SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Nikhil Garg
Cornell Tech
ngarg@cornell.edu
Algorithmic recommender systems such as Spotify and Netflix affect not only consumer behavior but also producer incentives. Producers seek to create content that will be shown by the recommendation algorithm, which can impact both the diversity and quality of their content. In this work, we investigate the resulting supply-side equilibria in personalized content recommender systems. We model the decisions of producers as choosing multi-dimensional content vectors and users as having heterogenous preferences, which contrasts with classical low-dimensional models. Multi-dimensionality and heterogeneity creates the potential for specialization, where different producers create different types of content at equilibrium. Using a duality argument, we derive necessary and sufficient conditions for whether specialization occurs. Then, we characterize the distribution of content at equilibrium in concrete settings with two populations of users. Lastly, we show that specialization can enable producers to achieve positive profit at equilibrium, which means that specialization can reduce the competitiveness of the marketplace. At a conceptual level, our analysis of supply-side competition takes a step towards elucidating how personalized recommendations shape the marketplace of digital goods.
Algorithms, recommender systems have disrupted the production of digital goods such as movies, music, and news. In the music industry, artists have changed the length and structure of songs in response to Spotify's algorithm and payment structure [Hodgson, 2021]. In the movie industry, personalization has led to low-budget films catering to specific audiences [McDonald, 2019], in some cases constructing data-driven "taste communities" [Adalian, 2018]. Across industries, recommender systems shape how producers decide what content to create, influencing the supply side of the digital goods market. This raises the questions: What factors drive and influence the supply-side marketplace? What content will be produced at equilibrium?
Intuitively, supply-side effects are induced by the multi-sided interaction between producers, the recommendation algorithm, and users. Users tend to follow recommendations when deciding what content to consume [Ursu, 2018]—thus, recommendations influence how many users consume each digital good and impact the profit (or utility) generated by each content producer. As a result, content producers shape their content to maximize appearance in recommendations; this creates competition between the producers, which can be modeled as a game. However, understanding such producer-side effects has been difficult, both empirically and theoretically. This is a pressing problem, as these gaps in understanding have hindered the regulation of digital marketplaces [Stigler Committee, 2019].
Figure 1: A symmetric equilibrium for different settings of  $\beta$ , for 2 users located at the standard basis vectors  $e_1$  and  $e_2$ ,  $P = 2$  producers, and producer cost function  $c(p) = \|p\|_2^\beta$ . The first 4 plots show the support of the equilibrium  $\mu$ . As  $\beta$  increases, there is a phase transition from a single-genre equilibrium to an equilibrium with infinitely many genres (Theorem 2). The profit also transitions from zero to positive, demonstrating how specialization reduces the competitiveness of the marketplace (Propositions 2-3). The last plot shows the cumulative distribution function of  $\|p\|$ .
At a high level, there are two primary challenges that complicate theoretical analyses of these supply-side effects. (1) Digital goods such as movies have many attributes and thus must be embedded in a multi-dimensional continuous space, leading to a large producer action space. This multidimensionality is a departure from traditional economic models of price and spatial competition. (2) A core aspect of such marketplaces is the potential for specialization: that is, different producers may produce different items at equilibrium. Incentives to specialize depend on the level of heterogeneity of user preferences and the cost structure for producing goods (whether it is more expensive to produce items that are good in multiple dimensions). As a result, supply-side equilibria have the potential to exhibit rich economic phenomena, but pose a challenge to both modeling and analysis.
We introduce a simple game-theoretic model for supply-side competition in personalized recommender systems, that captures the multi-dimensional space of producer decisions, rich structures of production costs, and general configurations of users. Users and digital goods are represented as  $D$ -dimensional vectors, and the value that a user with vector  $u$  derives from a digital good  $p$  is equal to the inner product  $\langle u, p \rangle$ . The platform offers personalized recommendations to each user, showing them the digital good with maximum value to them. Each producer chooses a single digital good  $p \in \mathbb{R}_{\geq 0}^{D}$  to create. The goal of a producer to maximize their profit, which is equal to the number of users who are recommended their content minus the (one-time) cost of producing the content. We study the Nash equilibria of this game between producers. We defer model specifics to Section 2.
