AbstractWe consider combinatorial principles based on playing several two person games simultaneously. We call strategies for playing two or more games simultaneously parallel. The principles are easy consequences of the determinacy of games, in particular they are true for all finite games. We shall show that the principles fail for infinite games. The statements of these principles are of lower logical complexity than the sentence expressing the determinacy of games, therefore, they can be studied in weak axiomatic systems for arithmetic (Bounded Arithmetic). We pose several open problems about the provability of these statements in Bounded Arithmetic and related computational problems.