A Unique Mixed Equilibrium Payoff in Quantum Bimatrix Games

Consider a quantum bimatrix game where each player has knowledge of the initial (quantum) state $$\alpha $$ α and sends an identical completely mixed strategy for measuring the final state $$\omega $$ ω to a judge, who then performs the measurement (as a combination of strategies). The strategies take on the form of general unitary operations and are associated with a pair of payoffs in the matrix A, contained within an arbitrary affine space of matrices. Let $${\textbf{1}}$$ 1 be the vector with all entries equal to one. Suppose (i) player one takes on a strategy that produces a Nash equilibrium and (ii) there exists a $${\textbf{q}}$$ q such that the dot (scalar) product $${\textbf{q}} \cdot {\textbf{1}}$$ q · 1 is equal to the dimension of the underlying space describing the game. Now let the reciprocal $$\left( {\textbf{q}} \cdot {\textbf{1}} \right) ^{- 1}$$ q · 1 - 1 denote the unique equilibrium payoff. We show that when $$A {\textbf{q}} = {\textbf{1}}$$ A q = 1 the mapping $$\alpha \mapsto \omega = \left( {\textbf{q}} \cdot {\textbf{1}} \right) ^{- 1}$$ α ↦ ω = q · 1 - 1 .