---
title: PNL-Haircut Domain in Perpetual Futures
url: https://www.emergentmind.com/topics/pnl-haircut-domain
type: topic
---

# PNL-Haircut Domain in Perpetual Futures

The **PNL-haircut domain** is the formal action space introduced for autodeleveraging (ADL) in perpetual futures markets, where an exchange restores solvency by haircutting **positive unrealized profit-and-loss (PNL)** of profitable accounts rather than posted collateral principal [2602.15182]. In this formulation, each ADL round selects a solvency budget and an allocation of account-level PNL seizures subject to per-account capacity constraints, so that ADL becomes a sequential control problem with explicit objectives for solvency recovery, overshoot control, and burden concentration. The domain is therefore not a generic “haircut” framework, but a specific online-learning abstraction for last-resort loss socialization on winner-side gains [2602.15182].

## 1. Scope and terminological boundaries

Within the supplied literature, the phrase **“PNL-haircut domain”** is formalized in the ADL setting of perpetual futures exchanges. In that setting, “PNL” refers to **positive unrealized gains**, and the haircut is imposed on those gains as part of a solvency-restoration mechanism [2602.15182]. The paper is explicit that “ADL haircuts apply to positive PNL (unrealized gains), not to posted collateral principal,” and also that it uses “strict PNL-only haircuts: ADL reallocates positive PNL and does not haircut protected collateral” [2602.15182].

This meaning should be distinguished from unrelated arXiv usages of the same acronym. In logic, **PNL** denotes **Permissive-Nominal Logic**, and the relevant papers explicitly state that they do **not** introduce a formally named “haircut domain”; the nearest constructions there concern permission sets, support restrictions, and restricted quantification domains rather than any exchange-loss-allocation mechanism [1111.4611] [2312.16480]. In another line of work, **PNL** denotes **positive and negative relations logic**, again unrelated to haircuting of unrealized gains [2405.01322]. Elsewhere, **PNL** denotes **Point-Neighborhood Learning** for image segmentation, which is also unrelated to ADL [2405.20044]. The term “haircut” is likewise polysemous across repo, securities lending, and literal hair or hairstyle modeling, but those uses refer to collateral valuation, grooming, or reconstruction rather than winner-side PNL seizure [1604.05404] [2111.13228] [2306.05872].

## 2. Formal definition of the domain

The domain is built on an account-level representation
\[
\mathfrak{p}_{i,t}=\big(q_{i,t},\bar p_{i,t},c_{i,t},\mathrm{PNL}_{i,t},e_{i,t}\big),
\]
with active positions
\[
\mathcal{P}_t=\{\mathfrak{p}_{i,t}: i\in\mathcal{I}_t\},
\]
and trader equity
\[
e_{i,t}=c_{i,t}+\mathrm{PNL}_{i,t}.
\]
Venue solvency is
\[
\mathsf{Solv}_t(\mathcal P_t)=A_t^x-L_t^x,
\]
equivalently
\[
\mathsf{Solv}_t(\mathcal P_t)=\sum_{\mathfrak p\in \mathcal P_t} e_t(\mathfrak p) =\sum_{i\in\mathcal I_t} e_{i,t},
\]
with residual shortfall
\[
S_t=\big(-\mathsf{Solv}_t(\mathcal P_t)\big)_+.
\]
For ADL, the operative loser-side deficit is
\[
D_t=\sum_{j\in L_t}\big(-e_{j,t}(p^{\mathrm{liq,exec}}(\mathfrak{p}_{j,t}, q_{j,t}))\big)_+.
\]
The profitable accounts are
\[
W_t=\{i\in\mathcal{I}_t:\mathfrak{p}_{i,t}\in\mathcal{P}_t,\ \mathrm{PNL}_{i,t}>0\},
\]
and each has PNL haircut capacity
\[
u_{i,t}=(\mathrm{PNL}_{i,t})_+,
\qquad
U_t=\sum_{i\in W_t}u_{i,t}.
\]

