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Synchronous Values of Games

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Abstract

We study synchronous values of games, especially synchronous games. It is known that a synchronous game has a perfect strategy if and only if it has a perfect synchronous strategy. However, we give examples of synchronous games, in particular graph colouring games, with synchronous value that is strictly smaller than their ordinary value. Thus, the optimal strategy for a synchronous game need not be synchronous. We derive a formula for the synchronous value of an XOR game as an optimization problem over a spectrahedron involving a matrix related to the cost matrix. We give an example of a game such that the synchronous value of repeated products of the game is strictly increasing. We show that the synchronous quantum bias of the XOR of two XOR games is not multiplicative. Finally, we derive geometric and algebraic conditions that a set of projections that yields the synchronous value of a game must satisfy.

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Notes

  1. Some authors refer to conditional probability densities as correlations.

  2. This can be seen directly by checking that \(\begin{pmatrix} 1&\omega _n^j&\dots&\omega _n^{j(n-1)} \end{pmatrix}^T\) is an eigenvector for each \(j \in \{0,1,\dots , n-1\}\).

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Acknowledgements

H.M. acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC). V.I.P. was supported by NSERC Grant 03784. All the authors wish to thank the American Institute of Mathematics (AIM) where this research originated.

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Authors and Affiliations

  1. Department of Mathematics, University of California San Diego, La Jolla, CA, 92093-0112, USA

    J. William Helton

  2. Department of Computer Science, Columbia University, New York, NY, 10027, USA

    Hamoon Mousavi

  3. Joint Center for Quantum Information and Computer Science and Department of Computer Science, University of Maryland, College Park, MD, 20742, USA

    Seyed Sajjad Nezhadi

  4. Institute for Quantum Computing and Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Vern I. Paulsen

  5. Army Cyber Institute and Department of Mathematical Sciences, United States Military Academy, West Point, NY, 10996, USA

    Travis B. Russell

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  1. J. William Helton
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  2. Hamoon Mousavi
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Correspondence to Vern I. Paulsen.

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Communicated by David Pérez-García.

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Helton, J.W., Mousavi, H., Nezhadi, S.S. et al. Synchronous Values of Games. Ann. Henri Poincaré (2024). https://doi.org/10.1007/s00023-024-01426-1

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