly define the utility for a user with embedding e as engagement expectation across all producers i.e, E P:(ck 5) ox St. The total user utility Ug is defined as
U
P.(ck. 5) c c.
Remark 2.1. The value of the total producer utility Up and the total user utility Uu are equal, and the average producer ulilily and the average user ufility are equal up to a multiplicativee factor. Remark 2.2. For the round-robin serving rule Q it is easy to see that the utilily for a producer i is zero if it is not being served. When it is served, the utility is given by max E CT. St achieved by setting 8, to the basis vector corresponding to the largest weight.
E Ck which is
3 Equilibrium Structure
In this section, derive main structural result for equilibrium in engagement games: namely, show that at equilibrium, each producer prefers largeting a single embeddded feature at time.
Our first main assumption is that for all users, their features are strictly positive:
Assumption 3.1. For every user l and feature f, we have e₁ (f) 0.
This assumption holds in practice when user representations are obtained via Non-Negative Matrir Factoriza- tion (NMF) (Lee and Seung. 2000 Luo et. al. 2014) as is observed in Jagadeesan et al. (002)) Hron et al. (2022) and in our experiments in Section We additionally make an assumption on the data distribution, guarantccing that user preferences CI,..., Ck are non-trivial:
Assumption 3.2. For all pairs of production strategies 8, s' E S such that 8 + 8', there must erist at least one user k such that cT. + s.
This is mild assumption that states that there is enough diversity in user preferences. Equivalently, this assumption states that user preferences non-trivial and diverse enough such that span(Ci,... CK) Rd i.e., the user preferences span all of IRd If the user preferences do not span all of Rdd this means that some
Indeed, there then exists subset. size of (ci.....cg) .g) that forms basis for Rd. of s', they must differ in least one coordinate this basis, there exist o& such that cf o.
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latent features are redundant. In practice, one may work with reduced embedding dimension and perform new matrix factorization until Assumption 3.2 holds.
We are now interested in understanding properties of the Nash equilibria (NE) of our engagementtbased content production game, defined as the game where each producer decides which content to produce to maximize their utility. Nash equilibria are a classical concept for solving games Nash 1951). We apply the standard definition to our formulation as follows.
Definition 3.3 (Nash Equilibrium). For any producer i, the strategy of S that solves u,(s), msx (4($;,8-1) is colled best response to s-i. A strategy profile (si,..., 8%) E S" is a pure-strategy Nosh equilibrium (pure NE) if and only if for every producer i E [n]]
u,(s; 8-,) = u4(8,, $2.).
#:68
A strategy profile (Di...., .D)) E AS", where AS denotes the probability simplez over S", is a mized- strotegy Nosh equilibrium (mized NE) if and only if for every producer i [m],
Be,
max Be,-Dae- [u)(s), D.EAS
Under either pure or mixed Nash equilibrium, all producers best respond to each other and do not want to change their strategy: ie., each producer maximizes its utility and gets the best utility it can by playing the Nash, assuming all remaining producers also play the Nash.
We now characterize the equilibria of our game, under both content serving rules of Equation 1 and Equation (3]. showing that producers prefer to focus on single embedding dimension at time:
Theorem 3.4. Suppose Assumptions 3.1 and 3.2 hold. Let B = (e1....,ca) Ca) be the standard basis for Rd, where each ej is the unit vector with value in coordinate E [d] and 0 all other coordinates. Under both types of senteng-serving rules, if there exists a NE, any pure strategy for producer i must satisfy sf B, and any mixed strategy must be distribution supported on B in this equilibrium.
Informally, the theorem shows that at equilibrium, producer strategies are supported on the standard unit basis rather than on the entire simplex = {s: ||s//| 1, s E R Each producer focuses on a single feature in the embedded space. Conceptually, the producers play a feature selection game where they trade-off i) choosing the "best" " features (those that align best with users' preferences) to improve the reward when showing content to users with ii) choosing potentially sub-optimal features to avoid competition with other producers over top features and improve the chance of showing content to users.
The remainder of the paper studies how the choice of content-serving rule affects the trade-off between effects i) and ii). i) pushes for more homogeneous content, production while ii) promotes specialization. Section 1 provides some thcoretical insights on this trade-off, and Section provides experimental results for both the proporopopootionall and the softmax serving rule.
3.1 Proof of Theorem 3.4
Preliminary properties producers' utilities We start by noting the convexity properties of producers" utilities. Everywhere this proof, we denote S(c. = Ejfs oT sj.
Claim 3.5 (Convexity for linear-proportionall serving rule). The function
x
f (x)=
x EØ c%
is conver in domain the Ryo. Purther, suppose Σj oT > 0. Then it is strictly conver in x on the domain Rso.
Proof. The first-order derivative is given by fh,0_, (x) =(28(c.0_1)+z) 2 0. The second order derivative [S(c.81))23)
of f is given by (x) 2S(c.s [=+ Note that on r E Rso, [l,_(()) 2 O, with fh._ (x) > when
5(c, 8-1) > 0. This concludes the proof.

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Claim 3.6 (Convexity for softmaz serving rulc). The function
exp (I/T) fk)) exp (a/T) + EEzs exp (cf 85/T)
is striclly conver in x for |x|] S T log(Seap)
In practice, we note that we expect this condition on x to hold when the game is large enough. Indeed, it is equivalent to Seep > exp (I/r). When the number of producers grows, Seap also grows while exp (a/r) remains bounded by exp (1/r) (we restrict attention in this entire proof to a representing an inner product of the form e which we know in [0, 1]).
Proof. Let overload the notation and write Sexp Ejzs exp sj/T). The first-order derivative is given by
(x) (1 x/r))xxpp (2//T Sexp + exp (2x/r)
(exp (x//T) Sexp)
The second order derivative is then given by
Seep exp (z/T)
(1)) (S + cxp (I/T))3
(2SerpT + Sexpat + 27 exp (x/T) exxp(x/1) (x/T)))
Seap exp (x/T)
Sexp - exp (x/TT
2T +
(2 + exp (I/T))²
8 Seup exp (I/T))
Notice that this is strictly positive so long as Seap exp (x/T), concluding the proof.
