Trim: Fee Level Matters More Than Skew in Lightning Routing

Trim: Fee Level Matters More Than Skew in Lightning Routing Huiping Sun1 ✉, Peilu Guo1 , Yiming Zhu2 , Yueqi Wu1 , and Zhong Chen2,3

1 School of Software & Microelectronics, Peking University, Beijing, China

sunhp@ss.pku.edu.cn, gpeilu@stu.pku.edu.cn, wuyueqi77@stu.pku.edu.cn 2 School of Computer Science, Peking University, Beijing, China ymzhu25@stu.pku.edu.cn, zhongchen@pku.edu.cn 3 School of AI and Liberal Art, Beijing Normal-Hong Kong Baptist University, Zhuhai, China

arXiv:2610.35296v1 [cs.NI] 26 Oct 2026

Abstract Lightning routing nodes lend their channel balance for a posted fee. Operators’ tools move that fee with the balance around a hand-set level, and which part of the fee drives revenue is unclear. We present Trim, which models a channel as a two-sided inventory with costly refills. Its optimal fee is a level, set by demand and replenishment, plus a balance skew, the shadow price of liquidity. The level is first order: no fee rule beats the ceiling demand and replenishment set, and on exactly solved channels the best flat fee comes within 1.6% of the optimum at the median, whereas a level off by a factor of two loses tens of percent. The skew is second-order insurance: it adds at most what the flow-balancing flat fee loses to refused forwards. So the level should be chosen: Trim’s rule traces the revenue–success frontier, posts one level on every channel, and adds a skew only where refills stay frequent and costly. On a 2026 snapshot with synthetic payments, no tuned tool beats that frontier by over a few percent, and a skew changes gross revenue by a few percent.

CCS Concepts • Applied computing → Electronic funds transfer; • Theory of computation → Stochastic control and optimization; • Networks → Network economics.

Keywords Lightning Network, payment channel networks, routing fees, dynamic pricing, inventory control, shadow price, liquidity

1

Introduction

Routing nodes and their fees. Payment channel networks (PCNs) such as the Lightning Network [39] settle payments off-chain and route them hop by hop through intermediary routing nodes, which lend their channel balance to strangers for a posted fee. About 5,500 self-interested nodes and 27,000 public channels post these prices, 150 ppm at the median channel direction; as Lightning carries small payments for online services and content [25, 34], they set what such payments cost and whether they succeed. Forwarding out of a channel draws down the node’s side, which is replenished only when payments flow back; a channel that runs dry stalls until reverse flow arrives or the operator pays for an on-chain refill [24]. How fees are set today. Operators who manage liquidity run rulebased tools (charge-lnd [14], feeadjuster [29], LNDg [30]) that raise a channel’s fee as its balance drains and lower it as the balance fills, around a level set by hand or inherited from the deployed fee. In 2019–2023 gossip the largest hubs reset that level 1.7 times a year, by a median 2.3×, most channels at once (our measurement, App. C). Which part of the fee drives a routing node’s revenue?

fee

refill balance 𝑏

skew

𝑐 −𝑏 replenishment

demand

level 𝐿

0

balance 𝑏

𝑏¯

𝑐

Figure 1: The Trim model. Inset: a channel as an inventory, drawn down by demand and restored by replenishment or a costly refill. Plot: the optimal fee, a level 𝐿 plus a skew. Trim. A routing node is a two-sided inventory manager (Fig. 1). Balance is inventory: demand depletes it, and replenishment, priced by other nodes, restores it, both sides holding the same asset [5, 20]. Its optimal fee has two layers that do different jobs. The level sets how much demand the node prices in and whether the channel drifts: at the flow-balancing fee demand matches replenishment and the balance neither drains nor fills, while a level set too low drains the channel and one set too high prices out demand. The skew, the marginal value of balance, moves with the balance and only absorbs its fluctuations around that point. Neither layer can be read off public data: both depend on how price-sensitive senders are and on flows a node does not observe. We call this account Trim: a model of the channel, the two-layer fee it implies, and the rule that follows. Contributions → (1) A two-layer fee whose level is first order (§4–§5). We model a channel as a two-sided inventory with costly refills and derive its optimal fee, a level plus a skew. Under our assumptions no fee rule beats the revenue ceiling that demand and replenishment set, for any decreasing demand (Prop. 4.3), and where demand outruns replenishment a level error costs in proportion to its size. At a hub the level stays uniform unless a channel drains (Prop. 5.1). → (2) The skew is second-order insurance (§4.3). A skew adds at most what the flow-balancing flat fee loses to refused forwards (Prop. 4.4) and pays only where refills are costly. → (3) A rule that makes the level a choice (§6). Trace the revenue– success frontier with a few levels, pick a point and post it on every channel; add a skew only where refills stay frequent and costly. → (4) Tuning mainly selects a level (§7). On a 2026 snapshot with synthetic payments, uniform levels trace a revenue–success frontier, and our tuned reimplementations of the deployed tools lie on or near it: their balance rules add little at equal success.

Results. On 144 exactly solved channels a well-chosen flat fee is nearly optimal, within 1.6% at the median (at most 9.6%), while a level off by a factor of two loses tens of percent; on 2,592 more with

Sun et al.

Table 1: Related work. Alone: deployable by one node. Family (work)

Fee moves with

Level

Alone

Deployed tools [14, 29, 30, 50] balance rules by hand ✓ Pricing imbalance [1, 17, 18, 40, 44, 46] imbalance, duals absent/implicit mostly Pricing inventory [2, 5, 15, 20, 22, 42] marginal value markup — Other [3, 4, 7, 13, 19, 21, 24, 33, 45, 48] fixed/strategies — — Trim (this paper) shadow price explicit ✓

other size laws and demand curves the median gap is 1.1–2.3%. On exactly solved hubs where a channel drains, per-channel levels, not a skew, recover half to three-quarters of what a uniform level loses. In the network, no tuned tool beats the frontier of uniform levels by more than a few percent, levels per hub or per channel class give no gain, and a skew moves gross revenue by a few percent at most. Tools win clearly only on 2019, with balance-aware senders, by pricing channels differently, not by a skew: deployed fees at their best uniform multiple do as well (§8). Code, raw outputs and a number checker will be made public upon acceptance.

2

Background and Related Work

Lightning fees. Each channel direction carries a fee set by the node on whose side a payment leaves, a base charge plus a proportional (ppm) rate [39]; since LND 0.18 a node can also post an inbound fee, at first only a discount [26]. Senders pick fee-minimizing routes [38], which LND weighs against the success probabilities its mission control learns from failures, so a node’s demand is pricesensitive. Deployed tools. Implementations ship static defaults (LND: 1,000 msat + 1 ppm [27]). charge-lnd interpolates linearly in the local share [14], feeadjuster scales a base rate by 500.5−𝑥 [29], LNDg steps fees by flow and liquidity thresholds [30], and CLBOSS scales competitors’ median fee by size, balance and price [50]. LND’s developers report that success falls with a channel’s fee, yet only 10% of nodes priced by liquidity in 2024 [11]. All tune the skew and inherit or search the level. Pricing imbalance. Proposed mechanisms surcharge forwards that worsen balance [18, 44], post the dual prices of an online packing program [46], or, in Debt [40], run network-wide dual descent toward the welfare-maximizing flow; node-profit controllers [1, 17] are not deployed. Debt’s prices are multipliers of flow conservation, our 𝑠ˆ one node’s marginal value of its own balance, which it can post alone; the incentive component Debt leaves to practice is the level. Throughout, the balance term is the lever and the level is given or implicit. Pricing an inventory. An inventory’s optimal price is the marginal value of a unit plus the markup 1/𝜅 [20, 42], with (𝑠, 𝑆, 𝑝) policies under fixed ordering costs [15], drift control [2, 49] and inventoryskewed quotes for market makers [5, 22]; static prices are nearoptimal in closed networks [6]. A channel’s inventory is bounded on both sides and replenished at prices set by others, and flow balance is its zero-drift point. Other related work. Rebalancing moves liquidity after imbalance forms [4, 21, 24, 33, 48] and admission control protects it [7], both with fees fixed; hub equilibria and channel games [3, 45], routing

profitability [13, 19] and depletion levers [37] concern network outcomes. We study one node’s split of its own fee; where levels settle when all reprice we leave open (App. C). Positioning. Trim is a two-layer account of one node’s fee and the rule that follows, not a routing, rebalancing or equilibrium design. No prior work we know derives the two layers and asks which one drives revenue (Table 1). Relative to inventory pricing, what we add is a revenue ceiling that binds every fee rule when replenishment is priced by others, a bound on what a skew adds over the best flat fee, and a diagnosis of the tools operators run.

3

Trim: A Channel as an Inventory

Scope and state. We model one channel of a routing node and its proportional rate, taking the node’s other channels as part of the network and its base fee as deployed (Fig. 1); it omits time-locks, HTLC limits, multi-part payments and channel closures (Simplifications below). The state is the node’s balance 𝑏 ∈ [0, 𝑐] on its side of a channel of capacity 𝑐, its inventory. Demand, the forwards that leave through that side, arrives at rate 𝜆(𝑓 ) that falls with the proportional fee 𝑓 , the control; replenishment, the forwards that enter, arrives at rate 𝜇. A node prices only payments that leave through its side; those that enter are priced by the peer and upstream nodes, so per channel 𝜇 is exogenous. The canonical demand is 𝜆(𝑓 ) = Λ𝑒 −𝜅 𝑓 , the logit form of least-cost route choice, with markup 1/𝜅. Offered demand in our simulator is close to it near deployed fees (𝑅 2 = 0.95, App. B). Far from them it mixes captive and competed demand, whose markup can grow faster than the fee (for our fitted mixture, between 52 and 1,052 ppm), so we do not assume log-concavity. Costs and objective. Unlike a market maker’s, the inventory carries no price risk: both sides hold the same asset. Risk enters only at the boundaries, where a forward larger than the balance is refused at a goodwill cost 𝑐 r and an on-chain refill of size Δ costs 𝐾 + 𝑘Δ. The node maximizes its long-run average net revenue: fees on served demand, less refusal penalties and refill costs. Formalization. Demand and replenishment are marked point processes 𝑁 ∓ with size laws 𝐺 and 𝐺 + , and refills add Δ𝑖 > 0 at times 𝜏𝑖 , so ∫ 𝑡∫ 𝑏𝑡 = 𝑏 0 − 𝜉 1{𝜉 ≤ 𝑏𝑢 − } 𝑁 − (𝑑𝑢, 𝑑𝜉) 0 ∫ 𝑡∫ (1) ∑︁ + 𝜂 1{𝜂 ≤ 𝑐 − 𝑏𝑢 − } 𝑁 + (𝑑𝑢, 𝑑𝜂) + Δ𝑖 . 0

𝜏𝑖 ≤𝑡

Assumption 1 (Demand). 𝜆 : [0, ∞) → (0, ∞) is 𝐶 1 with 𝜆 ′ < 0 and 𝑓 𝜆(𝑓 ) → 0 as 𝑓 → ∞; its markup is 𝜓 (𝑓 ) := −𝜆(𝑓 )/𝜆 ′ (𝑓 ). Assumption 2 (Depletion cost). (i) A refused forward of demand costs a goodwill penalty 𝑐 r ≥ 0 plus the forgone fee; refused replenishment costs nothing directly. (ii) A refill of size Δ costs 𝐾 + 𝑘Δ, with 𝐾 > 0 and 𝑘 ≥ 0. (iii) Service of feasible forwards is compulsory: the node prices demand away rather than refusing a feasible forward, so refusals occur only when a forward exceeds the balance. A policy 𝜋 is a predictable fee process together with a refill rule; a Markov policy sets 𝑓𝑡 = 𝜑 (𝑏𝑡 − ). Let 𝑁𝑡r count refused forwards of

Trim: Fee Level Matters More Than Skew in Lightning Routing

inventory value, less the goodwill cost of refusals; the second term is replenishment.

Table 2: Notation. 𝑏, 𝑐 𝜇 𝜉, 𝜂 𝑐r 𝐿, 𝑠ˆ (𝑏 ) 𝑔; 𝑅 ∗ 𝑏 ∗ ; 𝑠ˆ∗

balance, capacity 𝜆(𝑓 ) replenishment rate 𝜓 forward sizes, out / in 𝑚 1 , 𝑚 2 ; 𝑚 1+ refusal penalty 𝐾 + 𝑘Δ level, skew 𝑓¯ average revenue; its ceiling at 𝜈0 flow balance flow-balance point; its shadow Φ(𝑠 ), 𝐹 (𝑠 ) price

demand rate, Λ𝑒 −𝜅 𝑓 markup, 1/𝜅 moments of 𝜉 ; mean of 𝜂 refill cost flow-balancing fee variance rate of 𝑏

𝑓 ≥0

fee income at shadow price 𝑠 ; plus inventory value

demand. The expected discounted net revenue is i h∫ ∞
∑︁ −𝜌𝜏𝑖 − 𝑒 −𝜌𝑡 𝑓𝑡 𝜉 𝑑𝑁 served −𝑐 r 𝑑𝑁𝑡r − 𝑒 (𝐾 + 𝑘Δ𝑖 ) , 𝐽𝜌 (𝜋) = E𝑏𝜋0 0

Proposition 4.1 (Optimality equation; exact). Under Assumptions 1–2, 𝑉 is the unique bounded solution of the quasi-variational inequality  min 𝜌𝑉 (𝑏) − sup (L 𝑓 𝑉 )(𝑏), 𝑉 (𝑏) − (M𝑉 )(𝑏) = 0, 𝑏 ∈ [0, 𝑐].

𝑖

(2) with 𝜌 > 0 and value function 𝑉 (𝑏) = sup𝜋 𝐽𝜌 (𝜋). For the average net revenue 𝑔 = sup𝜋 lim inf𝑇 →∞ 𝑇1 E𝜋 [·] the optimality equation has the same form, with 𝜌𝑉 replaced by 𝑔 and 𝑉 by the bias ℎ [23]. We prove the structural results below for 𝜌 > 0 and take the average criterion as their limit 𝜌 → 0, which we assume exists (a numerical assumption: relative value iteration converges on every instance we solve). Simplifications. The model has no time-locks, HTLC limits, multipart payments or channel closures (the simulator has HTLC and MPP limits as options). A refill restores the balance at once, and demand depends on the channel’s own fee only. We do not model inbound fees, with which a node would also price its replenishment; in an exact extension the inbound fee is a second level, and level errors still cost more than state dependence adds (App. A).

(4) Only the demand bracket of (3) depends on 𝑓 . Define the shadow price of liquidity ∫𝑏
𝑉 (𝑏) − 𝑉 (𝑏 − 𝜉) 𝐺 (𝑑𝜉) + 𝑐 r𝐺¯ (𝑏) 0 𝑠ˆ(𝑏) = , (5) 𝑚(𝑏) the inventory value the node gives up per unit forwarded, including the expected refusal cost. Proposition 4.2 (Optimal fee structure; exact). In the continuation region the optimal fees maximize 𝜆(𝑓 ) (𝑓 − 𝑠ˆ(𝑏)); each positive one solves
𝑓 ∗ (𝑏) = 𝑠ˆ(𝑏) + 𝜓 𝑓 ∗ (𝑏) , (6) and the largest, 𝑓 ∗ (𝑏), is nondecreasing in 𝑠ˆ(𝑏); it is the only root if 𝑓 − 𝜓 (𝑓 ) increases (log-concave 𝜆), else it can jump. For 𝜆 = Λ𝑒 −𝜅 𝑓 , 𝑓 ∗ (𝑏) = 𝑠ˆ(𝑏) + 1/𝜅. This is the classical structure of inventory pricing [20], congestiondependent pricing of queues [31, 32, 36] and risk-neutral market making [22]; what is specific to a channel is how the shadow price depends on the balance.

