Learning in Proportional Allocation Auctions Games

TL;DR

This paper analyzes the repeated Kelly auction game with logarithmic utilities, proving convergence to Nash equilibrium under various learning dynamics and validating findings through extensive simulations.

Contribution

It introduces a logarithmic utility model derived from wireless network fairness, proves convergence to NE under multiple learning algorithms, and compares their performance via simulations.

Findings

Convergence to NE under OGD, DAQ, and BR algorithms.

BR achieves fastest convergence and highest utility.

Heterogeneous update rules may prevent convergence.

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.