Strong isomorphism in Marinatto-Weber type quantum games

Our purpose is to focus attention on a new criterion for quantum schemes by bringing together the notions of quantum game and game isomorphism. A quantum game scheme is required to generate the classical game as a special case. Now, given a quantum game scheme and two isomorphic classical games, we additionally require the resulting quantum games to be isomorphic as well. We show how this isomorphism condition influences the players' strategy sets. We are concerned with the Marinatto-Weber type quantum game scheme and the strong isomorphism between games in strategic form.