When I use Google Authenticator two-factor authentication, Google will tell me if I get my password wrong before prompting me for an authentication code.
This goes against at least one intuitive line of reasoning -- it allows an attacker to check whether they have the right password before trying to attack two-factor authentication.
What are the trade-offs that make this a reasonable choice in Google's case, and in general? How should this inform similar questions in other spaces?
This question is about 2FA in general but Google is just a widely used example.