On Pay-off induced Quantum Games

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.