OK, so we need to talk about how far back you’re talking about.
The concept of the “debit card” didn’t exist when I was growing up. I remember getting issued my first one somewhere around the year 2000, which was well after the internet became a thing.
We had ATM cards, but they all used their own networks (similar to the Visa / Mastercard / American Express / Diner’s Club / Discover networks for credit cards). They pretty much only worked at ATMs, and weren’t integrated into Point of Sale systems. They also only worked at ATMs that were part of their network – and there were a bunch of networks.
When you used the ATM, it would dial up the network’s central computer via a modem and a phone line, and it would query the network to see if you had enough cash to let you withdraw. The network would then contact your bank’s computers to see if you had the cash in your account, and if it did, the bank would tell the network, which would then tell the ATM to give you the money.
Today, a debit card actually has two modes: one that uses the credit card network, and one that uses the old ATM network. If you look at the back of your card, you probably show two network symbols – a Visa symbol, and one of the big 3 ATM networks (STAR, NYCE, or Pulse).
If you use your PIN, then it processes with the ATM network, and if you sign, it uses the credit card network. Vendors prefer than you use the PIN, because the fees to them are lower than if you sign.
The actual process of querying the network that then queries your account is pretty much identical to how it worked back in the pre-internet days. It’s just a series of servers between the place that has your card and the bank’s central computer asking one another if you have enough money to allow the transaction to go through.
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