SlowGuess/NeurIPS_2023_OCR huggingface dataset:
3.2 Corollaries of Theorem 1
To further understand the condition in equation (4), we consider a series of special cases that provide intuition for when single-genre equilibria exist (proofs deferred to Section F.2). First, let us consider
$\beta = 1$ , in which case the cost function is a norm. Then  $\mathcal{S}^1 = \mathcal{S}$  is convex, so a single-genre equilibrium always exists.
Corollary 1. The threshold  $\beta^{*}$  is always at least 1. That is, if  $\beta = 1$ , there exists a single-genre equilibrium.
The economic intuition is that norms incentivize averaging rather than specialization.
We next take a closer look at how the choice of norm affects the emergence of genres. For cost functions  $c(p) = \| p\| _q^\beta$ , we show that  $\beta^{*}\geq q$  for any set of user vectors, with equality achieved at the standard basis vectors.
Corollary 2. Let the cost function be  $c(p) = \| p\| _q^\beta$ . For any set of user vectors, it holds that  $\beta^{*}\geq q$ . If the user vectors are equal to the standard basis vectors  $\{e_1,\ldots ,e_D\}$ , then  $\beta^{*}$  is equal to  $q$ .
Corollary 2 illustrates that the threshold  $\beta^{*}$  relates closely to the convexity of the cost function and whether the cost function is superlinear. In particular, the cost function must be sufficiently nonconvex for all equilibria to be multi-genre. For example, for the  $\ell_{\infty}$ -norm, where producers only pay for the highest magnitude coordinate, it is never possible to incentivize specialization: there exists a single-genre equilibrium regardless of  $\beta$ . On the other hand, for norms where costs aggregate nontrivially across dimensions, specialization is possible.
In addition to the choice of norm, the geometry of the user vectors also influences whether multiple genres emerge. To illustrate this, we first show that in a concrete market instance with 2 equally sized populations of users, the threshold depends on the cosine similarity between the two user vectors:
Corollary 3. Suppose that there are  $N$  users split equally between two linearly independently vectors  $u_{1}, u_{2} \in \mathbb{R}_{\geq 0}^{D}$ , and let  $\theta^{*} \coloneqq \cos^{-1}\left(\frac{\langle u_{1}, u_{2} \rangle}{\|u_{1}\|_{2}\|u_{2}\|}\right)$ . If the cost function is  $c(p) = \|p\|_2^\beta$ , then,
$$
\beta^ {*} = \frac {2}{1 - \cos (\theta^ {*})}.
$$
Corollary 3 demonstrates the threshold  $\beta^{*}$  increases as the angle  $\theta^{*}$  between the users decreases (i.e. as the users become closer), because it is easier to simultaneously cater to all users. In particular,  $\beta^{*}$  interpolates from 2 when the users are orthogonal to  $\infty$  when the users point in the same direction.
Finally, we consider general configurations of users and cost functions, and we upper bound  $\beta^{*}$ :
Corollary 4. Let  $\| \cdot \|_*$  denote the dual norm of  $\| \cdot \|$ , defined to be  $\| p\|_* = \max_{\| p\| = 1, p \in \mathbb{R}_{\geq 0}^D} \langle q, p \rangle$ . Let  $Z := \|\sum_{n=1}^{N} \frac{u_n}{\|u_n\|_*}\|_*$ . Then,
$$
\beta^ {*} \leq \frac {\log (N)}{\log (N) - \log (Z)}. \tag {5}
$$
In equation (5), the upper bound on the threshold  $\beta^{*}$  increases as  $Z$  increases. As an example, consider the cost function  $c(p) = \| p\| _2^\beta$ . We see that if the user vectors point in the same direction, then  $Z = N$  and the right-hand side of (5) is  $\infty$ . On the other hand, if  $u_{1},\ldots ,u_{n}$  are orthogonal, then  $Z = \sqrt{N}$  and the right-side of (5) is 2, which exactly matches the bound in Corollary 2. In fact, for random vectors  $u_{1},\dots,u_{N}$  drawn from a truncated gaussian distribution, we see that  $Z = \tilde{O} (\sqrt{N})$  in expectation, in which case the right-hand side of (5) is close to 2 as long as  $N$  is large. Thus, for many (but not all) choices of user vectors, even small values of  $\beta$  are enough to induce multiple genres. In Section 3.4, we compute the right-hand side of (5) on user embeddings generated from the MovieLens dataset for different cost functions.