4.2 4

For one channel, the level is 𝐿 = 𝑓 ∗ (𝑏 ∗ ), the optimal fee at the balance 𝑏 ∗ where the drift vanishes (𝑐 if none), and the skew is the rest, 𝑓 ∗ (𝑏) − 𝐿; a flat fee is a level alone. This split is a definition; what we prove is what a wrong level costs and what any balancedependent fee can add over the best flat one. We derive the two layers (§4.1), then the level (§4.2) and the skew (§4.3), check both on exactly solved instances and map where the skew pays (§4.4). Each proposition’s title says whether it is exact or uses the diffusion approximation, which we check against exact solutions. Proofs are in App. A; Table 2 lists the notation. Write 𝐺¯ (𝑥) = P(𝜉 > 𝑥), 𝑚(𝑥) = E[𝜉 1{𝜉 ≤ 𝑥 }], 𝑚 1 = E𝜉, 𝑚 2 = E𝜉 2 and 𝑚 +1 = E𝜂.

4.1

The level is first order

The Channel Fee: A Level plus a Skew

Two layers

For a fee 𝑓 and a value function 𝑤 on [0, 𝑐], the controlled generator is h∫ 𝑏 i
(L 𝑓 𝑤) (𝑏) = 𝜆(𝑓 ) 𝑓 𝜉 + 𝑤 (𝑏 − 𝜉) − 𝑤 (𝑏) 𝐺 (𝑑𝜉) − 𝑐 r𝐺¯ (𝑏) 0

∫ 𝑐 −𝑏 +𝜇 0

𝑤 (𝑏 + 𝜂) − 𝑤 (𝑏) 𝐺 + (𝑑𝜂),

(3) and the refill operator is (M𝑤)(𝑏) = sup0<Δ≤𝑐 −𝑏 {𝑤 (𝑏 + Δ) − 𝐾 − 𝑘Δ}. The first bracket is a served forward’s fee plus its change in

By first order we mean that a level off by a factor costs more than the skew can add (§4.3): in proportion to the error where flow balance binds, and 1 − 𝑟𝑒 1−𝑟 (26% at 𝑟 = 2) at the markup peak. Let 𝑓¯ be the flow-balancing fee, at which demand balances replenishment in Λ𝑚 1 volume, 𝑚 1 𝜆( 𝑓¯) = 𝜇𝑚 +1 ; under exponential demand 𝑓¯ = 𝜅1 ln 𝜇𝑚 +. 1 When flow balance lies above the refill trigger, in the diffusion approximation the level is max{1/𝜅, 𝑓¯}: the shadow price there is max{0, 𝑓¯−1/𝜅}, so the level is the markup 1/𝜅 when replenishment keeps up with demand at that fee, else 𝑓¯. Either way it is set by demand and replenishment alone and scales as 1/𝜅. A revenue ceiling at flow balance. Write Φ(𝑠) := 𝑚 1 sup 𝑓 ≥0 𝜆(𝑓 ) (𝑓 − 𝑠) for the fee income of a node that values liquidity at 𝑠, and 𝐹 (𝑠) := Φ(𝑠) + 𝜇𝑚 +1 𝑠 for its fees plus inventory value. 𝐹 is convex for any demand, Φ being a supremum of affine functions of 𝑠 (for exponential demand Φ(𝑠) = 𝑚𝜅𝑒1 Λ 𝑒 −𝜅𝑠 ); where some 𝑠ˆ∗ ∈ [0, 𝑘] has 𝑓¯ as its best fee, its minimum over [0, 𝑘] is attained there, where demand balances replenishment (ˆ𝑠 ∗ = 𝑓¯ − 1/𝜅 for exponential demand), and equals 𝑅 ∗ := 𝜇𝑚 +1 𝑓¯, the fee income of a channel held exactly at flow balance. Proposition 4.3 (Revenue ceiling; exact). Let the long-run rate of replenishment volume served be at most 𝜇𝑚 +1 , whether 𝜇 is constant, state-dependent or set by the network. Under Assumptions 1–2, every policy, with any fee rule and any refills, earns 𝑔 ≤ 𝐹 (𝑠) for every 𝑠 ∈ [0, 𝑘], so 𝑔 ≤ 𝐹𝑘 := min0≤𝑠 ≤𝑘 𝐹 (𝑠). For any decreasing demand

b The level is first order

A level plus a skew

0.4 fee

0.3 0.2 0.1

exact optimum f * (b) diffusion fee curve skew s(̂ b) − s(̂ b * ) level L = f * (b * )

six instances; a flat fee vs its level

revenue, % of optimum

one instance; exact vs diffusion

0.0 0.00 0.25 0.50 0.75 1.00 balance b

The skew is second order

12 10 8 6 4 2 0

144 instances; skew gain vs level error

level + skew (optimum)

100 80 60 40

flat, demand/inflow 3 flat, demand/inflow 5 flat at 1/κ best flat fee

20 0

c

0.5 1.0 1.5 flat fee / flow-balancing fee f ̄

no refusal penalty refusal penalty cr 1/8 of the level loss

0 25 50 75 loss of a level off by 2× (%)

d The skew insures a low level

anchored skew at half the level (pts)

demand / replenishment

a

most a skew adds (%)

Sun et al.

cheap refills

costly refills

8

0

-1

-1

98

10

4

5

0

0

-3

97

18

4

3

0

-1

-4

76

18

0

1.5

1

2

0

0

2

0

0.02 0.08 0.2 0.02 0.08 0.2

forward size / capacity

Figure 2: The exact single channel: the level is first order, the skew second-order insurance. (a) One instance: the optimal fee, a level 𝐿 = 𝑓 ∗ (𝑏 ∗ ) plus a skew, against the diffusion curve. (b) Six instances: a flat fee’s revenue against its level, relative to the optimum (filled: best flat fee; hollow: the markup 1/𝜅). (c) 144 instances: the most any balance-dependent fee adds over the best flat fee, against the loss of a level off by a factor of two. (d) What the anchored skew (8) recovers at half the best level, by ratio of demand to replenishment, forward size and refill cost (cell medians; dot: a cell where the half level drains some channel into a net loss). under which 𝑓¯ maximizes 𝜆(𝑓 )(𝑓 − 𝑠) for some 𝑠 = 𝑠ˆ∗ ∈ [0, 𝑘] (for exponential demand, Λ𝑚 1 ≥ 𝑒 𝜇𝑚 +1 and 𝑘 ≥ 𝑠ˆ∗ ), 𝐹𝑘 = 𝑅 ∗ : no policy earns more than the fee income of a channel held at flow balance. In words: no fee rule, flat or balance-dependent, beats the ceiling that demand and replenishment set. A wrong level. It drains the channel or starves it. Let 𝑑 (𝑓 ) = 𝜇𝑚 +1 − 𝑚 1 𝜆(𝑓 ) be the drift of the balance under a constant fee 𝑓 , so 𝑑 ( 𝑓¯) = 0. Below 𝑓¯ a static fee drains the channel at rate |𝑑 (𝑓 )|, above it starves it. A wrong level is also costly. When demand outruns replenishment, the markup is the wrong level: on six exactly solved instances a flat fee there loses 6–99% (Fig. 2(b)).

4.3

The skew is second-order insurance

Why its revenue value is small. Let 𝜒 be the scale over which the marginal value changes and 𝜀 := 𝑚 2 /(𝑚 1 𝜒). When 𝜀 ≪ 1 and the balance lies well above 0, 𝑠ˆ = ℎ ′ + 𝑂 (𝜀) and, in the diffusion approximation [2, 12] (checked against the exact model in App. A), the average-reward optimality equation reduces to 𝑔 = Φ(ℎ ′ ) + 𝜇𝑚 +1 ℎ ′ + 𝜈20 ℎ ′′,

(7) 2 ¯ where 𝜈 0 = 𝜆( 𝑓 )𝑚 2 + 𝜇 E𝜂 is the variance rate of the balance at flow balance, held constant. Refused replenishment at a full channel is costless, so ℎ ′ (𝑐) = 0. In the diffusion limit the shadow price falls with the balance (Proposition A.1). What it can add. In words: over the best flat fee, a skew adds at most what that fee loses to refused forwards (part (i)); its job is to keep a wrong level from draining the channel (part (ii)). Proposition 4.4 (Value of state dependence; partly diffusion). Under exponential demand with Λ𝑚 1 > 𝑒 𝜇𝑚 +1 : (i) let 𝑘 ≥ 𝑠ˆ∗ ; then the optimal policy earns 𝑔 < 𝑅 ∗ . If demand and replenishment forwards share one size law, the static fee 𝑓¯ without refills
earns exactly 𝑅 ∗ − Γ, with Γ := 𝜆( 𝑓¯) 𝑓¯ E[𝜉 (𝜉 ∧ 𝑐)] + 𝑐 r E[𝜉 ∧ 𝑐] /𝑐, which

is ( 𝑓¯ + 𝑐 r𝑚 1 /𝑚 2 ) 𝜈 0 /(2𝑐) when 𝜉 ≤ 𝑐, as it is for any size laws in the diffusion regime; the optimal policy gains at most Γ over the best static fee, and in the diffusion regime, if 𝑏 ∗ exists, the skew slope 𝑎 = 2(𝑅 ∗ − 𝑔)/𝜈 0 at flow balance satisfies 𝑎 𝑐 ≤ 𝑓¯ + 𝑐 r𝑚 1 /𝑚 2 . (ii) A static fee 𝑓0 loses (or must refill) net volume at rate at least |𝑑 (𝑓0 )|, by flow conservation, and at rate |𝑑 (𝑓0 )| (1 + 𝑜 (1)) in the diffusion regime with |𝑑 (𝑓0 )| 𝑐 ≫ 𝜈 0 . An affine skew with the same level, 𝑓 (𝑏) = 𝑓0 − 𝑎(𝑏 − 𝑏 0 ), has zero drift at 𝑏 eq = 𝑏 0 + (𝑓0 − 𝑓¯)/𝑎; if 𝑏 eq ∈ (0, 𝑐) its loss rate is exponentially small in 𝜃 min(𝑏 eq, 𝑐 − 𝑏 eq ) 2 /𝜈 0 , with 𝜃 = 𝜅𝜇𝑚 +1 𝑎 its rate of mean reversion (App. A). Against the best static fee, state dependence adds little when payments are small relative to capacity (𝜈 0 /𝑐) and refusals cost little (𝑐 r𝑚 1 /𝑚 2 ), as in revenue management [6, 9, 20]; on six exactly solved instances the bound is 4% of 𝑅 ∗ without 𝑐 r and 7–22% with it, while the best constant fee loses 0.7–5.4%. The slack is of higher order in 𝛿 = 𝜈 0 /(2𝑐𝜇𝑚 +1 ): the best flat fee loses 𝛿 times a bracket, the optimum 𝛿 2 times one (App. A). What a skew does is part (ii): a static fee that misses 𝑓¯ drains or starves the channel, while a skew turns the level error into a shift of the operating balance, seen as fewer refills. The anchored skew. With the level held at 𝐿, a node can post ¯ 𝑓 (𝑏) = 𝐿 + 𝑠ˆ𝐿 (𝑏) − 𝑠ˆ𝐿 (𝑏), (8) where 𝑠ˆ𝐿 is the shadow price under the flat fee 𝐿, computable from the channel’s own flows at 𝐿, and 𝑏¯ the channel’s time-average balance. The anchor keeps the fee at 𝐿 in the typical state: the skew cannot move the level. §4.4 checks it on the exact instances.

4.4

Exact check

We solve the jump model exactly by relative value iteration (400 balance steps) on 144 instances that vary price sensitivity, demand over replenishment, forward size, refill cost and refusal penalty (App. A). The bound of Proposition 4.4(i) applies to the 72 that meet 0 < 𝑠ˆ∗ ≤ 𝑘; the computations below cover all 144. The optimal

Trim: Fee Level Matters More Than Skew in Lightning Routing

fee is a level plus a skew (Fig. 2(a); 𝜅 = 10, demand five times replenishment, 𝐾 = 0.05), and the level carries the revenue (Fig. 2(b)). Over the 144 instances the best flat fee comes within 1.6% of the optimum at the median and 9.6% at most. A flat fee at half or twice the best level loses at least 17.6% (40.5% at the median), at least 8 times what any balance-dependent fee adds on the same channel (Fig. 2(c); worst ratio 1/8.05, median 1/25). Below 𝑒 times replenishment the peak is smooth: revenue at 𝑟 times the peak level is 𝑟𝑒 1−𝑟 of it. Above, the level sits at the kink 𝑓¯ and errors cost first order in fee income, about 1 − 𝑟 below it, where the channel drains, and ¯ 1 − 𝑟𝑒 −𝜅 𝑓 (𝑟 −1) above; on the grid a flat fee 20% below or 25% above the best one loses 9.5 and 8.9% of the optimum at the median, and 17.0 and 21.5% with demand eight times replenishment. Beyond exponential sizes and demand, on 2,592 exact instances with lognormal or Pareto forward sizes and mixed or linear demand (App. A), the best flat fee stays within 1.1–2.3% of the optimum at the median, and a level off by a factor of two loses 20.6–61.1%; only under the mixed demand does it lose less than the flat fee’s gap, on 144 of 864 instances. The anchored skew (8) does not repair a wrong level. At the best flat fee it cuts the gap from 1.6 to 0.5% at the median, but at twice that fee it changes nothing at the median and at half it gains 0.3 points. It pays only on the 28 instances where the half level drains the channel into a net loss, cutting that net loss from 154 to 80% of the optimum at the median. It is insurance against a level set too low where refills are costly. Scaling its shadow price by 𝛽 = 0.5, 0.8 or 1.25 changes the loss at the best level by at most 1.1 points from 𝛽 = 1 on ten instances (App. A). The exact solutions thus support both roles: the best flat fee is near the optimum, and a factor-two level error costs far more than any skew adds. When does the skew pay? 𝜅 sets the level’s scale and leaves the ordering; forward size sets the skew’s bound, and the ratio of demand to replenishment sets 𝑓¯. What a skew adds grows with the ratio of demand to replenishment, from 0.0 to 3.2% at the median, and with forward size, from 0.8 to 2.7%; with a refusal penalty it reaches 9.6%, without one 3.8%. The skew’s insurance has a region of its own: at half the best level it recovers most of the optimum only where refills are costly and forwards small against capacity (Fig. 2(d)); elsewhere it recovers at most 18 points or loses a few.

