Do all PS1 Controllers Fully Work With All PS2 Games?

Since the controller plugs are the same for both the Sony PS1 and PS2 game consoles, they are multi compatible with each other which is very convenient. Most PlayStation 2 games are fully supported by any PS1 analog type of controller that is plugged in. However, there are a few minority that don't work since they require dualshock 2 support.