The Strategic Form of Quantum Prisoners' Dilemma

In a normal form, prisoners' dilemma (PD) is represented by a payoff matrix showing players' strategies and payoffs. To obtain the distinguishing trait and strategic form of PD, certain constraints are imposed on the elements of its payoff matrix. We quantize PD by a generalized quantization scheme to analyze its strategic behavior in the quantum domain. The game starts with a general entangled state of the form |ψ>=cos(ξ/2)|00>+sin(ξ/2)|11> and the measurement for payoffs is performed in entangled and product bases. We show that for both measurements, there exist respective cutoff values of entanglement of the initial quantum state up to which the strategic form of the game remains intact. Beyond these cutoffs the quantized PD behaves like the chicken game (CG) up to another cutoff value. For the measurement in the entangled basis the dilemma is resolved for sinξ >1/7 with Q⊗Q as a Nash Equilibrium (NE). However, the quantized game behaves like PD when sinξ >1/3; whereas in the range 1/7ξ <1/3 it behaves like CG with Q⊗Q as an NE. For the measurement in the product basis the quantized PD behaves like classical PD for sin²(ξ/2) <1/3 with D⊗D as an NE. In region 1/32(ξ/2)<3/7, the quantized PD behaves like classical CG with C⊗D and D⊗C as NEs.