A new model for quantum games based on the Marinatto-Weber approach

The Marinatto-Weber approach to quantum game is a straightforward way to apply the power of quantum mechanics to classical game theory. In the simplest case, the quantum scheme is that players manipulate their own qubits of a two-qubit state either with the identity operator or the Pauli operator $\sigma_{x}$. However, such a simplification of the scheme raises doubt as to whether it could really reflect a quantum game. In this paper we put forward examples which may constitute arguments against the present form of the Marinatto-Weber scheme. Next, we modify the scheme to eliminate the undesirable properties of the protocol by extending the players' strategy sets.