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Strategic Multi-Armed Bandit Problems Under Debt-Free Reporting

Ahmed Ben Yahmed, Clément Calauzènes, Vianney Perchet

TL;DR

This work addresses strategic multi-armed bandit problems under debt-free reporting, where arms may withhold part of observed rewards to maximize private utilities. It proposes an incentive-aware algorithm, S-SE, that combines mechanism design with adaptive elimination to induce a dominant-strategy SPE in which arms report truthfully. The learner achieves near-classical regret levels, with problem-dependent bounds of $O(\log(T)/\Delta)$ and a worst-case bound of $O(\sqrt{KT\log(T)})$, while ensuring the second-best arm's reward as the revenue benchmark $\mu_2 T$ up to sublinear terms. The results demonstrate that strategically designed bonuses, together with elimination dynamics, can sustain truthful reporting and enable effective learning in adversarial economic environments, with practical implications for revenue management and ad allocation where budget-balance constraints apply.

Abstract

We consider the classical multi-armed bandit problem, but with strategic arms. In this context, each arm is characterized by a bounded support reward distribution and strategically aims to maximize its own utility by potentially retaining a portion of its reward, and disclosing only a fraction of it to the learning agent. This scenario unfolds as a game over $T$ rounds, leading to a competition of objectives between the learning agent, aiming to minimize their regret, and the arms, motivated by the desire to maximize their individual utilities. To address these dynamics, we introduce a new mechanism that establishes an equilibrium wherein each arm behaves truthfully and discloses as much of its rewards as possible. With this mechanism, the agent can attain the second-highest average (true) reward among arms, with a cumulative regret bounded by $O(\log(T)/Δ)$ (problem-dependent) or $O(\sqrt{T\log(T)})$ (worst-case).

Strategic Multi-Armed Bandit Problems Under Debt-Free Reporting

TL;DR

This work addresses strategic multi-armed bandit problems under debt-free reporting, where arms may withhold part of observed rewards to maximize private utilities. It proposes an incentive-aware algorithm, S-SE, that combines mechanism design with adaptive elimination to induce a dominant-strategy SPE in which arms report truthfully. The learner achieves near-classical regret levels, with problem-dependent bounds of and a worst-case bound of , while ensuring the second-best arm's reward as the revenue benchmark up to sublinear terms. The results demonstrate that strategically designed bonuses, together with elimination dynamics, can sustain truthful reporting and enable effective learning in adversarial economic environments, with practical implications for revenue management and ad allocation where budget-balance constraints apply.

Abstract

We consider the classical multi-armed bandit problem, but with strategic arms. In this context, each arm is characterized by a bounded support reward distribution and strategically aims to maximize its own utility by potentially retaining a portion of its reward, and disclosing only a fraction of it to the learning agent. This scenario unfolds as a game over rounds, leading to a competition of objectives between the learning agent, aiming to minimize their regret, and the arms, motivated by the desire to maximize their individual utilities. To address these dynamics, we introduce a new mechanism that establishes an equilibrium wherein each arm behaves truthfully and discloses as much of its rewards as possible. With this mechanism, the agent can attain the second-highest average (true) reward among arms, with a cumulative regret bounded by (problem-dependent) or (worst-case).

Paper Structure

This paper contains 24 sections, 11 theorems, 17 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Arm Utility and Subgame Perfect Equilibrium Among Arms
  4. Player's Strategic Regret
  5. Related Work
  6. Objectives and Challenges
  7. Contributions
  8. Dominant-Strategy SPE
  9. Regret Bound.
  10. Tightness of Regret.
  11. Utilities Bounds.
  12. Regret Analysis Across Arbitrary Strategy Profiles
  13. Sensitivity of Algorithm \ref{['Strategic ETC with eliminations']} around truthful SPE.
  14. Limitations and Future Work
  15. Conclusion
  16. ...and 9 more sections

Key Result

Theorem 3.1

Under Algorithm Strategic ETC with eliminations and using the bonus function in Definition bonusDef, for any arm $k$, any strategy $\boldsymbol{\pi}_{k}$, any strategy profile of other arms $\boldsymbol{\pi}_{-k}$, at any time $t$, and given any history $h_{k,t-1}$, the truthful reporting strategy $

Figures (2)

  • Figure 1: Cumulative regret under different strategies: 1. Under untruthful arbitrary reporting, where arms arbitrarily choose to keep a portion of their reward. 2. Under truthful reporting, where arms adhere to the dominant truthful SPE. 3. Under "optimal" reporting, where only the two best arms report truthfully and the remaining suboptimal arms withhold the entirety of their observed reward.
  • Figure 2: Comparison of arms' utilities between truthful reporting and arbitrary untruthful strategies

Theorems & Definitions (11)

  • Theorem 3.1: Incentive-Compatibility
  • Theorem 3.2: Regret Bound
  • Corollary 3.1: Utilities Upper Bound
  • Theorem 4.1
  • Theorem A.1: Hoeffding's Inequality
  • Theorem B.1
  • Theorem C.1: Incentive-Compatibility
  • Theorem D.1: Regret Bound
  • Corollary D.1: Utilities Upper Bound
  • Lemma E.1
  • ...and 1 more