As the other poster said, there's time, possibility/likelihood and social relations ("politeness" is a limited concept) involved in choosing which to use.
We thought we might have won. DISTANCE IN TIME (and possibility)
We might go to the beach this weekend. DISTANCE OF POSSIBILITY/LIKELIHOOD
Might I help you, sir? DISTANCE IN SOCIAL RELATIONS
We may win today. CLOSENESS IN TIME (and possibility)
We may go to the beach this weekend. CLOSE POSSIBILITY/LIKELIHOOD
May I help you, sir? CLOSENESS IN SOCIAL RELATIONS