To ground our investigation of the complex producer choices, we focus on one particular economic phenomena—the potential for specialization. Specialization, which occurs when different producers create different genres (i.e., directions) of content at equilibrium, has several economic consequences. For example, whether specialization occurs, as well as the form that specialization takes, determines the diversity of content available on the platform. Moreover, specialization influences market competitiveness by reducing the amount of competition in each genre. This raises the questions:
Under what conditions does specialization occur at equilibrium? What is its impact on market competitiveness? What form does specialization take?
We investigate specialization and its consequences on the supply-side market. We analyze how the specialization exhibited at equilibrium varies with user geometry and producer cost function parameters. Figure 1 and 3 depict the equilibria in our model across several concrete instantations. Our main results provide insight into each of the questions from above: we characterize when specialization occurs (Section 3) and investigate the impact of specialization on market competitiveness (Section 4). We provide an empirical analysis of supply-side equilibria using the MovieLens-100K dataset [Harper and Konstan, 2015] (Figure 4), and we analyze the particular form that specialization takes in the case of 2 equally sized populations (deferred to Appendix D).
Our simple model yields a nuanced picture of supply-side equilibria in recommender systems. Our results provide insight into specialization and its implications, and en route to proving these results, we develop a technical toolkit to analyzing the multi-dimensional behavior of producers (see Appendix C.1 for a summary). More broadly, our model and results open the door to investigating how recommender systems shape the diversity and quality of content created by producers.
Our work is related to research on societal effects in recommender systems, models of competition in economics, and strategic effects induced by algorithmic decisions.
Supply-side effects of recommender systems. A line of work in the machine learning literature has studied supply-side effects from a theoretical perspective, but existing models do not capture the infinite, multi-dimensional decision space of producers. Ben-Porat and Tennenholtz [2018] study supply-side effects with a focus on mitigating strategic effects by content producers; Ben-Porat et al. [2020], building on Basat et al. [2017], also studied supply-side equilibria with a focus on convergence of learning dynamics for producers. The main difference from our work is that producers in these models choose a topic from a finite set of options; in contrast, our model captures the infinite, multi-dimensional producer decision space that drives the emergence of genres. Moreover, we focus on the structure of equilibria rather than the convergence of learning.
In concurrent and independent work, Hron et al. [2022] study a related model for supply-side competition in recommender systems where producers choose content embeddings in  $\mathbb{R}^D$ . One main difference is that, rather than having a cost on producer content, they constrain producer vectors to the  $\ell_2$  unit ball (this corresponds to our model when  $\beta \rightarrow \infty$  and the norm is the  $\ell_2$ -norm, although the limit behaves differently than finite  $\beta$ ). Additionally, Hron et al. incorporate a softmax decision rule to capture exploration and user non-determinism, whereas we focus entirely on hardmax recommendations. Thus, our model focuses on the role of producer costs while Hron et al.'s focuses on the role of the recommender environment. Technically, Hron et al. study the existence of different types of equilibria and the use of behaviour models for auditing, whereas we analyze the economic phenomena exhibited by symmetric mixed strategy Nash equilibria, with a focus on specialization.
Other work has studied the emergence of filter bubbles [Flaxman et al., 2016], the ability of users to reach different content [Dean et al., 2020], the shaping of user preferences [Adomavicius et al., 2013], and stereotyping [Guo et al., 2021].
Models of competition in microeconomics. Our model and research questions relate to classical models of competition in economics; however, particular aspects of recommender systems—high-dimensionality of digital goods, rich structure of producer costs, and user geometry—are not captured by these classical models. For example, in price competition, producers set a price, but do not decide what good to produce (e.g. Bertrand competition, see [Baye and Kovenock, 2008] for a textbook treatment). Price is one-dimensional, but producer decisions in our model are multi-dimensional.
Another line of work on product selection has investigated how producers choose goods (i.e., content) at equilibrium (see Anderson et al. [1992] for a textbook treatment). For example, in spatial location models, producers choose a direction in a low-dimensional space (e.g.,  $\mathbb{R}^1$  in [Hotelling, 1929, d'Aspremont et al., 1979] and  $\mathbb{S}^1$  in [Salop, 1979]), and users typically receive utility based on the negative of the Euclidean distance. In contrast, producers in our model jointly select direction and magnitude, and users receive utility based on inner product. Since some spatial location models allow producers to set prices, it may be tempting to draw an analogy between  $\| p\|$  in our model and the price in these models. However, this analogy breaks down because costs can be nonlinear in  $\| p\|$ .