Now, note that for every producer j, in each strategy 8j supported in mixed strategy profile at equilibrium, must have 8j 0. Indeed, if any producer sets sj 0 and produce no content, they get utility of O; however, setting 8 O leads to 8. Ca > 0 for all users k. by Assumption 3.1 and yields strictly positive utility. For a mixed strategy profile to constitute an equilibrium, every action on the support of each player's strategy must have the same utility, and this utility then has to be strictly positive, so it must be that 8y > 0 on the entire mixed stratey''s 's support. Then, Sie, 8-;) > on any strategy profile on the support of a mixed Nash equilibrium. Therefore, we can restrict attention to S' S/O0 We know that ft..., (x) is then strictly convex in x. Now, let's examine the function (s)) fE.._ (ck s;). Clearly, Ik.s_, is convex and for all k,
I (28; + (1 -))ss) & (s) ) + (1 x)gk,s,($/))
Further, pick any X E (0,1) 1) and 8; # s/c There must exist k, by Assumption 3.2 such that oj s # of s/ Then, we have that, for that k;
gk. (Aa (1 (1-))s) = Sh.s-, (A 8i + - -1)c((
< <ifn,e, 81) + (1 - fi. (c st)
>ggc., + (1 A)gk,a., (8%);
where the inequality follows by strict convexity of fk.e_, (z). Summing over all k's, we then get that
K
k
K
Th.a Ti..,(Ass + (1-A)s] < A Σ Σ
Ik.s (8i) + (1- A)
Σ Th
1=1
as at least one of the inequality in the sum has to be strict. This shows that while each Ik.e_, is not necessarily strictly convex, Σn Ik.s_, is. Now, note that both the linear case as per Equation 3 and in the softmax case as per Equation D we have that producer i's utility for playing 8,, under a mixed strategy profile 8-5 D for
the remaining agents, given by
K Σ
(s;,; D) = EA_IND
(si)
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Then, (s; D) is also strictly convex in s; (this follows immediately by writing the definition of strict convexity and by linearity of the expectation). Finally, now; suppose by contradiction that producer i's best response is not unit vector ey. Then there exists f with 0 < si(f) < 1. In this case, note that
ui(8;, D) = ui Σ (:De, [Eonner,
<> 8: se(f)uzees
<
by strict convexity of s; + us (s,, D). This is only possible there exists f such that u,(s, D) < u(ep, D). This concludes the proof.
Below, we provide our heuristic for computing Nash equilibria, under Algorithm | Our heuristic relies the structural result of Theorem 3.4 if pure equilibrium exists, then it must be supported the standard basis B, which simplifies the best-response computation.
Algorithm Best Response Dynamics for Pure Equilibrium Computation
Inputs: User embeddings (ci,..., Cx). Utility (1(8..8-) for producer i. Max iterations Nmaz. Output: Pure Nash equilibrium of the engagement game.
Initialize termination variable fin and iteration variable iter Initialize producer i's strategy 8t uniformly at random in B (e1.-.., ed)- while fin 0 and iter < Nmax do Produce random permutation vector of producers 1 to n.
for each producer i in the above permutation do Set fin= 1;
Compule o7 arg maxs,es us (81, 8-1;
if ( u, s-:) then
8i = 8/;
Set fin 0 and exit the for loop.
end end for end while
return ($1....,5n) sn) fin 1, and if fin = 0.
4 Equilibria under Simplified User Behavior
To provide theoretical insights about engagement equilibria, we first consider a simple, special case of our framework where cach user is single-minded, and is only interested in a single type of content. We also focus on the linear-proportional content serving rule of Equation 3 for tractability.
Assumption 4.1 (Single-minded users). For any user k, Cj e for some f E [d], where (ei...., ea) is the standard basis.
Note that we only make this assumption in the current section, and that this is an assumption on user and not producer behavior. This simplified assumption and setting allow us to derive our first insights towards characterizing the equilibria of our recommender system, and how producers decide what type of content. produce as a function of the total user weight on each feature. Our experiments in Scction show that the insights we derive in this simple single-minded user setting hold for general types of users under the right. choice of content-serving rule.
Under this assumption, we provide a simplified characterization of the equilibria of our game. We consider equilibria supported on the standard basis B, following the insights of Theorem 3.4 and let. (m1,... ma) be the number of users interested in content type e ed, respectively. We assume my > 0 without loss of generality; otherwise, feature f brings utility to no user nor producer, and can be removed. On the procodure side, we use the notation (n....., na) denote an aggregate strategy profile where for all [d], number nf of producers pick action ey-
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Lemma 4.2. Suppose Assumption 4.1 holds. For all f E [d], let my 0 be the number of users with es. Under the sinea-prrppottooaa contend serving rule of Equation 3 there erists pure NE supported on B and given by (Ni,..., Ra) if and only if
f " +1 mp
for all f. [d]
(6)
mf
The full proof is given in Appendix B.1
Interpretation of the equilibrium conditions Let 8, _ E be the fraction of users that are
interested in feature f. Consider strategy profile n, = Syn, and assume ny is integer We show that this pure strategy profile is an equilibrium of our engagement game, by verifying that Condition (6) in Lemma
4.2
holds for this construction. To see this, note that the equilibrium condition is equivalent to " 2 8y 8y
On
the left-hand side, we have of n. On the right-hand side, we have Rpt1 always holds.
8pm+1 n, and the inequality
6y
by
In this equilibrium, the number of producers that pick feature f is (ignoring rounding) proportional to the number of users interested feature f- This aligns with intuition: at equilibrium, the probability with which each producer is recommended to user should be the same across all features f; otherwise, producer that deviates to feature f" with higher probability will be shown more often and gain more utility. We note that Hron et al. (2022) made related observation in a different setting they note that producer content aligns with the average user weight on cach embedded fcature, with the average producer content being ē= K EKEK k Ck at an approximate equilibrium. However, we note that the insight of Hron el al. (2022 remains different, in that all producers produce the same homogeneous content aligned with e we show specialization and heterogencous content production, which we believe are commonplace in practice.
5 Experiments
We now provide experiments that expand our undorstanding of producer equilibria and utilitics at equilibrium for general user incentives.
