Quantum-Prisoners-Dilemma

We investigate the quantization of non-zero sum game called Prisoners’ Dilemma.

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Introduction:

1. The introduction of quantum strategies in Prisoner’s dilemma can remove the dilemma as well as ensure absolute reward against any classical strategy.
2. The classical strategies cooperate |C⟩ and defect |D⟩ are used to represent the game’s quantum state as a vector in the space with |CC⟩ , |CD⟩ , |DC⟩ , and |DD⟩ as bases, the first alphabet in each representing Alice’s move.
3. To model (any) entanglement between Alice and Bob’s moves, the game starts with a Unitary Transform Ĵ applied to |CC⟩.
4. Now, Alice and Bob play their strategies which are 2x2 Unitary operators denoted by ÛA and ÛB respectively.
5. Finally, to determine the players’ individual pay-off the qubits are forwarded to a measurement consisting of a Unitary operator Ĵ^†^ followed by a state collapse. Final state: |Ѱf⟩ = Ĵ^†^ (ÛA⊗ÛB)Ĵ |CC⟩.
6. Due to intrinsically probabilistic outcomes, we measure the expected payoff, that is $A=rPCC+pPDD+tPDC+sPCD where r,p,t and s are from the standard payoff matrix and the P’s are the measured probabilities of the respective outcome.
7. The quantum strategies are simulated as a 2-parameter set of Unitary Matrices.
8. Now we run the game for several settings of , and derive the 𝜙 𝜃 payoff statistics for Alice.

Circuit:

Results: