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Spatial Quantum Game Simulation

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Quantum Game Simulation

Part of the book series: Emergence, Complexity and Computation ((ECC,volume 36))

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Abstract

This chapter deals with collective games in which the players are arranged in a spatially structured two-dimensional lattice as explained in Sect. 3.1. Sections 3.2 and 3.3 deal with two-parameter and three-parameter strategy simulations respectively. A variant of the canonical EWL model is considered in Sect. 3.4.

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Notes

  1. 1.

    In the five simulations of Fig. 3.6 at \(\gamma =0\) it is: \((\overline{\theta }_A,\overline{\theta }_B)= (1.568,1.710), (0.741,2.068), (1.239,2.256), (1.240 1.906), (1.606,2.086)\), with associated mean-field payoffs: \((p^*_A,p^*_B)=(0.147,1.503), (0.006,2.414), (-0.238,1.928), (0.113,1.878)\), \((-0.125,1.456)\), very close to the actual payoffs: \((\overline{p}_A,\overline{p}_B)=(0.149,1.502)\), \((-0.054,2.201), (-0.130,1.813), (0.027,1.798), (-0.087,1.456)\), shown in the left frame of the figure.

  2. 2.

    The reference [14] is also relevant at this respect, but the occasional reader should be warned about the variation of the \(\alpha \) and \(\beta \) parameters in the \([-\pi ,\pi ]\) interval instead of in \([0,\pi /2]\), as proposed for \(\alpha \) in the seminal EWL paper.

  3. 3.

    \((\hat{U}_{\!A}\otimes \hat{U}_{\!B})\vert \psi _i\rangle = \left( \begin{matrix} 0 &{} 0 &{} 0 &{} e^{i\pi /2}=i \\ 0 &{} 0 &{} -1 &{} 0 \\ 0 &{} -1 &{} 0 &{} 0 \\ e^{-i\pi /2}=-i &{} 0 &{} 0 &{} 0 \end{matrix}\right) \left( {\begin{matrix}\cos \frac{\gamma }{2}\\ 0\\ 0\\ i\sin \frac{\gamma }{2}\\ \end{matrix}}\right) \!=\! \left( {\begin{matrix} -\sin \frac{\gamma }{2}\\ 0\\ 0\\ -i\cos \frac{\gamma }{2} \end{matrix}}\right) \), \(\vert \psi _f\rangle \!=\!\) \( \left( \begin{matrix} \cos \frac{\gamma }{2}&{}0&{}0 &{} -i\sin \frac{\gamma }{2}\\ 0&{}\cos \frac{\gamma }{2}&{} i\sin \frac{\gamma }{2}&{}0\\ 0&{}i\sin \frac{\gamma }{2}&{}\cos \frac{\gamma }{2}&{}0\\ -i\sin \frac{\gamma }{2}&{}0&{}0&{}\cos \frac{\gamma }{2} \end{matrix}\right) \left( \begin{matrix} -\sin \frac{\gamma }{2}\\ 0\\ 0\\ -i\cos \frac{\gamma }{2} \end{matrix}\right) = \left( \begin{matrix} -\cos \frac{\gamma }{2}\sin \frac{\gamma }{2} -\sin \frac{\gamma }{2}\cos \frac{\gamma }{2}\\ 0\\ 0\\ i\sin ^2\frac{\gamma }{2}-i\cos ^2\frac{\gamma }{2} \end{matrix}\right) = \left( \begin{matrix} -\sin \gamma \\ 0\\ 0\\ -i\cos \gamma \end{matrix}\right) \).

  4. 4.

    \(\vert \psi _f\rangle = J^\dag i\left( {\begin{matrix} i\sin (\gamma /2)\\ 0\\ 0\\ -\cos (\gamma /2)\\ \end{matrix}}\right) =i\left( \begin{matrix} i\cos (\gamma /2)\sin (\gamma /2)+i\sin (\gamma /2)\cos (\gamma /2)\\ 0\\ 0\\ \sin ^2(\gamma /2) - \cos ^2(\gamma /2) \end{matrix} \right) \).

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  1. Technical Universty of Madrid, Madrid, Spain

    Ramón Alonso-Sanz

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Correspondence to Ramón Alonso-Sanz .

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Alonso-Sanz, R. (2019). Spatial Quantum Game Simulation. In: Quantum Game Simulation. Emergence, Complexity and Computation, vol 36. Springer, Cham. https://doi.org/10.1007/978-3-030-19634-9_3

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