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Learning in Proportional Allocation Auctions Games

Younes Ben Mazziane, Cleque-Marlain Mboulou Moutoubi, Eitan Altman, Francesco De Pellegrini

Abstract

The Kelly or proportional allocation mechanism is a simple and efficient auction-based scheme that distributes an infinitely divisible resource proportionally to the agents bids. When agents are aware of the allocation rule, their interactions form a game extensively studied in the literature. This paper examines the less explored repeated Kelly game, focusing mainly on utilities that are logarithmic in the allocated resource fraction. We first derive this logarithmic form from fairness-throughput trade-offs in wireless network slicing, and then prove that the induced stage game admits a unique Nash equilibrium NE. For the repeated play, we prove convergence to this NE under three behavioral models: (i) all agents use Online Gradient Descent (OGD), (ii) all agents use Dual Averaging with a quadratic regularizer (DAQ) (a variant of the Follow-the-Regularized leader algorithm), and (iii) all agents play myopic best responses (BR). Our convergence results hold even when agents use personalized learning rates in OGD and DAQ (e.g., tuned to optimize individual regret bounds), and they extend to a broader class of utilities that meet a certain sufficient condition. Finally, we complement our theoretical results with extensive simulations of the repeated Kelly game under several behavioral models, comparing them in terms of convergence speed to the NE, and per-agent time-average utility. The results suggest that BR achieves the fastest convergence and the highest time-average utility, and that convergence to the stage-game NE may fail under heterogeneous update rules.

Learning in Proportional Allocation Auctions Games

Abstract

The Kelly or proportional allocation mechanism is a simple and efficient auction-based scheme that distributes an infinitely divisible resource proportionally to the agents bids. When agents are aware of the allocation rule, their interactions form a game extensively studied in the literature. This paper examines the less explored repeated Kelly game, focusing mainly on utilities that are logarithmic in the allocated resource fraction. We first derive this logarithmic form from fairness-throughput trade-offs in wireless network slicing, and then prove that the induced stage game admits a unique Nash equilibrium NE. For the repeated play, we prove convergence to this NE under three behavioral models: (i) all agents use Online Gradient Descent (OGD), (ii) all agents use Dual Averaging with a quadratic regularizer (DAQ) (a variant of the Follow-the-Regularized leader algorithm), and (iii) all agents play myopic best responses (BR). Our convergence results hold even when agents use personalized learning rates in OGD and DAQ (e.g., tuned to optimize individual regret bounds), and they extend to a broader class of utilities that meet a certain sufficient condition. Finally, we complement our theoretical results with extensive simulations of the repeated Kelly game under several behavioral models, comparing them in terms of convergence speed to the NE, and per-agent time-average utility. The results suggest that BR achieves the fastest convergence and the highest time-average utility, and that convergence to the stage-game NE may fail under heterogeneous update rules.

Paper Structure

This paper contains 16 sections, 8 theorems, 44 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Contributions
  3. Paper outline
  4. Problem Formulation
  5. Repeated Kelly Game
  6. Motivation: Bandwidth allocation in wireless networks
  7. Single-Agent Formulation of the Online Bidding Problem
  8. Bidding algorithms
  9. Convergence guarantees
  10. Numerical Simulations
  11. Homogeneous dynamics
  12. Heterogeneous dynamics
  13. Conclusion
  14. Supplementary Material
  15. Proof of Theorem \ref{['thm:SDSC_log']}
  16. ...and 1 more sections

Key Result

Theorem 1

The set of Nash equillibria of $\mathcal{G}$, denoted $NE(\mathcal{G})$ is non-empty, i.e., $NE(\mathcal{G})\neq \emptyset$.

Figures (7)

  • Figure 1: Repeated resource allocation game.
  • Figure 2: Bandwidth Allocation between tenants and users
  • Figure 3: Convergence speed under homogeneous dynamics and varying payoff's heterogeneity levels.
  • Figure 4: Agent's payoff, $\gamma=0$.
  • Figure 5: Heterogeneous dynamics. In each sub-figure: instantaneous payoff (top) and bids (bottom).
  • ...and 2 more figures

Theorems & Definitions (12)

  • Definition 1: Nash Equilibrium
  • Theorem 1
  • Remark 1
  • Remark 2
  • Definition 2: Strong Diagonal Strict Concavity
  • Theorem 2
  • Corollary 1
  • Theorem 3: Convergence of OGD
  • Theorem 4: Convergence of RMQ and DAQ
  • Theorem 5
  • ...and 2 more