Comment on “Quantum Games and Quantum Strategies''

In a recent Letter, Eisert et al. [1] presented a quantum mechanical generalization of Prisoner’s Dilemma. In the classical form of this game, rational analysis leads the two players to ‘defect ’ against one-another in a mutually destructive fashion [2]. A central result of Eisert et al.’s Letter is the observation that their quantum variant, illustrated in Figure 1, can avoid the ‘dilemma’: the mutually destructive outcome is replaced with an effectively cooperative one. Specifically, it is asserted that the maximally entangled game’s unique Nash equilibrium [2] occurs when both players apply the strategy ˆ Q = iσz, yielding a pay-off equivalent to cooperative behaviour in the classical game. In this Comment we show that their observation is incorrect. The mistake follows from the following erroneous assertion: It proves to be sufficient to restrict the strategic space to the 2-parameter set of unitary 2 x 2 matrices, iφ e cosθ/2 sinθ/2 Û(θ, φ) = −sinθ/2 e−iφ) cosθ/2