5

The Level at a Hub

In a hub the channels feed each other, so a channel’s replenishment is no longer exogenous; this section couples them. A forward enters a hub through one channel and leaves, and is priced, through another: a channel’s replenishment 𝜇𝑖 is exogenous to its own fee but falls with the hub’s other fees. Let channel 𝑗 face demand 𝜆 𝑗 (𝑓 ) = Λ 𝑗 ℓ (𝑓 ) with a common shape ℓ, and let a share 𝜋𝑖 𝑗 of the forwards leaving through 𝑗 enter through 𝑖, so Í 𝜇𝑖 = 𝑗≠𝑖 𝜋𝑖 𝑗 𝜆 𝑗 (𝑓 𝑗 ). We replace each channel’s replenishment by its time average under the posted fees, a mean-field decoupling that the exactly solved hubs below test. Proposition 5.1 (Hub level; mean-field, (ii) exact, (v) conjectured under coupling). Let the hub post one level 𝐿 on every channel and a skew on each, with 𝜇𝑖 decoupled as above. Then (i) its Í average revenue is 𝑖 𝑔𝑖 (𝐿; 𝜇𝑖 (𝐿)) and, given 𝐿, channel 𝑖’s best skew is that of Proposition 4.2 at 𝜇𝑖 (𝐿); (ii) under flat fees at 𝐿, 𝜇𝑖 (𝐿)/𝜆𝑖 (𝐿)

At a hub the level comes first

exactly solved two-channel hubs, 48 instances uniform level

+ skew

per-channel levels

balanced, equal shapes draining, equal shapes balanced, shapes differ draining, shapes differ

per-channel + skew

13 42 32 23

0

10

20

30

40

median loss against the exact optimum (%)

Figure 3: At a hub the level comes first. Exactly solved twochannel hubs: the median loss against the exact optimum of the best uniform level, and where a skew on it (orange) or per-channel levels (blue; diamond: with each channel’s own skew) take it, by flow balance and demand shape. does not depend on 𝐿: the level scales a channel’s inflow and outflow alike and leaves the sign of its drift, relative to its flow, unchanged; Í (iii) where refills and refusals are rare at 𝐿, 𝑖 𝑔𝑖 = 𝐿 𝑚 1 Λ ℓ (𝐿) +𝑜 (1) Í with Λ = 𝑖 Λ𝑖 , so the best level 𝐿 ∗ = arg max𝐿 𝐿ℓ (𝐿) is uniform and is each channel’s own best level, whatever the Λ𝑖 , 𝜋 and 𝜇𝑖 ; for ℓ (𝑓 ) = 𝑒 −𝜅 𝑓 , 𝐿 ∗ = 1/𝜅 and a level 𝑟𝐿 ∗ earns the fraction 𝑟 𝑒 1−𝑟 of the peak, 0.74 at 𝑟 = 2 and 0.82 at 𝑟 = 1/2; (iv) levels differ across channels only where refills bind, through 𝑓¯𝑖 at 𝜇𝑖 (𝑓−𝑖 ) (channel 𝑖 drains when 𝑓¯𝑖 > 𝐿 ∗ ) (a fixed point exists, not necessarily unique, with fees capped where demand is negligible), or where the shapes differ; (v) the Í skews gain at most 𝑖 ( 𝑓¯𝑖 + 𝑐 r𝑚 1 /𝑚 2 ) 𝜈 0,𝑖 /(2𝑐𝑖 ) (Proposition 4.4(i) per channel, under its conditions; with coupled replenishment a conjecture, which we test). Part (iv) gives the two cases for per-channel levels, binding refills and price discrimination across channels; neither is a skew. Exactly solved hubs. Where a channel drains or demand shapes differ, per-channel levels, not a skew on a uniform level, recover what a uniform level loses (Fig. 3). Setup. We check this on hubs small enough to solve exactly: two or three channels and the rest of the hub, a deep channel, exchange forwards, each moving its size from the exit’s balance to the entry’s, so the state is the joint balance. Part of each channel’s traffic goes to the rest of the hub, so total balance is not conserved. Channel 1 drains and channel 2 fills, in a ratio no uniform level changes. Against the optimum (policy iteration) we evaluate the best uniform level, per-channel levels and each with the anchored skew (App. A, Table 5). These instances test the mechanism, not real hubs. Balanced flows, equal shapes. A uniform level is right: the channels’ own best levels lie within 3% of the uniform level and add at most 2.0 points. The uniformity of (iii) thus holds although refills are not rare, but its 𝑜 (1) does not: state dependence is worth 11–16% here, as each forward moves two balances and is 5–10% of capacity. A draining channel or different shapes. The uniform level loses 22–42% of the optimum at the median and per-channel levels cut that to 11–13%. A skew on the uniform level does not recover it (at worst the loss reaches 163% of the optimum, a net loss), as (iv) says. On per-channel levels the skew adds a further 7 points, more than on one channel, as the two balances move together. A factor-two

Sun et al.

a The level is first order

b Tuned tools mainly choose a level

150 125 100 75 50 25 0

L*

best tuned tool vs frontier, every setting

main

κ = 1000 κ = 2000

1000

0.50

0.55

per run, 2026, 40 hubs, 10 seeds

κ = 100 κ = 400

L*

levels, shared levels, MC tools, ppm tools, × skew at dep. level + skew

c Levels cut refills and refusals

1000 dep.

0.60 0.65 0.70 payment success

dep.

wallet limits persistent senders 20-wallet memory heavy-tailed WTP LND route cost all realism options full capacities economic refills

0.75

count per run

hub revenue (k sats)

levels trace a frontier; tools on or below it

shared MC

−20 −10 0 10 gap to frontier at equal success (%)

102 101 100

refills refused forwards shared MC

dep. 250 1000 L* uniform level (ppm)

Figure 4: Network evaluation on the 2026 snapshot, 40 largest hubs: the level is first order, and tuned tools mainly choose a level. (a) Hub revenue against payment success: uniform levels (lines; hollow: above 𝐿 ∗ ), tuned tools (diamonds: ppm ranges; squares: multiples) and a skew at deployed fees or on a level (circles, triangles). (b) The best tuned tool against the frontier at its own success, in each of the 19 settings and routing models (95% intervals; WTP: willingness to pay). (c) Refills (solid) and refused forwards (dashed) per run against the uniform level; hollow: deployed fees; filled: 𝐿 ∗ . error in the uniform level costs a further 18% at the median (at least 10%), 1.5 to 2.6 times (minimum, median) what that skew adds. Decoupling and the bound. Per-channel levels with each channel’s own skew come within 4.7% of the optimum at the median, the cost of the decoupling in (i). Part (v) applies to the 24 two-channel and 4 three-channel hubs with equal shapes: the optimum’s gain over per-channel levels reaches 0.41–0.83 of its bound on the 8 twochannel ones that meet 𝑘 ≥ 𝑠ˆ∗ , exceeds it on 2 of the 16 that do not (1.18 and 1.51), and all four three-channel ones hold. Three-channel hubs repeat these cases (App. D). At a hub too the level, set per channel where one drains, is the first-order choice; §6 turns this into a rule.

6

The Trim Rule: Choosing the Level

The level is first order and the skew second-order insurance (§4–§5), so the rule makes the level an explicit, measured choice and the skew an exception; §7 tests the claims behind it. The level cannot be computed from public data alone, as 𝜅 and the senders’ demand are not observed before a scan (given them, a model predicts it, §7.1); it also sets how many payments the node carries. Choosing the level. The operator first stops any other tool that writes the ppm field (charge-lnd, feeadjuster, LNDg’s auto-fees, CLBOSS). It then posts one uniform level 𝐿, which moves the node along a revenue–success frontier (§7). Levels above the revenue peak 𝐿 ∗ earn no more at lower success, so they are dominated; senders’ wallet fee limits cap the level where they bind (App. B.1). The operator traces its own frontier by posting a few uniform levels, skew off, and picks a point by revenue, success and welfare. One choice is the highest level that keeps network payment success within a chosen margin: on our frontier 1,000 ppm costs 4 points of network success. A large node, one that can hold each level for tens of thousands of its own forwards, traces its frontier: comparing revenue, each level must be held long enough to tell it from the next, about 32–40 thousand own forwards per level √ for levels √ 4% apart [10]. Counting forwards is cheaper: probe 𝐿/ 2 and 𝐿 2,

ˆ estimate the local elasticity 𝜅ˆ of the forward count and move to 1/𝜅, the first-order condition 𝐿 = 1/𝜅 (𝐿), which needs a few hundred pooled forwards per probe. In our network it finds the peak when the hubs pool their forwards, but a lone hub’s probes hold too few and it overshoots (§7.3). A small node, or any node searching alone, takes the level a model of its captive demand predicts (§7.1), or a similar hub’s, and keeps the skew off. Adding the skew. By default the rule posts no skew: with refills priced per amount it has no measurable net value in our network. Only where refills are frequent and costly at a level already chosen does it post the anchored skew (8). Net of the fixed-trigger refills, the skew at deployed fees gains 34.2 ± 6.3k (shared memory) and 1.2 ± 0.6k (mission control) sat per run at 𝐾 = 1,000 (skew tables fitted for 𝐾 = 1,000), and 688k and 22k at 𝐾 = 20,000; these are upper bounds, since a node would not refill at that cost on a fixed trigger. With refills priced per amount it changes net revenue by at most 0.1k (App. B.1). Added to 1,000 ppm or 𝐿 ∗ it changes net revenue by at most 0.6k, within the noise of the gain above, at either cost: a measurable net gain needs refills on a fixed trigger at 𝐾 ≥ 1,000 sat. Deployment. The level needs no node changes, only the ppm field; the skew needs a small program, written by the operator, that reads each channel’s balance and posts the skew (8). The Trim rule at a node. Input: local balances and forwards; the operator’s weight on revenue, success and welfare. Choose: a large node traces the frontier by posting a few uniform levels, each long enough in own forwards to tell them apart, and picks 𝐿; a small node takes a model’s or a similar hub’s level. Run: post 𝐿 on every channel. Only if refills remain frequent and costly, fit each busy channel’s flows, compute 𝑠ˆ𝐿 , and post the skew (8) whenever the balance moves by 5% of capacity. Guard: stop other fee tools; probe within [𝐿/2, 2𝐿]; to hold a chosen level, revert a move that loses more than twice the spread of the last

Trim: Fee Level Matters More Than Skew in Lightning Routing

eight probe pairs (it does not find one, §7.3); skew only channels with enough traffic.

Scope. The rule is for one hub choosing its own level: a hub repricing alone gains less than the 40 together (App. E), and 𝐿 ∗ is not an equilibrium, since with every hub at 𝐿 ∗ most gain by cutting (App. C). The peak is the revenue end of the frontier, not a recommendation: senders pay for it (§8).

7

Table 3: Policies on the 2026 snapshot under shared memory (S) and mission control (M): gross revenue (k sats per run), payment success, and gap to the frontier at the policy’s success (%, 95% interval; negative: below it); tools tuned over ppm ranges or multiples (×).

Network Evaluation

In a network the level depends on senders and other hubs, so we test the results of §4–§6 by simulation, with every other node at its deployed fee. We stated predictions for the level, the skew and the tools before the runs; those for the skew and the tools are confirmatory (§7.2, §7.4). Setup. The snapshot is the Lightning network of 4 August 2026 from public gossip [28, 41], 5,507 nodes and 27,063 channels with their posted fees. Topology, fees and the tools’ rules are real; only the payments are synthetic, as no public dataset carries forwarding traces. Each run routes 12,000 payments (mean 41k sat) with capacities proxied by the maximum HTLC size, scaled to 30% so that depletion occurs within a run. Senders accept a route with probability exp(−𝜅𝑝/𝑣), 𝜅 = 200, for path fee 𝑝 and amount 𝑣, take the cheapest path blind to balances, and retry up to 10 routes. They remember failures in one of two ways: with shared memory, all senders avoid a failed channel for 300 payments; mission control is kept per sender, as in LND; most senders pay once or twice, so it learns mostly within a payment’s retries (500 persistent senders exercise it, Fig. 4(b)). Channels start balanced. The 40 highest-degree hubs (the largest) run a candidate policy; a hub channel below 5% of capacity is refilled to 50% at 𝐾 = 100 sats. We run charge-lnd, feeadjuster and LNDg, implemented from their source and tuned on calibration seeds over absolute ranges and over multiples of deployed rates. We also run uniform levels from 250 to 8,000 ppm, among them 𝐿 ∗ , the revenue peak of a calibration scan (4,900 ppm; App. B), and the anchored skew (8). Frontier and tests. Results are on 10 disjoint evaluation seeds with paired 95% intervals. The frontier is the per-seed upper envelope of the revenue–success points of the deployed fees and the seven uniform levels run (250–8,000 ppm; Table 3 shows three), piecewise linear in success; the confirmatory tests add the tools’ centers (2,450, 6,125 ppm) and 12,000 ppm; choosing the levels on other seeds moves the tools’ gaps by at most 0.8 points (App. B). Two claims on the main setting are confirmatory. Both use onesided paired 𝑡-tests over the seeds, with a Holm correction over the 12 tests (three tools and three skew comparisons per routing model; family-wise 𝛼 = 0.05). First, no ppm-tuned tool beats the frontier by more than 3% (𝑝 ≤ 0.009 after Holm). Second, the skew adds less than 3% under mission control and 5% with shared memory, whose seed noise is larger. Everything else, including every robustness setting, is exploratory (App. B.1).

Policy

Gross S M

deployed fee

7.0

Success S M

Gap (%) S

M

38.6 0.75 0.70

0 −39.5±1.9

LNDg, ppm 115.0 130.3 0.62 0.55 charge-lnd, ppm 106.9 118.5 0.63 0.54 feeadjuster, ppm 94.2 113.3 0.66 0.60

−3.2±3.5 −1.5±2.3 −4.2±4.0 −10.5±2.8 −4.0±5.4 −7.6±2.5

charge-lnd, × feeadjuster, ×

37.1 35.6

55.8 0.68 0.63 −52.7±4.2 −50.2±1.6 57.7 0.68 0.63 −57.5±3.4 −48.8±2.0

skew at deployed 6.8 38.7 0.75 0.70 −21.0±6.5 −39.3±1.9 level 1,000 56.4 100.9 0.71 0.66 – – level 2,000 88.1 120.9 0.68 0.61 – – ∗ level 𝐿 118.9 130.1 0.62 0.54 – –

7.1

The level is first order

Finding 1 Under both routing models the level traces a revenue–success frontier. → From deployed fees to 𝐿 ∗ , hub revenue rises from 7.0k to 118.9k with shared memory and from 38.6k to 130.1k under mission control, the 40 hubs repricing together. At 1,000 ppm revenue is 2.6× the deployed fees’ under mission control and 8.1× with shared memory.

Each step of the level trades payment success for revenue (Table 3, Fig. 4(a)) and cuts refills and refused forwards (Fig. 4(c)): at 𝐿 ∗ their counts per run fall by 84–98%. As Proposition 5.1(ii) predicts, uniform levels from 1,000 to 8,000 ppm keep the hubs’ refills per thousand forwards at 3.1–3.9 (shared memory) and 1.6–2.3 (mission control), against 12.6 and 3.9 under the deployed, non-uniform fees. Much of the cut thus comes from posting one level on every channel, and the rest from carrying fewer forwards. Net of refills at 𝐾 = 100 the ordering is unchanged: refills cost 0.2–1.4% of revenue on a level, and turn the deployed fees’ net revenue negative with shared memory. Nor do levels that differ across hubs or channel classes beat the uniform one: per-hub levels found by coordinate ascent and the best split of draining and filling channels earn no more than 𝐿 ∗ on the evaluation seeds (App. F). Where the peak sits. The peak is set by the senders’ acceptance rule and by how much of the hubs’ demand has no way around them. We route each payment at deployed fees around all 40 hubs. Then 35% of the hubs’ forwarded volume has no route around them, 99% of it through one hub and 88% because its sender or receiver reaches only hubs (App. B). Above a few hundred ppm the hubs thus earn from captive demand alone, 𝐿 𝑒 −𝜅𝐿 per hub hop, which peaks at 1/𝜅 = 5,000 ppm (Proposition 5.1(iii)). Predictions frozen before the runs held for the 10, 20 and 80 largest hubs (App. B). Posted fees sit below 𝐿 ∗ on 97% of channel directions, so real senders are likely more price-sensitive than ours, or competition binds. The peak’s location and the size of the multiples thus belong to the simulated senders; what we claim beyond them is the ordering.

Sun et al.

