I was reading the "Mastering The Lightning Network" book when I came across this part about Multipart payments which mean dividing a payment and sending it through multiple paths to increase the chances of the payment succeeding.
My question is if a payment is going to be sent through multiple nodes, isn't that going to increase the routing fees? If yes, wouldn't it be cheaper to send that transaction through the standard bitcoin network and not through Lightning Network?
Both onchain fees and Lightning routing fees are subject to market forces i.e. supply and demand. A key difference between onchain fees and Lightning routing fees is that onchain fees are driven by the size of the transaction rather than the amount being sent while Lightning routing fees are driven by the amount being sent. Large amounts sent through the Lightning Network either require large channel capacities, the use of multiple paths or both.
My question is if a payment is going to be sent through multiple nodes, isn't that going to increase the routing fees?
Possibly but not necessarily. It depends on the market forces. If there is lots of demand for using channels with large capacities they can charge a higher fee and then using a number of channels with smaller capacities may end up being cheaper. Similarly if there is little demand for routing payments across the whole network those channels with large capacities may charge a very low fee similar to the channels with small capacities and it may work out cheaper to use the channels with large capacities.
It is worth noting that there are currently two routing fees on the Lightning Network: a base fee and a proportional fee. If the proportional fee is the same across channels with large capacities as channels with small capacities then the total proportional fee will be exactly the same whether you use one path or multiple paths (assuming the number of hops is the same on each path). To work out which is cheaper would depend on a comparison of the base fees.
If yes, wouldn't it be cheaper to send that transaction through the standard bitcoin network and not through Lightning Network?
Again possibly but not necessarily. If it was a huge amount, say 10,000 Bitcoin, it would be cheaper to send that transaction onchain versus via the Lightning Network as today you probably wouldn't be able to send it through the Lightning Network and it successfully reach its destination (without channels being opened for the purpose of routing this particular payment). But you'd expect sending smaller amounts to be cheaper using the Lightning Network versus onchain regardless of whether that be using a single path or multiple paths.
