The subtle distinctions between these (and between other so-called modal verbs) are gradually disappearing, but they still matter to many readers and can be useful.

may implies that the possibility remains open: The Mies van der Rohe tower may have changed the face of British architecture for ever (it has been built); might suggests that the possibility remains open no longer: The Mies tower might have changed the face of architecture for ever (if only they had built it). Similarly, they may have played tennis, or they may have gone boating suggests I don't know what they did; they might have played tennis if the weather had been dry means they didn't, because it wasn't.

may also has the meaning of "having permission", so be careful: does Megawatt Corp may bid for TransElectric Inc mean that it is considering a bid, or that the competition authorities have allowed it to bid?