7.2

The skew is second-order insurance

Finding 2 The skew changes gross revenue by at most a few percent, as Proposition 4.4 allows here; it pays only where refills are frequent. → Alone at deployed fees or added to a level of 1,000 ppm or 𝐿 ∗ , the skew changes gross revenue within the noise under both routing models (below the confirmatory bounds); at deployed fees it cuts forced refills by 21% (shared memory) and 5% (mission control), raising net revenue by 3.5 ± 0.7k and 0.2 ± 0.1k per run at 𝐾 = 100. On a level the hubs refill only 2–9 times per run, and the skew has little left to save.

At deployed fees its −21% gap in Table 3 comes from less success on the frontier’s steep low end, not from less revenue. The skew has room to matter here: forwards are not small against the scaled capacities, as the size-biased mean forward 𝑚 2 /𝑚 1 is 6% of capacity at the volume-weighted median hub channel, so Proposition 4.4 allows a skew a few percent of revenue, as observed. At deployed fees, scaling the skew by 0.5–8× keeps its refill cut at 19–23% (shared memory) and 6–8% (mission control) and gross revenue within 1.4% of the deployed fees’ (5 seeds; App. G).

7.3

Choosing the level explicitly

The forward-counting search of §6 finds the level when the 40 hubs pool their forwards: probing for 1,500 payments each, it ends at 4,300–5,600 ppm (shared memory) and 3,300–4,000 ppm (mission control) from starts of 350–8,000 ppm, within 24,000 payments. Per hub it overshoots (8,700–10,500 ppm), and the probes cost 4–29% of revenue against posting 𝐿 ∗ throughout (App. H). Probes sized in network payments do not identify the level: at 600 payments a probe holds only 1–6 forwards of the average hub (App. B), so searches started at 1,000 ppm, per hub or with one level for all, end 4–10% below the frontier at their success, near where they start; when all hubs search at once, the end depends on the start. A guard that undoes losing moves holds a level but does not find one: from 350 ppm it reaches only 400–450 ppm and earns 16–27% less than the unguarded search.

7.4

Table 4: From the exact model to the network, per parameter. Exact: single-channel medians; skew adds: best balancedependent over best flat fee; level error: flat fee at half or twice the best level. MC: mission control. † Upper bound: best level on the evaluation seeds. Parameter

Exact model

skew adds 1.0–2.2% (𝜅 = 5–20) size / capacity skew adds 0.8 → 2.7% (size 0.02 → 0.2 of capacity)

𝜅

Network

𝜅 = 100–2,000: 𝐿 ∗ from 8,000 to 500 ppm, as 1/𝜅 capacity 15–100%, payments 5× : multiples 3.3–20.9× † ; tools at most 2.9% above at full capacities uniform levels: 3.1–3.9 refills per 1,000 forwards, deployed fees 12.6

demand / skew adds 0.0 → 3.2%; level replenishment error 22 → 50% (medians, ratio 1.5 → 8) 𝐾 = 100, 1,000, 20,000, economic refill cost skew adds 1.3 → 2.2%; insures a low level only where refills: the skew’s net gain scales refills are costly with 𝐾 hub coupling draining hubs: per-channel per-hub or per-class levels: no levels cut the loss from 22–42 gain ( −7.8% per hub, MC) to 11–13%, a skew on the uniform level does not senders, – two memory models (multiples routing 3.4 vs 17.1× ), memory pooled by 1–100 wallets (3.2–3.5× ), wallet limits, persistent senders, heavier tails; 2019 balance-aware senders: tools +14–24%

uniform level on both the 2026 and the 2019 snapshot (App. B.1). Tuned instead to the success of 1,000 ppm, the tools lie 8.1–10.0% below the frontier with shared memory and −4.3 to +2.9% off it under mission control (App. I). Across settings. We repeat the test in the 19 settings of Fig. 4(b) (Table 4) and on the 2019 snapshot. The ordering holds in every 2026 setting: in every row of Fig. 4(b) the best tool’s estimate is at most +2.9%. Its interval excludes 0 in three cases, all under mission control: wallet fee limits (+0.9 ± 0.4%), LNDg at full capacities (+2.8±2.0%; App. B.1) and, outside Fig. 4(b), the success-constrained picks (+2.7–2.9%). Tools win clearly only on 2019, with balanceaware senders, and not through a skew (§8).

Tuned tools mainly choose a level 8

Finding 3 On the 2026 snapshot no tool lies more than a few percent above the frontier. → Centered near 𝐿 ∗ , the tools come within 1.5–10.5% of the frontier at equal success and earn no more than the flat level at their center, within the noise.

Tuned over absolute ranges, each tool’s best range posts 2,450– 6,125 ppm at a balanced channel: tuning a tool chooses a level. On charge-lnd’s calibration grid (six centers by three spreads), the center moves revenue up to 3.1× and 1.5× at a fixed spread. At the same effective level, the other spreads earn at most 1.8% more and up to 46% less (App. B). LNDg, near-flat, earns 3.2% (shared memory) and 1.5% (mission control) less than the frontier at its own success, inside the noise; charge-lnd and feeadjuster, which spread the fee over the balance, earn 4.0–10.5% less. Tuned as multiples of deployed fees, which keep the deployed spread, the tools lie 48.8–57.5% below the frontier; both picks sit on the grid’s top edge. Mechanisms that price imbalance, Fence [46], Di Stasi et al. [18] and the Merchant [44], calibrated to the deployed rate, earn less than a

Limitations and Conclusion

Limitations. Traffic: payments are synthetic, acceptance is exponential and demand is stationary. The peak’s location reflects the simulated senders (§7.1); the ordering holds across settings (Table 4). Routing: senders take the cheapest single path, blind to balances except through mission control. Scaled capacities put network success (0.70–0.75 at deployed fees) below the 99.7% a large node reports [47], and closer to full capacities tools edge above the frontier (Fig. 4(b)). The frontier is an empirical envelope of uniform levels, not the optimum over all policies. The tools are our reimplementations of their published rules, tuned on our seeds; on a live node’s traffic they could do better. Model: with an inbound fee the skew adds more, 3.8–8.3%, still below what a 20–25% level error costs (App. A); multi-part payments would shrink forwards, tightening the bound of §4.3. The 2019 exception: on the 2019 snapshot the level still earns 3.6–15.5× the deployed fees’ revenue (App. B.1), but with 𝜅 = 100 and senders that see balances, tuned charge-lnd earns 14–24% more than the frontier at its success. So do the deployed fees at their best uniform multiple on the calibration seeds

Trim: Fee Level Matters More Than Skew in Lightning Routing

(12×), which have no balance term and earn within 0.9% of it: the gain is price discrimination across channels, the case of Proposition 5.1(iv), not a skew; even there a skew on a uniform level changes revenue by −1.7 to +2.6%. Welfare. Senders pay: from deployed fees, 𝐿 ∗ lowers total surplus by 7.3 ± 1.3% and 6.7 ± 0.9%, against 2.0 ± 0.6% and 0.5 ± 0.3% at 1,000 ppm; at 𝐿 ∗ the surplus lost is 0.78 and 0.99 of the hubs’ net gain. Counting a refused sender as a total loss, these are upper bounds (App. J). Conclusion. A routing node’s level carries most of its revenue and should be chosen on a revenue–success frontier; the skew’s value lies in fewer refills, where refills are costly. Tuned tools beat a flat level at equal success by at most a few percent, except with 2019’s balance-aware senders. How real senders respond, and where levels settle when all hubs reprice, remain open. Use of generative AI. An LLM assistant helped write code and edit text; the authors take full responsibility.

Sun et al.

References [1] Kiana Asgari, Aida Afshar Mohammadian, and Mojtaba Tefagh. 2022. DyFEn: Agent-Based Fee Setting in Payment Channel Networks. arXiv:2210.08197. [2] Barış Ata, J. Michael Harrison, and Lawrence A. Shepp. 2005. Drift Rate Control of a Brownian Processing System. The Annals of Applied Probability 15, 2 (2005), 1145–1160. doi:10.1214/105051604000000855 [3] Zeta Avarikioti, Lioba Heimbach, Yuyi Wang, and Roger Wattenhofer. 2020. Ride the Lightning: The Game Theory of Payment Channels. In Financial Cryptography and Data Security (FC 2020) (LNCS, Vol. 12059). 264–283. [4] Zeta Avarikioti, Stefan Schmid, and Samarth Tiwari. 2024. Musketeer: IncentiveCompatible Rebalancing for Payment Channel Networks. In Advances in Financial Technologies (AFT) (LIPIcs, Vol. 316). 13:1–13:22. [5] Marco Avellaneda and Sasha Stoikov. 2008. High-Frequency Trading in a Limit Order Book. Quantitative Finance 8, 3 (2008), 217–224. [6] Siddhartha Banerjee, Daniel Freund, and Thodoris Lykouris. 2022. Pricing and Optimization in Shared Vehicle Systems: An Approximation Framework. Operations Research 70, 3 (2022), 1783–1805. doi:10.1287/opre.2021.2165 [7] Mahsa Bastankhah, Krishnendu Chatterjee, Mohammad Ali Maddah-Ali, Stefan Schmid, Jakub Svoboda, and Michelle Yeo. 2023. R2: Boosting Liquidity in Payment Channel Networks with Online Admission Control. In Financial Cryptography and Data Security (FC 2023) (LNCS). 309–325. doi:10.1007/978-3031-47754-6_18 [8] Ferenc Béres, István A. Seres, and András A. Benczúr. 2021. A Cryptoeconomic Traffic Analysis of Bitcoin’s Lightning Network. Cryptoeconomic Systems 1, 1 (2021). DOI: 10.21428/58320208.d4cd697e. Fee-policy distribution: Sec. 5 and Fig. 17 (67.5% of channels at the default base fee, 66.7% at the default fee rate). [9] Omar Besbes, Adam N. Elmachtoub, and Yunjie Sun. 2022. Static Pricing: Universal Guarantees for Reusable Resources. Operations Research 70, 2 (2022), 1143–1152. doi:10.1287/opre.2020.2054 [10] Omar Besbes and Assaf Zeevi. 2009. Dynamic Pricing Without Knowing the Demand Function: Risk Bounds and Near-Optimal Algorithms. Operations Research 57, 6 (2009), 1407–1420. doi:10.1287/opre.1080.0640 [11] bitromortac. 2024. Blazing the Trails: Improving LND Pathfinding Reliability. Lightning Labs blog, https://lightning.engineering/posts/2024-05-23pathfinding-2/. [12] Ping Cao and Dacheng Yao. 2018. Optimal Drift Rate Control and Impulse Control for a Stochastic Inventory/Production System. SIAM Journal on Control and Optimization 56, 3 (2018), 1856–1883. doi:10.1137/16M110246X [13] Andrea Carotti, Cosimo Sguanci, and Anastasios Sidiropoulos. 2024. Rational Economic Behaviours in the Bitcoin Lightning Network. In IEEE ICBC. 594–601. doi:10.1109/ICBC59979.2024.10634468 [14] charge-lnd contributors. 2026. charge-lnd: Policy-Based Fee Manager for LND. https://github.com/accumulator/charge-lnd. Strategy proportional in charge_lnd/strategy.py. Accessed 2026-09. [15] Xin Chen and David Simchi-Levi. 2004. Coordinating Inventory Control and Pricing Strategies with Random Demand and Fixed Ordering Cost: The Infinite Horizon Case. Mathematics of Operations Research 29, 3 (2004), 698–723. doi:10. 1287/moor.1040.0093 [16] Mark H. A. Davis. 1993. Markov Models and Optimization. Chapman & Hall. [17] Anupa De Silva, Subhasis Thakur, and John Breslin. 2023. Profitable Fee Controller for Payment Channel Networks. In Blockchain and Applications, 4th International Congress (BLOCKCHAIN 2022) (LNNS). Springer, 88–99. doi:10.1007/9783-031-21229-1_9 [18] Giovanni Di Stasi, Stefano Avallone, Roberto Canonico, and Giorgio Ventre. 2018. Routing Payments on the Lightning Network. In IEEE iThings/GreenCom/CPSCom/SmartData. 1161–1170. doi:10.1109/Cybermatics_2018.2018.00209 [19] Oğuzhan Ersoy, Stefanie Roos, and Zekeriya Erkin. 2020. How to Profit from Payments Channels. In Financial Cryptography and Data Security (FC 2020) (LNCS, Vol. 12059). 284–303. [20] Guillermo Gallego and Garrett van Ryzin. 1994. Optimal Dynamic Pricing of Inventories with Stochastic Demand over Finite Horizons. Management Science 40, 8 (1994), 999–1020. doi:10.1287/mnsc.40.8.999 [21] Zhonghui Ge, Yi Zhang, Yu Long, and Dawu Gu. 2022. Shaduf: Non-Cycle Payment Channel Rebalancing. In Network and Distributed System Security Symposium (NDSS). [22] Olivier Guéant, Charles-Albert Lehalle, and Joaquin Fernandez-Tapia. 2013. Dealing with the Inventory Risk: A Solution to the Market Making Problem. Mathematics and Financial Economics 7, 4 (2013), 477–507. [23] Onésimo Hernández-Lerma and Jean B. Lasserre. 1996. Discrete-Time Markov Control Processes: Basic Optimality Criteria. Springer. [24] Rami Khalil and Arthur Gervais. 2017. Revive: Rebalancing Off-Blockchain Payment Networks. In ACM Conference on Computer and Communications Security (CCS). 439–453. [25] Lightning Labs. 2024. L402: Lightning HTTP 402 Protocol. https://github.com/ lightninglabs/L402. [26] Lightning Labs. 2024. lnd v0.18.0 Release Notes: Inbound Routing Fees. https://github.com/lightningnetwork/lnd/blob/master/docs/release-notes/