5.1 Experimental setup
Synthetic Data We generate three types of user distributions for our synthetic data. In the uniform distribution, we generate user embeddings uniformly at random over the probability simplex. Hence, each feature is equally represented in the data in expectation. In the skewed distribution, positive weights w1 .... S wa (sampled from the probability simplex and then sorted). We then generate uniformly distributed user embedding, but re-weight each feature f by wy to obtain This creates a non-symmetric, skewed distribution where the total user weight for each feature is proportional to wy, leading to differences across features. We also generate a sparse distribution for which we generate user embeddings e E Rd uniformly at random over the probability simplex and then apply an element- masking operation. This operation uses random boolean vectors {0, 13d, with 90% of its values being zero. As per our modeling assumptions, normalize user features to have -n-or 1. All synthetic experiments use K 10, 000 users. We vary the dimension and the number of producers
first randomly sample
Real Data Following prior work on producer-side competition, we use the NMF implementation in the scikit-surprise package (Hug) 2020) to obtain user embeddings for the MovieLenns-00kk dataset (Harper and Konstan 2015); this dataset contains 100k ratings, 943 users and 1682 movies. We also run experiments with two other, larger-scale ratings datasets: AmazonMusic rating (Ni et al. 2019) and Rent RueRuy clothing rating (Misra et al. 2018). These datasets have around 840k and 100k unique users respectively.
WWhen 8yn is not integral, can instead round carefully that ny's 's still add n, and the proposed strategy remains approximate NE, that the equilibrium conditions satisfied up to small additive slack factor oThey use exposure instead engagement, different assumptions on user preferences, and the softmax content-serving rule

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The insights from experiments on these datasets turn out to be similar to the Movielens-100k dataset and are deferred to Appendix F
Equilibrium computation: a best-response based heuristic We provide a simple heuristic based on best-response dynamics to compute pure-strategy Nash equilibria in Algorithm [] In each iteration, it goes through the list of producers in randomly sorted order. For each producer, it computes the best basis vector response of the producer. Tf the producer is already playing best response, the algorithm goes to the next producer; otherwise, the algorithm notes the producer is not currently best responding, updates the producer's strategy to his best response, and starts the next iteration. If at the end of the for loop, we realize that all producers are playing best response, we have found a NE, hence we stop and output the current strategy profile. If after Nmaz iterations, we still have not found a NE, we output I. to signal that our dynamics did not converge.
We will see that our heuristic terminates the large majority of the time in our experiments; in that case, it must output pure Nash Equilibrium, as the algorithm can only terminate when all producers best respond to each other
5.2 Experimental results
Convergence of Algorithm and rate of convergence For our engagement game, we consider different numbers of producers (2 10, 20,.... 100) and 6 different embedding dimensions (5, 10, 15, 20, 50, 100). Each of these 12 6 configurations is instantiated with 5 random seeds to account for the randomness of the NMF algorithm. Across all 360 instances of the (producers, dimension, seed) configuration, the softmax content-serving rule with temperatures E {100, 10, 1, 0.1} always converges to unique Nash Equilibrium (NF) on the Movielens-1000 dataset. Even with a low softmax temperature of r = 0.01, Algorithm [ stiil converges to a unique NE in a large number instances. Additionally, for the top-k softmax serving rule reducing the top-k malue-making the serving rule "greedier" rreuulssss in fewer converged instances. The Iinear-rroportional serving rule converges in many instances, and the round-robin serving rule as highlighted in Remark 2.2
always converges to the dimension with maximum weight.
Serving Rule
100
10
T = 0.1
0.01
Serving Rule Converged
(Full) Softmax
360
360
360
360
342
Greedy
269
(Top-20) Softmax
253
247
232
182
326
Linear
357
10) Softmax
170
171
165
147
299
Round-Robin
360
(a) Converged instances for Softmax- serving rules
(b)
Converged instances for
lempperaturc-indeeeedee rules.
Table 1: Convergence to Nash Equilibrium (NE) across serving rules the Movielens-100k dataset. (a)
Softmax serving rules with varying temperature and top k values, (b) Teunperature-iddeeendenn rules.
Further, we examine the rate of convergence of the best response dynamics in Algorithm to Nash Equilibrium. In Figure [ we plot the number of iterations (averaged over 40 runs) of Algorithm with increasing number of producers and across varying embedding dimensions on the Movielens- 100k dataset. Figures la and 1b plot the number of iterations to convergence with the lincar and the softmax content-serving rule. In Appendix D.I we provide figures for the synthetic datasets and observe insights similar to Figure [
Based on the observations above, we conclude that Algorithm ! empirically seems be reliable and computationally efficient heuristic to find pure Nash equilibria for our engagement game, with performance scaling well with the number of producers. In contrast, naive brute-force approach enumerating all best. responses takes exponential time over the number of producers.
"We do not provide theoretical guarantocs the existence of pure NE convergence of dyuamics: in fact, Algorithm does not converge in few instances (Table g confirming that our game is not what called "potential" game. When game is not potential, existence pure NE and convergence dynamics are guaranteed and equilibrium existence certification and computation generally computationalyy hard problem. Heuristics often the best that we can hope for.
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200
(a) the emear-proportional serving rule Movielens-
(b) the softmax serving rule Movielens-100k
Figure 1: Number of iterations Algorithm 1 until convergence to a Nash Equilibrium on the Movielens- 100k dataset. The different curves represent different embedding dimensions the game E (5, 10, 15, 20, 50, 100); the error bars represent standard error over 40 runs.
In the following, we study how changing the softmax temperature affects producer specialization and the producer utility. We consider 5 different values for the softmax temperature {0.01, 0.1, 1, 10, 1003 and use the linear-proportional serving rule as a benchmark.
Equilibrium results In Figures 2 and 3 highlight the impact of the content-serving rule and softmax temperature the degree of specialization the instance-level. We show single (but representative) instance of the problem in each figure to provide visual representation of producer specialization at equilibrium; our insights are consistent across our generated instances.
(a) Softmax 7 100
(b) Softmax r 10
(c) Softmax = 0.01 (d) the linear- prouu-proportional serving rule
Figure 2: Average user weight on each feature (blue, left bar) and fraction of producers going for each feature (red, right bar) 100 producers, embodding dimension d 13. Lower softmax temperature leads to more producer specialization. User embeddings obtained from NMF MovieLens-1000
(a) Softmax
(b) Softmax 7 0.1
(c) Softmax 0.01 (d) the linear- pecer-proportional serving rule
Figure 3: Average user weight on each feature (blue, left bar) and fraction of producers going for each feature (red, right bar) n 100 producers, embedding dimension d = 15. Lower softmax temperature leads to more producer specialization. Skewed-uniforn distribution of users.