3.3 Location of single-genre equilibrium
We next study where the single-genre equilibrium is located, in cases where it exists. As a consequence of the proof of Theorem 1, we can show that the location of the single-genre equilibrium maximizes the Nash social welfare Nash [1950] of the users.
Corollary 5. If there exists  $\mu$  with  $|\mathrm{Genre}(\mu)| = 1$ , then the corresponding producer direction maximizes Nash social welfare of the users:
$$
\operatorname {G e n r e} (\mu) = \underset {\| p \| = 1 | p \in \mathbb {R} _ {\geq 0} ^ {D}} {\arg \max } \sum_ {i = 1} ^ {N} \log \left(\langle p, u _ {i} \rangle\right). \tag {6}
$$
Figure 3: A symmetric equilibrium for different settings of  $\theta^{*}$ , for 2 users located at  $u_{1}$  and  $u_{2}$  such that  $\theta^{*} = \cos^{-1}\left(\frac{\langle u_{1}, u_{2} \rangle}{\|u_{1}\|\|u_{2}\|}\right)$ , for producer cost function  $c(p) = \|p\|_2^\beta$  with  $\beta = 7$ , and for  $P = \infty$  producers (see Theorem 3). The first 4 plots show the support of the equilibrium in a reparameterized space: note that the x-axis is  $\langle u_{1}, p \rangle$  and the y-axis is  $\langle u_{2}, p \rangle$ , i.e., the values that the users derive from good  $p$ .
Corollary 5 demonstrates that the single-genre equilibrium directions maximizes the Nash social welfare Nash [1950] for users. Interestingly, this measure of welfare for users is implicitly maximized by producers competing with each other in the marketplace. Properties of the Nash social welfare are thus inherited by single-genre equilibria.
3.4 Empirical analysis on the MovieLens dataset
We provide an empirical analysis of supply-side equilibria using the MovieLens-100K dataset and recommendations based on nonnegative matrix factorization (NMF). In particular, we compute the single-genre equilibrium direction for different cost functions as well as an upper bound on  $\beta^{*}$  (i.e., the threshold where specialization starts to occur) for different values of the dimension  $D$ . These experiments provide qualitative insights that offer additional intuition for our theoretical results.
We focus on the rich family of cost functions  $c_{q,\alpha,\beta}(p) = \left\| [p_1 \cdot \alpha_1, \dots, p_D \cdot \alpha_D] \right\|_q^\beta$  parameterized by weights  $\alpha \in \mathbb{R}_{\geq 0}^D$ , parameter  $q \geq 1$ , and cost function exponent  $\beta \geq 1$ . The weights  $\alpha \in \mathbb{R}_{\geq 0}^D$  capture asymmetries in the costs of different dimensions (a higher value of  $\alpha_i$  means that dimension  $i$  is more costly). The parameters  $q$  and  $\beta$  together capture the tradeoffs between improving along a single dimension versus simultaneously improving among many dimensions. To isolate the impact of each parameter, we either fix  $q = 2$  and vary  $\alpha$  (and  $\beta$ ) or we fix  $\alpha = [1,1,\dots,1]$  and vary  $q$  (and  $\beta$ ).
Setup. The MovieLens 100K dataset consists of 943 users, 1682 movies, and 100,000 ratings [Harper and Konstan, 2015]. For  $D \in \{2, 3, 5, 10, 50\}$ , we obtain  $D$ -dimensional user embeddings by running NMF (with  $D$  factors) using the scikit-surprise library. We calculate the single-genre equilibrium genre  $p^* = \arg \max_{\|p\|=1} |p \in \mathbb{R}_{\geq 0}^D \sum_{i=1}^{N} \log(\langle p, u_i \rangle)$  (Corollary 5) by solving the optimization program, using the cvxpy library for  $q = 2$  and projected gradient descent for  $q \neq 2$ . We calculate the upper bound  $\beta_u := \frac{\log(N)}{\log(N) - \log\left( \| \sum_{n=1}^{N} \frac{u_n}{\| u_n \|_*} \|_* \right)} \geq \beta^*$  from Corollary 4.
Single-genre equilibrium direction  $p^*$ . Figures 4a-4d show the direction of the single-genre equilibrium  $p^*$  across different cost functions. These plots uncover several properties of the genre  $p^*$ . First, the genre generally does not coincide with the arithmetic mean of the users. Moreover, the genre varies significantly with the weights  $\alpha$ . In particular, the magnitude of the dimension  $p_i$  is higher if  $\alpha_i$  is lower, which aligns with the intuition that producers invest more in cheaper dimensions. In contrast, the genre turns out to not change significantly with the norm parameter. Altogether, these insights illustrate that how the genre can be influenced by specific aspects of producer costs.