release-notes-0.18.0.md. [27] Lightning Labs. 2026. lnd Sample Configuration (sample-lnd.conf) and chainreg Defaults. https://github.com/lightningnetwork/lnd/blob/master/ sample-lnd.conf. [28] Lightning RFC contributors. 2023. BOLT #7: P2P Node and Channel Discovery. https://github.com/lightning/bolts/blob/master/07-routing-gossip.md. [29] lightningd/plugins contributors. 2026. feeadjuster: Balance-Based Fee Adjustment Plugin for Core Lightning. https://github.com/lightningd/plugins/tree/ master/feeadjuster. [30] LNDg contributors. 2026. LNDg: Lite GUI and Auto-Fees for LND (af.py). https://github.com/cryptosharks131/lndg. [31] David W. Low. 1974. Optimal Dynamic Pricing Policies for an M/M/s Queue. Operations Research 22, 3 (1974), 545–561. [32] Constantinos Maglaras and Assaf Zeevi. 2003. Pricing and Capacity Sizing for Systems with Shared Resources: Approximate Solutions and Scaling Relations. Management Science 49, 8 (2003), 1018–1038. [33] Shruti Mishra, Shubham Kapoor, and Sujata Pal. 2026. FeeDeR: Fee-Based BiDirectional Rebalancing in Payment Channel Networks. In IEEE International Conference on Blockchain and Cryptocurrency (ICBC). doi:10.1109/ICBC67748. 2026.11575561 [34] Nostr Protocol. 2023. NIP-57: Lightning Zaps. https://github.com/nostr-protocol/ nips/blob/master/57.md. [35] Bernt Øksendal and Agnès Sulem. 2019. Applied Stochastic Control of Jump Diffusions (3rd ed.). Springer. [36] Ioannis Ch. Paschalidis and John N. Tsitsiklis. 2000. Congestion-Dependent Pricing of Network Services. IEEE/ACM Transactions on Networking 8, 2 (2000), 171–184. [37] Rene Pickhardt. 2026. A Mathematical Theory of Payment Channel Networks. arXiv:2601.04835. [38] Rene Pickhardt and Stefan Richter. 2021. Optimally Reliable and Cheap Payment Flows on the Lightning Network. arXiv:2107.05322. [39] Joseph Poon and Thaddeus Dryja. 2016. The Bitcoin Lightning Network: Scalable Off-Chain Instant Payments. https://lightning.network/lightning-network-paper. pdf. [40] Suryanarayana Sankagiri and Bruce Hajek. 2025. Pricing for Routing and FlowControl in Payment Channel Networks. IEEE Transactions on Networking 33, 6 (2025), 3455–3470. doi:10.1109/TON.2025.3588049 [41] Arik Sosman. 2022. Announcing Rapid Gossip Sync. https://lightningdevkit.org/ blog/announcing-rapid-gossip-sync. Lightning Dev Kit blog, August 30, 2022. [42] Kalyan T. Talluri and Garrett J. van Ryzin. 2004. The Theory and Practice of Revenue Management. Springer. doi:10.1007/b139000 [43] Danila Valko and Jorge Marx Gómez. 2025. Geolocated Lightning Network Topology Snapshots: A Dataset Covering 2019–2023. Scientific Data 12 (2025), 1939. doi:10.1038/s41597-025-06413-7 Dataset: Harvard Dataverse, doi:10.7910/DVN/2OAVO6. [44] Yuup van Engelshoven and Stefanie Roos. 2021. The Merchant: Avoiding Payment Channel Depletion through Incentives. In IEEE International Conference on Decentralized Applications and Infrastructures (DAPPS). 59–68. [45] Xiaojian Wang, Huayue Gu, Zhouyu Li, Fangtong Zhou, Ruozhou Yu, and Dejun Yang. 2022. Why Riding the Lightning? Equilibrium Analysis for Payment Hub Pricing. In IEEE ICC. 5409–5414. doi:10.1109/ICC45855.2022.9839171 [46] Xiaojian Wang, Ruozhou Yu, Dejun Yang, Guoliang Xue, Huayue Gu, Zhouyu Li, and Fangtong Zhou. 2024. Fence: Fee-Based Online Balance-Aware Routing in Payment Channel Networks. IEEE/ACM Transactions on Networking 32, 2 (2024), 1661–1676. doi:10.1109/TNET.2023.3324136 [47] Sam Wouters. 2023. The Lightning Network Grew by 1212% in 2 Years. River Financial research report, https://www.river.com/learn/files/river-lightningreport-2023.pdf. Data for August 2023. [48] Yueqi Wu, Huiping Sun, Peilu Guo, Yiming Zhu, and Zhong Chen. 2026. Sluice: Global Invariant, Local Enforcement for Pooled Payment-Channel Liquidity. arXiv:2609.29975 [cs.CR] https://arxiv.org/abs/2609.29975 [49] Dacheng Yao. 2017. Joint Pricing and Inventory Control for a Stochastic Inventory System with Brownian Motion Demand. IISE Transactions 49, 12 (2017), 1101– 1111. [50] ZmnSCPxj and CLBOSS contributors. 2026. CLBOSS: The Core Lightning Node Manager (fee modifiers). https://github.com/ksedgwic/clboss.

Trim: Fee Level Matters More Than Skew in Lightning Routing

A Proofs and Exact Solutions A.1 Proofs for Section 4 We prove the results of §4 in their order. One more result is needed first: the diffusion fee curve and its slope at flow balance, which §4.3 cites and Fig. 2(a) compares with the exact optimum. Proposition A.1 (Fee curve and skew slope; diffusion). In the diffusion regime with constant variance rate 𝜈 0 and exponential demand, let 𝑔 be the optimal average reward, taken from the exact jump model, and assume 𝑔 < 𝐹 min := min𝑠 ≥0 𝐹 (𝑠), which is 𝑅 ∗ if Λ𝑚 1 ≥ 𝑒 𝜇𝑚 +1 and Φ(0) otherwise (Proposition 4.3 gives 𝑔 ≤ 𝑅 ∗ when 𝑘 ≥ 𝑠ˆ∗ ). Then: (i) (cf. [2, eqs. (20)–(22)]) above the refill trigger the optimal shadow price solves
𝑠ˆ′ (𝑏) = − 𝜈20 𝐹 (𝑠ˆ(𝑏)) − 𝑔 , 𝑠ˆ(𝑐) = 0, (9) and the optimal fee 𝑓 ∗ (𝑏) = 𝑠ˆ(𝑏) + 1/𝜅 is strictly decreasing in 𝑏; (ii)

if Λ𝑚 1 > 𝑒 𝜇𝑚 +1 and the curve reaches 𝑠ˆ∗ , then at the flow-balance point 𝑏 ∗ , where 𝑠ˆ(𝑏 ∗ ) = 𝑠ˆ∗ and the fee is 𝑓¯, the skew slope is 2 (𝑅 ∗ − 𝑔) 𝑎 := −ˆ𝑠 ′ (𝑏 ∗ ) = , (10) 𝜈0 and the skew is at least this steep everywhere. Proof of Proposition 4.1. Between events the balance does not move, and every fee gives total event rate 𝜆(𝑓 )+𝜇 ≤ 𝑞 := 𝜆(0)+𝜇 (Assumption 1). Uniformizing at rate 𝑞, the   continuation operator (𝑇𝑤) (𝑏) = sup 𝑓 ≥0 (L 𝑓 𝑤)(𝑏) + 𝑞 𝑤 (𝑏) /(𝜌 + 𝑞) has bounded rewards (𝑓 𝜆(𝑓 ) is continuous and vanishes as 𝑓 → ∞; 𝜉 ≤ 𝑐; sup ∅ = −∞ in M at 𝑏 = 𝑐) and is a contraction with modulus 𝑞/(𝜌 + 𝑞) on bounded functions. Two consecutive refills cost at least as much as one of the same total size (𝐾 > 0, linear 𝑘), so refilling twice in a row is never optimal, and 𝑉 is a fixed point of 𝑤 ↦→ max{𝑇𝑤, M𝑇𝑤 }. M is nonexpansive (a supremum of translates less constants), so this map is again a contraction with modulus 𝑞/(𝜌 + 𝑞), and its unique bounded fixed point is 𝑉 . Rearranged, the fixed-point equation is (4), and any bounded solution of (4) is such a fixed point [16, 35]. □ Proof of Proposition 4.2. The 𝑓 -dependent term of (3) is 𝑢 (𝑓 ) = ∫𝑏 𝜆(𝑓 ) [𝑓 𝑚(𝑏) + 𝐷 (𝑏)] with 𝐷 (𝑏) = 0 (𝑉 (𝑏 − 𝜉) − 𝑉 (𝑏))𝐺 (𝑑𝜉) − 𝑐 r𝐺¯ (𝑏) = −ˆ𝑠 (𝑏)𝑚(𝑏). So 𝑢 = 𝑚(𝑏)𝜆(𝑓 )(𝑓 − 𝑠ˆ(𝑏)) is continuous, positive exactly for 𝑓 > 𝑠ˆ(𝑏) (or everywhere if 𝑠ˆ(𝑏) < 0) and vanishes as 𝑓 → ∞ (Assumption 1): a maximizer exists, and at a positive one 𝑢 ′ (𝑓 ) = 0; dividing by −𝜆 ′ (𝑓 )𝑚(𝑏) > 0 gives 𝑓 − 𝑠ˆ(𝑏) = 𝜓 (𝑓 ). Where 𝑢 > 0, log 𝑢 = log 𝑚(𝑏) + log 𝜆(𝑓 ) + log(𝑓 − 𝑠) with 𝑠 = 𝑠ˆ(𝑏) has 𝜕 2 /𝜕𝑓 𝜕𝑠 = (𝑓 − 𝑠) −2 > 0. If 𝑠 < 𝑠 ′ , 𝑓 maximizes at 𝑠, 𝑓 ′ at 𝑠 ′ and 𝑓 ′ < 𝑓 , then 𝑓 > 𝑓 ′ > 𝑠 ′ and log 𝑢 (𝑓 , 𝑠 ′ ) − log 𝑢 (𝑓 ′, 𝑠 ′ ) > log 𝑢 (𝑓 , 𝑠) − log 𝑢 (𝑓 ′, 𝑠) ≥ 0, so 𝑓 ′ is not a maximizer: every maximizer at 𝑠 ′ is at least every maximizer at 𝑠, and the largest is nondecreasing. If 𝑓 − 𝜓 (𝑓 ) is strictly increasing the root is unique. For exponential demand 𝜓 ≡ 1/𝜅. □ Lemma A.2 (Monotone, concave value). In the diffusion regime of §4.3, 𝑉 is nondecreasing and concave and 𝑠ˆ is nonincreasing, strictly under the average criterion (Proposition A.1). Corollary A.3 (Reservation-fee skew). Wherever 𝑠ˆ is nonincreasing in the balance, so is 𝑓 ∗ (Proposition 4.2): the node charges more on a depleted channel and less on a flush one.

Proof of Lemma A.2. In the diffusion regime (7) with discounting, 𝑠 = 𝑉 ′ solves 𝜌𝑠 = [Φ′ (𝑠) + 𝜇𝑚 +1 ]𝑠 ′ + 𝜈20 𝑠 ′′ with 𝑠 = 𝑘 ≥ 0 at the refill boundary (or, without refills, 𝑠 (0) ≥ 0: 𝑉 is nondecreasing, since a copy started with more balance, under the same fees, refuses no more forwards until the two meet) and 𝑠 (𝑐) = 0. Monotonicity. At an interior minimum, 𝑠 ′ = 0 and 𝑠 ′′ ≥ 0 force 𝜌𝑠 ≥ 0; with the boundary values, 𝑠 ≥ 0. Concavity. At an interior local maximum 𝑠 ′′ ≤ 0 forces 𝑠 ≤ 0, so 𝑠 = 0 there. If 𝑠 ′ > 0 somewhere, then, since 𝑠 ≥ 0 and 𝑠 (𝑐) = 0, 𝑠 has a later interior local maximum with a positive value, a contradiction; hence 𝑠 ′ ≤ 0 and 𝑉 is concave; in the diffusion regime 𝑠ˆ is identified with 𝑠, so it is nonincreasing. Strict decrease under the average criterion is Proposition A.1(i), under its hypotheses. □ Proof of Proposition 4.3. Fix 𝑠 ∈ [0, 𝑘]. By the definition of Φ, 𝑚 1 𝜆(𝑓 )(𝑓 − 𝑠) ≤ Φ(𝑠) for every fee 𝑓 . Under any policy let 𝑄 and 𝑈 be the long-run average rates (along horizons attaining the lim inf in 𝑔) of served demand volume and of refilled volume; the balance being bounded, 𝑄 ≤ 𝜇𝑚 +1 + 𝑈 . At fee 𝑓 volume is served at a rate 𝑞 ≤ 𝑚 1 𝜆(𝑓 ), and 𝑓 𝑞 = (𝑓 −𝑠)𝑞 +𝑠𝑞 ≤ Φ(𝑠) +𝑠𝑞 (the first term is negative if 𝑓 < 𝑠, and Φ ≥ 0), so fee income is at most Φ(𝑠) + 𝑠𝑄. Refills cost at least 𝑘𝑈 and refusal penalties are nonnegative, so 𝑔 ≤ Φ(𝑠)+𝑠 𝜇𝑚 +1 +(𝑠 −𝑘)𝑈 ≤ 𝐹 (𝑠). 𝐹 is convex, Φ being a supremum of functions affine in 𝑠. Since Φ(𝑠) ≥ 𝑚 1 𝜆( 𝑓¯)( 𝑓¯ − 𝑠), 𝐹 (𝑠) ≥ 𝑅 ∗ for every 𝑠, with equality at an 𝑠ˆ∗ where 𝑓¯ attains the supremum (assumed: with demand that is not log-concave the maximizer can jump over 𝑓¯); there 𝐹 (𝑠ˆ∗ ) = 𝑚 1 𝜆( 𝑓¯)( 𝑓¯ − 𝑠ˆ∗ ) + 𝜇𝑚 +1 𝑠ˆ∗ = 𝜇𝑚 +1 𝑓¯ = 𝑅 ∗ . 𝑠ˆ∗ ≥ 0 when the fee maximizing 𝜆(𝑓 )𝑓 is at most 𝑓¯, and 𝑠ˆ∗ ∈ [0, 𝑘] when also 𝑘 ≥ 𝑠ˆ∗ . □ Proof of Proposition A.1. Diffusion reduction. Expanding ℎ(𝑏)− ℎ(𝑏 − 𝜉) = 𝜉ℎ ′ (𝑏) − 12 𝜉 2ℎ ′′ (𝑏) + 𝑂 (𝜉 3 ) in (5) (with ℎ in place of 𝑉 ),   𝑚3 2 𝑠ˆ(𝑏) = ℎ ′ (𝑏) − 12 ℎ ′′ (𝑏) 𝑚 𝑚 1 + 𝑂 𝑚 1 + 𝑅(𝑏), where 𝑅(𝑏) collects the refusal term and the truncation of the integral at 𝑏; both are bounded by a constant times 𝐺¯ (𝑏)/𝑚(𝑏), which is 𝑂 (𝜀) in the region considered. Since |ℎ ′′ |/|ℎ ′ | = 𝑂 (1/𝜒), 𝑠ˆ = ℎ ′ + 𝑂 (𝜀). Expanding the replenishment term the same way and maximizing the demand term over 𝑓 turns the average-reward optimality equation into (7). The curve. Put 𝑠 = ℎ ′ . Then (7) reads 𝑔 = 𝐹 (𝑠) + 𝜈20 𝑠 ′ , which is (9); 𝑠 (𝑐) = 0 because refused replenishment at 𝑏 = 𝑐 is costless (reflection without cost). This is the reduction of [2, §3]: with drift 𝑑 (𝑓 ) = 𝜇𝑚 +1 − 𝑚 1 𝜆(𝑓 ), reflected coordinate 𝑧 = 𝑐 −𝑏, and cost rate 𝑚𝜅𝑒1 Λ −𝑚 1 𝜆(𝑓 )𝑓 (restricting 𝑓 ≥ 1/𝜅, which is without loss since 𝑠 ≥ 0), their 𝜑 and 𝛾 become 𝐹 − 𝑚𝜅𝑒1 Λ and 𝑚𝜅𝑒1 Λ −𝑔, so their eqs. (20)–(22) are (9) and the quadrature. The equation holds for the optimal 𝑔, which the lower (depletion or refill) boundary pins; in the diffusion limit the fee at 𝑏 = 0 is a degenerate control, so we take 𝑔 from the jump model. For exponential demand the supremum in Φ is attained at 𝑓 = 𝑠 + 1/𝜅, so 𝑓 ∗ = 𝑠ˆ + 1/𝜅 and Φ(𝑠) = 𝑚𝜅𝑒1 Λ 𝑒 −𝜅𝑠 . Then 𝐹 ′ (𝑠) = 𝜇𝑚 +1 − 𝑚 1 𝜆(𝑠 + 1/𝜅) and 𝐹 ′′ (𝑠) = 𝜅𝑚 1 𝜆(𝑠 + 1/𝜅) > 0, so 𝐹 is strictly convex with its minimum where 𝑚 1 𝜆(𝑠 +1/𝜅) = 𝜇𝑚 +1 , i.e. at 𝑠ˆ∗ , and 𝐹 (𝑠ˆ∗ ) = 𝜅1 𝜇𝑚 +1 + 𝜇𝑚 +1 𝑠ˆ∗ = 𝑅 ∗ . (i) Since 𝐹 (𝑠) ≥ 𝐹 min > 𝑔 on 𝑠 ≥ 0, (9) gives 𝑠 ′ < 0 everywhere. (ii) At 𝑏 ∗ , 𝐹 (𝑠 (𝑏 ∗ )) = 𝑅 ∗ , which gives (10); 𝑏 ∗ is the flow-balance point because the controlled drift 𝜇𝑚 +1 − 𝑚 1 𝜆(𝑓 ∗ (𝑏)) vanishes exactly where 𝑠 = 𝑠ˆ∗ ; and −𝑠 ′ (𝑏) = 𝜈20 (𝐹 (𝑠 (𝑏)) −𝑔) ≥ 𝜈20 (𝑅 ∗ −𝑔) = 𝑎. Remark. If Λ𝑚 1 ≤ 𝑒 𝜇𝑚 +1 ,