Other related work has investigated supply function equilibria (e.g. [Grossman, 1981]), where the producer chooses a function from quantity to prices, rather than what content to produce, and the pure characteristics model (e.g. [Berry, 1994]), where attributes of users and producers are also embedded in  $\mathbb{R}^D$  like in our model, but which focuses on demand estimation for a fixed set of content, rather than analyzing the content that arises at equilibrium in the marketplace. Recent work in economics has extended Berry [1994] to allow for endogenous product choice (e.g. [Wollmann, 2018]) and also studied specialization (e.g. [Vogel, 2008, Peregno and Yuksel, 2022]), though with different modeling assumptions than our work.
Strategic classification. A line of work of strategic classification [Brückner et al., 2012, Hardt et al., 2016] has studied how classification can induce participants to strategically change their features to improve their outcomes. Models for participant behavior (e.g. Kleinberg and Raghavan [2019], Jagadeesan et al. [2021], Ghalme et al. [2021]) generally do not capture competition between participants. One exception is Liu et al. [2022], where participants compete to appear higher in a ranking; in contrast, the participants in our model compete for users with heterogeneous preferences.
We introduce a game-theoretic model for supply-side competition in recommender systems. Consider a platform with  $N \geq 1$  heterogeneous users who are offered personalized recommendations and  $P \geq 2$  producers who strategically decide what digital good to create.
Embeddings of users and digital goods. Each user  $i$  is associated with a  $D$ -dimensional embedding  $u_{i}$  that captures their preferences. We assume that  $u_{i} \in \mathbb{R}_{\geq 0}^{D} \setminus \{\vec{0}\}$  —i.e., the coordinates of each embedding are nonnegative and each embedding is nonzero. While user vectors are fixed, producers choose what content to create. Each producer  $j$  creates a single digital good, which is associated with a content vector  $p_{j} \in \mathbb{R}_{\geq 0}^{D}$ . The value of good  $p$  to user  $u$  is  $\langle u, p \rangle$ .
Personalized recommendations. After the producers decide what content to create, the platform offers personalized recommendations to each user. We consider a stylized model where the platform has complete knowledge of the user and content vectors. The platform recommends to each user the content (and thus producer) of maximal value to them. Mathematically, the platform assigns a user  $u$  to the producer  $j^*$ , where  $j^*(u; p_{1:P}) = \arg \max_{1 \leq j \leq P} \langle u, p_j \rangle$ . If there are ties, the platform sets  $j^*(u; p_{1:P})$  to be a producer chosen uniformly at random from the argmax.
Producer cost function. Each producer faces a fixed (one-time) cost for producing content  $p$ , which depends on the magnitude of  $p$ . Since the good is digital and thus cheap to replicate, the production cost does not scale with the number of users. We assume that the cost function  $c(p)$  takes the form  $\| p\|^{\beta}$ , where  $\| \cdot \|$  is any norm and the exponent  $\beta$  is at least 1. The magnitude  $\| p\|$  captures the quality of the content: in particular, if a producer chooses content  $\lambda p$ , they win at least as many users as if they choose  $\lambda'p$  for  $\lambda' < \lambda$ . (This relies on the fact that all vectors are in the positive orthant.) The norm and  $\beta$  together encode the cost of producing a content vector  $v$ , and reflect cost tradeoffs for excelling in different dimensions (for example, producing a movie that is both a drama and a comedy). Large  $\beta$ , for instance, means that this cost grows superlinearly. In Section 3, we will see that these tradeoffs capture the extent to which producers are incentivized to specialize.
Producer profit. A producer receives profit equal to the number of users who are recommended their content minus the cost of producing the content. The profit of producer  $j$  is equal to:
$$
\mathcal {P} \left(p _ {j}; p _ {- j}\right) = \mathbb {E} \left[ \left(\sum_ {i = 1} ^ {n} 1 \left[ j ^ {*} \left(u _ {i}; p _ {1: P}\right) = j \right]\right) - \| p _ {j} \| ^ {\beta} \right], \tag {1}
$$
where  $p_{-j} = [p_1, \dots, p_{j-1}, p_{j+1}, \dots, p_P]$  denotes the content produced by all of the other producers; the expectation is over the randomness over platform recommendations in the case of ties.
We defer further discussion of our model to Appendix B.1.