The defining feature of the domain is that the action can only reallocate these positive-PNL capacities. Let \(x_{i,t}\) denote the amount haircut from winner \(i\) in round \(t\). Then
\[
0\le x_{i,t}\le u_{i,t}.
\]
The exchange’s round action is a pair \(a_t=(B_t,x_t)\), where \(B_t\) is the aggregate solvency budget and \(x_t=(x_{i,t})_{i\in W_t}\) is the account-level allocation. The feasible action set is
\[
\mathcal{A}(s_t)= \left\{(B,x):\ 0\le B\le U_t,\ 0\le x_i\le u_{i,t},\ \sum_{i\in W_t}x_i=B\right\}.
\]
This is the formal PNL-haircut domain: a budgeted redistribution problem over winner-side unrealized gains [2602.15182].

| Symbol | Meaning |
|---|---|
| \(D_t\) | residual loser-side deficit |
| \(W_t\) | profitable accounts with \(\mathrm{PNL}_{i,t}>0\) |
| \(u_{i,t}\) | account \(i\)’s positive-PNL haircut capacity |
| \(B_t\) | aggregate solvency budget chosen for the round |
| \(x_{i,t}\) | haircut allocated to winner \(i\) |
| \(\mathcal{A}(s_t)\) | feasible budget-allocation set |

The paper also gives an equivalent severity parameterization:
\[
B_t=\theta_t D_t,\qquad \theta_t\in[0,1],
\]
and
\[
x_{i,t}=h_{i,t}(u_{i,t}+\varepsilon),\qquad h_{i,t}\in[0,1].
\]
Operationally, the venue first chooses how much of the observed deficit to socialize this round, then chooses how that burden is distributed across winners.

## 3. Sequential online-learning formulation

The state observed at round \(t\) is
\[
s_t=\big(\mathcal{P}_t,D_t,W_t,u_t,\zeta_t\big)\in\mathcal{S},
\]
where \(\zeta_t\) collects auxiliary round-start observables such as “price and volatility snapshots, spread/depth summaries, and recent liquidation-flow aggregates” [2602.15182]. A policy is history-dependent:
\[
\pi_t:\mathcal{H}_t\to\mathcal{A}(s_t), \qquad \mathcal{H}_t=(s_1,a_1,\dots,s_{t-1},a_{t-1},s_t).
\]
State transitions follow
\[
s_{t+1}=F_t(s_t,a_t,\omega_t),
\]
with shock \(\omega_t\).

A central modeling distinction is between the **ex ante** estimate of required ADL and the **ex post** amount actually needed after liquidation execution prices are realized. The ex ante target is
\[
\widehat B_t^{\mathrm{needed}}
=
\sum_{k\in t}\left|\hat p_k^{\mathrm{liq,exec}}-p_k^{\mathrm{bk}}\right|\,|\hat q_k^{\mathrm{ADL}}|,
\]
whereas the replay benchmark is
\[
B_t^{\mathrm{needed}}
=
\sum_{k\in t}\left|p_k^{\mathrm{liq,exec}}-p_k^{\mathrm{bk}}\right|\,|q_k^{\mathrm{ADL}}|.
\]
This distinction is what makes the problem online rather than static: the venue chooses \(B_t\) and \(x_t\) using only round-start information, but performance is evaluated using ex post realized execution.

The paper first introduces an asymmetric loss
\[
\tilde\ell_t(x_t)=
\lambda_{\mathrm{under}}[B_t^{\mathrm{needed}}-H_t]_+
+\lambda_{\mathrm{over}}[H_t-B_t^{\mathrm{needed}}]_+
+\lambda_{\mathrm{fair}}\,\Gamma_t(x_t,u_t),
\]
with
\[
H_t=\sum_i x_{i,t}.
\]
Its deployed convex surrogate is
\[
\ell_t(x_t)=
\lambda_{\mathrm{track}}\left|H_t-B_t^{\mathrm{needed}}\right|
+\lambda_{\mathrm{fair}}\max_{i\in W_t}\frac{x_{i,t}}{u_{i,t}+\varepsilon}.
\]
The first term penalizes tracking error in solvency restoration; the second penalizes concentration of burden on the most heavily hit winner. The paper notes that, under exact execution, \(H_t=B_t\), so tracking error is mostly a **severity-selection** problem, whereas queue-versus-pro-rata design primarily affects the **fairness/concentration** term [2602.15182].

## 4. Mechanism classes on the domain

A single ADL round has a three-part execution lifecycle. First, the venue measures residual loser-side deficit \(D_t\) and chooses a budget \(B_t\), or equivalently a severity \(\theta_t\in[0,1]\) with \(B_t=\theta_t D_t\). Second, it chooses winner-side reductions \(x_{i,t}\). Third, it matches winners and losers at loser bankruptcy transfer prices \(p_k^{\mathrm{bk}}\) [2602.15182]. The PNL-haircut domain accommodates several mechanism families.

**Queue-based mechanisms** assign a score \(s_{i,t}\), sort winners, and exhaust the budget greedily from the front of the queue. If \(\sigma_t\) sorts scores in decreasing order, then the top-ranked accounts are fully haircutted,
\[
x_{\sigma_t(j),t}=u_{\sigma_t(j),t},
\]
until the budget is nearly filled; the marginal account may be partially haircutted, and all later accounts remain untouched. The paper emphasizes that this geometry can produce effectively \(100\%\) haircuts for early queue positions [2602.15182].

**Partial-haircut policies** distribute budget across many winners without fully exhausting each touched account. Writing
\[
h_{i,t}=x_{i,t}/(u_{i,t}+\varepsilon),
\]
such policies aim for \(h_{i,t}<1\), or enforce a cap \(h_{i,t}\le \bar h<1\). This formulation assumes divisibility of PNL; the paper notes that integer contract granularity can complicate exact implementation [2602.15182].

**Pro-rata allocation** is the canonical continuous fairness benchmark:
\[
x^{\mathrm{PR}}_{i,t} = \frac{u_{i,t}}{U_t}\,B_t.
\]
It equalizes normalized burdens \(x_{i,t}/u_{i,t}\) across all winners. The discrete counterpart is the **integer min-max ILP**
\[
\min_{x,z}\ z
\quad\text{s.t.}\quad
\sum_{i\in W_t}x_i=B_t,\ \
0\le x_i\le u_{i,t},\ \
\frac{x_i}{u_{i,t}+\varepsilon}\le z,\ \
x_i\in\mathcal G_{i,t},
\]
which minimizes the worst normalized burden subject to budget exactness and lot-feasible execution [2602.15182].

This mechanism taxonomy matters because the domain itself does not impose queueing, pro-rata, or any specific fairness criterion. It only imposes the PNL-capacity constraints. Mechanism design then specifies how a feasible haircut vector \(x_t\) is selected inside that domain.

## 5. Theoretical properties

The paper’s main performance theorem concerns the one-dimensional severity controller
\[
B_t=\theta_t D_t,\qquad \theta_t\in[0,1],
\]
with loss
\[
\ell_t^{\theta}(\theta_t)=D_t\left|\theta_t-\theta_t^{\mathrm{needed}}\right|,
\qquad
\theta_t^{\mathrm{needed}}=\min\!\left\{1,\frac{B_t^{\mathrm{needed}}}{D_t+\varepsilon}\right\}.
\]
If the comparator path variation is
\[
P_T^{\theta}=\sum_{t=2}^{T}\left|\theta_t^\star-\theta_{t-1}^\star\right|,
\]
projected OGD on \([0,1]\) yields
\[
\mathrm{Reg}^{\mathrm{dyn},\theta}_T
\le
\frac{1+2P_T^{\theta}}{2\eta}
+
\frac{\eta}{2}\sum_{t=1}^{T}D_t^2,
\]
and, with
\[
\eta^\star=\sqrt{(1+2P_T^\theta)/\sum_t D_t^2},
\]
\[
\mathrm{Reg}^{\mathrm{dyn},\theta}_T
\le
\sqrt{(1+2P_T^\theta)\sum_{t=1}^{T}D_t^2}.
\]
The result identifies the two core hardness parameters of repeated ADL: the scale of deficits and the path variation of the latent optimal severity [2602.15182].

