If Player A has face-up monsters on the field with different attributes, he just says something like "I choose WATER attributes to stay on the field" and sends all other face-up non-WATER monsters to the graveyard.
If he has no face-up monsters, the next time a monster is face-up on the field, the attribute of the card is applied for Gozen Match (I don't quite know what'd happen with cards like Dark Simorgh or so, but fuck that).
While Gozen Match is active, neither player can summon a monster with a different attribute than what attribute is applied for Gozen Match, even if it means that the new monster would be the only one of its attribute! Example, you control 2 face-up Fluff Tokens (WIND, as a reminder), you now cannot tribute these two to summon Blue-Eyes White Dragon, you are onpy able to Tribute Set it , and then you can flip it up, first.
Synchro, Fusion and Xyz Summons aren't possible unless all the materials and the monster attempted to summon have the same attribute (for example, overlaying 2 Brotherhood of the Fire Fist- Bear's to Xyz Summon BotFF - Tiger King is perfectly legal, bit you cannot dummon Number 39: Utopia, for example).
By the way, everything mentioned for Player A in above paragraph applies to Player B, too, of course.