REFLEX: Reflexive Equilibrium Fixed-point Learning for Endogenous eXchanges
Authors: Vignesh Nagarajan, Shriraghav Ashok
Organizations: Texas A&M University Brown Foundation Scholar College Station, TX, USA · University of California, Berkeley Haas School of Business Berkeley, CA, USA
In over-the-counter corporate bond markets, dealers compete for client trades by quoting bid and ask prices. Tighter quotes attract more business, but also informed customers more likely to trade ahead of adverse price moves, leaving the dealer holding the risk. As dealers increasingly use machine learning to set quotes, they retrain these models on the trades their own quotes attract, creating a feedback loop in which each model reshapes the market that generates its next training data. The question is therefore not only whether a quoting model performs well, but whether the market it creates stays stable as the model learns from it. Existing performative prediction theory gives a sharp stability condition, yet expresses it through abstract properties of the learning objective a trading desk cannot measure before deployment. We introduce REFLEX, a framework that replaces those unobservable quantities with three measurable features of dealer behavior: how strongly trading volume responds to tighter quotes, how sharply the dealer's objective bends around its optimum, and how quickly informed flow increases as spreads narrow. REFLEX combines these into a single retraining modulus, a pre-deployment stability margin estimated from a desk's own quote and execution history that predicts whether repeated retraining will converge or amplify itself. In simulation, predicted and measured stability agree within 8%, and competing dealers increase instability by 1.74x with two and 3.16x with three, as predicted. Where ordinary retraining becomes unstable at modulus 1.21, a structurally anchored correction converges as blind retraining collapses. Calibrated over 36 years of public market data, stability headroom falls roughly 4.4x for investment grade and 4.3x for high yield from calm to crisis regimes. Ultimately, REFLEX turns an abstract convergence theorem into a market-level safety margin.
Figures & tables
Certificate family
raw
calib.
worst resid.
tol.
R1 analytic boundary
5/5
5/5
6.8×10−2
8×10−2
R2 PerfGD identities
4/4
4/4
1.3×10−6
2×10−2
R2 PerfGD demo dynamics
4/4
–
1.4×10−11
1×10−3
R3 multi-dealer
6/6
6/6
2.2×10−16
1×10−9
R4 robust boundary
7/7
7/7
3.2×10−2
1×10−1
R5 factor scaling
3/3
3/3
0.0
1×10−9
Table 1. Numerical proof certificates (raw + calibrated real-unit configurations). Each family’s worst residual sits inside its stated tolerance.
Figure 1. The daily market-fragility index, 1990–2026: the R1 closed forms evaluated on real data. Top: fragility(t)=median(ε∗)/ε∗(t) for IG (solid) and HY (dashed), with the Lehman and COVID-freeze dates marked. Bottom: the VIX spine, colored by volatility regime. The crisis plateau reflects the degenerate crisis-cell fit and is flagged, not hidden.
regime
εIG∗
εHY∗
mIG(hobs)
mHY(hobs)
calm
1207.7
93.5
0.847
0.592
normal
739.2
61.3
0.870
0.563
elevated
594.4
45.9
0.520
0.367
stress
477.3
35.6
0.223
0.163
crisis
275.9
21.8
0.139
0.091
Table 2. Per-regime stability headroom ε∗=γ/β and modulus at observed spreads, evaluated on the 1990–2026 panel.
Figure 2. Predict-then-verify phase diagram over the feedback gain f : measured modulus (median, IQR band, and R4 robust band over 8 seeds) against the a-priori closed form at the probe spread and the saturating fixed-point curve, with the measured ( f∗≈3.17 ) and predicted ( 4.70 ) crossings marked.
instrument
ε^
vs. realized
closed form, a-priori state ( ρ=1 )
0.764
0.44×
closed form, realized state
1.726
1.00
CRN best-response slope
0.508
0.29×
Sinkhorn/ W1
4.578
2.65×
Fitted informed-flow curve
4.011
2.32×
Table 3. Three-way ε triangulation at the operating spread, against the closed form at the a-priori (A2) and at the realized deployment state ( ρ=2.32 , ∣g∣=0.446 ).
Figure 3. The R3 systemic surface: joint modulus mN=Neffm1 over the (f,N) grid at full spillover ( κ=1 ), with the mN=1 boundary contour. Gains that are comfortably stable for a single dealer cross the systemic boundary as the dealer count grows; the probes of Table 4 measure the first rows of this surface on the simulated market.
