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Robustness of Online Proportional Response in Stochastic Online Fisher Markets: a Decentralized Approach

Yongge Yang, Yu-Ching Lee, Po-An Chen, Chuang-Chieh Lin

TL;DR

This work addresses online Fisher markets where item values fluctuate stochastically and buyers must decide bids without revealing private valuations. It introduces online proportional response, a decentralized bidding rule that aligns with online mirror descent when analyzed through the Shmyrev convex program, yielding a closed-form bid update and guaranteed performance. The paper proves regret bounds under stationary noise ($ ext{Fairness regret} \\[\mathcal{F}\] \\le \\log MN$; individual regret \\[\mathcal{R}_i\] = O(\\sqrt{T})$) and extends to several non-stationary models (independent corruptions, ergodic, periodic) with explicit bounds, demonstrating robustness. The approach preserves buyer privacy, ensures continuous market clearance, and generalizes across scenarios, offering practical decentralized solutions for dynamic, real-world markets.

Abstract

This study is focused on periodic Fisher markets where items with time-dependent and stochastic values are regularly replenished and buyers aim to maximize their utilities by spending budgets on these items. Traditional approaches of finding a market equilibrium in the single-period Fisher market rely on complete information about buyers' utility functions and budgets. However, it is impractical to consistently enforce buyers to disclose this private information in a periodic setting. We introduce a distributed auction algorithm, online proportional response, wherein buyers update bids solely based on the randomly fluctuating values of items in each period. The market then allocates items based on the bids provided by the buyers. Utilizing the known Shmyrev convex program that characterizes market equilibrium of a Fisher market, two performance metrics are proposed: the fairness regret is the cumulative difference in the objective value of a stochastic Shmyrev convex program between an online algorithm and an offline optimum, and the individual buyer's regret gauges the deviation in terms of utility for each buyer between the online algorithm and the offline optimum. Our algorithm attains a problem-dependent upper bound contingent on the number of items and buyers under stationary inputs in fairness regret. Additionally, we conduct analysis of regret under various non-stationary stochastic input models to demonstrate the algorithm's efficiency across diverse scenarios. The online proportional response algorithm addresses privacy concerns by allowing buyers to update bids without revealing sensitive information and ensures decentralized decision-making, fostering autonomy and potential improvements in buyer satisfaction. Furthermore, our algorithm is universally applicable to many worlds and shows the robust performance guarantees.

Robustness of Online Proportional Response in Stochastic Online Fisher Markets: a Decentralized Approach

TL;DR

This work addresses online Fisher markets where item values fluctuate stochastically and buyers must decide bids without revealing private valuations. It introduces online proportional response, a decentralized bidding rule that aligns with online mirror descent when analyzed through the Shmyrev convex program, yielding a closed-form bid update and guaranteed performance. The paper proves regret bounds under stationary noise (; individual regret \ = O(\\sqrt{T})$) and extends to several non-stationary models (independent corruptions, ergodic, periodic) with explicit bounds, demonstrating robustness. The approach preserves buyer privacy, ensures continuous market clearance, and generalizes across scenarios, offering practical decentralized solutions for dynamic, real-world markets.

Abstract

This study is focused on periodic Fisher markets where items with time-dependent and stochastic values are regularly replenished and buyers aim to maximize their utilities by spending budgets on these items. Traditional approaches of finding a market equilibrium in the single-period Fisher market rely on complete information about buyers' utility functions and budgets. However, it is impractical to consistently enforce buyers to disclose this private information in a periodic setting. We introduce a distributed auction algorithm, online proportional response, wherein buyers update bids solely based on the randomly fluctuating values of items in each period. The market then allocates items based on the bids provided by the buyers. Utilizing the known Shmyrev convex program that characterizes market equilibrium of a Fisher market, two performance metrics are proposed: the fairness regret is the cumulative difference in the objective value of a stochastic Shmyrev convex program between an online algorithm and an offline optimum, and the individual buyer's regret gauges the deviation in terms of utility for each buyer between the online algorithm and the offline optimum. Our algorithm attains a problem-dependent upper bound contingent on the number of items and buyers under stationary inputs in fairness regret. Additionally, we conduct analysis of regret under various non-stationary stochastic input models to demonstrate the algorithm's efficiency across diverse scenarios. The online proportional response algorithm addresses privacy concerns by allowing buyers to update bids without revealing sensitive information and ensures decentralized decision-making, fostering autonomy and potential improvements in buyer satisfaction. Furthermore, our algorithm is universally applicable to many worlds and shows the robust performance guarantees.
Paper Structure (41 sections, 15 theorems, 148 equations, 3 figures, 1 algorithm)

This paper contains 41 sections, 15 theorems, 148 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main Contributions
  3. Related Work
  4. Distributed algorithms for Fisher markets.
  5. Online Fisher markets.
  6. Online resource allocation problems.
  7. Fair allocation.
  8. Online learning algorithms.
  9. Notations
  10. Preliminaries
  11. Static (Offline) Linear Fisher Markets
  12. Online Linear Fisher Markets
  13. Proportional Response for Online Fisher Markets
  14. Warm Up: Mirror Descent
  15. Online Proportional Response
  16. ...and 26 more sections

Key Result

Lemma 3.1

Let $h(\mathbf{p})=\sum_{j \in \mathcal{M}}(p_j\log p_j -p_j)$, and its Bregman divergence is $D_h(\mathbf{p}, \mathbf{q})=\sum_{j \in \mathcal{M}}p_{j}\log(\frac{p_j}{q_j})$, then for any $\mathbf{b}_t,\mathbf{b}_{t-1} \in K$, it has that $\varphi(\mathbf{b}_t,\varepsilon_t)=\ell_\varphi(\mathbf{b}

Figures (3)

  • Figure 1: The fairness regret of the online proportional response (ALGORITHM \ref{['alg:one']}) under various inputs.
  • Figure 2: The log-log plots of individual buyer's regret of the online proportional response under various inputs ($N=20$, $M=30$).
  • Figure 3: The performance of bid updates with time-varying step-size

Theorems & Definitions (25)

  • Definition 2.1: Market equilibrium
  • Lemma 3.1
  • Definition 3.1: Relative smoothness
  • Proposition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Proposition 3.2
  • Theorem 4.1
  • Corollary 4.1
  • Proposition 4.1
  • ...and 15 more