2.1 Equilibrium concept and existence of equilibrium
We study the Nash equilibria of the game between producers. Each producer  $j$  chooses a (random) strategy over content, given by a probability measure  $\mu_{j}$  over the content embedding space  $\mathbb{R}_{\geq 0}^{D}$ . Recall that strategies  $(\mu_1,\ldots ,\mu_P)$  form a Nash equilibrium if no producer—given the strategies of other producers—can choose a different strategy where they achieve higher expected profit.<sup>1</sup>
A salient feature of our model is that there are discontinuities in the producer utility function in equation (1), since the function  $\arg \max_{1\leq i\leq P}\langle u_i,p_j\rangle$  changes discontinuously with the producer vectors  $p_j$ . Due to these discontinuities, pure strategy equilibria do not exist (Proposition 4 in Appendix B.2). Nonetheless, we show that a symmetric mixed strategy equilibrium (i.e. an equilibrium where  $\mu_1 = \ldots = \mu_P$ ) is nonetheless guaranteed to exist.
Proposition 1. For any set of users and any  $\beta \geq 1$ , a symmetric mixed equilibrium exists.
These equilibria are interestingly atomless (Proposition 5 in Appendix B.2), so there is randomization over quality  $\| p\|$  as well as randomness over genres  $p / \| p\|$ . We defer an example of the equilibria
That is,  $p_j \in \arg \max_{p \in \mathbb{R}_{>0}^D} \mathbb{E}_{p_{-j} \sim \mu_{-j}}[\mathcal{P}(p_j; p_{-j})]$  for every  $j \in [P]$  and every  $p_j \in \operatorname{supp}(\mu_j)$ .
in our model for homogenous users to Appendix B.3. We take the symmetric mixed equilibria of this game as the main object of our study, since they are both tractable to analyze and rich enough to capture asymmetric solution concepts such as specialization.
2.2 Specialization and the formation of genres
When users are heterogeneous, there are inherent tensions between catering to one user and catering to other users. As a result, the producer make nontrivial choices not only about the quality (vector norm  $\| p\|$ ) of their content, but also the genre (direction  $p / \| p\|$ ) of their content. This can lead to specialization, which is when different producers create goods tailored to different users; alternatively, all producers might still produce the same genre of content at equilibrium and thus only exhibit differentiation on the axis of quality.
We define genres as the set of directions that arise at a symmetric mixed Nash equilibrium  $\mu$ :
$$
\operatorname {G e n r e} (\mu) := \left\{\frac {p}{\| p \|} \mid p \in \operatorname {s u p p} (\mu) \right\}, \tag {2}
$$
where we normalize by  $\| p\|$  to separate out the quality (norm) from the genre (direction). The set of genres  $\mathrm{Genre}(\mu)$  captures the set of content that may arise on the platform in some realization of randomness of the producers' strategies. When an equilibrium has a single genre, all producers cater to an average user, and only a single type of content appears. On the other hand, when an equilibrium has multiple genres, many types of digital content are likely to appear on the platform.
We thus say that specialization occurs at an equilibrium  $\mu$  if and only if the support of  $\mu$  has more than one genre (direction). The particular form of specialization exhibited by  $\mu$  is further captured by the number and set of genres in the support of  $\mu$ . See Figure 1 for a depiction of markets with a single- and multi-genre equilibria, respectively.
3 When does specialization occur?
In order to investigate whether specialization occurs in a given marketplace, we distinguish between two regimes of marketplaces based on whether or not a single-genre equilibrium exists:
1. A marketplace is in the single-genre regime if there exists an equilibrium  $\mu$  such that  $|\mathrm{Genre}(\mu)| = 1$ . All producers thus create content of the same genre.
2. A marketplace is in the multi-genre regime if all equilibria  $\mu$  satisfy  $|\mathrm{Genre}(\mu)| > 1$ . Producers thus necessarily differentiate in the genre of content that they produce.
To understand these regimes, we ask: what conditions on the user vectors  $u_{1}, \ldots, u_{N}$  and the cost function parameters  $\| \cdot \|$  and  $\beta$  determine which regime the marketplace is in?
In Section 3.1, we give necessary and sufficient conditions for all equilibria to have multiple genres (Theorem 1). In Section 3.2, we show several corollaries of Theorem 1. In Section 3.3, we show that the location of the single-genre equilibrium (in cases where it exists) maximizes the Nash social welfare. In Section 3.4, we provide an empirical analysis using the MovieLens-100K dataset [Harper and Konstan, 2015] that validate our theoretical results. We defer proofs of these results to Appendix F.
