We are seeing that if a customer is checking out, and they select an International Billing Address but a US Shipping Address, then there is no Payment Method available.
In this case, a customer should be able to use PayPal to pay for their order.
In your case, you have two payment options: authorize.net and Paypal.
Each payment method must be assigned the countries that you want to accept orders from.
If, for example, the customer puts in a country that is not allowed, there will be no options shown at checkout.
To do this - go to System > Configuration > Payment Methods. Then:
Tip: You should use the Ctrl button on your keyboard while clicking with your left mouse button in order to to mark several countries.