Sun et al.

then for 𝑠 ≥ 0, 𝐹 ′ (𝑠) = 𝜇𝑚 +1 − 𝑚 1 𝜆(𝑠 + 1/𝜅) ≥ 𝜇𝑚 +1 − 𝑚 1 Λ/𝑒 ≥ 0, and under the optimal fee 𝑓 ∗ ≥ 1/𝜅 the drift 𝜇𝑚 +1 − 𝑚 1 𝜆(𝑓 ∗ ) ≥ 𝜇𝑚 +1 − 𝑚 1 Λ/𝑒 ≥ 0, so the balance drifts toward 𝑐, where 𝑠 (𝑐) = 0 and 𝑓 ∗ = 1/𝜅: the fee there is the markup alone. With discounting, 𝜌𝑉 replaces 𝑔 in (7) and the same steps 𝑎 = 2(𝑅 ∗ − 𝜌𝑉 (𝑏 ∗ ))/𝜈 0 . ∫ give 𝜈0 𝑠 𝑑𝑣 Integrating (9) gives 𝑏 (𝑠) = 𝑐 − 2 0 𝐹 (𝑣) −𝑔 . □ Proof of Proposition 4.4. (i) By Proposition 4.3, 𝑔 ≤ 𝑅 ∗ . Equality 𝑔 = 𝑅 ∗ would need the fee to equal 𝑓¯ almost surely (the supremum defining Φ(𝑠ˆ∗ ) is attained only there), and that constant fee refuses demand at a positive rate (below), so 𝑔 < 𝑅 ∗ . Under the constant fee 𝑓¯ without refills, let both sides’ sizes have law 𝐺; then 𝑚 1 = 𝑚 +1 and flow balance give 𝜆( 𝑓¯) = 𝜇, and the uniform law on [0, 𝑐] satisfies detailed balance: a jump from 𝑏 to 𝑏 − 𝑥, allowed iff 𝑥 ≤ 𝑏, has rate 𝜆( 𝑓¯)𝐺 (𝑑𝑥), and the reverse jump from 𝑏 − 𝑥 to 𝑏, allowed iff 𝑏 ≤ 𝑐, has rate 𝜇𝐺 (𝑑𝑥). Under it the ∫ 𝑐refused demand volume and count per unit time are 𝜆( 𝑓¯)𝑐 −1 0 E[𝜉1{𝜉 > 𝑏}] 𝑑𝑏 = 𝜆( 𝑓¯)E[𝜉 (𝜉 ∧ 𝑐)]/𝑐 and 𝜆( 𝑓¯)E[𝜉 ∧ 𝑐]/𝑐; for 𝜉 ≤ 𝑐, with 𝜈 0 = 2𝜆( 𝑓¯)𝑚 2 , these are 𝜈 0 /(2𝑐) and (𝑚 1 /𝑚 2 ) 𝜈 0 /(2𝑐). For any size laws, in the diffusion regime the balance is a driftless Brownian motion with variance rate 𝜈 0 reflected at 0 and 𝑐, again uniform, with refused volume rate 𝜈20 · 𝑐1 (the local-time rate at 0). Each unit of refused volume costs the fee 𝑓¯ and each refused forward 𝑐 r , which gives the stated revenue. Subtracting gives the gain bound, and (10) with 𝑔 ≥ 𝑅 ∗ − ( 𝑓¯ + 𝑐 r𝑚 1 /𝑚 2 )𝜈 0 /(2𝑐) gives the slope bound. (ii) With drift 𝑑 < 0 the local-time rate at 0 is |𝑑 |/(1 − 𝑒 −2|𝑑 |𝑐/𝜈0 ) = |𝑑 |(1 + 𝑜 (1)) when |𝑑 |𝑐 ≫ 𝜈 0 ; under the affine skew the balance is an Ornstein–Uhlenbeck process around 𝑏 eq with variance 𝜈 0 /2𝜃 , whose boundary densities are Gaussian tails. The exponent holds while 𝜅𝑎 min(𝑏 eq, 𝑐 − 𝑏 eq ) ≲ 1; beyond, the drift toward depletion saturates at 𝜇𝑚 +1 and the exponent grows linearly, as 2𝜇𝑚 +1 𝑏 eq /𝜈 0 . □

A.2

Exact single-channel solutions

We solve the jump-model optimality equation exactly on 18 instances, 𝜅 ∈ {5, 10, 20}, two demand-to-replenishment ratios and three refill costs. On all 18 instances ℎ is nondecreasing and concave except on at most 5% of grid points, at the refill kink and the boundaries; 𝑓 ∗ is nonincreasing except on one instance, on 7.5% of grid steps just above the refill trigger. The 16 of the 18 with an interior flow-balance point satisfy the slope identity (10) within 0.12% (0.13% with the 8 binding grid instances whose flow-balance point lies well above the trigger). Table 6 (App. K, 𝐾 = 0.05) gives the jump model’s loss under the ODE curve (9), the best constant fee (with and without 𝑐 r ) and 1/𝜅. Given the exact 𝑔, on the 6 instances of that table the curve (9) loses at most 0.3% of optimal revenue, against 0.7–5.4% for the best constant fee (0.7–2.4% without 𝑐 r ). The grid of §4.4 has 144 instances (𝜅 ∈ {5, 10, 20}, demand-to-replenishment ratios 1.5–8, forwards of 2–20% of capacity, cheap and costly refills, with and without 𝑐 r ). The anchored skew (8) at 1/2, 1 and 2 times the best constant fee takes 𝑠ˆ𝐿 from the flat fee’s value function and 𝑏¯ under the posted fee (t1_skew_insurance.py); the 28 instances where the half level drains the channel into a net loss all have 𝑐 r > 0. Small forwards and estimation error. With forwards scanned from 16 to 1% of capacity (𝜅 = 10, demand 3 or 8 times replenishment,

with and without 𝑐 r , no refills; t1_asymptotic.py), the balance under a flat fee is a reflected diffusion whose boundary local times give the best flat fee’s loss 𝜇𝑚 +1 𝛿 min𝑥 [ 𝑓¯𝜗 (𝑥) + 𝑐 r′𝜗 (−𝑥) − 𝑥/𝜅] + 𝑂 (𝛿 2 ), with 𝛿 = 𝜈 0 /(2𝑐𝜇𝑚 +1 ) the share of volume refused, 𝜗 (𝑥) = 𝑥/(1 − 𝑒 −𝑥 ) and 𝑐 r′ = 𝑐 r𝑚 1 /𝑚 2 (𝑥 = 0 gives Γ); with demand eight times replenishment the exact loss is within 2.2% of it at every size, and within 0.4% of Γ without 𝑐 r . In the small-forward (widechannel) limit the optimum loses 2𝜋 2 𝜇𝑚 +1 𝛿 2 /𝜅, from the Riccati form of (7): the exact ratio rises from 0.33 to 0.91 as forwards shrink, and the gain reaches 0.82 of Γ at 1%. With demand three times replenishment, nearer the markup regime, convergence is slow: the first-order term is 1.3–2.3 times too small and the 𝛿 2 term reaches 0.15–0.42 of the exact loss. Scaling the anchored skew’s shadow price by 𝛽 ∈ {0.5, 0.8, 1.25} on ten instances (t1_skew_robust.py) changes the loss at the best flat fee by at most 1.1 points; at half that fee, where six of them drain into a net loss, any 𝛽 > 0 cuts that loss (from 101–262 to 57–122%). Other size laws and demand curves. The 2,592 instances (t1_general.py) cross exponential, lognormal and Pareto forward sizes with exponential, mixed (captive plus competed, 𝑤 = 0.44) and linear demand, six demand-to-replenishment ratios, four forward sizes, four refill costs and three refusal penalties; the flat-fee gap is 1.1–2.3% at the median and at most 15.8, 24.1 and 21.5% by demand. On the 684 with 𝑓¯ above the fee that maximizes 𝜆(𝑓 )𝑓 and 𝑘 ≥ 𝑠ˆ∗ , the optimum never exceeds 𝑅 ∗ (at most 0.999𝑅 ∗ ). Inbound fees. When the node also posts an inbound fee 𝑔in on replenishment, accepted with probability 𝑒 −𝜅𝑔in (t1_inbound.py; six instances, forwards of 8% of capacity), its best inbound fee is 1/𝜅 less the marginal value of incoming balance, so the skew enters both fees with opposite signs. A balance-dependent pair adds 3.8–8.3% over the best flat pair (0.1–7.2% for the outbound fee alone), while a flat fee 20% below or 25% above the best one, with the inbound fee re-optimized, loses 10.8–18.8%. At a hub the entering forward also pays an outbound fee elsewhere, so we report the structure, not the inbound income.

A.3

The hub

Proof of Proposition 5.1. (i) Channel 𝑖’s generator (3) involves the Í others only through 𝜇𝑖 = 𝑗≠𝑖 𝜋𝑖 𝑗 𝜆 𝑗 (𝐿 + 𝑠 𝑗 (𝑏 𝑗 )). Its time average Í 𝜇𝑖 (𝐿) = 𝑗≠𝑖 𝜋𝑖 𝑗 Λ 𝑗 ℓ (𝐿) ℓ¯𝑗 , with ℓ¯𝑗 the mean of ℓ (𝐿 + 𝑠 𝑗 )/ℓ (𝐿) (1 at the anchor of (8)), decouples the channels into problems of §4; Proposition 4.2 gives each skew. (ii) Under flat fees ℓ¯𝑗 = 1, so Í 𝜇𝑖 (𝐿)/𝜆𝑖 (𝐿) = 𝑗≠𝑖 𝜋𝑖 𝑗 Λ 𝑗 /Λ𝑖 , free of 𝐿, so 𝑑𝑖 (𝐿) = 𝜆𝑖 (𝐿) 𝑚 +1 𝜇𝑖 /𝜆𝑖 −
𝑚 1 has a sign free of 𝐿 and scales with the forward rate. (iii) 𝑔𝑖 = 𝐿 𝑚 1 𝜆𝑖 (𝐿)(1 − 𝑝𝑖 ) less refusal and refill costs, with 𝑝𝑖 the refused share; where these are rare, 𝑔𝑖 = 𝐿 𝑚 1 Λ𝑖 ℓ (𝐿) + 𝑜 (1), and the sum’s maximizer is free of the Λ𝑖 . For ℓ = 𝑒 −𝜅 𝑓 , 𝐿ℓ (𝐿) peaks at 1/𝜅 and 𝑟𝐿 ∗ ℓ (𝑟𝐿 ∗ )/𝐿 ∗ ℓ (𝐿 ∗ ) = 𝑟𝑒 1−𝑟 . (iv) By Proposition A.1 the level is max{𝐿 ∗, 𝑓¯𝑖 } with 𝑚 1 𝜆𝑖 ( 𝑓¯𝑖 ) = 𝑚 +1 𝜇𝑖 . Raising 𝑓 𝑗 lowers 𝜇𝑖 and so raises 𝑓¯𝑖 : the map 𝑓 ↦→ (min{max{𝐿 ∗, 𝑓¯𝑖 (𝑓−𝑖 )}, 𝑓max })𝑖 is nondecreasing on the complete lattice [0, 𝑓max ] 𝑛 , with 𝑓max a fee above which demand is negligible, so Tarski’s theorem gives a fixed point; with shapes ℓ 𝑗 that differ, so does the markup arg max𝐿 𝐿ℓ 𝑗 (𝐿). (v) Under the decoupling of (i), it sums Proposition 4.4(i) over the channels. □

Trim: Fee Level Matters More Than Skew in Lightning Routing

Exactly solved hubs. The hub of §5 has 𝑛 = 2 modelled channels (capacity 1, a grid of 40 balance steps each) and a deep channel 𝑅, the rest of the hub, with no liquidity limit (t1_hub_exact.py). A forward from 𝑖 to 𝑜 (𝑖 ≠ 𝑜 in {1, 2, 𝑅}) arrives at potential rate Λ𝑖𝑜 , is accepted with probability 𝑒 −𝜅𝑜 𝑓𝑜 , has an exponential size of mean 𝑚 1 ∈ {0.05, 0.1} on the grid, needs its size on the peer’s side of 𝑖 and on the hub’s side of 𝑜 (else it is refused, at cost 𝑐 r = 0.01), and moves its size from 𝑏𝑜 to 𝑏𝑖 . Rates, costs and solver details are in App. D, the grid in Table 5. Table 5: Exactly solved two-channel hubs (48 instances: forwards of 5 and 10% of capacity, drain ratios 1, 1.5 and 3, two traffic shares, equal or different demand shapes, two refill costs): loss against the exact optimum (%, median and range; above 100% a net loss). Hub

Uniform

Per-ch. Unif. + skew Per-ch. + skew

bal., equal shapes (8) 12.6 11–17 11.4 11–16 5.8 3–12 drain., equal shapes (16) 42.3 21–84 11.0 8–16 39.7 19–163 bal., shapes differ (8) 31.6 27–36 12.7 12–17 27.8 25–31 drain., shapes differ (16) 22.7 17–34 12.0 11–16 19.8 11–30

B

3.9 3–7 4.8 3–8 4.2 3–6 4.8 4–8

Evaluation Design

Parameters and code. Distributions, seeds, grids, tool rules and solver settings are in App. L; code, raw outputs and a number checker will be made public upon acceptance; null tests check that each extension reproduces the base simulator [8] bit for bit. Traffic and calibration. Offered focal demand falls exponentially in the posted fee near the deployed level (𝑅 2 = 0.95 over four exogenous fee scalings). A policy’s gap interpolates the frontier per seed at the policy’s own success. 𝐿 ∗ comes from a calibration scan of 15 uniform levels (25–12,000 ppm) with balances frozen, never on the grid edge in any setting. The captive share is computed from the snapshot (analysis/param_topo.py). The captivedemand prediction of §7.1 peaks at 4,990 ppm, stays within 0.02 of the scan’s revenue relative to its peak at every level, and moves to 1/𝜅 for 𝜅 from 100 to 2,000, where the evaluated best levels lie at 0.6–1.0 times it. The demand mixture of §3 is a two-part fit 𝜆(𝐿) = Λ[(1−𝑤)𝑒 − (𝜅𝑐 +200)𝐿 +𝑤𝑒 −200𝐿 ] to the calibration scan, with captive share 𝑤 = 0.44. The frozen predictions for 10, 20 and 80 hubs and their test are in App. M. With 160 and 320 hubs repricing (5 seeds, not predicted in advance) the best evaluated level stays at 3,500–4,900 ppm, earning 15.5 and 19.1× the deployed fees’ revenue with shared memory and 4.3 and 5.3× under mission control, while network success at it falls to 0.48 and 0.45 with 320 hubs (from 0.75 and 0.70 at deployed fees). Tools. A “day” of LNDg is 300 payments; with a day of 100 or 30 payments or its default cap, LNDg lies 1.1–16.6% below the frontier, and charge-lnd re-evaluated on every balance change 7.3% and 11.1%. We reimplement charge-lnd’s linear rule in the local share, feeadjuster’s 500.5−𝑥 multiplier and LNDg’s daily tiered adjustment. On a grid extended to 12,000 ppm feeadjuster’s score keeps rising to the edge, but the pick there lies 13.1 ± 6.6% below the frontier on the evaluation seeds, against 4.0±5.4% for the pick we report. Choosing the frontier’s levels on the other seeds changes the tools’ gaps by at

most 0.8 points (loo_frontier.py). Paired by seed, no tool earns significantly more than the flat level at its center (per-tool intervals in App. M). Power. With 10 seeds, a tool tied with the frontier would be shown not to exceed it by more than 4–5% (mission control) and 7–10% (shared memory) with 80% power at the strictest Holm level; the tuned tools lie 2–11% below it (on a wider frontier that also includes 2,450, 6,125 and 12,000 ppm). Per-test power and probe sizes: App. M.

