Win-score promotion gates in aggregator-routed RFQ markets: A two-tier stochastic control model

Abstract

We study market making in aggregator-routed RFQ markets where platform routing depends on slowly varying dealer performance scores. We propose a two-tier stochastic control model that separates RFQ-level price competition from a macro routing layer: tier A represents aggregator flow whose opportunity intensity is multiplied by a promotion gate driven by the dealer's win score, while tier B captures background flow that is not gated and does not update the score. RFQs arrive in multiple sizes and the dealer chooses a size-ladder of bid/ask offsets; conditional on winning, trades earn spread minus an adverse selection correction and contribute to inventory risk. The resulting Hamilton-Jacobi-Bellman equation admits a reduced Bergault-Guéant operator form with explicit win/lose branches for the score on tier A. Using the envelope-theorem argument, we express optimal controls through derivatives of the one-dimensional reduced Hamiltonians, yielding an interpretable mapping from optimal win probabilities to optimal offsets. In the long-memory regime, we derive an adiabatic approximation that separates fast inventory dynamics from slow score dynamics. A quadratic inventory ansatz and quadratic Hamiltonian expansion lead to a quasi-stationarity inventory-curvature scaling and a one-dimensional score drift field. For steep (logistic) promotion gates, the score dynamics can exhibit fold bifurcations, bistability, and hysteresis, producing an endogenous "campaign vs. harvest" pattern in optimal quoting. Numerical experiments confirm this behaviour and highlight the stabilizing role of background flow in maintaining inventory-mixing capacity even when the dealer is weakly promoted.

Figures (8)

• Figure 1: Heatmap of the stationary value function $v(0, q, R)$ for a default parameter set defined in the text. Horizontal dotted line corresponds to $R_0$.
• Figure 2: Tier A optimal quote offset $\hat{\delta}_z^A$ for $z = 10$ at zero inventory as a function of score $R$ for $\alpha = 0.01$ (feedback) and $\alpha =0$ (no feedback). Other parameters are defined in the text. Due to bid/ask parameter symmetry, side index is omitted. Vertical dotted line corresponds to $R_0$.
• Figure 3: Tier A optimal quote offset $\hat{\delta}_z^A$ for $z = 10$ as a function of inventory $q$ for several values of score $R$. Parameters are defined in the text. Due to bid/ask parameter symmetry, side index is omitted.
• Figure 4: Logistic gate $G(R)$ (dashed line) and instant PnL of the gated tier at zero inventory $\Pi_A(0, R)$ (solid line) as functions of score $R$. Parameters are defined in the text. Vertical dotted line corresponds to $R_0$.
• Figure 5: Instant PnL of the gated tier at zero inventory $\Pi_A(0, R)$ as a function of score drift $\dot R$ color coded by the value of $R$. Parameters are defined in the text. Larger circle on the curve corresponds to $R_0$. Horizontal dotted line corresponds to $\Pi_A(0, 0)$.