In Figure [ we show the Nash equilibria for the linear prrear-rrpootionall serving rule and the softmax serving rule with varying temperatures on the MovieLens-1001 dataset, with user emboddings obtained NMF. For the softmax serving rule, we observe that the degree of specialization decreasing the temperature т. We see specialization occurring the temperature drops from 100 to 0.01. A high temperature (r 100) incentivizes
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homogencous content production: this is expected, the content-serving rule becomes largely independent of producers' decisions (content is shown with probability converging to 1/n), and producers maximize their utility by homogeneously targeting the highest-utilitty content (arg max jedxxe c+(f)). When is of the order of 10 is when we first start sceing specialization; and the lowest temperature we experiment with (r = 0.01) leads to the most. specialization.
In Figure g we replicate our experiment on the skewed-uniform dataset. Here too, the degree of producer specialization is decreasing in the softmax temperature. However, the levels of specialization seem to decrease for the skewed-uniform dataset and specialization seems to first occur at lower temperature. This can be explained by the fact that low-weight features may not be worth targeting and ignored by the producers.
Further, we note that with the linear-proportional serving rule we observe a high level of specialization on the Movielens- 100k and skewed-uniform datasets as seen in Figures 2d and 3d respectively.
In Appendix D.2 we present producer distribution figures for the uniform dataset, which exhibit insights
similar those in Figure 2 Additionally, we provide extended figures with few additional softmax
temperatures for the Movielens- 100k and Skewed datasets (see Figures and [0 Appendix B includes producer distribution figures for the sparse synthetic dataset.
Producer utility equilibrium In Figure we plot the average producer utility with increasing softmax-serving temperature and with varying numbers of producers a Nash Equilibrium. We observe that the producer utility is decreasing with temperature, and temperature = 0.01 (near-hardmax) has the highest utility. We believe this provides an argument in favour of using low temperatures in the softmax content-serving rule. Recall that since the average user utility is identical to the average producer utility (up to a multiplicative factor), the benefit, of low temperature in the softmax-serving also extends to user utilities.
we compare the average producer utility across different serving rules while varying the softmax temperature. For softmax serving, lower temperature leads to higher producer utility, similarly, in the top-k softmax rules, reducing k results in higher utility at the same temperature. This indicates that "greedier" serving stralegies improve producer utility. However, there is a trade-off between increasing producer utility by making the serving rule greedier and the potential for decreased convergence to a Nash Equilibrium, as observed in the Nash convergence table Note that the lincar and round-robin serving rules are independent of uemperature; we plot the same mean and standard error across all lemperature values. Among these, round-robin serving yields the lowest producer utility, while linear serving remains competitive.
In Figure 4b
producers 10 producers
Liner
20
2.6
RoundRobin
Average producer utility
50 producers
Average producer utility
Soltmax
2.4
Top- softmwt
Top 20 softmax
15-
2.2
2.0
10
1.a
1.4
2
10
10f
10a
Temperature
Temperature
(a) Producer utility with varying producers, softmax serving
(b) Producer utility across serving rules, n = 50
producers
Figure 4: Average producer utility the Movielens-100k. dataset. (a) Varying the number of producers E (5, 10, 50} with embedding dimension d = 15. (b) Comparing producer utility across serving rules: Linear (bluc), RoundRobin (red), Softmax (orango), Top-10/20 Softmax (green/ (green/purple) with n = 50 producers and d 15. Error bars represent standard error over 5 seeds.
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6 Conclusion
In this paper, studied engagement games, a game-theoretic model of producers competing for user engagement in recommender system. Our main structural result showed that each producer targets a single feature in embedded space at equilibrium. We then leveraged this structural result to study content- specialization by producers and showed both thcoretically and via extensive experiments that specialization arises at a pure Nash Equilibriumn of our game, and as a result of natural game dynamics. We also observe that lower temperatures in the softmax content-serving rule incentivize specialization and improve producer utility. Our lincar-popportinnl serving rule serves competitive benchmark still demonstrating high levels of specialization and producer utility.
Limitations As in previous work on producer competition in recommender systems, we assume that cach producer is rational, and fully controls placement of their stralegy 8;. Rationality is common assumplion in systems with strong profit motives for the producers. Full control may be less realistic, as producers can modify content features, but they do not. know exactly how these changes affect the content embedding. However, this model provides a first-order approximation to real-life behavior, and the same assumptions are typical related work (Hron al. 2022 Jagadeesan et al. 2022 Yao et. al. 2023a: Hu et al. 2023
Our main structural result holds only for non-negative embeddings. While we make no assumption on how these non-negative embeddings are obtained, an interesting direction for futurr work is to study engagement games with potentially negative embeddings. There, it is not clear whether our best-response dynamics still converge, and propose to study no-regret dynamics as direction for equilibrium computation.
Finally, we caution treating the equilibria of our engagement games as definitive; we rather present them as insights to competition recommender systems, given the significant complexities of wal-world recommender systems and environments in which they operate.
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A Extended comparison with related work
Producer
Producer
User
Producer
Paper
Reward
Dynamics?
Dynamics?
Equilibrium
Ours
Engagement
Yes
No
Pure NE
Ben-Porat et al.
2019b:
2020)
Exposure
Yes
No
Pure NE
Hron et al. 20222)
Exposure
Yes
No
Local NE
Jagadeesan et al.
2022)
Exposure
Yes
No
Mixed NE
Hu et al. (0023)
Mechanism design
Yes
No
Mixed NE
Yao et al.
[2023a]
Engagement
Yes
No
CCE
Yao et al.
[2023b
2024
Mechanism design
Yes
No
Local NE
Dean and
Morgenstern (0022)
N/A
N/A
Yes
N/A
Lin et. al.
2024
Engagement
Yes
Yes
N/A
Table 2: Comparing our paper to related work
al. 2019a 2020) study exposure games with finite strategy spaces (producers selecting content from a finite catalog) and search for Pure NE using graph algorithms. Hron et al. (2022) focus on the concept of e-local Nash Equilibria: i.e,, given joint strategy profile (-1,8 each producer's 's strategy 8; is optimal in the open ball at (s,, 8-1) with radius g this is a weaker notion of equilibrium than pure NE, which holds globally. The work of Jagadeesaa
al. (2022) characterizes the support of the mixed Nash Equilibrium of their game, but do not provide a closed-form form expression for said mixed NE-we instead provide closod-form characterization in the restricted setting of Section 4 Hu et al. (2023) builds on the exposure model in
Jagadeesan et al. (2022) but studies the mechanism design problem with the platform modeled by linear contextual bandit. Yao et al. (2023a) focus on engagement games with the top-k serving rule, studying no-regret dynamics and the Coarse Correlated Equilibria (CCE) to which these dynamics converge. Yao et al. (2023b 2024h address the mechanism design problem, aiming to incentivize welfare-maximizing local Nash equilibria. The works of Dean and Morgenstern) (2022); Lin et al.] (2024) model user dynamics, studying user polarization and do not model producer competition and equilibrium.
