A Simple PSPACE-Complete Problem

Back in the typecast last month I promised a simple PSPACE-complete game in a future post. Here it is:

The SET GAME

Given: A collection of finite sets S₁,...,S_k.

The Game: Each player takes turns picking a non-empty set S_i. Remove the elements of S_i from all the sets S_j. The player who empties all the sets wins.

This game came up in a discussion I had with Steve Fenner trying to extend his student's work that Poset Games were PSPACE-complete. The PSPACE-completeness of determining a winner of the SET GAME is an easy reduction from from Poset Games.

An open question: Do we still get PSPACE-completeness if the size of the S_i are bounded? I don't even know the answer if the sets have size at most two.