Lightning Network Economics: Channels

1. Motivation Model Structural Implications Conclusion The Cost of Lightning Network Channels and Implications for Network Structure Paolo Guasoni1 Gur Huberman2 Clara Shikhelman3 Dublin City University1 Columbia Business School2 Chaincode Labs3 SIAM Activity Group on Financial Mathematics and Engineering Virtual Seminar, August 20th , 2020

2. Motivation Model Structural Implications Conclusion Outline • Motivation: Bitcoin’s Limitations. Scale and Immediacy. • Model: Stationary Payment Flows. • Results: Channel Costs and Structural Implications

5. Motivation Model Structural Implications Conclusion Bitcoin • Bitcoin: anonymous decentralized payment system. • Public ledger (“blockchain”). No trusted parties. • Any user can help processing payments (“mining”). • New block of transactions added to blockchain about every 10 minutes. • Each block contains about 2000 transactions. • Lottery among miners decides who will add the next block. • Winning miner rewarded with Bitcoin.

12. Motivation Model Structural Implications Conclusion Limits of Bitcoin • Scale: No more than 3 to 4 transactions per second, regardless of size. Limit imposed by design. • Immediacy: At least ten minutes required to confirm transaction. Transaction secure typically in an hour (to rule out potential forks).

13. Motivation Model Structural Implications Conclusion Limits of Bitcoin • Scale: No more than 3 to 4 transactions per second, regardless of size. Limit imposed by design. • Immediacy: At least ten minutes required to confirm transaction. Transaction secure typically in an hour (to rule out potential forks).

14. Motivation Model Structural Implications Conclusion Lightning Network: Channels • Main Idea: Not all transactions need to happen on the blockchain. • If Alice and Bob transact often, they establish a private ledger (channel). • Each of them deposits some Bitcoin as escrow in the channel. • Alice pays Bob by signing a message agreeing to an updated balance. • When the deposit of either party is depleted, the balance is transferred to the other party – in the blockchain. • Either party can retrieve its balance early, unilaterally closing the channel. • These features are implemented through a cryptographic protocol.

21. Motivation Model Structural Implications Conclusion Lightning Network: Links • Alice has a channel with Bob. Bob has a channel with Charlie. • Alice can pay Charlie through Bob. • Cryptographic protocol. Can handle longer paths. • Bob may be better or worse off after transaction. • Bob could charge a fee for facilitating the transaction.

26. Motivation Model Structural Implications Conclusion Questions • Which channels should be opened? • How much should each party deposit? • What is the minimal cost of a channel? • What are the implications for network structure?

30. Motivation Model Structural Implications Conclusion Channel Model (i) Two users, 1 and 2. (ii) User 1 pays user 2 at Poisson arrival times with rate λ1. User 2 pays user 1 at Poisson arrival times with rate λ2. (iii) The two Poisson processes are independent. All payments are of one unit. (Inessential but simplifying assumption.) (iv) User 1 commits l1 to the channel. User 2 commits l2. (v) Stationary: when one channel is depleted, an identical one replaces it. • Transaction cost: An on-chain transaction entails a cost of B, regardless of size. • Opportunity cost: Locking l1 + l2 in the channel for dt foregoes r(l1 + l2)dt in interest. • unidirectional channel if λ1 = 0 or λ2 = 0.

38. Motivation Model Structural Implications Conclusion Tradeoff • Higher deposit implies lower costs from on-chain transactions. Lower deposit implies lower opportunity costs (lost interest rate). • Demand for cash models: Baumol (1952), Tobin (1956): deterministic spending (unidirectional). Miller and Orr (1966): random balance fluctuations (bidirectional). • Portfolio choice with transaction costs: Constantinides (1986), Davis and Norman (1990), and many others. Costs of transactions vs. displacement from Merton proportion. • Fixed cost of on-chain transaction. Impulse control problem. • Related work in computer science: Brânzei, Segal-Halevi, Zohar (2017) on economics of LN, Avarikioti et al. (2018, 2019, 2020) on the game-theory of fees. Seres et al. (2019) and Schmid (2017) on simulations. Jianhong et al. (2020), and Seres et al. (2020) on state of the network.