B Omitted Proofs
B.1 Proof of Lemma 4.2
First, note that producer i's utility is convex in s; (following the same proof as in Appendix 3.1 is generally not strictly convex. This implies (as for the proof in Appendix 3.1) that each producer i has a best response in the standard basis B. There may however be best responses supported outside of B, and not exist an equilibrium supported on B. Yet, note that if there exists a st rategy profile in which each producer's strategy is supported on B and all producers best respond to each other, on B: no producer can improve their utility by deviating within B, and no producer who cannot improve by deviating on B improve by deviating outside of B since there is a best response B. In turn, there exists an equilibrium supported on B if and only if there exists an equilibrium when producers' strategies are restricted to B.
however
in fact an equilibrium supported
Now, let us restrict ourselves to producers playing on B. Pick any f such that ny > O. We start by computing the utility of producer who plays ey:
K 1/cx =ey
ma(e),8-1) = Σ plcce, .fe cee==
ny
where the second equality follows from the fact that ((, -Te n and piler,ef, = 0 for I' # f. Therefore, the engagement utility from user e if et # ej, and 1/n, ct = Cy. Since there is a number my of users with e = ey, it follows that the total engagement utility for producer i, if picks s; es, is given by my m//ng.
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Now, let us study the deviations for any given producer with s; = ey. Suppose the producer deviates to ef where I' # f. Then, the new number of producers picking I' is ny + 1, and producer i now obtains utility mp/(ng + 1). Therefore, producer does not deviate if and only if for all f", we have
mi
my
R
n +1
Then (n1....,,al ma) is an equilibrium iff no producer wants to deviate, ic iff for all f.f' such that ny > 0,
f
n
my
my
my
If n 0, there no deviation for any producer that picks es as such producers do not exist, and the inequality trivially holds. This concludes the proof.
C Supplementary results on the Movielens-1000 dataset
C.1 Producer distribution
Figure shows the producer distribution for the Movielens- 100k dataset with softmax serving, illustrating increased specialization the temperature decreases. Similarly, Figure G shows similar insights for top-20 softmax serving, however, Unlike full softmax, top-20 serving shows specialization even at high temperatures
(T = 100), as it retains the top 20 producers and serves those producers almost randomly (high r). Figure 7 presents producer distributions for the three non-tempeerauure-deeeeeee rules: greedy, linear, and round-robin. Greedy and linear serving result in producer specialization, while round-robin, as highlighted in remark 2.2 leads all producers to target the highest-weight feature.
Frocdtin heetinnnhhhriiiii Penauoies
Fridtio of Troaaaaa
Teaturo
(a) Softmax== 100
(b) Softmax T 10
(c) Softmax T ==
Asrogg
11 7014
(d) Softmax 0.1
(e) Softmax 0.01
Figure 5: (Full) Softmax serving: Average user weight each feature (blue, left bar) and fraction of producers going for each feature (red, right bar) 100 producers, embedding dimension d = 15. User embeddings obtained from NMF on MovieLens-1000
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X
Norrgg Ui porgge
Jenrge Use W003
(a) Softmax T 100
(b) Softmax 7
(c) Softmax T
Age
feature
(d) Softmax 0.1
(e) Softmax 7 = 0.01
Figure 6: (Top-20) Softmax serving: Average user weight on each feature (blue, left bar) and fraction of producers going for each feature (red, right bar) n = 100 producers, embedding dimension d = 15. User embeddings obtained from NMF on MovieTens--100k 100k, Note: unlike full softmax, top-20 serving shows spocialization even a high temperature.
Dorap Digrre Banveet
Zoorge Usee wridk
SOSTLER
(a) Greedy
(b) Linear
(c) Round robin
Figure Temperature independent serving rules: Greedy, Linear and Round-robin serving, Average user weight on each feature (blue, left bar) and fraction of producers going for each feature (red, right bar) n 100 producers, embedding dimension d 15. User embeddings obtained from NMF on MovieLens-100k.
D Supplementary results on the uniform and skewed synthetic datasets
D.1 Number of iterations until convergence
In Figure we plot the number of iterations (averaged over 40 runs) of Algorithm with increasing number of producers and across varying embedding dimensions. We do so on the Uniform and Skewed-uniforn datasets. With both the linear-proportional and softmax content serving rules, we observe that Algorithm seems scale linearly in the number of producers, further highlighting the computational efficiency of our heuristic.
D.2 Producer distribution
Figure and Figures 10 provide producer distribution plots for softmax serving on the synthetic uniform and skewed datasets. These plots further highlight how the degree of specialization increases as the temperature decreases, few more softmax temperatures (7 10 and 0.1).
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130
26
100
(a) the Ehear-proportional (b) the linear- linear-proportes proportional (c) the softmax serving rule (d) the softmax serving rule skewed-unifor dataset
serving rule uniform dataset serving rule skewed-uniform uniform dataset dalaset
Figure 8: Number of iterations of Algorithm | until convergence to Nash Equilibrium the uniform, skewed- uniform datasets. The different curves represent different embedding dimensions in the game d E {5, 10, 15, 20); the error bars represent standard error over 40 runs.
Softmax 100
(b) Softmax 10
Softmax
(d) Softmax 7 0.1
Softmax 0.01
Linear- Ahnaa--poopottioal serving rule
Figure 9: Average user weight on each feature (blue, left bar) and fraction producers going for each feature (red, right bar) n 100 producers, embedding dimension 15. Lower softmax temperature leads to more producer specialization. Uniform distribution of users.
Fouttim of mwwdtto
Frutte of Piot
feaburo
(a) Softmax T = 100
(b) Softmax 7 10
(c) Softmax T =
Tracsiv reotion
Stataze
(d) Softmax F
(e) Sofumax
(f) the linear- pooccaaproportinalll serving rule
Figure 10: Average user weight on each feature (blue, left bar) and fraction of producers going for each feature (red, right bar) n = 100 producers, embedding dimension d = 15. Lower softmax temperature leads to more producer specialization. Exewed-uniform distribution of users.

