Uniswap's USDC-WETH 0.05% pool assigned 86% of lag-one trade-sign autocovariance to different-wallet pairs in one June 2023 week, despite little correlation between wallets. For a liquidity provider, that gap changes how to read claims of herding in DeFi. Bai and Kan's decomposition explains why.

What does the herding share measure?

The method starts with Tóth et al. Split the covariance of consecutive order signs between pairs from the same account and pairs from different accounts. The fraction contributed by different accounts is the herding share, h. Applying this to Uniswap v3, Di Nosse et al. found that h dominated. They interpreted the result as a reversal of equities, where splitting dominates.

Bai and Kan express h exactly in terms of two inputs. The first, p, is the probability that consecutive orders come from the same account. The second, κ, divides cross-account sign correlation by same-account sign correlation. κ equals one when every trader responds to a common public signal and zero when accounts split orders independently. Thousands of wallets make p tiny. A large h can therefore coexist with κ close to zero.

For the USDC-WETH 0.05% pool, the samples cover one June week each in 2024 and 2023. They produce h of 0.65 and 0.86, alongside κ of 0.05 and 0.10. Consecutive same-wallet swaps have correlations of 0.73 to 0.81; cross-wallet correlations are 0.04 to 0.07. Cross-wallet pairs account for 97 to 98% of consecutive pairs.

The other venue is an unnamed, fully on-chain EVM order book for a BTC perpetual. Its main sample spans 116-hour trading in autumn 2026, with an earlier 72-hour sample. Here h is comparable, at 0.61 and 0.87. But κ reaches 0.76 and 0.72. Uniswap swaps of $100k or more fall between the two venues, with κ of 0.34 and 0.52. Those swaps account for 76 to 82% of pool volume.

The paper then asks what drives co-movement when κ is high. A gradient-boosted model predicts each order's sign using only Binance perpetual data sampled on a 100 ms grid. The authors repeat the decomposition on the residuals. For the book, the filter removes 84% of cross-account covariance in the main sample and 73% in the earlier sample. For large Uniswap swaps, removal reaches 109% (2024) and 91% (2023). A result above 100% means residual cross-wallet covariance has turned negative.

The authors interpret high-κ co-movement as a race against venue prices that lag Binance, at liquidity providers' expense. In an earlier note, we argued that one-period statistics cannot distinguish herding from arbitrage on an automated market maker (AMM). Here, account labels and a centralized-exchange (CEX) signal supply the information needed to make that distinction.

Table 1 passes the arithmetic check

A calculator reproduces much of the decomposition. For Uniswap 2024, p = 0.025 and κ = 0.05 yield h = 0.975 × 0.05 / (0.025 + 0.975 × 0.05). That gives about 0.66, compared with the reported 0.65. For the book's main sample, p = 0.331 and κ = 0.76 produce 0.606, rounding to the reported 0.61.

The booster specification is complete: 300 iterations, a learning rate of 0.05, 31 leaves and an L2 penalty of 1. Minimum leaf size is 200 orders for the book and 50 swaps for Uniswap. Cross-fitting assigns folds by clock-hour parity. The intervals use hour-block bootstraps with 2,000 resamples.

Venue anonymity and other judgment calls

The order-book venue is identified only as EVM contracts on a chain whose blocks arrive in under a second. Rebuilding its 94,469 and 142,659 orders first requires identifying the venue. Timestamp treatment also varies between samples. The main sample takes first receipt on the public feed. For the earlier sample, the authors interpolate whole-second timestamps and apply a fitted shift of 0.65 s.

The race window remains a free parameter. An order qualifies as racing when another account submits an order in the same direction within three blocks on the book or two on Uniswap. We did not find a sensitivity check for these windows. The Uniswap filter is fitted to swaps of at least $10k, although the reported removal shares cover $100k and up. Hour-parity folds also leave every test hour beside two training hours. An implementer concerned about serial dependence could leave gaps.

Account concentration deserves attention. Two takers generate 75% of book volume in the main sample, compared with 24% in the earlier sample. The authors examine this directly: merging those two takers gives κ = 0.73; merging addresses with a common sender gives 0.85.

Size cuts pull the venues in opposite directions. On the book, κ declines from 0.73 for orders of at least $100 to 0.51 for the 22% above $500. On Uniswap in 2024, it increases from 0.05 across all swaps to 0.34 at $100k and up. Daily κ on the book goes above one, reaching 1.47 and 1.56 on the main sample's last two days. Same-account correlation falls to 0.41 on those days, while cross-account correlation remains at 0.61 to 0.65.

Could the Binance-only filter see ahead?

The authors acknowledge the timing leak: the latest feature cell can contain up to 100 ms of Binance data arriving after the block. A 0.5 s delay to every input changes R² from 69% to 68%, and removal from 84% to 83%. They also acknowledge the limit of that check. It covers only the book's main sample; the earlier book sample and both Uniswap weeks have no such check.

Timing supplies the argument against imitation. Across accounts, book orders in the same block correlate at 0.84 and 0.80. Both orders were submitted before either trader could observe the other's fill. Pairs separated by no more than ten blocks account for 90% and 96% of cross-account covariance.

Large Uniswap swaps have a different pattern. Within-block covariance is −0.32 in 2024 and +0.24 in 2023. Co-movement of 0.42 to 0.56 appears one to five blocks apart. The authors explicitly leave an explanation based on AMM design for future work. Even on the book, the filter leaves residual cross-account correlations of 0.33 and 0.40, compared with raw correlations of 0.63 and 0.69.

Maker markouts, with different fee treatment

Alignment multiplies an order's sign by the filter's prediction. Sorting on this measure takes makers' 60 s book markouts from −0.51 to −2.31 bp in the main sample, and from +0.08 to −2.48 bp in the earlier sample. These figures exclude trading fees and gas. Race orders account for 54% and 74% of book volume, yet generate 70% and 91% of maker losses. They cost 1.87 and 1.70 bp, compared with 0.94 and 0.46 bp for other orders.

Uniswap markouts include the 5 bp pool fee and cover swaps of $10k or more. Across alignment terciles, they fall from +4.57 to −4.23 bp in 2024 and from +5.29 to −5.08 bp in 2023. Before the fee, the most aligned third costs LPs 9 to 10 bp. Race swaps make up 66 to 78% of volume and lose LPs 2.38 and 2.44 bp after the fee. Other swaps earn 2.43 and 5.57 bp.

The ladders cannot be compared like for like.

The paper describes its measure as a pre-trade toxicity score and ends with the ranking. We found no tested fee or quoting rule based on it.

We could not run any of this ourselves. Each quantity requires wallet-level swap and fill events, together with 100 ms Binance quotes and signed volume. Our crypto data consist of minute bars without account identities. They cannot produce κ or identify a race lasting a few blocks.

Anyone measuring a venue of their own should report κ beside h. At p = 0.016, a herding share of 0.86 remains consistent with κ of 0.10.