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Why there aren't negative fees on lightning network protocol and the ability to set incoming fees? Would there be any benefit? Let me describe the situation. There are nodes where the most of the liquidity goes to (the terminal type) and the nodes where the most of the liquidity goes from (the initial type). To rebalance it back is sometimes costly. A lightning node operator would maybe be willing to pay people to send satoshis from certain terminal nodes (setting incoming fees on negative) and send them to the initial type nodes (setting outcoming fees on negative).

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I am not sure if it is possible to give a satisfying / definite answer to your question. As far as I can tell lightning developers have been very cautious in the early days (and still are) with everything that might make people loose funds. Negative fees are an obvious possibility to loose money as people would route along that channel and you would have to pay for that if you don't actively manage your node and know what you do.

On a bit more opinionated level and from a mathematical point of view I agree that it is a little bit strange that the cost function is just not permitted to have negative values and I agree that a lot of things with respect to liquidity management, routing and payments would probably be much easier if we did not have to make that somewhat artificial distinction.

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