Hm, this is getting more elaborate than I had anticipated. I guess Purvis (or at least Purvis' Yukari) doesn't enjoy the idea of an Old Maids Alliance.
For Reimu, we can tell her it's a tournament for all Gensokyo, and no location represents all of Gensokyo more than the Hakurei Shrine.
I think we're doing Mahjongg because it was mentioned, and because it has the illusion of letting everyone participate at the same level, unlike physical sports. I would prefer poker for various reasons, but it may be too late narratively to switch games.
Poker can have a different number of players: many for the earlier rounds, one-on-one for the later rounds. The rules of poker are simpler, and we can assume all the voices in her head know the rules.