A second major result shows that fixed queue policies can incur **linear regret**. In a two-winner construction with alternating capacities
\[
(u_{1,t},u_{2,t})=
\begin{cases}
(1,M), & t\ \text{odd},\\
(M,1), & t\ \text{even},
\end{cases}
\]
budget \(B_t=1\), and fairness-only loss
\[
\ell_t(x)=\lambda_{\mathrm{fair}}\max_i \frac{x_i}{u_{i,t}},
\]
a fixed queue always serving account 1 first suffers
\[
\mathrm{Reg}^{\mathrm{dyn}}_T(\mathrm{queue})
=
\frac{T}{2}\lambda_{\mathrm{fair}}\!\left(1-\frac{1}{M}\right)
=
\Omega(T).
\]
This establishes that queue mechanisms can be structurally poor on the PNL-haircut domain even when total budget is exact.

The paper also decomposes total realized loss into online-control error and execution-estimation error:
\[
\sum_{t=1}^T \ell_t(x_t)
\le
\min_{\pi'\in\mathcal{P}}\sum_{t=1}^T \ell_t(x^{\pi'}_t)
+
\mathrm{Reg}^{\mathcal{P}}_{T}(\pi;\hat\ell)
+
2\lambda_{\mathrm{track}}\sum_{t=1}^T \big|B_t^{\mathrm{needed}}-\widehat B_t^{\mathrm{needed}}\big|.
\]
This makes robustness depend not only on the online algorithm but also on the quality of liquidation-execution estimation. The associated ex post failure metric is
\[
V_T := \sum_{t=1}^T \big[B_t^{\mathrm{needed}} - H_t\big]_+,
\]
which can remain large even when regret is small.

At the allocation level, the appendix formalizes the feasible haircut set for a fixed budget \(B\) and capacities \(u\) as the polytope
\[
X(B,u) := \left\{ x \in \mathbb{R}^n : 0 \le x_i \le u_i,\ \forall i,\ \sum_{i=1}^n x_i = B \right\}.
\]
Queue allocations are shown to be **extreme points** of this polytope and to coincide with linear optimization over it. The paper then proves a sharp instability result: for every \(\delta>0\), there exist score vectors \(s,s'\) with \(\|s-s'\|_\infty\le\delta\) but
\[
\|Q(s)-Q(s')\|_1 = 2B.
\]
By contrast, the min-max burden problem
\[
\min_{x\in X(B,u)}\ \max_i \frac{x_i}{u_i}
\]
has unique optimizer
\[
x^{\mathrm{MM}}_i = \frac{B}{U}\,u_i,
\]
the pro-rata allocation. Every queue has strictly worse worst-burden,
\[
z(x^{Q}) > z(x^{\mathrm{MM}})=\frac{B}{U}.
\]
In the language of the paper, this makes pro-rata the unique continuous optimizer of max-normalized fairness on the PNL-haircut domain [2602.15182].

## 6. Hyperliquid stress episode

The empirical case study reconstructs the **October 10, 2025 Hyperliquid stress episode** over the window 21:16–21:27 UTC. The replay uses
\[
T=16
\]
ADL rounds and a total liquidation volume of
\[
\$2{,}103{,}111{,}431
\]
[2602.15182]. The replay holds fixed the realized round boundaries, loser deficits \(D_t\), winner sets \(W_t\), capacities \(u_{i,t}\), observable context \(\zeta_t\), bankruptcy transfer prices \(p_k^{\mathrm{bk}}\), and the realized market path. Counterfactual mechanisms therefore vary only in severity and allocation.

