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

This paper proposes a two-tier stochastic control model for market making in aggregator-routed RFQ markets that links dealer quoting strategies to slowly varying win-score promotion gates, revealing how the interplay between fast inventory dynamics and slow score evolution can lead to bistability, hysteresis, and endogenous "campaign vs. harvest" quoting patterns.

The Gist

Imagine you are a high-stakes food truck owner operating in a busy city square. You have two types of customers, and your business strategy depends on a mysterious "Popularity Score" that the city council uses to decide who gets to serve customers.

This paper is a mathematical guide for how you should price your food (your "quotes") to maximize your profit, while keeping in mind that your pricing affects your Popularity Score, which in turn affects how many customers you see tomorrow.

Here is the breakdown of the paper's complex ideas using simple analogies:

1. The Two Types of Customers (The Two Tiers)

The paper splits your customers into two groups:

• Tier A (The VIPs): These are customers who only come to food trucks that are on the "City Council's Top List."
  • The Catch: To stay on this list, you need a high Popularity Score. This score is based on how often you win the "best price" vote against other trucks.
  • The Gate: If your score is high, the Council sends you many VIP customers. If your score drops, you get almost none.
• Tier B (The Locals): These are customers who wander by and buy from anyone, regardless of your score.
  • The Benefit: They don't care about your score, and your score doesn't change when you serve them. They are your "safety net."

The Problem: You want to win the VIPs (Tier A) because they are more profitable, but to win them, you have to offer very cheap prices. Offering cheap prices hurts your immediate profit margin. If you get too greedy and charge too much, you lose the VIPs, your score drops, and you lose access to the VIP list entirely.

2. The "Campaign vs. Harvest" Strategy

The paper discovers that smart food truck owners naturally fall into two distinct modes, depending on their current Popularity Score:

• Mode 1: The Campaign (The Grind)
  • When: Your score is just below the "promotion line" (the threshold to get on the Top List).
  • Action: You lower your prices significantly, even if it means making less money per burger right now.
  • Why: You are "campaigning" to boost your score. You are sacrificing short-term profit to buy a ticket to the VIP list. It's like a politician spending money on ads to get elected, hoping for a bigger salary later.
• Mode 2: The Harvest (The Cash Out)
  • When: Your score is safely high on the Top List.
  • Action: You raise your prices. You make more profit per burger, even though you might lose a few customers.
  • Why: You are "harvesting" the rewards of your hard work. You know you are safe on the list for now, so you milk the profits.

The Twist (Hysteresis): The paper shows that this isn't a smooth line. It's like a light switch.

• If you are just below the line, you must campaign aggressively to jump up.
• Once you are just above the line, you can relax and harvest.
• But if you get lazy and your score drops just a tiny bit below the line, you might fall all the way down to a "bad zone" where it's impossible to climb back up without a massive effort. This is called bistability: you can get stuck in a "low score" trap or a "high score" paradise, and it's hard to switch between them.

3. The "Safety Net" (Tier B)

Why doesn't the business collapse when the score is low?

• Tier B (Locals) keeps the truck moving. Even if you lose the VIPs, you still have locals to serve.
• This prevents the "inventory risk" (having too much or too little food) from becoming a disaster.
• The Metaphor: Think of Tier B as a backup generator. When the main power (Tier A VIPs) is cut off because your score is low, the backup generator keeps the lights on so you don't go completely bankrupt. This allows you to survive the "campaigning" phase without going out of business.

4. The "Memory" Factor

The Popularity Score doesn't update instantly. It's like a slow-moving average.

• If you win one VIP customer today, your score goes up a tiny bit.
• If you lose one, it goes down a tiny bit.
• Because it moves slowly, you can't panic. You have time to plan. The paper uses math to show that because the score moves slowly, you can treat your daily pricing (fast) and your long-term reputation (slow) as two separate problems that influence each other.

5. The "Adverse Selection" (The Hidden Trap)

The paper also mentions that sometimes, the customers who really want your food are the ones who know the market is about to crash.

