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Online Resource Sharing: Better Robust Guarantees via Randomized Strategies

David X. Lin, Daniel Hall, Giannis Fikioris, Siddhartha Banerjee, Éva Tardos

TL;DR

The paper tackles robust online resource sharing without monetary transfers, modeling repeated first-price auctions with artificial currency and agent-specific fair shares. It introduces Randomized Robust Bidding (RRB), which bids from a uniform distribution when values fall in the top $\alpha$-quantile and where budget allows, achieving a robustness of $2-\sqrt{2}-O\left(\sqrt{\frac{\log T}{T}}\right)$ relative to each agent's ideal utility, nearly matching the known $1-1/e$ upper bound. A key contribution is the reduction from general value distributions to Bernoulli cases, establishing Bernoulli as the worst-case for robustness; upper bounds show that fixed bidding cannot surpass $3/5$ robustness while an adaptive adversary can cap robustness at $1-1/e+\alpha/e+O\left(\sqrt{\log T/T}\right)$. Experiments confirm that all agents using RRB achieve near-optimal fractions of ideal utility, especially approaching $1-1/e$ as the number of agents grows, highlighting the practical potential of randomized strategies for fair online resource sharing.

Abstract

We study the problem of fair online resource allocation via non-monetary mechanisms, where multiple agents repeatedly share a resource without monetary transfers. Previous work has shown that every agent can guarantee $1/2$ of their ideal utility (the highest achievable utility given their fair share of resources) robustly, i.e., under arbitrary behavior by the other agents. While this $1/2$-robustness guarantee has now been established under very different mechanisms, including pseudo-markets and dynamic max-min allocation, improving on it has appeared difficult. In this work, we obtain the first significant improvement on the robustness of online resource sharing. In more detail, we consider the widely-studied repeated first-price auction with artificial currencies. Our main contribution is to show that a simple randomized bidding strategy can guarantee each agent a $2 - \sqrt 2 \approx 0.59$ fraction of her ideal utility, irrespective of others' bids. Specifically, our strategy requires each agent with fair share $α$ to use a uniformly distributed bid whenever her value is in the top $α$-quantile of her value distribution. Our work almost closes the gap to the known $1 - 1/e \approx 0.63$ hardness for robust resource sharing; we also show that any static (i.e., budget independent) bidding policy cannot guarantee more than a $0.6$-fraction of the ideal utility, showing our technique is almost tight.

Online Resource Sharing: Better Robust Guarantees via Randomized Strategies

TL;DR

The paper tackles robust online resource sharing without monetary transfers, modeling repeated first-price auctions with artificial currency and agent-specific fair shares. It introduces Randomized Robust Bidding (RRB), which bids from a uniform distribution when values fall in the top -quantile and where budget allows, achieving a robustness of relative to each agent's ideal utility, nearly matching the known upper bound. A key contribution is the reduction from general value distributions to Bernoulli cases, establishing Bernoulli as the worst-case for robustness; upper bounds show that fixed bidding cannot surpass robustness while an adaptive adversary can cap robustness at . Experiments confirm that all agents using RRB achieve near-optimal fractions of ideal utility, especially approaching as the number of agents grows, highlighting the practical potential of randomized strategies for fair online resource sharing.

Abstract

We study the problem of fair online resource allocation via non-monetary mechanisms, where multiple agents repeatedly share a resource without monetary transfers. Previous work has shown that every agent can guarantee of their ideal utility (the highest achievable utility given their fair share of resources) robustly, i.e., under arbitrary behavior by the other agents. While this -robustness guarantee has now been established under very different mechanisms, including pseudo-markets and dynamic max-min allocation, improving on it has appeared difficult. In this work, we obtain the first significant improvement on the robustness of online resource sharing. In more detail, we consider the widely-studied repeated first-price auction with artificial currencies. Our main contribution is to show that a simple randomized bidding strategy can guarantee each agent a fraction of her ideal utility, irrespective of others' bids. Specifically, our strategy requires each agent with fair share to use a uniformly distributed bid whenever her value is in the top -quantile of her value distribution. Our work almost closes the gap to the known hardness for robust resource sharing; we also show that any static (i.e., budget independent) bidding policy cannot guarantee more than a -fraction of the ideal utility, showing our technique is almost tight.

Paper Structure

This paper contains 19 sections, 12 theorems, 84 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our results
  3. Related Work
  4. Preliminaries
  5. Model and Ideal Utility
  6. Robust Strategies, Equilibria and Price of Anarchy
  7. Pseudo-market mechanism
  8. Formulation as a two-player zero-sum game
  9. Robust Strategy Lower Bound of 0.59
  10. Reduction to Bernoulli value distributions
  11. The Randomized Robust Bidding strategy
  12. Robust Strategy Upper Bounds
  13. Fixed bidding distribution 3/5 upper bound
  14. Constructive proof for an upper bound of 1 - 1/e
  15. Experimental Evaluation of Randomized Robust Bidding
  16. ...and 4 more sections

Key Result

Lemma 1

Fix an arbitrary mechanism and an agent with fair share $\alpha$. Assume there is a $\beta$-robust policy $\hat{\pi}$ for that agent when she has a $\hat{\mathcal{F}} = \mathrm{Bernoulli}(\alpha)$ value distribution. Then for any value distribution $\mathcal{F}$, we can construct a $\beta$-robust po

Figures (2)

  • Figure 1: CDF of the adversary's bid distribution used in \ref{['thm:arbitrarystrategyupperbound']} when $\alpha\to 0$. The expected bid under this distribution is $1$, so the adversary will not run out of budget. The CDF is carefully chosen such that the agent cannot win more than $1-1/e$ fraction of the rounds regardless of strategy.
  • Figure 2: Fraction of ideal utility that an agent obtains under differing strategy profiles. We compare the agents' utility when they all use the previously best-known robust strategy from gorokh2019remarkable, labeled Deterministic Robust Bidding, with the agents' utility when they all use \ref{['str:RRB']}. We also plot an agent's utility when they use \ref{['str:RRB']} but the other agents adversarially always bid $1$ regardless of their values, labeled Randomized Robust Bidding against Adversary. When all agents use \ref{['str:RRB']}, they achieve $\approx1 - (1-1/n)^n$ fraction of their ideal utility, the theoretical maximum for any allocation procedure. When one agent uses \ref{['str:RRB']} but the other agents behave adversarially, the agent using \ref{['str:RRB']} achieves at least a $2-\sqrt2$ fraction, the guarantee of \ref{['thm:lowerBound']}.

Theorems & Definitions (20)

  • Definition 1
  • Lemma 1
  • proof : Proof of Lemma \ref{['lem:bernoullivaluedistributionisworstcase']}
  • Theorem 1
  • Theorem 2
  • Lemma 2
  • Theorem 3
  • Theorem 3
  • proof : Proof of \ref{['thm:lowerBound']}
  • Theorem 3
  • ...and 10 more