  We quantize prisoner dilemma in presence of collective dephasing with
dephasing rate $\gamma $. It is shown that for two parameters set of strategies
$Q\otimes Q$ is Nash equilibrium below a cut-off value of time. Beyond this
cut-off it bifurcates into two new Nash equilibria $Q\otimes D$ and $D\otimes
Q$. Furthermore for maximum value of decoherence \ $C\otimes D$ and $D\otimes
C$ also become Nash equilibria. At this stage the game has four Nash
equilibria. On the other hand for three parameters set of strategies there is
no pure strategy Nash equilibrium however there is mixed strategy (non unique)
Nash equilibrium that is not affected by collective dephasing..
