  Strong placement games (SP-games) are a class of combinatorial games whose
structure allows one to describe the game via simplicial complexes. A natural
question is whether well-known invariants of combinatorial games, such as "game
value", appear as invariants of the simplicial complexes. This paper is the
first step in that direction. We show that every simplicial complex encodes a
certain type of SP-game (called an "invariant SP-game") whose ruleset is
independent of the board it is played on. We also show that in the class of
SP-games isomorphic simplicial complexes correspond to isomorphic game trees,
and hence equal game values. We also study a subclass of SP-games corresponding
to flag complexes, showing that there is always a game whose corresponding
complex is a flag complex no matter which board it is played on.