• The Metaphor: Imagine a customer who knows the price of tomatoes is about to drop. They rush to buy your tomatoes at today's high price. If you sell to them, you lose money later.
• The model accounts for this by saying: "If I win a customer, I might be getting a 'bad' customer." So, you have to adjust your prices to protect yourself, which makes the "Campaign vs. Harvest" strategy even more delicate.

Summary: What is the Big Idea?

This paper proves that in markets where your future opportunities depend on your past performance (like a "win score"), the smartest strategy isn't just to maximize profit today.

Instead, you must play a two-level game:

1. Short-term: Manage your inventory and risk.
2. Long-term: Decide whether to sacrifice today's profit to climb the ladder (Campaign) or enjoy the view from the top (Harvest).

The math shows that this creates a "tipping point" effect. Once you cross a certain score threshold, your behavior changes completely. And having a steady stream of "non-scored" customers (Tier B) is crucial to keep you from falling off a cliff if you stumble.

In one sentence: To win in a rigged game where your past performance dictates your future opportunities, you must sometimes lose money today to buy a better seat for tomorrow, but you need a safety net to ensure you don't starve while you wait for the promotion.

Technical Summary

Here is a detailed technical summary of the paper "Win-score promotion gates in aggregator-routed RFQ markets: A two-tier stochastic control model" by Alexander Barzykin.

1. Problem Statement

The paper addresses the challenge of market making in aggregator-routed Request-for-Quote (RFQ) markets, particularly in Foreign Exchange (FX). In these markets:

• Micro-level: Dealers compete on price (bid/ask offsets) for specific RFQs of varying sizes.
• Macro-level: Aggregators route RFQs to a subset of dealers based on a performance score (e.g., win ratio).
• The Feedback Loop: A dealer's aggressive quoting improves their win score, which increases their probability of being invited to future RFQs (the "promotion gate"). Conversely, losing or quoting passively lowers the score, reducing future flow.

The Core Gap: Existing literature treats either micro-level inventory management or macro-level routing separately. There is a lack of stochastic control models that explicitly treat the slowly varying win score as a state variable that feeds back into the intensity of future trading opportunities, creating a dynamic trade-off between immediate profit ("harvesting") and future access ("campaigning").

2. Methodology

The author proposes a two-tier stochastic control model to separate the problem into distinct time scales and flow types.

A. Model Structure

1. Two-Tier Flow Decomposition:
   • Tier A (Aggregator Flow): Opportunities arrive at an intensity $\lambda^A_z(R) = \Lambda^A_z G(R)$, where $G(R)$ is a "gate" function dependent on the dealer's score $R$. Winning or losing updates the score $R$.
   • Tier B (Background Flow): Residual flow arrives at a constant intensity $\Lambda^B_z$. It does not update the score $R$ and is not gated. This tier acts as a stabilizer for inventory risk.
2. State Variables:
   • $q_t$: Inventory.
   • $R_t$: Long-term win score (modeled as an Exponential Moving Average of wins).
   • $S_t$: Mid-price (martingale diffusion).
3. Dynamics:
   • Score Update: $R \to (1-\alpha)R + \alpha w$, where $w \in \{0,1\}$ is the win indicator for Tier A. $\alpha$ is small, implying slow score dynamics relative to inventory.
   • Objective: Maximize expected terminal wealth minus quadratic inventory penalties (risk aversion) and adverse selection costs.

B. Mathematical Framework

• Hamilton-Jacobi-Bellman (HJB) Equation: The problem is formulated as a stochastic control problem leading to a reduced HJB equation.
• Bergault-Gu'eant Operator Form: The HJB is reduced to a form separating "win" and "lose" branches for Tier A.
  • Tier A involves a mixture of trading (moving to new inventory/score) and losing (only score downgrades).
  • Tier B involves standard jump operators without score updates.
• Envelope Theorem Application: The author derives a mapping where optimal bid/ask offsets are determined by the derivatives of a one-dimensional reduced Hamiltonian. This provides an explicit inverse mapping from optimal win probabilities to optimal price offsets.

