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Producers Equilibria and Dynamics in Engagement-Driven Recommender Systems

Krishna Acharya, Varun Vangala, Jingyan Wang, Juba Ziani

TL;DR

This work analyzes producers competing to maximize engagement in engagement-driven recommender systems under fixed content-serving rules. It formalizes an engagement game where producers choose embedded-feature content vectors $s_i$ to maximize $u_i(s) = \sum_{k=1}^K p_i(c_k, \vec{s}) \cdot c_k^\top s_i$, with serving rules including the softmax $p_i(c,s)$ and the linear-proportional rule, and proves a structural result: at any Nash equilibrium, each producer concentrates on a single embedded feature, i.e., $s_i$ is supported on the standard basis. It further analyzes equilibria under a simplified single-minded-user scenario, derives a closed-form condition for a pure NE, and develops a best-response dynamics heuristic that converges to a NE on real data. Empirical results on Movielens-100k and other datasets show that specialization increases as the softmax temperature decreases and that both producer and user utilities decline with higher temperatures, providing guidance on selecting content-serving rules to balance engagement and welfare in practice.

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.

Producers Equilibria and Dynamics in Engagement-Driven Recommender Systems

TL;DR

This work analyzes producers competing to maximize engagement in engagement-driven recommender systems under fixed content-serving rules. It formalizes an engagement game where producers choose embedded-feature content vectors to maximize , with serving rules including the softmax and the linear-proportional rule, and proves a structural result: at any Nash equilibrium, each producer concentrates on a single embedded feature, i.e., is supported on the standard basis. It further analyzes equilibria under a simplified single-minded-user scenario, derives a closed-form condition for a pure NE, and develops a best-response dynamics heuristic that converges to a NE on real data. Empirical results on Movielens-100k and other datasets show that specialization increases as the softmax temperature decreases and that both producer and user utilities decline with higher temperatures, providing guidance on selecting content-serving rules to balance engagement and welfare in practice.

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.
Paper Structure (49 sections, 2 theorems, 18 equations, 17 figures, 3 tables, 1 algorithm)

This paper contains 49 sections, 2 theorems, 18 equations, 17 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Summary of contributions
  3. Related work
  4. Exposure games
  5. Engagement games
  6. Mechanism design
  7. User dynamics
  8. Our model
  9. Producer model
  10. User model
  11. Recommender system's content-serving rule
  12. Producer utilities
  13. User utilities
  14. Equilibrium Structure
  15. Proof of Theorem \ref{['thm:pure_eq']}
  16. ...and 34 more sections

Key Result

Theorem 3.4

Suppose Assumptions as:positive_c and as:nontrivial_c hold. Let $\mathcal{B} := (e_1,\ldots,e_d)$ be the standard basis for $\mathbb{R}^d$, where each $e_j$ is the unit vector with value $1$ in coordinate $j \in [d]$ and $0$ in all other coordinates. Under both types of content-serving rules, if the

Figures (17)

  • Figure 1: Number of iterations of Algorithm \ref{['alg:bestrep_dynamics']} until convergence to a Nash Equilibrium on the Movielens-100k dataset. The different curves represent different embedding dimensions in the game $d \in \{5,10,15,20, {50, 100}\}$; the error bars represent standard error over $40$ runs.
  • Figure 2: 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. User embeddings obtained from NMF on MovieLens-100k.
  • 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-uniform distribution of users.
  • Figure 4: Average producer utility on the Movielens-100k dataset. (a) Varying the number of producers $n \in \{5,10,50\}$ with embedding dimension $d=15$. (b) Comparing producer utility across serving rules: Linear (blue), RoundRobin (red), Softmax (orange), Top-10/20 Softmax (green/purple) with $n=50$ producers and $d=15$. Error bars represent standard error over 5 seeds.
  • Figure 5: (Full) 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 MovieLens-100k.
  • ...and 12 more figures

Theorems & Definitions (9)

  • Remark 2.1
  • Remark 2.2
  • Definition 3.3: Nash Equilibrium
  • Theorem 3.4
  • Claim 3.5: Convexity for linear-proportional serving rule
  • proof
  • Claim 3.6: Convexity for softmax serving rule
  • proof
  • Lemma 4.2