N
measured
predicted
differential mode
1
0.786
0.786
–
2
1.369
1.571
0.0034
3
2.480
2.357
0.0032
Table 4. Measured common-mode retraining slope on the shared-pool market versus the R3 prediction Neffm1 (full spillover, κ=1 ).
Figure 4. Left: half-spread trajectories of the four learned retraining loops from a common seed in the RRM-unstable regime (the frozen-reference hPO and hSP lines are context, not the benchmark; Sec. 6.5 ). Blind RRM collapses; the structurally anchored loop converges to the realized performative optimum. Right: the three-way ML ↔ theory seam, showing the free-form learned, structurally fitted, and analytic toxic slopes logged at every deployment. The free-form slope flips sign and hovers near zero; the structural fit tracks the analytic shape.
mode
final h
outcome
blind RRM
0.16
echo-chamber collapse
PerfGD-analytic
0.36
no settle (operator gradient)
PerfGD-learned
0.66
no settle (negative result)
PerfGD-structural
3.19
converges to realized optimum
Table 5. Loop outcomes in the RRM-unstable demo regime (10 deployments, common seed; convergence verified against independent structural fits).
Figure 5. Lazy deployment at a beyond-boundary configuration ( m^=1.205 ). Left: the measured signed K -step map (medians over seeds; faint dots are per-seed readings) against the one-parameter fit μ(K)=−m+cK(1+m) with c=0.661 , its K→∞ asymptote −m , and the predicted deadbeat cadence. Right: the implied effective curvature γeff(K)/γ , predicted against measured, with the stiff branch decaying through the exact-RRM line between K=5 and K=8 .
Figure 6. Factor-model scaling (R5) with data-calibrated per-bond dispersion. Left: ρ(M) is flat in universe size from d=8 to 128 bonds and pinned to the worst scalar modulus, far below the ρ=1 boundary (dotted). Right: the truncation bound O(λk+1(C)) against the measured error ∣ρ−ρk∣ at d=128 , showing three to four orders of magnitude of slack at every retained-factor count k .
We describe an adaptive market-making architecture that preserves the analytical structure of the Avellaneda--Stoikov framework while introducing a successor measure-style adaptation mechanism. In our paper we keep Avellaneda--Stoikov fast Hamilton--Jacobi--Bellman structure and make it adaptive to changing market regimes and trading objectives. The central idea is to separate market dynamics from the trading objective. The market state determines a low-dimensional set of Avellaneda--Stoikov parameters, while recent realized rewards determine a low-dimensional objective vector. The HJB forward map then converts this objective into optimal bid and ask quotes through a scalarization of future reward features.
We propose a market-making game played among N≥2 strategic market makers and a single non-strategic representative trader. In a continuous quote space, standard finite-game arguments do not apply, while non-convex and discontinuous payoffs rule out classical continuous existence results and complicate learning. Nonetheless, we characterize pure-strategy Nash equilibria (PSNEs) and show that at least one PSNE always exists under minimal assumptions on the underlying asset value distribution, and that the equilibrium bid-ask spread widens strictly with asset value volatility. To obtain a tractable learning framework, we discretize prices and, under an additional regularity condition, show that the time-averaged play of no-internal-regret learners approaches the continuous Nash price envelope as the tick size vanishes. Finally, we empirically validate our theoretical results through numerical simulations.
Matteo Onger, Luigi Foscari
Department of Computer Science, University of Milan
Many learning systems are performative: once deployed, their predictions change the data distribution on which future models are trained. In multi-agent settings, this feedback is routed through a network of interacting learners, so stability depends not only on the strength of performativity but also on who affects whom. We study repeated retraining in multi-agent performative prediction and identify a spectral threshold that separates stable learning from worst-case learning obstruction. Prior work often collapses cross-agent feedback into a scalar contraction constant, missing this network structure. We instead introduce the reflexivity matrix Γ, which records how strongly one agent's deployment can move another agent's best response, defined from per-agent gradient sensitivities and curvature parameters. The threshold occurs when the spectral radius of this matrix crosses one. Below the threshold, repeated retraining converges geometrically to a unique performatively stable equilibrium, with cumulative excess loss bounded uniformly in the time horizon. Above the threshold, we construct noisy supercritical instances in which every local-information learner incurs expected cumulative excess loss growing linearly for horizons up to order 1/(ρ(Γ)−1). Together, these results characterize the threshold behavior through upper and lower bounds and recover the single-agent scalar threshold from prior work as a special case.
Ahmed Mahrous, Roberto Di Pietro
King Abdullah University of Science and Technology (KAUST), Saudi Arabia