They do, sort of. What happens IME is that the terminal is linked to a gateway, and then the gateway links to a processor, and then the processor contacts the i

They do, sort of. What happens IME is that the terminal is linked to a gateway, and then the gateway links to a processor, and then the processor contacts the issuer through the payment network. Most gateways can contact many processors, and many processors can contact any issuer through several payment networks. What seems to be happening is that the payment networks are making exclusive deals with big issuers like banks and these deals are allowing specific payment networks to have a lot of pricing power with merchants, who have to pay a fee to everyone involved in the transaction(gateway, processor, etc).