  Quantum game theory is a rapidly evolving subject that extends beyond
physics. In this research work, a schematic picture of quantum game theory has
been provided with the help of the famous game Prisoners' Dilemma. It has been
considered that the shared qubits of the prisoners may interact with the
environment which is a bath of simple harmonic oscillators. This interaction
introduces decoherence and this model has been compared with phase damping.
Calculation of the decoherence factor shows that decoherence reduces the payoff
of the prisoners. The factor has been calculated based on whether the
decoherence occurred once or twice. A comparative discussion establishes that
the process of decoherence is faster for the case when it occurs twice. It
needs to be mentioned here that the total time of decoherence is the same in
both cases.
