A Unique Mixed Equilibrium Payoff in Quantum Bimatrix Games

Abstract

Consider a quantum bimatrix game where each player has knowledge of the initial (quantum) state $\alpha$ and sends an identical completely mixed strategy for measuring the final state $\omega$ 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 ${\textbf{1}}$ 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 ${\textbf{q}}$ such that the dot (scalar) product ${\textbf{q}} \cdot {\textbf{1}}$ is equal to the dimension of the underlying space describing the game. Now let the reciprocal $\left( {\textbf{q}} \cdot {\textbf{1}} \right) ^{- 1}$ denote the unique equilibrium payoff. We show that when $A {\textbf{q}} = {\textbf{1}}$ the mapping $\alpha \mapsto \omega = \left( {\textbf{q}} \cdot {\textbf{1}} \right) ^{- 1}$.