B.1

Full design and results

The tuned picks are, with shared memory and mission control, LNDg starting at 4,900 ppm, charge-lnd over (2,450, 9,800) ppm and feeadjuster soft at 4,900 / 2,450 ppm, and as multiples charge-lnd (16, 1024)× and feeadjuster default at 512×. Ordering across settings. Per-setting results for every row of Fig. 4(b) (full capacities with 10 seeds, others as listed there) are in App. N; Table 4 summarizes them. Refills and the skew. With economic refills (𝐾 = 100 sat plus 300 ppm of the amount, only when the liquidity sold covers them), hubs at deployed fees refill under once per run, since refills rarely pay there, and the skew changes their net revenue by at most 0.1k. The 2019 snapshot. On the 2019 snapshot of Beres et al. [8] (real capacities; 9,000 payments, shared memory, 𝐾 ∈ {100, 1,000}) twothirds of the channel policies post LND’s default 1 ppm. Across hub tiers, destination skews, capacity scales and 𝜅 ∈ {100, 400}, the level earns 3.6–15.5× the deployed fees’ revenue at 𝐿 ∗ = 2,406– 9,671 ppm, with 59–97% fewer refills, and the skew cuts refills at deployed fees by 10–43%. Fence, the Merchant and Di Stasi et al., calibrated to each channel’s deployed rate, earn 2.1–13.7k on 2019, below the deployed fees’ 20.3k, and on 2026 lie 12–89% below the frontier at their success. The tools’ gains of §8 come from runs with 𝜅 = 100 and senders that route on known balances.

C

The Level

Levels in public gossip. In the 336 gossip snapshots of [43] (2019– 2023, posted fees) we follow the nodes ever among the 40 largest. Their level moves 1.7 times a year, by a median 2.3×, most channels at once. Pooled over fee setters (an operator’s nodes counted as one), 50% of the posted log-fee variation is the level over time, 33% persistent differences across channels and 16% the time-varying rest. Raises stick: after six months 83% are kept or extended, against 60% of cuts (Fisher 𝑝 = 0.006), and the hubs’ median level rose from 1 ppm in 2019–20 to 149–197 ppm in mid-2023. Competition. When all reprice, 𝐿 ∗ is not an equilibrium: in an exploratory best-response check (5 hubs, 10 seeds), with every hub at 𝐿 ∗ none gains by raising to 8,000 ppm and most gain by cutting, taking revenue from the other hubs, as the 40 together earn no more.

Sun et al.

D

Exactly Solved Hubs: Rates, Solver and Three-Channel Hubs

Exactly solved hubs, rates and costs. With 𝑔 = (𝜌 1/2, 𝜌 −1/2 ), Λ𝑖𝑜 = 𝑤𝐹𝑔𝑜 /𝑔𝑖 between the modelled channels, Λ𝑅𝑜 = (1 − 𝑤)𝐹𝑔𝑜 and Λ𝑖𝑅 = (1 − 𝑤)𝐹 /𝑔𝑖 , 𝐹 = 0.4/𝑚 1 , 𝜌 ∈ {1, 1.5, 3} and 𝑤 ∈ {0.5, 0.8}; 𝜅 = 10 on every exit, or 20 on channel 1 when the shapes differ; refills top any set of channels up at cost 𝐾 + 𝑘Δ per channel, 𝐾 ∈ {0.05, 0.5}, 𝑘 = 0.05. Exactly solved hubs: solver and per-instance results. Policy iteration with exact linear solves gives the optimum, every exit’s fee set state by state in closed form; for a fixed fee rule the same iteration optimizes the refills. The best uniform level is found by a bounded scalar search, the per-channel levels (both channels and 𝑅) by coordinate ascent from it; the anchored skew uses each channel’s shadow price averaged over the other balance under the flat policy’s stationary law, anchored at the time-average balance. Part (v) applies to the 24 hubs with equal shapes. Per instance, in the 8 two-channel hubs with balanced flows and equal shapes the channels’ best levels differ from the uniform level by at most 2.3% and add 0.1–2.0 points; the 8 that meet 𝑘 ≥ 𝑠ˆ∗ reach 0.41–0.83 of the bound of (v), and of the 16 that do not, 2 exceed it (1.18 and 1.51). With three modelled channels (4 instances, forwards of 20% of capacity, drain ratios 1 and 3) the cases repeat: with balanced flows the channels’ own best levels match the uniform level and add at most 1.6 points; with a draining channel the uniform level loses 33–39%, per-channel levels 9–12%, and a skew on the uniform level 36–67%; the bound of (v) holds in all four.

degree strata (App. B). Under mission control it kept seven, whose separate gains of 8.3% shrank to 0.7% together, as the hubs took traffic from each other; on the evaluation seeds the seven lost 7.8 ± 4.4% of revenue against 𝐿 ∗ . Nor does a level per channel class, which the exact hubs of §5 favor. We class each hub’s channels as draining (6–8% of them) or filling by net flow at 𝐿 ∗ . The best split over {𝐿 ∗ /2, 𝐿 ∗, 2𝐿 ∗ }2 on the calibration seeds puts 2𝐿 ∗ on draining channels in both models, as predicted. But it earns no more than 𝐿 ∗ there, and −0.6 ± 7.2% and −4.5 ± 4.7% on the evaluation seeds.

G

Skew amplitude. The anchored skew at deployed fees with its shadow-price term scaled by 𝑎 (skew_amp; 𝑎 = 1 is the paper’s skew), on the skew tables of 𝐾 = 100 and 𝐾 = 1,000, seeds 0-4, paired with the deployed fees; refills on the fixed trigger, net at the table’s 𝐾. The refill cut and gross revenue do not move with 𝑎.

Exactly solved hubs, global check. On three instances a global grid over the three levels (73 cells) followed by coordinate ascent reaches the same optimum as coordinate ascent from the best uniform level (t1_hub_global.py).

E

Hubs Repricing Alone and Competition

A hub alone. A hub repricing alone, all other nodes at deployed fees (10 seeds; the largest hub and, drawn at random, one per quartile of deployed revenue, two under mission control), gains less: under mission control 1,000 ppm raises the revenue of seven of nine hubs, by 1.2–2.9× (one more from a 29-sat base), and lowers two; 𝐿 ∗ raises four significantly and lowers three. With shared memory 1,000 ppm raises each of the five hubs tried, by 2.5× or more. Competition. When all reprice, 𝐿 ∗ is not an equilibrium. In an exploratory best-response check (5 hubs: the largest, 𝐻 , and one per quartile of deployed revenue; 10 seeds), with every hub at 𝐿 ∗ none gains by raising to 8,000 ppm and most gain by cutting: 𝐻 most at 2,000 ppm of the levels tried (+21 ± 9% shared memory, +44 ± 15% mission control), taken from the other hubs, as the 40 together earn no more.

F

Per-Hub and Per-Class Levels

Per-hub and per-class levels. Nor do levels that differ across hubs beat the uniform one. From 𝐿 ∗ we ran coordinate ascent on the calibration seeds, halving or doubling each hub’s level. With shared memory it kept none of the 80 moves, as none gained more than twice its paired standard error, nor any of the 8 moves of four

Skew Amplitude and Forward Sizes

model

𝐾

𝑎

gross (%) refills (%) net at 𝐾 (k)

shared shared shared shared shared shared shared shared shared shared mc mc mc mc mc mc mc mc mc mc

100 100 100 100 100 1000 1000 1000 1000 1000 100 100 100 100 100 1000 1000 1000 1000 1000

0.5 1 2 4 8 0.5 1 2 4 8 0.5 1 2 4 8 0.5 1 2 4 8

-0.6 -0.9 -1.3 -0.8 -0.2 -0.9 -0.5 +1.4 -0.9 +1.3 +0.1 +0.2 +0.2 +0.2 +0.2 +0.1 +0.1 +0.1 +0.2 +0.1

-22.3 -21.1 -21.7 -23.0 -19.5 -21.2 -20.5 -20.6 -20.7 -20.1 -6.0 -6.0 -7.4 -8.1 -6.0 -7.4 -6.0 -6.6 -6.0 -8.1

+3.8 ± 1.2 +3.6 ± 0.8 +3.6 ± 0.9 +3.9 ± 1.1 +3.3 ± 0.6 +36.3 ± 6.1 +35.2 ± 6.1 +35.5 ± 4.7 +35.3 ± 4.8 +34.5 ± 4.1 +0.2 ± 0.0 +0.2 ± 0.0 +0.2 ± 0.1 +0.3 ± 0.1 +0.2 ± 0.1 +1.6 ± 0.7 +1.5 ± 1.0 +1.6 ± 1.4 +1.5 ± 1.1 +1.9 ± 1.0

Forward sizes. On 2026 the size-biased mean forward 𝑚 2 /𝑚 1 is 6% of capacity at the volume-weighted median hub channel (3–13% interquartile; 18% at the median hub channel that carries traffic).

H

Level Search Details

√ √Forward-counting level search. Each iteration probes 𝐿/ 2 and 𝐿 2 for an epoch of payments each, counts the √ key’s√forwards (not its revenue), estimates 𝜅ˆ = ln(𝑛𝑙𝑜 /𝑛ℎ𝑖 )/(𝐿 2 − 𝐿/ 2), moves to 1/𝜅ˆ (at most a factor 2 per iteration, inside [𝐿0 /16, 16𝐿0 ]; no fall in forwards moves up by 2), and holds the level once a move is under 10%. "joint": one level for the 40 hubs, forwards pooled; "hub": each hub on its own forwards. 24,000 payments, seeds 0-4; revenue against uniform 4,900 ppm (guard26, same seeds). model

arm

end (ppm) rev. vs 𝐿 ∗ (%)

shared shared shared shared shared shared mc mc mc mc mc mc

elast_hub_e1500_L1000 elast_joint_e1500_L1000 elast_joint_e1500_L350 elast_joint_e1500_L8000 elast_joint_e3000_L1000 levelsearch_joint_L1000 elast_hub_e1500_L1000 elast_joint_e1500_L1000 elast_joint_e1500_L350 elast_joint_e1500_L8000 elast_joint_e3000_L1000 levelsearch_joint_L1000

10548 4282 4702 5630 5091 3380 8747 3607 3320 3977 3488 2297

-32.6 ± 6.9 -15.3 ± 6.8 -29.0 ± 8.7 -5.9 ± 7.4 -27.6 ± 3.4 -18.9 ± 15.9 -11.6 ± 10.4 -3.5 ± 8.3 -17.0 ± 8.6 -5.3 ± 12.1 -11.4 ± 10.4 -7.5 ± 10.5

Trim: Fee Level Matters More Than Skew in Lightning Routing

Per-hub forward counting with long probes. Each hub searches on its own forwards with probes of 6,000 payments, four times those above, over 60,000 payments from 1,000 or 8,000 ppm, seeds 0-4. On average only 9–15 of the 40 hubs settle, and the hubs earn 12–35% less than posting 𝐿 ∗ throughout: a hub’s probe holds too few forwards for the count ratio, so the estimated elasticity is biased low and the level drifts up. Noise of revenue against forward counts. On evaluation seeds 0-9 at uniform 1,000 and 4,900 ppm, the squared coefficient of variation of a hub’s revenue across seeds is a median 3.2–5.6 times that of its forward count (hubs with at least 5 forwards per seed), and 4.9–18.5 times for the 40 hubs pooled: comparing forward counts needs proportionally fewer forwards than comparing revenue. Level search, details. When all 40 hubs search at once (24,000 payments, 5 seeds), the mean level moves from 350 to about 700, from 1,000 to 1,300–1,400 and from 𝐿 ∗ to 3,500–3,900 ppm, still falling: where the search stops depends on where it starts. With the guard, from 𝐿 ∗ and 8,000 ppm it earns 5–10% more than the unguarded search (significant with shared memory), and from every start it stays within 8% of holding the start level. • Each hub posts one level 𝐿, probes 𝐿𝑢 and 𝐿/𝑢 for 600 payments each and moves 𝐿 by 𝑢 toward the probe that earned more on its own forwards; 24,000 payments, 5 seeds. The step 𝑢 starts at 1.5 and is multiplied by 0.7 after each reversal, down to 1.05. Probes should be sized by own forwards until two probes differ by twice their noise. • Guard: a move is undone when the next pair loses more than twice the standard deviation of the hub’s last eight pairs; probes stay within [𝐿0 /2, 2𝐿0 ]. • A fixed 10% threshold, inside a pair’s noise, undoes 150–240 moves per run, not 50, and freezes the search.

I

Success-Constrained Tool Picks

Success-constrained tool picks. On the calibration seeds each tool family’s best ppm configuration whose mean success reaches that of uniform 1,000 ppm (shared 0.709, mission control 0.662), scored by gross less 100 sat per refill; evaluated on seeds 0-9 against the Table 3 frontier.

J

model

pick

success gap (%)

shared shared shared mc mc mc

clndabs_150_2400 feeadjabs_soft_1200 lndgabs_1200 clndabs_300_1200 feeadjabs_soft_600 lndgabs_600

0.713 0.714 0.707 0.676 0.687 0.701

-8.3 ± 5.1 -8.1 ± 4.7 -10.0 ± 2.2 +2.9 ± 2.2 +2.7 ± 1.6 -4.3 ± 1.3

Welfare and Who Pays

Welfare. An accepted payment gives its sender an expected surplus of 𝑣/𝜅 and fees are transfers; a refused sender, who may still pay later, counts as a total loss, so the losses below are upper bounds. On 2026, from deployed fees, 1,000 ppm lowers total surplus, net of the hubs’ refills, by 2.0 ± 0.6% (shared memory) and 0.5 ± 0.3% (mission control), and 𝐿 ∗ by 7.3 ± 1.3% and 6.7 ± 0.9%, while sender surplus falls by 16–18%. The surplus lost is 0.42 and 0.11 of the hubs’ net gain at 1,000 ppm, and 0.78 and 0.99 at 𝐿 ∗ : at the peak the network loses about what the hubs gain.

Who pays. Senders pay for the level: posting 𝐿 ∗ together lowers success, and payments with a detour route around the hubs (at 𝐿 ∗ the hubs forward 80–93% less, the network only 16–18%). • We scale revenue to the 6.6 million routed payments of August 2023 (River): the 40 hubs (80% of public capacity) earn 0.03–0.17% a year on their side of their channels at deployed fees and 0.5–0.6% at 𝐿 ∗ : the multiples start from a low base. This is a scale check, not a calibration: it counts idle channels, at 2023 volumes.