43. Motivation Model Structural Implications Conclusion Worth it? • In contrast to cash demand models, Lightning channel are optional. • All transactions could be settled on-chain instead. Lemma The on-chain settlement cost for a transaction stream with rate λ is µ = λB/r. • Intuitive formula. Price of annuity paying λB per unit of time. • Denote by µ = λB/r the weight of the channel. • When should you open a channel? • Find alternative cost Ll1,l2 (λ1, λ2) with Lightning.

50. Motivation Model Structural Implications Conclusion Lightning Cost Theorem (i) A unidirectional channel costs L0,l2 (λ2, 0) = l2 + B r + λ2 λ2 l2 − 1 !−1 . (ii) A bidirectional channel costs Ll1,l2 (λ2, λ1) = l1 + l2 − B αl1 −(1 − αl1+l2 + ) − αl1 +(1 − αl1+l2 − ) αl1 −(1 − αl1+l2 + ) − αl1 +(1 − αl1+l2 − ) + αl1+l2 + − αl1+l2 − , where α± = λ2 + λ1 + r ± p (λ2 − λ1)2 + r2 + 2r(λ2 + λ1) 2λ2 . • Explicit formulas for given choice of deposits l1 and l2. • Optimal choice l1 and l2 not explicit in general.

55. Motivation Model Structural Implications Conclusion No Lightning for Turtles Theorem (i) If a unidirectional channel with rate λ costs less than on-chain transactions, then λ r/B. (ii) If a bidirectional channel with rates λ1, λ2 costs less than on-chain transactions, then √ λ1λ2 r/(2B). • Necessary conditions. Not sufficient. • unidirectional intuition: a slow trickle of transactions should stay on-chain. • Bidirectional trickier: geometric average of payment rates should be higher than threshold. Needs both rates to be large enough. • Justifies use of asymptotics for r near zero. • Conditions depend only on weights: µ 1 and µ1µ2 1/4 respectively. (A bidirectional channel has two weights.)

63. Motivation Model Structural Implications Conclusion Asymptotics Proposition For r near zero: (i) The minimal cost of a unidirectional channel with rate λ is Lopt (λ, 0) = 2 Bλ r 1/2 − B 2 + O(r1/2 ) and is achieved for the channel size l = Bλ r 1/2 + O(r1/2 ). (ii) The minimal cost of a symmetric, bidirectional channel with both rates λ is Lopt (λ, λ) = 3 2Bλ r 1/3 − B 6 + O(r1/3 ) and is achieved for channel sizes l1 = l2 = 2Bλ r 1/3 + O(r1/3 ). • Baumol-Tobin rate 1/2 for unidirectional. Miller-Orr 1/3 for symmetric. • No explicit solution for bidirectional asymmetric.

68. Motivation Model Structural Implications Conclusion Traffic Reduction Proposition In a bidirectional channel with sizes l1, l2 and rates λ1, λ2, the long-term ratio between channel transactions and on-chain transactions equals (q + 1) l2 ql1 − 1 − l1 ql2 + l1 (q − 1) (ql1+l2 − 1) where q = λ2/λ1. In particular: (i) In a unidirectional channel (q ↓ 0) the ratio simplifies to l1. (ii) In a symmetric bidirectional channel (q → 1) the ratio simplifies to l1l2. • Traffic reduction is linear for unidirectional channels. • Quadratic for symmetric. Somewhere in between for bidirectional. • Netting effect largest with symmetry.

74. Motivation Model Structural Implications Conclusion Channel Graph • Several users (vertices) v1, . . . , vn. • Set of payment flows F. (vi , vj , λij ) ∈ F if vi pays vj with frequency λij . • A directed graph G with weighted edges is a valid LN with respect to F if: (i) For every (vi , vj , λij ) ∈ F there is a path from vi to vj . (ii) The (directed) weight of each edge equals the sum of the weights of the paths passing through that edge (in that direction). • Intuitively, a valid LN is a graph and a set of paths that allow all necessary payments to be carried out. • Denote the cost of a graph G as f(G). • Small groups of users can change the network to reduce their costs. • What is the resulting structure of the payment network?

