  This work, based on the author's MA thesis, concentrates on simultaneous move
quantum games of two players. A numerical algorithm based on the method of best
response functions, designed to search for pure strategy Nash equilibrium in
quantum games is constructed. It is then used show that there is a critical
degree of entanglement above which a pure strategy Nash Equilibrium does not
exist. The behavior of the two payoffs approach the cooperative classical ones.
Bayesian quantum games are defined, and it is shown that under certain
conditions, there {\it is} a pure strategy Nash equilibrium in such games even
when entanglement is maximal. The basic ingredients of a quantum game based on
a two-players {\it three} decisions classical games are presented. This
requires the definition of trits (instead of bits) and quantum trits (instead
of quantum bits). It is shown that in this quantum game, there is no classical
commensurability in the sense that the classical strategies are not obtained as
a special case of the quantum strategies. An entanglement operator transforming
a pure two qutrit state into a maximally entangled one is explicitly
constructed.
