HOW TO CHARGE
LIGHTNING
The Economics of Bitcoin Transaction Channels
Simina Brânzei1, Erel Segal-Halevi2, Aviv Zohar1,3
1Hebrew University , 2Ariel University , 3QED-itGoals
We want to understand■ How to manage payment channel networks– (channel funding, when to reset them, etc.)■ The effect of channels on the economics of Bitcoin– on-chain & off-chain volume and fees
We lack data. Requires lots of assumptions ontransaction size distributions, usage patterns.

Managing a single channel

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3

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We assume a random walk (biased or unbiased) on thechannel state10Basic facts about single
channels■ If Alice sends 1 unit to Bob at Poisson rate 𝜆𝐴 ,■ Bob sends 1 unit back at rate 𝜆𝐵 ,■ there is 𝑤 units of capacity in the channel,■ 𝑚 in Alice’s hands,
expected channel lifetime:Basic facts about single
channels■ If Alice and Bob exchange 1 coin with the sameprobability, then they will do
Xm = 𝑤𝑚 − 𝑚2transfers in expectation before hitting the boundary.
■ So, if we start a channel funded equally, its
expected number of transfers is:Transfers 𝑤

𝑤2=4

How much do people
transact?
■ We make small payments often■ We make large payments infrequently■ Data usually shows: power law behavior, e.g.
1𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑝𝑎𝑦𝑚𝑒𝑛𝑡𝑠 ∝𝑎𝑚𝑜𝑢𝑛𝑡 2

Channel resets
With varying payments sizes:■ When we count the number of blockchain hits, it is profitableto reset the channel near the borderHow much do channels cost?
Two main costs:■ Setup and settlement cost of channels (blockchain fee)

■ Interest rate payments to fund the channel itselfFees■ Fixed fees on the blockchain■ But on lightning, larger transactions shorten channel lifetime
more. Charge by transfer volume.
■ At least to cover costs

Choice■ Txs choose cheapest route (lightning / blockchain)■ Or no route if both are too expensiveSteps in our approach■ Pattern of payment flows and distribution ofamounts■ Channel management (topology + funding +reset policy■ Market equilibrium for feesSome intuition■ Large transfers prefer the blockchain (fixed fee)
■ Small transfers split the cost of channel creation
Both compete for blockchain records.

Parameters(for running example)■ Each person does 10 txs per day (in expectation)– Drawn from a power law distribution.
■ Willing to pay 1% of transfer size as fees
■ 4% yearly interest rate
■ 288,000 on-chain records per day (pre segwit)

Topologies
Pairs

Uniform Payments(with central hub)
Hub: Lots of flexibility, double the channel costs

Walkthrough example (in the pairs model)■ Given the on-chain fee we derive:– Optimal reset radius– Optimal channel funding– Fee for lightning transactions– Demand for blockchain records at that price■ Then find market clearing fees that sell exactly allblockchain records.

BEHAVIOR OF
INDIVIDUALSFunding the channelinterest■ Channel capacity grows proportionally to■ So do interest payments𝑓𝑒𝑒Demand for blockchain records
1
In a world without lightning: demand is ∝feeDemand for blockchain records
In a world with lightning:■ demand ∝1
feeSCALING UP

Fees

without lightning

Fees are lower
with lightning
up to ~20M
Above 20M
lightning fees
are higher

Number of users

with lightning
Miner revenue
is proportional
to fees (fixedsupply)

Number of users

With lightning

Fees
Blochchain
Fees decline
more than x2

Number of usersSupply grew x2
Miner Revenue
decreasedwith lightning &2x blocksize

Number of users

Num tx

Nearly all transactions go through lightning

Num users

100M

Tx volume

But, More volume goes directly via the blockchain

Num users

100M

Conclusion■ Lightning helps (by a lot? Not by a lot?)■ 2X Block increases help (not by a lot)

What I did not show
■ Heterogenous populations■ More complex patterns of flow
■ Other transfer distributions

Concerns■ How fragile / viable is the lightning model?
■ Will changes in interest rates & fees have high impact?