Instead of putting your money in a bank account, you will put it into "payment channels" with businesses that you pay every so often. Sometimes, you'll decide not to use that business any more. Sometimes you'll start using and paying a new business. Sometimes, someone will pay you through one of those businesses with which you have a payment channel.
How would all that work? It works in something called the "Lightning Network" (LN) which runs on top of Bitcoin. Bitcoin works because anyone with a cryptographic key (some seemingly random data that represents a very large number that no one else has) can use that key to spend bitcoin that has been locked to that key. To spend it, they need only to make a packet of data called a transaction which proves they have the key and assigns the bitcoin to some other key, or to a little program ("script") that can be used to spend it again.
If you often buy gas at Arco, as many do, you might sign a bitcoin transaction with Arco for $500 in which you get $475 and Arco gets $25, and you get $25 worth of gas. This initial channel-opening transaction might cost $10, which you would pay. Why? Because after that, there are no processing fees. You could get gas 35 times and pay the $0.30 debit card processing fee each time, losing $10.50 over, say, 30 weeks, or you could pay $10 once. "Ahh, but Dave," you say, "that's $875 and you said I only put $500 in there."
Well played, my friend, but the company you work for has a channel open with Arco too, and they keep way more money in their channel with Arco than you do. Since Arco has agreed to charge zero fees to relay their payments, whenever you get paid, part of that goes into a balance update between your company and Arco so that Arco has more money, and then you and Arco update your channel balance so you have that same amount more.
Also, your company has a large channel with the same IT consulting firm that Arco uses, and that consulting company wants to be nice to its clients and suppliers, so they don't charge relay fees either. Arco pays the consulting company for software updates through a channel, and the consulting company pays your company for, let's say paper products, by sending payments back to Arco, which then sends them back to your company. Lastly, the consulting company uses the same ISP as you do, so part of your paycheck goes back to the consulting company, who sends it back to your ISP, who then forwards it to you, replenishing your channel with your ISP.
Some channels have more than two parties, and whenever there is a balance update between two of them, none of the others need to worry about it. Most channels on the LN charge zero fees because they want to continue avoiding the costs and headaches of moving money around.
Sometimes, you will want to spend bitcoin in a place only once. Maybe you're trying out a new restaurant. They have a LN node, but you don't want to spend an extra $10 for a channel. Your Bitcoin Lightning wallet comes to the rescue because it can see that the restaurant's node has incoming liquidity from Sam's Club, who has incoming liquidity from your company. Maybe Sam's Club pays the restaurant to prepare some of the ready-made food that Sam's Club sells. It doesn't matter. What matters is that your wallet can shift money around so that you have less and the restaurant has more. Or maybe it can't.
When it can't "find a path" to pay a new place you're trying out, it requests a discount from the restaurant's node to cover the fee. In the same way that a restaurant risks the cost of each meal it serves to a customer who might get up and leave without paying, it risks the cost of this meal, and gives up the cost of the network fee, in order to create a large payment channel with you, telling your wallet "Yes, here's the discount, and I'm committing $200 to our payment channel." A restaurant that expects to serve its customers well would take that risk under the assumption that customers will return for many more meals. Maybe one of your future paychecks will fill that channel up.
Another possibility is that the new place does have a path: you send to Arco and Arco sends to the new place, but this doesn't work because nearly all the money in your channel with Arco already belongs to Arco. Maybe Arco's wallet, your wallet, and the new place's wallet will work together to increase the size of your channel with Arco while the new place pays the fee. Maybe it will be a three way channel.
Lastly, while fiat currency is still a thing, lying around losing purchasing power, the new place might just accept cash. They might even offer to sell you bitcoin and open a channel with you if you have a lot of cash that you want to convert to bitcoin. Cash, has, so far, persisted even though we have credit cards and debit cards. I believe checks are dying faster than cash. Cards are not yet dying, and they may never die. I have a card from which I can spend precious metals, and I know there are similar cards through which people can spend cryptocurrency. My main prediction here is that Lightning Channels will ultimately drive transaction fees to zero, although I suspect fees may be used to encourage relays to go in one direction or another in order to balance channels. For example, if you get close to spending all $500 in your channel with Arco before your paycheck comes in and shifts that balance back to you, you might set your fees to negative for people who send payments through Arco to you to someone else with whom you have some balance, and thus shift balance between you and Arco back to your side.
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