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E Experiments on the Sparse synthetic dataset
E.1 Average producer utility
Figure 11 illustrates the average producer utility across serving rules for the spanse synthetic dataset, revealing trends similar to those observed in Figure 4b in the main text. Specifically, lower & values and lower temperatures lead to higher producer utility; however, this comes with a decreased likelihood of convergence to Nash equilibrium.
Linse
140
RoundBobia
Average producor utility
Sottma
Top 10 stftms
120
100
60
20-
Temperature
Figure 11: Average producer utility the Sparse Synthetic dataset: comparing producer utility across serving rules: Linear (blue), RoundRobin (red), Softmax (orange), Top-10/20 Softmax (green/purpll with n n= 50 producers and d = 15. Error bars represent standard error over seeds.
E.2 Producer distribution at Nash Equilibrium
Pructio Phodtor
Teature
Softmax 100
Softmax T = 10
(c) Softmax T
604
600-
(d) Softmax T 0.1
(e) Softmax T = 0.01
(f) the linear- chear-proportional serving rule
Average user weight on each feature (blue, left bar) and fraction of producers going for each feature (red, right. bar) n 100 producers, embedding dimension d = 15. Lower softmax temperature leads to more producer specialization. Sparse distribution of users.
Figure 12:
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Published in Transactions on Machine Learning Research (02/2025)
F Experiments on the AmazonMusic and Rent TheRunway datasets
F.1 Number of iterations till convergence
Figure 13 plots the number of iterations of Algorithm 1 until convergence to NE, averaged over 40 runs of best response dynamics, on the MuszonMusic and Rent TheRunway datasets. Similar to Figure 1 we note a fast time to convergence that seems to scale lincarly in the number of producers.
Beverin
Easeri 5
daverin
dracece
28
25
16
16
(a) Linear serving AmazonMusic
Softmax serving
(c) Linear serving
(d) Softmax serving RentTheRunway
AmazonMusic
Rent TheRunway
Figure 13: Number iterations Algorithm until convergence to NE on the AmazonMusic and Rent Runwwayy datasets. The different curves represent different embedding dimensions in the game d (5, 10, 15, 20}, error bars represent standard error over 40 runs.
F.2 Producer distribution at Nash equilibrium
14 and provide plots for the producer distribution at Nash equilibrium on the AmazonMusic and Rent TheRuuway dataset respectively. We note that our insights on how the softmax temperature affects specialization at equilibrium also arise in these two additional datasets: namely, lower temperatures lead to higher degrees of specialization.
Figures
Daee Waree
of Thudicans
(a) Softmax T 100
(b) Softmax T = 10
Softmax T =
coa
0515
(d) Softmax 7 0.1
(e) Softmax 0.01 0.01
(f) the finear-rrppottionll serving rule
Figure 14: Average user weight on each feature (blue, left bar) and fraction of producers going for each feature (red, right bar) n 100 producers, embedding dimension 15. Lower softmax temperature leads to more producer specialization. User embeddings obtained from NMF the AmazonMusic dataset.
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Published in Transactions on Machine Learning Research (02/2025)
14
1230
(a) Softmax T 100
(b) Softmax T 10
(c) Softmax T=T
00.1
30.34 34
0515
(d) Softmax T 0.1
(e) Softmax = 0.01
(f) the Ineea--roportional serving rule
Average user weight on each feature (blue, left bar) and fraction of producers going for each feature (red, right bar) n 100 producers, embodding dimension = 15. Lower softmax temperature leads to more producer specialization. User embeddings oblained from NMF the RenlTheRunway dataset.
Figure 15:
F.3 Average producer utility
we plot the average producer utility with increasing soflmax-serving temperature and with varying numbers of producers on the AmazonMusic and RentTheRunway datasets. As in Figure [4a] we observe that the producer utility is decreasing with temperature, and temperature 0.01 (pear-hardmax) has the highest utility. This further supports using low temperatures in the softmax content-serving rule.
In Figure 16
producaa
1912
1200e
10:0
2000
10
(a) Producer utility AmazonMusic
(b) Producer utility
Rent Tenttwhyyy
Figure 16: Average producer utility is decreasing in the softmax temperature, results on the Amazon and Rent TheRunway datascts. The different curves represent different number of producers in the game n E (5, 10, 50), embedding dimension = 15, error bars represent standard error over seeds
In Figure 17
compare the average producer utility with the softmax serving rule (across increasing temperatures) v.s with the Iinear-proportional serving rule. We observe that across both datasets, the lowest. softmax temperature we experiment with (r 0.01) leads to greater utility when compared to lincar serving. However, linear serving still obtains a competitive utility, greater than that with softmax temperatures T {0.1, 10, 100}.
G Tables for Nash Equilibrium convergence
For our engagement game, we consider different number of producers (2, 5, 10, 20, 100) and 4 different embedding dimensions (5, 10, 15, 20). Fach of these 12 4 games instantiated with random seeds for the
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Published in Transactions on Machine Learning Research (02/2025)
130
(a) Producer utility
(b) Producer utility RentTheRunway
Music
Figure 17: Average producer utility with lincar serving (in blue) v.s with softmax-sorving (in red) across increasing temperatures on the Muszo and Renl TheRRwwayyy datasels. 50 producers, embedding dimension d = 15, error bars represent standard error over 5 seeds.
randomness in the draws of the synthetic data, and the randomness of NMF algorithm on the real datasets. We observe that on each of these 240 instances of prrducee dimension, seed), the softmax content-serving rule with temperatures т E {100, 10, 0.13 always converges to unique NE on all the datasets. With still converges to a unique NE in a large number instances. The lincar- phoar-propottional serving also converges to unique NE almost always.
softmax temperature T = 0.01, Algorithm []
Linear
Softmax
Dataset \
Serving Rule
100
T
10
T = 1
T= 0.1
== 0.01
Uniform
239
240
240
240
240
232
Skewed-uniform
240
240
240
240
240
240
Movielens-lee 100k
240
240
240
240
240
233
AmazonMusic
240
240
240
240
240
236
Rent TenntTheauwwy
240
240
240
240
240
237
Table Number of instances which Algorithm
1 converges to a Nash equilibrium
Reproducibility
Here we briefly describe the datasets, embedding generation, and experiments. The code and further details are available in the supplementary.
