An utterly mad, or entirely brilliant idea...

[robus]

To reflect the influence of PC abilities on those competitions, you give the PCs character-specific advantages or disadvantages in those games. A particularly fast RPG character might get bonuses to rolls in a board game. An insightful one might get an additional draw for a card game. Etc.

That is interesting, but for my tastes the gear switch would be too much and not give the campaign feel I'm looking for. I'm more interested in the possibility of interweaving a story through the three settings.

My current sketchy idea is that the prize money from the tournament is needed to get the teens in the real world out of some jam - perhaps they're being black-mailed or loan-sharked in some manner and need the money to pay off the debts? And then there's cheating going on in the tournament via some virtual world shenanigans that is going to cause them trouble there.