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A Parametric Contextual Online Learning Theory of Brokerage

François Bachoc, Tommaso Cesari, Roberto Colomboni

TL;DR

Algorithms for the online learning problem of brokerage between traders are designed and optimal theoretical regret guarantees under various standard assumptions are proved.

Abstract

We study the role of contextual information in the online learning problem of brokerage between traders. In this sequential problem, at each time step, two traders arrive with secret valuations about an asset they wish to trade. The learner (a broker) suggests a trading (or brokerage) price based on contextual data about the asset and the market conditions. Then, the traders reveal their willingness to buy or sell based on whether their valuations are higher or lower than the brokerage price. A trade occurs if one of the two traders decides to buy and the other to sell, i.e., if the broker's proposed price falls between the smallest and the largest of their two valuations. We design algorithms for this problem and prove optimal theoretical regret guarantees under various standard assumptions.

A Parametric Contextual Online Learning Theory of Brokerage

TL;DR

Algorithms for the online learning problem of brokerage between traders are designed and optimal theoretical regret guarantees under various standard assumptions are proved.

Abstract

We study the role of contextual information in the online learning problem of brokerage between traders. In this sequential problem, at each time step, two traders arrive with secret valuations about an asset they wish to trade. The learner (a broker) suggests a trading (or brokerage) price based on contextual data about the asset and the market conditions. Then, the traders reveal their willingness to buy or sell based on whether their valuations are higher or lower than the brokerage price. A trade occurs if one of the two traders decides to buy and the other to sell, i.e., if the broker's proposed price falls between the smallest and the largest of their two valuations. We design algorithms for this problem and prove optimal theoretical regret guarantees under various standard assumptions.
Paper Structure (21 sections, 8 theorems, 44 equations, 1 table, 2 algorithms)

This paper contains 21 sections, 8 theorems, 44 equations, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Setting
  3. Online protocol.
  4. Market value.
  5. Contexts.
  6. Gain from trade and Regret.
  7. Challenges and contributions
  8. Related Works
  9. Structural and Technical Results
  10. Learning in Contextual Brokerage
  11. Full Feedback
  12. Beyond Bounded Densities
  13. Conclusions
  14. Missing Proofs of Structural and Technical Results (Section \ref{['s:structural']})
  15. Proof of Lemma \ref{['l:structural']}
  16. ...and 6 more sections

Key Result

Lemma 2.1

Suppose that $V$ and $W$ are two $[0,1]$-valued independent random variables, with possibly different densities bounded by some constant $L\ge 1$, and such that $\mathbb{E}[V] = \mathbb{E}[W] \eqqcolon m$. Then, for each $p \in[0,1]$, it holds that

Theorems & Definitions (12)

  • Lemma 2.1
  • Corollary 2.2
  • proof
  • Lemma 2.3
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • Theorem 4.1
  • Theorem 4.2
  • Example 5.1
  • ...and 2 more