  In recent years methods have been proposed to extend classical game theory
into the quantum domain. This paper explores further extensions of these ideas
that may have a substantial potential for further research. Upon reformulating
quantum game theory as a theory of classical games played by "quantum players"
I take a constructive approach. The roles of the players and the arbiter are
investigated for clues on the nature of the quantum game space.
  Upon examination of the role of the arbiter, a possible non-commutative
nature of pay-off operators can be deduced. I investigate a sub-class of games
in which the pay-off operators satisfy non-trivial commutation relations.
Non-abelian pay-off operators can be used to generate whole families of quantum
games.