Datascts: We import the Movielens-1008 dataset from the scikit-surprise package. For the Amazon- Music ratings we use the "ratings only" Digital Music. from index.htm] this dataset has approximately 1.5 million ratings, 840k unique users and 450k unique items. For Renn'TheRunee we use has around 190k ratings, 100k unique users and 5.8k unique items.
fit this dataset.
Embedding gencration: We consider the following embedding sceds {13, 17, 19, 23, 29} for randomness
the embedding generation. These 5 embeddings seeds are used in the random draws for the synthetic uniform and skewed embeddings, and for the randomness in the Non-negative matrix factorization embeddings for the Movielens-100k, AmazonMusic and Rent TheRuwway datascts. Note that the real data embeddings are generated using the NMF implementation in scikit-surprise where we pick 4 different. factors d e (5, 10, 15, 20). The synthetic and real data embeddings take 2.6 CPU hours in total to generate and are saved for further use in the experiments.
Note that to parallelize workloads for embedding generation and for the following experiments we use Slurm job arrays.
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Published in Transactions on Machine Learning Research (02/2025)
1 For engagement game, consider 12 different number of producers n {2, 10, 20, 30, 40, 40,50,00000 60, 70, 80, 90, 100} and 4 different embedding dimensions d (0, 10, 15, 20). Each of these 12 x 4 engagement games are instantiated with the 5 embedding seeds described above which determine the user embeddings for a dataset. We then run Algorithm I 40 times with the maximum number of iterations Nmax set. to 500. In each of the 40 trials of Algorithm | the producer strategies are randomly initialized using the sequential seeds (1. .0).......... We then plot the mean and standard error across these in Figure and 13
Experiment Convergence of Algorithm
All plots use softmax temperature of t 1 for simplicity
of exposition.
This experiment on the Uniform, skewed and Moviclens- 100k datasets take approximately 50, 74 and 3 total CPU hours respectively. With the larger scale AmazonMusic and TheRuwway it takes ~ 4.5k and 1k total CPU hours.
Experiment 2 Producer distribution and utility
Producer distribution: Here fix the embedding seed to 17, and plot the producer distribu- tion for the linear serving rule and for the softmax-serving rule after Algorithm terminates. Note that here we don't report mean producer distribution across embedding seeds as this can hide lack of specialization, in the following manner: for a given embedding, only a few features are targeted by the producers, like in Figure 9b all producers go for index #2. However, the specific index of the feature could change as the embedding seed changes. that case, when averaging over the seeds, we may be under the impression that specialization occurs, when it does not.
Producer utility: Here for each of the 12 4 instances of #producers n and dimension d, instantiate the engagement game with linear-serving and 5 different softmax-serving temperatures E {0.01, 0.1, 1, 10, 100). We measure the average producer and user utility at the unique Nash equilibrium and report its mean across the embedding seeds. Average utilities are measured by taking the average across converging runs, and non-converging seeds are dropped.
This experiment on the Uniform, skewed and Movielens- 100k datasets take approximately 4.4, and 0.5 total CPU hours respectively. With the larger scale datascts, Muszcn and Rent RentTheRuwway it takes ~ 530 and 105 total CPU hours.
24


Reviews:
[{"review_id": "SbUU58fWGR", "reviewer": "Reviewer_w2au", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Strengths\n- Theoretical results consist of proofs with clear assumptions, technically sound derivations and compelling results. The proofs in the main paper are correct (checked in reasonable detail) whereas appendix proofs also seem correct but were checked less carefully.\n- The application of game theoretic analysis for producer strategy in recommendation systems is a very interesting topic that is bound to resonate with at least part of the TMLR research community\n- The paper is well motivated and reasonably easy to follow in the technical parts\n\n## Weaknesses\nThe main problems with the paper are in the experimental evaluation. \n- There seems to be a lack of comparison to baselines. Why not compare total utility obtained from the proposed technique for the various datasets with other methods or even a naive baseline? For example produce a figure similar to Figure 4 with the method as well some a selection of reasonable baselines.\n- The scaling of the proposed algorithm in Figure 1 requires some further analysis. I understand that theoretical complexity calculation for the Nash Equilibrium can be challenging and the bounds quite loose in the worst case. However, it would be nice to see how a very large embedding dimension affects number of iteration needed. What if the dimension is 100? Hopefully the pattern of diminishing increases in the iteration count to convergence continues which would indicate a low asymptotic complexity, a good result for this approach. On the other hand, if the empirical complexity is poor perhaps an argument as to the inherently low dimensional embedding space covering user interests could be made.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper investigates producer Nash Equilibria (NE) in recommendation systems. The proposed approach presents a theoretical case for producer specialization to maximize user engagement and presents an algorithm to obtain the NE. The algorithm is shown to converge on a synthetic dataset as well standard recommendation system datasets.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "In general, my outlook of the paper will be improved in terms of validating the contribution and ensuring the claims are verified if the authors address the weaknesses of the paper to strengthen the paper.\n\n- Include reasonable baseline methods, e.g., some of the methods from Table 2 in the experiments and show how the proposed methods improves any reasonable metric such as total utility.\n\n- Investigate scaling of the approach in asymptotic case as embedding dimension grows.\n\n- Move Algorithm 1 to main text to improve readability. It is a core contribution of the paper and referenced in the main text, it cannot be left to appendix.\n\n- Fix typos, e.g., Section 5.2 \"softmax content-serving rule rule\"\n\n- Minor Question: The single mindedness of the users is an important assumption. How to evaluate this key assumption? Is there a procedure to test how well this works in specific dataset that the authors recommend?", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "GtCDsj2bjh", "reviewer": "Reviewer_DPx2", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- The paper makes a nice contribution to the content producer incentives literature, incorporating that the producer reward is based on engagement and that the content-serving rule is continuous. (I found the comparison in Appendix A very helpful.) This clean model gives rise to interesting economic behaviors. \n- The paper clearly explains the conceptual insights emerging from the equilibrium structure analysis. The first conceptual insight (from Theorem 3.4 in Section 3) is that each producer targets a single type of content, even if consumers care about multiple types of content at once. The second conceptual insight (Lemma 4.2 in Section 3) is that the supply of content (up to rounding) matches consumer demand when consumers are single-minded (Section 4). The third conceptual insight is that low temperatures incentivize specialization and lead to higher producer utility (Section 5). \n- The paper has a nice mixture of empirical and theoretical results. \n\nWeaknesses:\n- The proof of the main result in Section 3 (i.e., Theorem 3.4) seems to make an implicit assumption for the softmax rule: that the temperature is sufficiently large. Otherwise, the function f might not be strictly convex, and strict convexity is used in the proof. Is this true? (See my question in the requested changes section.) \n- The main result in Section 3 (i.e., Theorem 3.4) analyzes the equilibrium structure assuming that equilibria exist, but does not establish equilibrium existence. This makes hard to interpret if there is a regime where equilibria do not exist. \n- The theoretical equilibrium characterization results in Section 4 (i.e., Lemma 4.2) are restricted to the case where the users are located at the standard basis vectors. As the paper notes, this case corresponds to \"single-minded\" consumers who are only interested in one dimension of content. This theoretical analysis does not capture the more realistic case where consumers care about multiple dimensions of content.