Presumably on a Visa transaction the payment provider takes a cut and Visa takes a cut. There's competition between payment providers to keep their cut down, but you can't charge Visa cards without Visa's help, meaning there's no competition on their cut. Unless a merchant is bold/foolhardy enough to decline Visa.

In the UK a lot of stores don't accept American Express due to their higher processing fees [1].

[1] http://www.theguardian.com/money/2009/nov/29/american-expres...