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Feature-Based Online Bilateral Trade

Solenne Gaucher, Martino Bernasconi, Matteo Castiglioni, Andrea Celli, Vianney Perchet

TL;DR

This work introduces the feature-based online bilateral trade model, where a learner prices a seller and buyer based on item features. It provides a multi-regime analysis: a noiseless two-bit, strong-budget setting achieving $O( abla \log T)$ regret; a noisy two-bit setting attaining $\tilde O(T^{3/4})$ via Explore-Or-Commit and $\tilde O(T^{2/3})$ with Scouting Bandits and information pooling; and a reduction to one-bit feedback under global budget balance that preserves sublinear regret. The results reveal a fundamental trade-off between feedback richness and budget constraints, and they establish near-optimal context-dependent rates in this online pricing game. The methodology combines ellipsoidal uncertainty for deterministic valuations, decompositions of EGFT, and scalable learning subroutines to handle partial feedback. The work thus advances understanding of contextual pricing in online bilateral trade with limited feedback and dynamic budgets, with implications for practical market design under privacy and information constraints.

Abstract

Bilateral trade models the problem of facilitating trades between a seller and a buyer having private valuations for the item being sold. In the online version of the problem, the learner faces a new seller and buyer at each time step, and has to post a price for each of the two parties without any knowledge of their valuations. We consider a scenario where, at each time step, before posting prices the learner observes a context vector containing information about the features of the item for sale. The valuations of both the seller and the buyer follow an unknown linear function of the context. In this setting, the learner could leverage previous transactions in an attempt to estimate private valuations. We characterize the regret regimes of different settings, taking as a baseline the best context-dependent prices in hindsight. First, in the setting in which the learner has two-bit feedback and strong budget balance constraints, we propose an algorithm with $O(\log T)$ regret. Then, we study the same set-up with noisy valuations, providing a tight $\widetilde O(T^{\frac23})$ regret upper bound. Finally, we show that loosening budget balance constraints allows the learner to operate under more restrictive feedback. Specifically, we show how to address the one-bit, global budget balance setting through a reduction from the two-bit, strong budget balance setup. This established a fundamental trade-off between the quality of the feedback and the strictness of the budget constraints.

Feature-Based Online Bilateral Trade

TL;DR

This work introduces the feature-based online bilateral trade model, where a learner prices a seller and buyer based on item features. It provides a multi-regime analysis: a noiseless two-bit, strong-budget setting achieving regret; a noisy two-bit setting attaining via Explore-Or-Commit and with Scouting Bandits and information pooling; and a reduction to one-bit feedback under global budget balance that preserves sublinear regret. The results reveal a fundamental trade-off between feedback richness and budget constraints, and they establish near-optimal context-dependent rates in this online pricing game. The methodology combines ellipsoidal uncertainty for deterministic valuations, decompositions of EGFT, and scalable learning subroutines to handle partial feedback. The work thus advances understanding of contextual pricing in online bilateral trade with limited feedback and dynamic budgets, with implications for practical market design under privacy and information constraints.

Abstract

Bilateral trade models the problem of facilitating trades between a seller and a buyer having private valuations for the item being sold. In the online version of the problem, the learner faces a new seller and buyer at each time step, and has to post a price for each of the two parties without any knowledge of their valuations. We consider a scenario where, at each time step, before posting prices the learner observes a context vector containing information about the features of the item for sale. The valuations of both the seller and the buyer follow an unknown linear function of the context. In this setting, the learner could leverage previous transactions in an attempt to estimate private valuations. We characterize the regret regimes of different settings, taking as a baseline the best context-dependent prices in hindsight. First, in the setting in which the learner has two-bit feedback and strong budget balance constraints, we propose an algorithm with regret. Then, we study the same set-up with noisy valuations, providing a tight regret upper bound. Finally, we show that loosening budget balance constraints allows the learner to operate under more restrictive feedback. Specifically, we show how to address the one-bit, global budget balance setting through a reduction from the two-bit, strong budget balance setup. This established a fundamental trade-off between the quality of the feedback and the strictness of the budget constraints.
Paper Structure (31 sections, 34 theorems, 159 equations, 2 figures, 6 algorithms)

This paper contains 31 sections, 34 theorems, 159 equations, 2 figures, 6 algorithms.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Related works
  4. Preliminaries
  5. Warm-up: two-bit feedback, strong budget balance, no noise
  6. Noisy valuations with two-bit feedback
  7. Building blocks: subroutines for learning parameters
  8. Explore-Or-Commit framework
  9. Closing the gap: scouting bandits with information pooling
  10. Trading off budget balance constraints for feedback
  11. Basic facts about ellipsoids
  12. Proof of Theorem \ref{['thm:no_noise_2_bits']}
  13. Proof of \ref{['thm:etc']}
  14. Proof of Theorem \ref{['thm:noise_2_bits']}
  15. Detailed Algorithm
  16. ...and 16 more sections

Key Result

Theorem 1

Assume that the seller's and buyer's valuations follow eq:2bit_no_noise. Under ass:bounded_context, alg:EPBT with $\epsilon = ABd^2/T$ has regret bounded by $R_T \leq 10ABd^2\log\left(20(d+1)Td^{-2}\right).$

Figures (2)

  • Figure 1: Illustration for the c.d.f. $f^s$ and $f^b$ corresponding to the example described in Section \ref{['sec:example']}. The c.d.f. $f^s$ is represented in red, the c.d.f. $f^b$ is represented in blue.
  • Figure 2: Expected gain from trade for different values of increment $\delta^s$ and difference in average valuations $\Delta$.

Theorems & Definitions (52)

  • Theorem 1
  • Lemma 1
  • Theorem 2
  • Theorem 3
  • Remark 1
  • Lemma 2
  • Lemma 3
  • Theorem 4
  • Remark 2
  • Theorem 4
  • ...and 42 more