\n- The model assumes that content production is free for producers, and that engagement and consumer utility are aligned. However, I think that these assumptions are fairly mild and also reasonable to make in a stylized model. \n- The mathematical analysis in the paper is fairly straightforward.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper investigates content producer incentives in recommender systems, building on a recent line of work from the machine learning community. \n\nThe paper analyzes a game-theoretic model for content producer competition facilitated by the recommender system. Consumer preferences and content are D-dimensional vectors. The recommender system uses a content-serving rule (i.e., linear proportional rule, or a softmax rule) that is a continuous version of maximizing engagement. Each producer (strategically) chooses a D-dimensional content vector, and producer utility is based on engagement from the recommendations that the producer wins. \n\nThe paper analyzes the equilibria in the game between producers, providing conceptual insights about the equilibrium structure. The paper proves one general property of the equilibria --- that the equilibrium support is at the standard basis vectors, even when users are not necessarily located at the standard basis vectors --- and also characterizes the equilibria when the users are located at the standard basis vectors. The paper empirically analyzes equilibria on synthetic data and on the MovieLens-100K dataset, analyzing the role of the temperature parameter in the content-serving rule.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Addressing the following questions/requests is critical to securing my recommendation for acceptance.\n- Does the proof of Theorem 3.4 implicitly assume that the temperature is sufficiently high for the softmax rule? Please clarify this.  \n- The paper does not prove equilibrium existence for general sets of users, which makes Theorem 3.4 a bit hard to interpret. Does a mixed Nash equilibrium always exist by Glicksberg's theorem, because the utility functions are continuous? It would be helpful to clarify this. \n- In Section 3, it seems like the assumption is that all of the features are strictly positive (Assumption 3.1), but the standard basis vectors in Section 4 do not seem to satisfy this assumption. This made it hard to compare the results in Sections 3 and 4. \n\nAddressing the following questions is not critical to securing my recommendation for acceptance, but could strength the work in my view. \n- The paper interchangeably uses \"consumers\" and \"users\". I think that it would be clear to stick to one of these. \n- When does a pure strategy equilibrium exist for general sets of users? I might expect that pure strategy equilibria would stop existing when $\\tau$ is sufficiently small (and this also appears to be the case in the experiments). \n- What would happen if producers had to pay a cost for creating content that depends on the content? Would this change the insight that producers always target one dimension?", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "B5Pw4pEVTe", "reviewer": "Reviewer_K3oA", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**\n\n- The focus of the paper on engagement (as opposed to exposure) is more aligned with real-world platforms like YouTube and Instagram.\n- The authors provide a clear structural characterization of producer equilibrium under stated assumptions, proving that producers specialize in a single feature in the embedded space at equilibrium.\n- The experiments utilize both synthetic and real-world datasets (e.g., MovieLens-100k, Amazon Music, Rent the Runway), providing diverse evidence to support the theoretical claims.\n- Overall, the characterization of how specialization naturally emerges in engagement-driven games is a valuable contribution, and once again aligns on a high level with my perceptions of the real-world dynamics of these platforms, making the results intuitive.\n\n**Weaknesses**\n\n- Key assumptions (e.g., strictly positive user preferences, fully diverse user embeddings, single-minded users) limit the practical takeaways of the results.\n- The heuristic algorithm does not guarantee convergence to a Nash equilibrium in all cases, particularly under extreme configurations such as very low softmax temperatures.\n- The experiments do not model noisy or adversarial user behaviors, which are present and critical to address in real-world recommender systems. Similarly, dynamic user and producer behaviors are not considered.\n- The model does not account for heterogeneity in producers' abilities to create different types of content, which is a significant factor in real-world platforms and could substantially impact equilibrium outcomes.\n- While the paper focuses on softmax and linear-proportional rules, it does not explore other plausible serving rules that might yield different equilibria or engagement patterns.\n- While the paper offers some discussion on practical implications, it does not offer much actionable advice on how platforms could use these findings to mitigate undesirable outcomes like homogenization or clickbait prevalence.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper investigates the equilibrium behavior of content producers in engagement-driven recommender systems. The authors propose a game-theoretic model where producers aim to maximize user engagement (rather than exposure, as many previous works have instead considered), considering two content-serving rules: the softmax and linear-proportional rules. They derive theoretical results (under key simplifying assumptions) showing that at equilibrium, producers specialize by focusing on a single feature in the embedded content space, driven by competition for engagement. \n\nUsing synthetic and real-world datasets, the authors examine these results experimentally.  Their findings demonstrate the emergence of producer specialization, and examine its dependence on the particular content-serving rule and softmax temperature. The paper also introduces a heuristic algorithm to compute pure Nash equilibria, which is shown to converge efficiently in most cases. The experimental results reveal that lower softmax temperatures promote specialization and maximize both producer and user utilities. These findings provide insights into how platform design choices influence content diversity and engagement outcomes.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- While it may not be feasible to explore theoretically, it would be very interesting to investigate the robustness of the equilibrium results when Assumptions 3.1 (strictly positive user preferences) and 3.2 (diverse user preferences) are relaxed. For instance, simulate scenarios with sparse or noisy user embeddings to assess whether specialization and equilibrium still hold.\n- It would be useful to provide some analysis of the failure cases where Algorithm 1 does not converge to a Nash equilibrium.\n- Though more of a stretch from the current version of the paper, it could be valuable to empirically investigate alternative content-serving rules beyond softmax and linear-proportional, to better understand their impact on producer specialization and engagement outcomes.", "broader_impact_concerns": "No concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: All the reviewers found the paper to be a nice contribution and suitable for publication at TMLR.
Decision: Accept (poster)