Volatility, VaR, loss frequency and turnover leave a risk dashboard unable to distinguish herding from arbitrage at equal size. Aldridge's proof takes only a few lines. It is the paper's most useful result.
Can the dashboard see herding?
Aldridge builds a deliberately small model. The pool's log return each period is a feedback coefficient β times the previous return's sign, plus symmetric news with scale s. When β is positive, herd-followers buy after an up move. The rule comes from a trend-chasing reward that includes a momentum bonus. Negative β makes traders lean against the previous move. The constant-product rule supplies price impact, linearised to twice the net flow.
A stressed period has a return below the threshold -c. Consecutive stressed periods form an episode. The model therefore counts back-to-back losing bars; it measures something different from the time a book needs to recover a drawdown.
Symmetry does the work. Reversing β exchanges the halves of a symmetric mixture and leaves the one-period return distribution unchanged. Equal herd and arbitrage masses also produce equal turnover. Episode lengths follow a geometric distribution, however, and their continuation probability increases with β. Under Gaussian news with c=0 and β=s, herding episodes average 6.3 periods against 1.19 for arbitrage. In the Monte Carlo check, β=±0.8 and c=1 over 2 million periods. Both markets have variance 1.6398; stress frequency is 0.2282 versus 0.2287. Their mean episode lengths are 1.7247 versus 1.0372.
The paper cites the Lo-MacKinlay variance ratio as an example of the serial statistics regulators need. Aldridge supplies a formal result: every functional of the return marginal and turnover agrees across the two markets, even as the persistence ratio increases without bound in β/s. There is a further twist. A herd-follower trading in half-unit steps can never have its own stress episode exceed two periods, on any path. The simulated mean is 1.08. Monitoring individual strategies thus makes herders look brief, even as the market-level runs they help sustain grow longer.
Counting matching signs
The proposed diagnostic counts the fraction of periods when the return sign agrees with the previous one. A fraction above one half signals herding. Aldridge proves that the sign of β can be identified with error at most δ once n ≥ 9(s/β)² log(1/δ). Even a test using the entire return path cannot improve on (1/2)(s/β)² log(1/(4δ)) for δ below 1/4. The scaling matches, with constants roughly 18x apart. This guarantee requires Gaussian news and |β| ≤ s.
A feedback test still leaves the trading question open. In the paper's game, riding the herd produces gross earnings determined solely by herd mass and sign persistence. The rider's own impact cancels across a round trip. With herd mass 0.3, a lone rider earns 0.1854 per unit per period before a fee of 0.2. Enough other riders must join before riding pays. At full participation, gross earnings reach 0.2164.
The basin boundary is about 0.21. Softmax policy-gradient learners that begin with 15% riders end at 0.9%; those beginning at 40% reach 98%. Down-runs stretch from 2.62 periods to 3.59.
Leaning against the herd is strictly dominated at every fee. Aldridge is explicit about the setting: "Without an external price anchor, negative feedback is never a rational response to herd flow." The paper identifies two outside sources of stabilising flow. Arbitrage can tie the pool to an external reference price, and agents can be paid to take the opposite side. Section 6 models the latter as market makers. A live pool receives that flow. The tipping result therefore applies to a pool isolated from its reference.
Hourly crypto shows clustered volatility, with no net feedback
The connection to data comes through a companion study co-authored by Aldridge. It trains the archetypes using DQN and REINFORCE on hourly AAVE, ETC and SUI bars from March 2024 to February 2026. Each agent acts as a price-taker in a separate environment. The paper says its results explain patterns in that study. The clearest match is that herd-followers experience the most frequent yet shortest stress episodes, as Proposition 4.6 explains.
The full hourly sample does not reveal the market-level feedback needed for the title result, as the paper acknowledges in Section 7. The two-hour variance ratio ranges from 1.010 to 1.017 and is statistically indistinguishable from one. Stress nevertheless clusters. Runs of hourly returns below -2% average 1.09 to 1.13 hours, compared with 1.02 to 1.04 under independence. Autocorrelation in absolute returns is 0.22 to 0.27.
Aldridge traces that persistence to volatility clustering, which lies outside the model's i.i.d. news assumption. The recommendation is to test sign persistence conditionally, controlling for volatility. Credit for acknowledging the problem. Run length separates the theoretical markets, yet volatility clustering inflates that same statistic in practice while the variance ratio stays near one. Counting sign agreement uses only signs and is consequently less exposed to clustering in scale. Its error bound still rests on i.i.d. noise; that bound does not carry over.
We did not backtest the diagnostic. Even a test using spot bars would assess only a return-sign risk overlay. Those bars contain no pool reserves, agent populations or maker absorption capacity. They cannot establish the collapse thresholds or test the fee policy.
The cliff depends on the calibration
The model's market makers absorb a share of herd flow at the batch midpoint. They hold the inventory for one period and receive a spread φ. In the price process, their role is exactly that of negative riders. A single net index then determines persistence: rider capital minus absorbed herd flow. Market makers on their own reduce expected down-runs from 2.62 periods to 2.00.
Set the fee to 0.12, spread to 0.2, absorption capacity to 1 and herd mass to 0.3. Makers leave when rider capital reaches 0.510 and re-enter only below 0.210. The resulting band has width 0.3, equal to absorption capacity times herd mass. That gives a clean closed form. At exit, the expected down-run jumps from 3.000 to 3.373. With learning dynamics, simulated exit occurs around 0.6 and recovery around 0.1, beyond the band. The paper attributes the overshoot to learning slowing near each threshold.
All those figures depend on midpoint fills, one-period inventory and a fixed spread. They also come from a single calibration with λ=s=1 and herd mass 0.3. There is no gas. Orders fill in random order within each batch, leaving transaction-ordering games outside the model. The pool's own liquidity providers never add or remove capital; entry and exit belong solely to the outside market makers.
The proposed policy pairs a maker rebate, which increases φ and raises both thresholds, with a taker fee that removes riders. It has no empirical test. The companion study cannot provide one: its agents never traded in a shared market.
Monitor sign agreement conditional on volatility; treat the cliff as a stress scenario.
The paper's third listed prediction would make the cliff a signal worth watching: maker participation drops discontinuously as trend-chasing capital increases, then returns only after a larger decline. Until that pattern appears in data, 0.510 and 0.210 remain thresholds for this calibration and nothing more.