Upper bound on the threshold  $\beta^{*}$  where specialization starts to occur. Figure 4e shows the value of  $\beta_{u}$  across different values of  $D$  and  $q$ . As the dimension  $D$  increases, the upper bound  $\beta_{u}$  decreases, indicating that specialization is more likely to occur. The intuition is that  $D$  amplifies the heterogeneity of user embeddings, subsequently increasing the likelihood of specialization. This insight has an interesting consequence for platform design: the platform can influence the level of specialization by tuning the number of factors  $D$  used in matrix factorization. Producer costs also impact whether specialization occurs: as the norm  $q$  increases, the value of  $\beta_{u}$  increases and specialization is less likely to occur.
3See Appendix A and https://github.com/mjagadeesan/supply-side-equilibria for details.
(a)  $p^*$  for  $D = 2$  and  $q = 2$  (varying  $\alpha$ )
(c)  $p^*$  for  $D = 3$  and  $q = 2$
(b)  $p^*$  for  $D = 2$  and  $\alpha = [1,1]$  (varying  $q$ )
(d)  $p^*$  for  $D = 3$  and  $\alpha = [1,1,1]$
(e)  $\beta_{u}$  for  $\alpha = [1,\dots ,1]$
Figure 4: Empirical analysis of supply-side equilibria on the MovieLens-100K dataset. Plots  $(a)-(d)$ : Single-genre equilibrium direction  $p^*$  (computed using Corollary 5) for different cost function weights  $\alpha \in \mathbb{R}_{\geq 0}^{D}$  and parameters  $q \geq 1$  as well as for different dimensions  $D \geq 1$ . Interestingly, the single-genre equilibrium direction is generally not aligned with the arithmetic mean and places a higher weight on cheaper dimensions. Plot  $(e)$ : Upper bound  $\beta_{u}$  on the threshold  $\beta^*$  where specialization starts to occur (computed using Corollary 4) for different values of  $q$ . Observe that higher values of  $D$  make specialization more likely to occur.
4 Impact of specialization on market competitiveness
Having studied the phenomenon of specialization, we next study its economic consequences on the resulting marketplace of digital goods. We show that producers can achieve positive profit at equilibrium, even though producers are competing with each other.
More formally, we can quantify the producer profit at equilibrium as follows. At a symmetric equilibria  $\mu$ , all producers receive the same expected profit given by:
$$
\mathcal {P} ^ {\mathrm {e q}} (\mu) := \mathbb {E} _ {p _ {1}, \dots , p _ {P} \sim \mu} [ \mathcal {P} (p _ {1}; p _ {- 1}) ] = \mathbb {E} \left[ \left(\sum_ {i = 1} ^ {N} 1 \left[ j ^ {*} \left(u _ {i}; p _ {1: P}\right) = 1 \right]\right) - \| p _ {1} \| ^ {\beta} \right], \tag {7}
$$
where expectation in the last term is taken over  $p_1, \ldots, p_P \sim \mu$  as well as randomness in recommendations. Intuitively, the equilibrium profit of a marketplace provides insight about market competitiveness. Zero profit suggests that competition has driven producers to expend their full cost budget on improving product quality. Positive profit, on the other hand, suggests that the market is not yet fully saturated and new producers have incentive to enter the marketplace.
We show a sufficient condition for positive profit in terms of the user geometry and the cost function, and we show this result relies on the equilibrium exhibiting specialization (Proposition 2).
To gain intuition, let us revisit two users located at the standard basis vectors and cost function  $c(p) = \| p\| _2^\beta$ . We can obtain the following characterization of profit.
Corollary 6. Let there be 2 users located at the standard basis vectors  $e_1, e_2 \in \mathbb{R}^2$ , let the cost function be  $c(p) = \|p\|_2^\beta$ . For  $P = 2$  and  $\beta \geq \beta^* = 2$ , there is an equilibrium  $\mu$  with  $\mathcal{P}^{eq}(\mu) = 1 - \frac{2}{\beta}$ .
