Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Convergence of No-Regret Dynamics in Information Retrieval Games with Proportional Ranking Functions

Omer Madmon, Idan Pipano, Itamar Reinman, Moshe Tennenholtz

TL;DR

This work addresses convergence of no-regret dynamics in information retrieval games where publishers compete over exposure under proportional ranking functions. It proves a tight equivalence: the activation function $g$ being concave is necessary and sufficient for the induced game to be concave and socially-concave, which in turn ensures convergence of any no-regret dynamics and, under bi-convex distance, unique Nash equilibria. The paper also provides a continuous embedding-based model and conducts extensive simulations using a state-of-the-art no-regret algorithm to explore welfare trade-offs between publishers and users as a function of $λ$, $n$, $s$, and $k$, showing clear trade-offs between welfare and convergence speed. These results offer practical guidance for ranking design in SEO-like systems while providing rigorous convergence guarantees under no-regret learning, and they outline key limitations and directions for future work.

Abstract

Publishers who publish their content on the web act strategically, in a behavior that can be modeled within the online learning framework. Regret, a central concept in machine learning, serves as a canonical measure for assessing the performance of learning agents within this framework. We prove that any proportional content ranking function with a concave activation function induces games in which no-regret learning dynamics converge. Moreover, for proportional ranking functions, we prove the equivalence of the concavity of the activation function, the social concavity of the induced games and the concavity of the induced games. We also study the empirical trade-offs between publishers' and users' welfare, under different choices of the activation function, using a state-of-the-art no-regret dynamics algorithm. Furthermore, we demonstrate how the choice of the ranking function and changes in the ecosystem structure affect these welfare measures, as well as the dynamics' convergence rate.

On the Convergence of No-Regret Dynamics in Information Retrieval Games with Proportional Ranking Functions

TL;DR

This work addresses convergence of no-regret dynamics in information retrieval games where publishers compete over exposure under proportional ranking functions. It proves a tight equivalence: the activation function being concave is necessary and sufficient for the induced game to be concave and socially-concave, which in turn ensures convergence of any no-regret dynamics and, under bi-convex distance, unique Nash equilibria. The paper also provides a continuous embedding-based model and conducts extensive simulations using a state-of-the-art no-regret algorithm to explore welfare trade-offs between publishers and users as a function of , , , and , showing clear trade-offs between welfare and convergence speed. These results offer practical guidance for ranking design in SEO-like systems while providing rigorous convergence guarantees under no-regret learning, and they outline key limitations and directions for future work.

Abstract

Publishers who publish their content on the web act strategically, in a behavior that can be modeled within the online learning framework. Regret, a central concept in machine learning, serves as a canonical measure for assessing the performance of learning agents within this framework. We prove that any proportional content ranking function with a concave activation function induces games in which no-regret learning dynamics converge. Moreover, for proportional ranking functions, we prove the equivalence of the concavity of the activation function, the social concavity of the induced games and the concavity of the induced games. We also study the empirical trade-offs between publishers' and users' welfare, under different choices of the activation function, using a state-of-the-art no-regret dynamics algorithm. Furthermore, we demonstrate how the choice of the ranking function and changes in the ecosystem structure affect these welfare measures, as well as the dynamics' convergence rate.
Paper Structure (27 sections, 3 theorems, 24 equations, 7 figures)

This paper contains 27 sections, 3 theorems, 24 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Related work
  4. Preliminaries
  5. Game theory
  6. No-regret dynamics
  7. Concave games
  8. The model
  9. Main result
  10. Experimental comparison of ranking functions
  11. Simulation details
  12. Evaluation criteria
  13. Simulations results
  14. Comparison between ranking function families
  15. The effect of the penalty factor $\lambda$
  16. ...and 12 more sections

Key Result

Lemma 1

A ranking function $r$ induces socially-concave games if and only if for every information need $x^*$, for every publisher $i \in N$ and for every $x_i \in X_i$, $r_i(x_i, x_{-i}; x^*)$ is convex in $x_{-i}$.

Figures (7)

  • Figure 1: The effect of the penalty factor $\lambda$, with $n=3, s=3$ and $k=3$.
  • Figure 2: The effect of the number of publishers $n$, with $\lambda = 0.5$, $s=3$ and $k=3$.
  • Figure 3: The effect of the demand distribution support size $s$, with $\lambda = 0.5, n=3$ and $k=3$.
  • Figure 4: The effect of the embedding space dimension $k$, with $\lambda = 0.5$, $n=3$ and $s=3$.
  • Figure 5: The effect of the intercept $b$ in the linear proportional ranking function, with $\lambda=0.5$, $n=3, s=3$ and $k=3$.
  • ...and 2 more figures

Theorems & Definitions (11)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Definition 3
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • proof
  • proof
  • ...and 1 more