A Parametric Contextual Online Learning Theory of Brokerage

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.

💡 Research Summary

The paper introduces a contextual online learning formulation for brokerage in over‑the‑counter (OTC) markets. At each round t, two traders arrive with private valuations Vₜ and Wₜ for an asset. The broker observes a d‑dimensional context vector cₜ (capturing asset‑specific and market‑wide information) and posts a price Pₜ. After the price is posted, the broker receives only two bits of feedback: whether each trader is willing to trade at that price (i.e., the indicators I{Pₜ ≤ Vₜ} and I{Pₜ ≤ Wₜ}). A trade occurs if the price lies between the lower and higher valuation, yielding a gain‑from‑trade (GFT) equal to |Vₜ − Wₜ|; otherwise the GFT is zero.

The authors assume that the hidden market value mₜ is a linear function of the context, mₜ = cₜᵀφ, where φ∈