83. Motivation Model Structural Implications Conclusion Mutations and Immutable Definition For k ≥ 1 a valid LN G is a k-immutable if There is no valid LN graph G0 and a set U ⊂ V, |U| = k such that G[V U] = G0 [V U] and f(G0 ) f(G). (G[V U] is the subgraph of G spanned by the vertices in V U.) k users cannot improve it by changing their channels. v1 v2 v3 v4 v5 (a) Before v1 v2 v3 v4 v5 (b) After

84. Motivation Model Structural Implications Conclusion Hermetization Definition A valid LN G is hermetic if no valid LN graph G0 with at most |V(G)| + 1 vertices has f(G0 ) f(G). Adding an intermediary to route payments does not reduce costs. v1 v2 v3 v4 (a) Before v1 v2 v3 v4 v5 (b) After

85. Motivation Model Structural Implications Conclusion Do not Pay a Payer Theorem A 3-immutable graph does not have a directed path with more than one edge. v1 v2 v3 µ1 µ2 (a) Before v1 v2 v3 µ1 µ1 µ2 (b) After • Paying a payer is bad because it is better to pay the final payee. The two payees can exchange payments through a bidirectional channel. • If one removes all bidirectional channels, the result is a family of stars. • Hub-and-spoke structure.

89. Motivation Model Structural Implications Conclusion No Bidirectional Clustering Theorem A valid LN satisfies the following: (i) A 3-immutable graph has no 3-cycle with bidirectional edges. (ii) A 4-immutable graph has no 4-cycle with bidirectional edges. • Bidirectional clustering is exactly zero. • If Alice is linked to Bob and Bob linked to Charlie, Alice and Charlie will not waste capital on another channel. • Incompatible with several models of random networks, such as Erdös-Renyi, small-world, and others. • To be considered when simulating Lightning Networks. • Statement not true replacing 3 or 4 with 5. Counterexample.

97. Motivation Model Structural Implications Conclusion Intermediation • Recall that the degree of a user is the number of adjacent edges. • s(v) and S(v) smallest and largest weights of edges adjacent to v. • Define R(G) = maxv S(v) s(v) as the discrepancy of the network. Theorem A graph G with average degree greater than 4 p 2R(G) is not hermetic. • In principle, an enterprising user could offer to act as a counterparty to any such set of users, charging part of their savings as fees. • If all edges have the same weight (R(G) = 1), the condition boils down to average degree greater than 6.

103. Motivation Model Structural Implications Conclusion Shooting for the Star? Theorem Assume that, for each user vi , the size of each channel (µk i )1≤k≤|N(i)| is distributed according to a law such that (i) there exist a 1, b 0 such that P(µk i x) bx−a for all 1 ≤ i ≤ |V|, 1 ≤ k ≤ |N(i)|, and x 0. (ii) there exists ε, such that P(µk i ε) = 1 for all 1 ≤ i ≤ |V|, 1 ≤ k ≤ |N(i)|. Then the graph is not hermetic with probability greater or equal than 1 − c X 1≤i≤|V| |N(i)|1−2a where c is a positive constant. • If payment flows follow a power law, then a star is more likely to emerge if the tail is thinner (less dispersion) and if the number of neighbors is higher.

107. Motivation Model Structural Implications Conclusion Open Questions • Tractability of asymmetric bidirectional channel. • Alternative cost objectives. Game-theoretical issues? • Fees for links in equilibrium? • Effect of fees on network structure?

111. Motivation Model Structural Implications Conclusion Conclusion • Lightning Network. Second layer on blockchain. Channels and links. • Use channels only above frequency threshold. • Channel costs: different orders for unidirectional and bidirectional. • Structural implications from channel costs. • No clustering. Hubs-and-spokes. • Potential for intermediaries in well-connected, homogenous groups. • Tradeoff between security and efficiency.

118. Motivation Model Structural Implications Conclusion Thank You! Questions?

Lightning Network Economics: Channels

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