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Payment Networks as Creation Games

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Pith's one-line read A stylized game-theoretic analysis finds fixed-fee conditions under which path, star, bipartite and clique networks are Nash equilibria, but the free-fee claims that the star is stable and bipartite graphs are unstable are contradicted by the paper's own lemma.

arxiv 1908.00436 v2 pith:Q3NQOPW2 submitted 2019-08-01 cs.GT

Georgia Avarikioti , Rolf Scheuner , Roger Wattenhofer This is my paper · ORCID
classification cs.GT
keywords transactionchannelnetworkpaymentchannelscreationequilibriumgame
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Payment networks like the Lightning Network let people send payments off the main blockchain through channels between pairs of users. Opening a channel costs a blockchain fee. Once channels exist, a payment can hop through several nodes, and each intermediate node can charge a forwarding fee. This creates a strategic question: which channels should a node open to reduce its own costs and collect fees from others?

The authors model this as a game. They assume every node sends the same number of payments to every other node, channels never run out of money, and all forwarding fees are equal in the first part of the paper. They then ask when common network shapes, a line, a star, a two-sided network, and a fully connected network, are Nash equilibria, meaning no single node can lower its cost by changing its channels. They derive fee ranges that make each shape stable. For example, a star is stable when the forwarding fee is low relative to the blockchain fee, and a fully connected network is stable when the fee is high.

The second half lets nodes set their own fees. The paper claims a star remains stable and that two-sided networks never are. This part is not rigorous. The main lemma says stability requires at least two independent zero-fee paths for every payment, which would rule out the star, yet the next sentence says the star is stable. The star claim also ignores the possibility that an outside node could open new channels and undercut the hub's fee to steal traffic. The model itself relies on an unstated rule for splitting traffic equally among equal-cost routes, and the revenue formulas depend on that rule.

Extended reading notes

Core claim

The abstract states: 'we show that the star is a Nash equilibrium when each channel party can freely decide the channel fee. On the other hand, we prove the complete bipartite graph can never be a Nash equilibrium, given a free fee policy.' If the paper is correct, a payment network with one central hub is stable even when nodes choose their own fees, while any two-sided hub structure collapses.

Load-bearing premise

The revenue and cost formulas in Sections 3.3 to 3.5 assume that when multiple cheapest routes exist, traffic is split uniformly among them (fractions 1/2, 1/d, 1/(N-2) appear in the cost functions). This uniform tie-breaking rule is never stated in Section 2, and it is load-bearing: the star's stability under a free fee policy in Corollary 1 would fail if a deviating node instead set a slightly lower fee and captured all traffic on a new route. Because the paper does not define tie-breaking, the NE conditions are not fully specified by the model.

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Editorial analysis

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Assumptions & free parameters 1 free parameters · 7 assumptions · 0 invented entities

The model is built on strong symmetry and idealization assumptions (unlimited capital, homogeneous payments, uniform fees). The most fragile is the unstated uniform tie-breaking rule, which is load-bearing for every revenue calculation. The fee game additionally suffers from an internal contradiction about whether nodes can open channels.

free parameters (1)
  • epsilon = arbitrarily small positive constant
    Introduced in Corollary 1 to make the star a strict (rather than weak) Nash equilibrium at the fee upper bound.
assumptions (7)
  • domain assumption All nodes have unlimited temporary capital, so channels are never depleted.
    Section 2. This idealization makes fees independent of payment value and removes liquidity constraints, which is a strong departure from real payment channels.
  • domain assumption The blockchain fee FB is fixed and uniform for all channel openings, closings, and on-chain transactions.
    Section 2. Assumed constant and stable over time; this is a modeling convenience.
  • domain assumption The forwarding fee f0 is fixed and identical for all nodes under the fixed-fee policy.
    Section 2. Used for the main NE analysis; the free-fee section later relaxes this but in a confused way.
  • domain assumption Payment scenario is homogeneous: each node sends exactly k payments to every other node.
    Section 3 before Section 3.2. The symmetric traffic pattern is required for the closed-form NE conditions.
  • domain assumption The game is simultaneous with full information.
    Section 2. Nodes choose channels at the start, before any transactions, and know the entire payment scenario.
  • ad hoc to paper When multiple cheapest routes exist, traffic is split uniformly among them.
    Implicit in the cost formulas (Sections 3.3 to 3.5): revenue terms contain fractions such as 1/2, 1/d, and 1/(N-2). This rule is never stated in Section 2, yet all NE conditions depend on it.
  • domain assumption In the fee game (Section 3.7), nodes cannot create new channels, only set fees.
    Section 3.7 states this explicitly, but the proof of Lemma 3 considers a node opening a new channel, contradicting the game definition.

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Pith. "Pith review of Payment Networks as Creation Games." pith.science (2026). https://pith.science/paper/Q3NQOPW2

@misc{pith2026190800436,
  author       = {Pith},
  title        = {Pith review of: Payment Networks as Creation Games},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q3NQOPW2}},
  note         = {Machine review of arXiv:1908.00436}
}
read the original abstract

Payment networks were introduced to address the limitation on the transaction throughput of popular blockchains. To open a payment channel one has to publish a transaction on-chain and pay the appropriate transaction fee. A transaction can be routed in the network, as long as there is a path of channels with the necessary capital. The intermediate nodes on this path can ask for a fee to forward the transaction. Hence, opening channels, although costly, can benefit a party, both by reducing the cost of the party for sending a transaction and by collecting the fees from forwarding transactions of other parties. This trade-off spawns a network creation game between the channel parties. In this work, we introduce the first game theoretic model for analyzing the network creation game on blockchain payment channels. Further, we examine various network structures (path, star, complete bipartite graph and clique) and determine for each one of them the constraints (fee value) under which they constitute a Nash equilibrium, given a fixed fee policy. Last, we show that the star is a Nash equilibrium when each channel party can freely decide the channel fee. On the other hand, we prove the complete bipartite graph can never be a Nash equilibrium, given a free fee policy.

Figures

Figures reproduced from arXiv: 1908.00436 by the authors.

Figure 1
Figure 1. Plots of the bounds for N = 103 with the upper bounds in red and the lower bounds in blue. (Deviation A) If the first node creates channels to a ∈ [1, N − 2] other nodes his cost function is c(µ,(a)) = a·FB + (N −1−a)·k ·f0. This cost function must be higher than c(µ,(N −1)) for all a. Thus, c(µ,(N −1)) < c(µ,(a)) ⇔ f0 > FB k . (Deviation B) If node i (not the first or last one) creates channels to a ∈ [0, N − i − 1… view at source ↗

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Works this paper leans on

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