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 \(\alpha \mapsto \omega = \left( {\textbf{q}} \cdot {\textbf{1}} \right) ^{- 1}\).
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Acknowledgements
We would like to thank the anonymous reviewers for the generous feedback that greatly improved this manuscript. Additionally, we are grateful to Dr. Matiur Rahman for the many helpful discussions, as well as the editor and Morgan Turpin for the constructive comments on the overall readability and structure of the paper.
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Communicated by Francesco Zirilli.
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Turpin, L. A Unique Mixed Equilibrium Payoff in Quantum Bimatrix Games. J Optim Theory Appl 196, 1119–1124 (2023). https://doi.org/10.1007/s10957-023-02170-y
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DOI: https://doi.org/10.1007/s10957-023-02170-y