Is quantum game theory reducible to classical game theory?

Question

Modnote: This question was manually migrated (closed and crossposted) to MathOverflow by request of the OP.

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Quantum game theory is an extension of classical game theory to the quantum domain. It differs from classical game theory in three primary ways:

1. Superposed initial states,
2. Quantum entanglement of initial states,
3. Superposition of strategies to be used on the initial states.

This theory is based on the physics of information much like quantum computing.

I wondered if QGT is reducible to classical GT, i.e., whether any quantum game can be transformed to some classical game.

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^{Related issues: To prove the opposite, would we need a space-like separation between players' acts? Would one need Bell's theorem? Should players' acts be outside each other's light-cones? Do we have to appeal to physics (e.g., QM itself and/or GR)? Would we need counterfactual definiteness? Would we need to dismiss superdeterminism? Are, some or all, such issues already covered (by hidden assumptions) in classical game theory or even economics?}

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Can anyone perhaps point to relevant literature that specifically deals with this (the title) question?