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Cellular automaton simulation of the quantum war of attrition game

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Abstract

The quantum war of attrition game is studied in this work via spatial numerical simulation. It is found that the implemented simulation converges to the Pareto optimal solution, i.e. no fighting at all, when the resign times of the players are entangled with higher factor, whereas larger resign times would be got with weak entanglement. This finding is shown to apply also in a fiercer war game, the war of extermination, in which game the non-entangled (or classical) simulation leads to very high resign times and consequently to very high negative payoffs.

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Notes

  1. \(w_1=\cosh 1.0, w_2=\sinh 1.0 \rightarrow 15.6w_1= 24.07, 15.6w_2=18.33 : (15.6, 8.7)\rightarrow (24.07+ 8.7w_2, 8.7w_1+18.33)=(24.07+10.22,13.42+18.33)=(34.29,31.75)\), \((15.6, 9.0)\rightarrow (24.07+ 9.0w_2, 9.0w_1+18.33)=(24.07+10.58,13.89+18.33)=(34.65,32.22)\), \((15.6, 3.6)\rightarrow (24.07+ 3.6w_2, 3.6w_1+18.33)=(24.07+ 4.23, 5.55+18.33)=(28.30,23.88)\), \((19.5,19.5)\rightarrow (24.07+19.5w_2,19.5w_1+18.33)=(24.07+22.92,30.09+18.33)=(46.99,48.42)\),

  2. Thus, replacing the extremely low intrinsic \(\epsilon \) used by the Fortran compiler. In GNU Fortran, it is \(\epsilon =2.22044604925031308E-016\).

  3. At \(\gamma =4.6342\), it is \(\overline{t_1}=0.0001\), \(\overline{t_2}=0.0000\), \(\overline{t_{1,2}^c}=0.0091\), \(\overline{u}_{1,2}=u^\#_{1,2}=4.9909=5.0000-0.0091\).

  4. In Table 8, at \(\gamma =0.0\) it is: \(6.3= (3(10-0)+(10/2-10 ))/4= 25/4\), \( 1.3=(3(10/2-0)+(-10))/4=5/4\), and at \(\gamma =1.0\) it is: \(-6.9= (3(10-10\sinh \,1)+(10/2-10e))/4=-27.44/4\), \(-0.1=(3(10/2-0)+(-10\cosh \,1))/4=-0.43/4\)

  5. In Table 9, at \(\gamma =0.0\) it is: \(-6.3 =(3(-10 )+(10/2-0))/4=-25/4\), \(-1.3= (3(10/2-10 +(10-0 ))/4=-5/4\), and at \(\gamma =1.0\) it is: \(-10.3=(3(-10\cosh \,1)+(10/2-0))/4=-41.29/4\), \(-17.1=(3(10/2-10e)+(10-10\sinh (1))/4=-68.3/4\).

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Acknowledgements

The RAS contribution to this work has been funded by the Spanish Grant PGC2018-093854-B-I00 and by the Quality Research allocation fund (FET, UWE, Bristol) during a stage in England.

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Alonso-Sanz, R., Adamatzky, A. Cellular automaton simulation of the quantum war of attrition game. Quantum Inf Process 19, 355 (2020). https://doi.org/10.1007/s11128-020-02860-w

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