C. Adiabatic Approximation (Long-Memory Regime)

Assuming $\alpha$ is small (slow score dynamics), the model is decoupled:

1. Fast Scale (Inventory): Inventory dynamics are solved conditionally on a quasi-static score $R$. A quadratic ansatz ($v \approx -\frac{1}{2}A(R)q^2 + B(R)$) is used.
2. Slow Scale (Score): The score evolves according to a drift field driven by the equilibrium win probability of the fast scale.
3. Riccati Scaling: The inventory curvature $A(R)$ is shown to scale inversely with the square root of total flow intensity. Crucially, the presence of Tier B prevents the curvature from blowing up when Tier A flow is gated off.

3. Key Contributions

1. Two-Tier Stochastic Control Model: Introduces a novel framework separating gated, score-updating flow from non-gated background flow, capturing the realistic structure of FX aggregator platforms.
2. Reduced HJB with Explicit Branching: Derives a tractable HJB equation with explicit win/lose branches for the score variable, solvable via envelope theorem arguments to yield interpretable controls.
3. Adiabatic Closure: Demonstrates that in the long-memory limit, the complex high-dimensional problem reduces to a one-dimensional score drift field coupled with a quasi-stationary inventory curvature.
4. Identification of Bistability and Hysteresis: Shows that under steep (logistic) promotion gates, the system exhibits fold bifurcations. This leads to:
   • Bistability: Two stable equilibrium states for the score (low vs. high).
   • Hysteresis: The path to the high-score state differs from the path to the low-score state.
5. "Campaign vs. Harvest" Strategy: Theoretically derives an optimal policy where dealers intentionally sacrifice immediate PnL (tighter quotes) to "campaign" for a higher score, or widen quotes to "harvest" profits once a high score is secured, even if it risks score decay.

4. Results

Numerical Findings

• Value Function Heatmaps: The stationary value function shows clear gradients where the dealer is willing to accept lower immediate edge to improve the score $R$ near a critical threshold ($R_0$).
• Quote Offsets:
  • Near the promotion threshold ($R_0$), optimal offsets for Tier A dip significantly (tighter spreads) to maximize win probability ("Campaigning").
  • Above $R_0$, offsets widen ("Harvesting"), but not fully to the no-feedback level, as the dealer must maintain the score.
• Instantaneous PnL: PnL drops during the "campaign" phase (low score, high effort) and spikes during the "harvest" phase (high score, wide spreads).
• Phase Portraits:
  • Without Feedback ($\alpha=0$): A single stable fixed point exists.
  • With Feedback: A S-shaped curve emerges in the score drift. For steep gates, there are three fixed points (two stable, one unstable).
  • Bistability: If the score starts below a lower fold point, the dealer cannot profitably campaign to reach the high state. If above the upper fold, the dealer can harvest safely. Between the folds, the outcome depends on initial conditions and noise.

Role of Tier B (Background Flow)

The background flow is critical for stability. Without it, if the score drops and the gate $G(R)$ closes, the inventory curvature would become singular (infinite risk). Tier B ensures a baseline of inventory mixing, keeping the curvature bounded and allowing the dealer to survive periods of low promotion.

5. Significance

• Practical Relevance: The model explains observed behaviors in FX markets where dealers aggressively quote to maintain "top list" status on aggregators, even when it reduces immediate margins. It quantifies the "cost of maintenance" for a high score.
• Strategic Insight: It provides a mathematical basis for the "Campaign vs. Harvest" cycle, suggesting that optimal quoting is not static but regime-dependent.
• Risk Management: The inclusion of the background tier highlights that "residual" client flow is not just noise; it is a vital risk management tool that prevents the system from collapsing when a dealer loses their ranking.
• Policy Implications: For platform designers, the model suggests that very steep promotion gates (high $\beta$) can lead to hysteresis and lock-in effects, where dealers get stuck in low-performance states or high-performance states regardless of marginal improvements, potentially reducing overall market efficiency.

In summary, the paper successfully bridges microstructure theory (inventory control) with platform economics (routing algorithms), revealing that optimal market making in aggregator markets is a dynamic game of reputation management characterized by non-linear phase transitions.