  Using formal tools in computer science to describe games is an interesting
problem. We give games, exactly two person games, an axiomatic foundation based
on the process algebra ACP (Algebra of Communicating Process). A fresh operator
called opponent's alternative composition operator (OA) is introduced into ACP
to describe game trees and game strategies, called GameACP. And its sound and
complete axiomatic system is naturally established. To model the outcomes of
games (the co-action of the player and the opponent), correspondingly in
GameACP, the execution of GameACP processes, another operator called playing
operator (PO) is extended into GameACP. We also establish a sound and complete
axiomatic system for PO. To overcome the new occurred non-determinacy
introduced by GameACP, we extend truly concurrent process algebra APTC for
games called GameAPTC. Finally, we give the correctness theorem between the
outcomes of games and the deductions of GameACP and GameAPTC processes.
