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If the fee rate increases then no one will save on fees. It will just get into a block faster or I guess you could aim for a low priority transaction and expect it to be fast.
I might have to look at it a bit more with a real world situation or use case. Payjoin I get as it's similar but obfuscates the change.
Fee rate has to be increased a bit to motivate nodes to broadcast the new transaction. But total fee can be still lower. See the schema where I'm removing B output, B input, and one header. Or the example from app print-screen is real case where I save 16% even with increased fee rate.
The bigger fees the bigger saving...
You are paying for data stored. Which is basically inputs, outputs, and headers.
Please have a look on the schema above. Output B (of original TX) is the same as input B (of CPFP TX). When you remove them you save on fees. Both are redundant. And you don't need even the second TX header since it will be one transaction now.
Interesting.
Why would the recipient use this? What's the advantage?
Maybe I'm not understanding in full but the part B would have to be the same amount for both parties. This is rare because few merchants have a fixed price in sats for their product.