K

Proposition A.1 Against the Exact Jump Model

Table 6: Proposition A.1 against the exact jump model (𝐾 = 0.05). ratio: Λ𝑚 1 /𝜇𝑚 +1 ; 𝑎: exact slope at 𝑏 ∗ ; id. err: error of (10); other columns: loss in % of optimal revenue. 𝜅 ratio 𝑎 exact id. err% ODE best const (𝑐 r =0) 1/𝜅 5 5 10 10 20 20

3 5 3 5 3 5

0.0655 0.0714 0.0331 0.0611 0.0168 0.0310

−0.07 −0.07 −0.07 −0.11 −0.08 −0.09

0.11 0.01 0.18 0.04 0.29 0.06

1.27 0.69 2.02 3.31 2.61 5.40

1.05 6.0 0.67 13.6 1.05 13.5 2.40 41.7 1.05 28.3 2.40 98.8

Slope identity. With 𝑔 from the jump model, which uses no evaluation data, the identity (10) is a consistency relation, not a prediction; where the flow-balance point lies well above the refill trigger (16 ergodic and 8 binding grid instances), the identity holds within 0.13%, and where refills supply the inventory it fails, even in sign (Table 6). A hub alone. A hub repricing alone, all other nodes at deployed fees (10 seeds; the largest hub and, drawn at random, one per quartile of deployed revenue, two under mission control), gains less: under mission control 1,000 ppm raises the revenue of seven of nine hubs, by 1.2–2.9× (one more from a 29-sat base), and lowers two; 𝐿 ∗ raises four significantly and lowers three. With shared memory 1,000 ppm raises each of the five hubs tried, by 2.5× or more. Welfare. An accepted payment gives its sender an expected surplus of 𝑣/𝜅 and fees are transfers; a refused sender, who may still pay later, counts as a total loss, so the losses below are upper bounds. On 2026, from deployed fees, 1,000 ppm lowers total surplus, net of the hubs’ refills, by 2.0 ± 0.6% (shared memory) and 0.5 ± 0.3% (mission control), and 𝐿 ∗ by 7.3 ± 1.3% and 6.7 ± 0.9%, while sender surplus falls by 16–18%. The surplus lost is 0.42 and 0.11 of the hubs’ net gain at 1,000 ppm, and 0.78 and 0.99 at 𝐿 ∗ : at the peak the network loses about what the hubs gain.

L

Simulation Parameters

Script and file names refer to the code release. Single-channel model. • Jump model solved exactly on a 400-point balance grid: 𝑐 = 1, exponential forward and replenishment sizes with 𝑚 1 = 𝑚 +1 = 0.02, 𝜇 = 20, 𝑐 r = 0.01, 𝑘 = 0.05.

Sun et al.

• The 18 verification instances: 𝜅 ∈ {5, 10, 20}, Λ𝑚 1 /𝜇𝑚 +1 ∈ {3, 5}, 𝐾 ∈ {0, 0.05, 0.2}; the six-instance table below uses 𝐾 = 0.05. • The 144-instance grid: 𝜅 ∈ {5, 10, 20}, demand-toreplenishment ratios 1.5–8, forwards of 2–20% of capacity, cheap and costly refills, with and without 𝑐 r . • Level-error ratios on that grid: at 0.8 or 1.25 times the best level a flat fee loses at least 2.3 times (median 5.3) what any skew can add, at 2/3 or 1.5 times at least 4.7 times. • The ODE curve with 𝑔 measured under the flat fee 1/𝜅 instead of the exact 𝑔: it loses 3.9–38%, and still 0.4–6.1% after five more updates of 𝑔.

Tools. • charge-lnd posts min +(1 − 𝑥)(max − min) for local share 𝑥, with min, max in ppm or as multiples of the deployed rate; checked when the share moves by 5%, it re-posts on changes of at least 10 ppm. • feeadjuster multiplies the base rate by 500.5−𝑥 (or its soft or hard variants) and re-posts on 5% share moves. • LNDg adjusts each channel once per "day" of 300 payments by the tier of its balance share and its one- and seven-day volume; its failed-HTLC rule is on in the ppm runs (default LNDg is unchanged), and its fee cap, 2,500 ppm by default, is 10,000 ppm, or 4𝐿 for starts 𝐿 ≥ 4,000 ppm. • ppm grids center on 𝐿 ∈ {600, 1,200, 2,450, 4,900} with ranges (𝐿/2, 2𝐿), (𝐿/4, 4𝐿), (0, 2𝐿) (feeadjuster soft and default); 2019: 𝐿 ∈ [250, 12,000]. Multiples on 2026: 24 configurations up to 1,024×. Extra centers of 9,800 and 12,000 ppm were added later (App. B). • Defaults (§7.4): charge-lnd from 0.25× to 4× the deployed rate, feeadjuster and LNDg as shipped, all starting from deployed fees.

Traffic and calibration. 2

• Payment amounts: log-normal, 𝑒 N (9.5,1.6 ) sat, capped at 466,359 sat; sources uniform, destinations drawn with probability proportional to degree1.5 ; 12,000 payments per run on 2026, 9,000 on 2019. • Every policy keeps each channel’s deployed base fee (0 or 1 sat on 99% of hub channels). • At 𝐾 = 100 the deployed fees’ refills cost 17.2k (shared memory) and 2.4k per run, at 𝐿 ∗ at most 0.3k. • Capacities: each channel’s maximum HTLC size, scaled to 30% (main setting), 15%, 50% or 100%. • Seeds. 2026: tools tuned on calibration seeds 1000–1002, evaluated on 0–9 (some settings 0–19); 2019: 1000–1004 and 0–19. Skew tables fitted on 1000–1004, refitted for the widercoverage test on those plus 2000–2044 (600k payments). • A report-only two-part fit 𝜆(𝐿) = Λ[(1 − 𝑤)𝑒 − (𝜅𝑐 +200)𝐿 + 𝑤𝑒 −200𝐿 ] to the scan gives a captive share 𝑤 = 0.44 (2026) and 0.42 (2019). • The calibration scan for 𝐿 ∗ posts 15 uniform levels (25– 12,000 ppm) with balances frozen; 𝐿 ∗ is the vertex of a parabola in log 𝐿 through the best grid point and its neighbours. Frontier gaps use per-seed linear interpolation between evaluated levels, never the parabola. • Skew tables: a 100-point balance grid, refill trigger 5% of capacity, refill target 50%, 𝐾 = 100 (refitted at 𝐾 = 1,000 for the 𝐾 = 1,000 runs), solved by relative value iteration; a channel gets a table with at least 20 outgoing and 10 incoming calibration forwards (10 and 5 in the wider-coverage refit). Routing models. • Shared memory: every sender avoids a failed channel, for that amount, during the next 300 payments. • Mission control: each sender records each node pair’s last result. A pair’s success probability for an amount is 0.95 up to a recorded success’s amount; from a recorded failure’s amount up, it recovers from 0 toward the prior with a halflife of 300 payments; otherwise it is the node prior, 0.6 with weight 2, mixed with the probabilities of the node’s recorded pairs. A route minimizes the sum of fees plus an attempt cost of 100 sat + 1,000 ppm times − ln of each hop’s probability (LND’s defaults). • Sensitivity: a half-life of 100 or 1,000 payments changes every policy’s revenue by less than 0.1% (5 seeds).

Realism options. • With every realism option on, removing refills changes revenue by −0.3% and the skew by +0.2%.

M

Confirmatory Tests, Frozen Predictions and Power

Two claims, both on the main 2026 setting: (1) no ppm-tuned tool beats the frontier by more than 3%; (2) the skew adds less than 3% under mission control and 5% with shared memory. One-sided paired t-tests over the 10 evaluation seeds, three tools and three skew comparisons per routing model, Holm correction over all 12, family-wise alpha = 0.05. All 12 null hypotheses are rejected; the largest Holm-adjusted p is 0.0086 (chan/holm12.json, all 12 rows with raw and adjusted p; the tool tests use the frontier over every uniform level run, including 2,450, 6,125 and 12,000 ppm, which Table 3’s frontier omits; on Table 3’s frontier the mission-control raw p change (4.8e-07 -> 7.5e-07, 8.5e-07 -> 2.5e-06, 6.7e-04 -> 7.4e04), the shared-memory ones do not, and no conclusion does; the 10 tool comparisons against H0 gap ≤ 0 are exploratory and not in the family). model

comparison

H0

mean

p

p (Holm, 12)

shared shared shared shared shared shared mc mc mc mc mc mc

charge-lnd (ppm) feeadjuster (ppm) LNDg (ppm) skew at deployed fees skew on 1,000 ppm skew on 𝐿 ∗ charge-lnd (ppm) feeadjuster (ppm) LNDg (ppm) skew at deployed fees skew on 1,000 ppm skew on 𝐿 ∗

gap ≥ +3% gap ≥ +3% gap ≥ +3% gain ≥ 5% gain ≥ 5% gain ≥ 5% gap ≥ +3% gap ≥ +3% gap ≥ +3% gain ≥ 3% gain ≥ 3% gain ≥ 3%

-4.17% -3.96% -3.17% -1.79% -0.74% +0.04% -10.55% -7.57% -1.50% +0.19% -0.04% +0.01%

0.0014 0.0086 0.0016 0.0013 3.4e-05 0.002 4.8e-07 8.5e-07 0.00067 1.8e-09 1.2e-13 1e-20

0.0065 0.0086 0.0065 0.0065 0.00024 0.0065 4.3e-06 6.8e-06 0.004 1.8e-08 1.3e-12 1.2e-19

Not part of the family: no tool, including those tuned as multiples of deployed fees, rejects H0 gap ≤ 0 (p ≥ 0.92 raw), so none lies significantly above the frontier.

Trim: Fee Level Matters More Than Skew in Lightning Routing

Frozen hub-count predictions. Frozen before the runs, its predictions for 10, 20 and 80 hubs held: under shared memory each peaks at 4,900 ppm, the grid level nearest the prediction, and stays within 0.07 of the predicted curve; under mission control the predicted level earns at least 94% of the best (predictions_hubk.json). Tools against their center. Against the flat level at its center, paired by seed, LNDg earns −3.1 ± 5.0% and +0.2 ± 2.3%, charge-lnd −5.2±4.6% and −2.2±3.3%, feeadjuster −20.4±4.8% and −8.8±1.8%, for at most 4.2 points more success. One-sided paired t-tests over the 10 evaluation seeds (chan/power.json, analysis/power_confirmatory.py). Power is for a true effect of 0; the last column is the smallest margin rejected with 80% power at the strictest Holm level (alpha/12). test

mean (%) se

margin power (𝛼 /12) power (𝛼 ) margin at 80%

shared/clndabs shared/feeadjabs shared/lndgabs shared/skew@deployed shared/skew@1000ppm shared/skew@𝐿 ∗ mc/clndabs mc/feeadjabs mc/lndgabs mc/skew@deployed mc/skew@1000ppm mc/skew@𝐿 ∗

-4.17 -3.96 -3.17 -1.79 -0.74 +0.04 -10.72 -8.04 -1.59 +0.19 -0.04 +0.01

3 3 3 5 5 5 3 3 3 3 3 3

1.77 2.39 1.56 1.65 0.83 1.29 1.17 1.01 1.00 0.13 0.05 0.01

0.11 0.05 0.14 0.43 0.98 0.68 0.29 0.41 0.42 1.00 1.00 1.00

0.47 0.31 0.55 0.88 1.00 0.97 0.76 0.86 0.87 1.00 1.00 1.00

7.7 10.4 6.8 7.1 3.6 5.6 5.1 4.4 4.3 0.5 0.2 0.0

Probe size: forwards a hub needs per level to tell two levels 4% apart (two-sided 5%, 80% power), from the seed-to-seed variation of its revenue at a fixed level. model/level

hubs CV

fwd/run needed (median) interquartile

shared/uniform1000 shared/uniform4900 mc/uniform1000 mc/uniform4900

39 37 40 39

21.8 11.0 53.8 19.1

0.38 0.62 0.24 0.51

32,186 35,473 32,934 39,672

16,597–41,067 26,056–44,234 24,660–42,042 25,034–56,721

N

Ordering Across 2026 Settings

Table 7: The ordering across 2026 settings, senders blind to balances, each setting with its own calibration (10 seeds): revenue peak, hub gross revenue and refills at 𝐿 ∗ relative to deployed fees, and the refill change from the skew at deployed fees. Refills are forced except with economic refills and all realism options (4.4, 6.6 and 5.0 per run at deployed fees, 𝐿 ∗ and with the skew). † Best level on the evaluation seeds, so the multiple is an upper bound (𝜅=100 and capacities 15, 50%: 5 seeds). –: no refills at deployed fees, or skew not run. 𝐿∗

Setting 2026, shared memory 2026, mission control 2026, economic refills, shared 2026, economic refills, MC 2026, all realism options 2026, MC, payments 5× 2026, shared, payments 5× 2026, MC, capacities 15; 50% 2026, shared, capacities 15; 50% 2026, MC, 𝜅=100 2026, MC, 𝜅=400 2026, MC, 𝜅=1000 2026, MC, 𝜅=2000 2026, shared, 𝜅=2000 2026, MC, heavy-tailed WTP 2026, shared, heavy-tailed WTP

Rev.

Refills Skew

4,900 17.1× −98% −21% 4,900 3.4× −91% −5% 4,900 13.1× – – 4,900 3.4× – – 4,900 3.2× +50% +14% † 4,000 4.3× −92% −7% 20.9× −98% −23% 4,900† † 4,000 3.3; 3.4× −88; −94% – † 4,900 17.6; 18.8× −99; −100% – † 8,000 5.7× −85% – 1,500† 2.5× −80% – † 600 2.3× −65% – 500† 2.2× −76% – 500† 4.3× −99% – 3,000† 2.7× −90% – 5,000† 13.6× −98% –

Ordering across settings. With LND’s route cost, mission control ranks routes by fee plus the attempt cost divided by the route’s success probability, as LND does. With wallet fee limits, senders accept a route only within their wallet’s default fee limit (CLN 0.5%, LND 5%; half the senders each), and the limits place the peak. With all realism options on, refills are made only when they pay (App. B.1). At full capacities (10 seeds) LNDg earns 2.1 ± 2.9% and 2.8 ± 2.0% more than the frontier at its success; the other tools are within the noise of it or below. With every realism option on, where refills are made only when they pay, the skew raises refills slightly (4.4 to 5.0 per run at deployed fees), since higher fees make refills pay. The best tools lie 2–11% below the frontier with 𝜅 = 2,000, where a uniform level still earns 2.2 and 4.3× the deployed fees, 5–6% with a heavier-tailed willingness to pay, 6–7% with payments five times larger, and 6–8% at capacity proxies of 15%, within 3% at 50%. At their defaults the tools keep the deployed level, earning −42 to +8% of its revenue, 40–49% below the frontier under mission control. Wallet-count sweep. Mission control kept per wallet, senders hashed to wallets; seeds 0-4 (20 wallets: 0-9). wallets best level multiple 1 5 20 100

4000 ppm 4000 ppm 4000 ppm 4000 ppm

3.54 3.34 3.18 3.15

Related documents

• Trim: Fee Level Matters More Than Skew in Lightning Routing#1458766
• Learning by Transacting: A General Theory of Bidirectional Knowledge Flows in Market Exchange#379288
• Personalized Pricing and Quality Design#942818
• Assortment Control Unlocks the Value of Dynamic Pricing in Mixed Last-Mile Delivery#1071319
• Media Frenzies in Markets for Financial Information#942741