You could base it off the version in Wedding of River Song, where the more you move a piece, the more voltage runs through it. Maybe you could face different characters, like Steven or Ian, before moving up to better in-universe players, like Fenric, Gantok, or 7 & 11.
I can see that working! Although, the idea of over four million volts zapping me if I make the wrong move in front of a restless crowd isn’t exactly accommodating…
That’s why it would have to be a video game - surely a little difficult to accommodate with regular pieces…

What about those little shock rings for pranks. It doesn’t have to be very high voltage.
I would love to play Live Chess.
An RPG like the Mario & Luigi series could work well, where there’s a good amount of problem solving.
The main issue would be what you would use for the attacks. The Doctor isn’t violent most times, after all. I suppose you could just make one of the main ones blasting with the Sonic Screwdriver, and the other main attack could be venusian aikido, then have something like the Mario & Luigi series where you have special attacks that require teamwork (for example, if River is a part of your party, there could be one where River fires her squareness gun to destroy some blocks keeping a wooden trolley in place, and the Doctor Sonics it to make it move and squish the enemies in the way).