3.1 Characterization of single-genre and multi-genre regimes
We first provide a tight geometric characterization of when a marketplace is in the single-genre regime versus in the multi-genre regime. More formally, let  $\mathbf{U} = [u_1;\dots ;u_N]$  be the  $N\times D$  matrix of user vectors, and let  $\mathcal{S}$  denote the image of the unit ball under  $\mathbf{U}$ :
$$
\mathcal {S} := \left\{\mathbf {U} p \mid \| p \| \leq 1, p \in \mathbb {R} _ {\geq 0} ^ {D} \right\} \tag {3}
$$
Each element of  $S$  is an  $N$ -dimensional vector, which represents the user values for some unit-norm producer  $p$ . Additionally, let  $S^{\beta}$  be the image of  $S$  under coordinate-wise powers, i.e. if
In this definition, we implicitly interpret the randomness in the producer strategies as differentiation between producers. This formalization of specialization obviates the need to reason about asymmetric equilibria, thus making the model much more tractable to analyze.
Figure 2: The sets  $S^{\beta}$  and  $\bar{S}^{\beta}$  for two configurations of user vectors (rows) and settings of  $\beta$  (columns). The user vectors are [1,0], [0,1] (top, same as Figure 1) and [1,0], [0.5,0.87] (bottom).  $S^{\beta}$  transitions from convex to non-convex as  $\beta$  increases, though the transition point depends on the user vectors. When  $S^{\beta}$  is convex, a single vector  $p$  can more easily satisfy both users at low cost.
$(z_{1},\ldots ,z_{N})\in \mathcal{S}$  then  $(z_1^\beta ,\dots ,z_N^\beta)\in S^\beta$  . We show that genres emerge when  $S^{\beta}$  is sufficiently different from its convex hull  $\bar{S}^{\beta}$  ..
Theorem 1. Let  $\mathbf{U} \coloneqq [u_1; \dots; u_N]$ , let  $\mathcal{S}$  be  $\{\mathbf{U}p \mid \|p\| \leq 1, p \in \mathbb{R}_{\geq 0}^D\}$ , and let  $\mathcal{S}^\beta$  be the image of  $\mathcal{S}$  under coordinate-wise powers. Then, there is a symmetric equilibrium  $\mu$  with  $|\mathrm{Genre}(\mu)| = 1$  if and only if
$$
\max  _ {y \in \mathcal {S} ^ {\beta}} \prod_ {i = 1} ^ {N} y _ {i} = \max  _ {y \in \bar {\mathcal {S}} ^ {\beta}} \prod_ {i = 1} ^ {N} y _ {i}. \tag {4}
$$
Otherwise, all symmetric equilibria have multiple genres. Moreover, if (4) holds for some  $\beta$ , it also holds for every  $\beta' \leq \beta$ .
Theorem 1 relates the existence of a single-genre equilibrium to the convexity of the set  $S^{\beta}$ . As a special case, the condition in Theorem 1 always holds if  $S^{\beta}$  is convex, but is strictly speaking weaker than convexity. Interestingly, the condition depends on the geometry of the user embeddings  $u_{1},\ldots ,u_{n}$  and the cost function  $c(p) = \| p\|^{\beta}$  but not on the number of producers  $P$ . Intuitively, convexity of  $S^{\beta}$  relates to the ease with which a vector  $p$  can satisfy all users simultaneously, at low cost—each dimension of  $S$  corresponds to a user's utility. In Figure 2, we display the sets  $S^{\beta}$  and  $\bar{S}^{\beta}$  for different configurations of user vectors and different settings of  $\beta$ .
Theorem 1 further shows that the boundary between the single-genre and multi-genre regimes can be represented by a threshold defined as follows
$$
\beta^ {*} := \sup  \left\{\beta \geq 1 \mid \max  _ {y \in \mathcal {S} ^ {\beta}} \prod_ {i = 1} ^ {N} y _ {i} = \max  _ {y \in \mathcal {S} ^ {\beta}} \prod_ {i = 1} ^ {N} y _ {i} \right\}.
$$
where single-genre equilibria exist exactly when  $\beta \leq \beta^{*}$ . Conceptually, larger  $\beta$  make producer costs more superlinear, which eventually disentivizes producers from attempting to perform well on all dimensions at once.
The proof of Theorem 1 draws a connection to minimax theory in optimization. In particular, we show that the existence of a single-genre equilibrium is equivalent to strong duality holding for a certain optimization program that we define (Lemma 2). This allows us to leverage techniques from optimization theory to analyze when specialization occurs. We defer a full proof to Appendix F.1.