We 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.