The paper reports the following aggregate quantities for the episode:
\[
\sum_t D_t \approx \$100.1\text{M},
\qquad
\sum_t B_t^{\mathrm{needed}} \approx \$15.1\text{M},
\qquad
\sum_t H_t^{\mathrm{prod}(0)} \approx \$60.1\text{M}.
\]
The production queue’s overshoot relative to needed budget is reported as
\[
O(0)=\$45{,}028{,}665.72,
\]
with a short-horizon sensitivity band of
\[
\$45.0\text{M} \text{ to } \$51.7\text{M}.
\]
The calibrated instance-level upper envelope from the severity-regret proposition is
\[
\mathcal{B}_{\mathrm{inst}}
:=
\sqrt{(1+2\widehat P_T^\theta)\sum_{t=1}^{T}D_t^2}
\approx
\$129.7\text{M},
\]
using
\[
\widehat P_T^\theta=7.06.
\]
Against that benchmark, the production ADL queue causes about
\[
\$64.86\text{M}
\]
of regret, which the paper reports as approximately
\[
50.0\%
\]
of the calibrated upper envelope. The best start-of-round baseline achieves
\[
\$3.40\text{M},
\]
or approximately
\[
2.6\%
\]
of the same bound [2602.15182].

| Policy | Total objective |
|---|---:|
| Production queue | \$64,859,522.21 |
| Integer pro-rata | \$3,404,185.00 |
| Vector mirror descent | \$4,413,367.61 |
| Min-max ILP | \$106,205.81 |
| Continuous pro-rata | \$2,732,437.29 |

The paper’s abstract states that “the best algorithm reduces overshoot to \$3M,” while the main text reports integer pro-rata at about \$3.40M total objective and vector mirror descent at \$4.41M. The supplied details note a mild discrepancy between the abstract/introduction and the body on this point, and the distinction is material because the best **deployable** start-of-round baseline in the table is integer pro-rata rather than vector mirror descent [2602.15182].

Substantively, the case study is used to argue that the production queue both **over-socialized** winner PNL and **concentrated** burden more heavily than smoother alternatives. In the paper’s interpretation, the domain therefore exposes a concrete exchange-design gap: queue-style ADL is feasible in the PNL-haircut domain, but it is not close to the best available use of that domain.

## 7. Relation to adjacent haircut literatures

The PNL-haircut domain belongs to the ADL literature, but it sits near several older haircut literatures in market microstructure and secured finance. In **repo pricing**, haircut is the discount from collateral market value used to determine cash lent, with economic capital identified as the main driver of repo spread and a negative linear relation between spread and haircut [1604.05404]. In **securities lending**, haircut is excess collateralization, with indemnification priced as the sum of a risk charge, a capital charge, and a funding charge when the transaction haircut falls short of a target credit standard [2111.13228]. In **repo fire-sale network models**, haircut or collateral valuation is made endogenous to aggregate liquidation, so that more sales worsen collateral value and force additional liquidation [2005.05364].

These adjacent literatures clarify what is distinctive about the ADL usage. In repo and securities lending, the haircut is applied to **collateral valuation** or **overcollateralization**. In the ADL domain, by contrast, the haircut applies to **positive unrealized PNL**, not to posted collateral principal [2602.15182]. A plausible implication is that the ADL formulation transfers the economic idea of a haircut from collateral-side credit protection to winner-side liability reduction: it is still a bounded haircut problem with budget, capacity, and fairness constraints, but the underlying object being haircut is a venue liability to profitable traders rather than collateral posted by a borrower.

That distinction also explains why the ADL paper formulates the problem as online learning. Repo and securities-lending haircut models center on loss distributions, economic capital, and pricing spreads [1604.05404] [2111.13228], whereas the PNL-haircut domain centers on repeated **severity selection**, **allocation across profitable accounts**, and **regret relative to an ex post solvency benchmark** [2602.15182]. The family resemblance is therefore economic rather than terminological: each literature studies how a haircut governs residual loss absorption, but the ADL domain is the one that formalizes this explicitly on **winner-side unrealized gains**.

In that sense, the PNL-haircut domain can be defined compactly as **the feasible budget-allocation space for haircuting positive unrealized PNL in repeated ADL rounds, together with the online decision problem of choosing severity and burden-sharing so as to recover exchange solvency with minimal tracking error and concentration** [2602.15182].

Source: https://www.emergentmind.com/topics/pnl-haircut-domain