Corollary 6 shows that there exist equilibria that exhibit strictly positive profit for any  $\beta \geq 2$ . The intuition is that (after sampling the randomness in  $\mu$ ), different producers often produce different genres of content. This reduces the amount of competition along any single genre. Producers are thus no longer forced to maximize quality, enabling them to generate a strictly positive profit.
We generalize this finding to sets of many users and producers and to arbitrary norms. In particular, we provide the following sufficient condition under which the profit at equilibrium is strictly positive.
Proposition 2. Suppose that
$$
\max  _ {\| p \| \leq 1} \min  _ {1 \leq i \leq N} \left\langle p, \frac {u _ {i}}{\| u _ {i} \|} \right\rangle <   N ^ {- P / \beta}. \tag {8}
$$
Then for any symmetric equilibrium  $\mu$ , the profit  $\mathcal{P}^{eq}(\mu)$  is strictly positive.
Proposition 2 provides insight into how the geometry of the users and structure of producer costs impact whether producers can achieve positive profit. To interpret Proposition 2, let us examine the quantity  $Q \coloneqq \max_{\| p \| \leq 1} \min_{i=1}^{N} \langle p, \frac{u_i}{\| u_i \|} \rangle$  that appears on the left-hand side of (8). Intuitively,  $Q$  captures how easy it is to produce content that appeals simultaneously to all users. It is larger when the users are close together and smaller when they are spread out. For any set of vectors we see that  $Q \leq 1$ , with strict inequality if the set of vectors is non-degenerate. The right-hand side of (8), on the other hand, goes to 1 as  $\beta \to \infty$ . Thus, for any non-degenerate set of users, if  $\beta$  is sufficiently large, the condition in Proposition 2 is met and producer profit is strictly positive.
Although Proposition 2 does not explicitly consider specialization, we show that specialization is nonetheless central to achieving positive profit at equilibrium. To illustrate this, we show that at a single-genre equilibrium, the profit is zero whenever there are at least  $P \geq 2$  producers.
Proposition 3. If  $\mu$  is a single-genre equilibrium, then the profit  $\mathcal{P}^{eq}(\mu)$  is equal to 0.
This draws a distinction between profit in the single-genre regime (where there is no specialization) and the multi-genre regime (where there is specialization). Interestingly, this distinction parallels between the distinction markets with homogeneous goods and markets with differentiated goods (as we describe in more detail in Appendix C.2).
Our results provide insight about the number of producers needed for a market to be saturated and fully competitive. Theorem 2 reveals that the marketplace of digital goods may need far more than 2 producers in order to be saturated. Nonetheless, the equilibrium profit does approach 0 as the number of producers in the marketplace goes to  $\infty$ : this is because the cumulative profit of all producers is at most  $N$  and producers achieve the same profit, so  $\mathcal{P}^{\mathrm{eq}}(\mu) \leq N / P$ . Perfect competition is therefore recovered in the infinite-producer limit.
We presented a model for supply-side competition in recommender systems. The rich structure of production costs and the heterogeneity of users enable us to capture marketplaces that exhibit a wide range of forms of specialization. Our main results characterize when specialization occurs, analyze the form of specialization, and show that specialization can reduce market competitiveness. More broadly, we hope that our work serves as a starting point to investigate how recommendations shape the supply-side market of digital goods.
One direction for future work is to further examine the economic consequences of specialization. Several of our results take a step towards this goal: Corollary 5 illustrates that single-genre equilibria occur at the direction that maximizes the Nash user welfare, and Proposition 2 shows that specialization can lead to positive producer profit. These results leave open the question of how the welfare of users and producers relate to one another. Characterizing the welfare at equilibrium would elucidate whether specialization helps producers at the expense of users or helps all market participants.
Another interesting direction for future work is to extend our model to incorporate additional aspects of content recommender systems. For example, although we focus on perfect recommendations that match each user to their favorite content, we envision that this assumption could be relaxed in several ways: e.g., the platform may have imperfect information about users, users may not always follow platform recommendations, and producers may learn their best-responses over repeated interactions with the platform. Moreover, although we assume that producers earn fixed per-user revenue, this assumption could be relaxed to let producers set prices.
Addressing these questions would further elucidate the market effects induced by supply-side competition, and inform our understanding of the societal effects of recommender systems.
We would like to thank Jean-Stanislas Denain, Frances Ding, Erik Jones, Quitze Valenzuela-Stookey, and Ruiqi Zhong for helpful comments on the paper. MJ acknowledges support from the Paul and Daisy Soros Fellowship and Open Phil AI Fellowship.
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