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The quantum Hotelling–Smithies game

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Abstract

This work studies the quantum Hotelling game with elastic demand by means of an ad hoc simulation technique that allows to scrutinize how the entanglement of the players induces the emergence of the Pareto optimal solution in Nash equilibrium (NE), even when NE does not exist in the classic game due to the proximity of the players.

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Data sharing is not applicable to this article as no external datasets were used during the current study and the outputs from the simulations are available through the figures.

Notes

  1. In multi-objective optimization problems, a solution is called Pareto optimal (or Pareto efficient), if none of the objective functions can be improved in value without degrading some of the other objective values.

  2. When a group of firms cooperates to maximize their profits in the marketplace instead of competing with each other, this is known as collusion. Collusion gives firms an unfair advantage in the marketplace and collusive practices like price fixing are designed to unfairly benefit firms at the expense of the consumer. Thus, antitrust laws are intended to prevent collusion between companies.

  3. \( p_2^c-p_1^c =p_2w_1(\gamma )+p_1w_2(\gamma )-(p_1w_1(\gamma )+p_2w_2(\gamma ))= p_2(w_1(\gamma )-w_2(\gamma ))-p_1(w_1(\gamma )-w_2(\gamma ))=(p_2-p_1)(w_1(\gamma )-w_2(\gamma ))=\) \((p_2-p_1)e^{-\gamma }\), because, \(w_1(\gamma )-w_2(\gamma )=e^{-\gamma }\).

  4. \(u_1=p_1^cQ_1\), \(Q_1=(\alpha -p_1^c){\overline{s}}+t(a{\overline{s}}-{\overline{s}}^2/2-a^2)\). \(u_2=p_2^cQ_2\), \(Q_2=(\alpha -p_2^c)(L-{\overline{s}})+t(b(L-{\overline{s}})-(L-{\overline{s}})^2/2-b^2)\)

    \(\frac{\partial u_1}{\partial p_1}=\cosh \gamma [(\alpha -p_1^c){\overline{s}}+t(a{\overline{s}}-{\overline{s}}^2/2-a^2)]+p_1^c[-\cosh \gamma {\overline{s}}-\frac{e^{-\gamma }}{2t}(\alpha -p_1^c)+t(-a\frac{e^{-\gamma }}{2t}-{\overline{s}}\frac{e^{-\gamma }}{2t})]\).

    \(a=b \rightarrow p_1=p_2=p \rightarrow p_1^c=p_2^c=pe^{\gamma } \rightarrow {\overline{s}}=L/2 \rightarrow \frac{\partial u_1}{\partial p_1}=\frac{\partial u_2}{\partial p_2}=0 \rightarrow \)

    \(\cosh \gamma [\alpha L+t(aL-(L/2)^2-2a^2)]+pe^{\gamma }[-2L\cosh \gamma +e^{-\gamma }(L/2-a)-\frac{e^{-\gamma }}{t}(\alpha -pe^{\gamma })]=0 \rightarrow \)

    \(e^{\gamma } p^2-[\alpha +t\left( a+2L\cosh \gamma e^{\gamma }-\frac{L}{2}\right) ]p+t\cosh \gamma [\alpha L+t(aL-L^2/4-2a^2)]=0\,.\)

  5. \((p^c_{1,2})^\star =\) \({\frac{(\lambda '{-}\sqrt{\lambda '^2{-}4te^{\gamma } \cosh \gamma \left( \alpha L{-}2c\right) })(\lambda '+\sqrt{\lambda '^2{-}4te^{\gamma }\cosh \gamma \left( \alpha L{-}2c\right) })}{2(\lambda '+\sqrt{\lambda '^2{-}4te^{\gamma }\cosh \gamma \left( \alpha L{-}2c\right) })} =\frac{2te^{\gamma }\cosh \gamma \left( \alpha L-2c\right) }{\lambda '+\sqrt{\lambda '^2-4te^{\gamma }\cosh \gamma \left( \alpha L-2c\right) }}=}\)

    =\(\frac{2t\left( \alpha L-2c\right) }{\frac{\lambda '}{e^{\gamma }\cosh \gamma }+\sqrt{\big (\frac{\lambda '}{e^{\gamma }\cosh \gamma }\big )^2-4t\frac{1}{e^{\gamma }\cosh \gamma }\left( \alpha L-2c\right) }}\). It is \(\lim _{\gamma \rightarrow \infty }\frac{\lambda '}{e^{\gamma }\cosh \gamma }=2Lt\). Thus, \(\lim _{\gamma \rightarrow \infty }(p^c_{1,2})^\star =\frac{2t\left( \alpha L-2c\right) }{2tL+\sqrt{4t^2L^2}}=\frac{2t(\alpha L-2c)}{4tL}=\frac{\alpha }{2}-\frac{c}{L}=p^{\bullet }\) .

  6. The authors themselves of the seminal paper [5] qualify its quantum model as a “minimal” extension of the classic Cournot’s duopoly game into the quantum domain.

  7. From Eq.(10), \(u^\bullet {=}(p_{1,2}^\bullet )^2\frac{L}{2}\) is maximized when \(p_{1,2}^\bullet =\alpha /2-c/L\) is maximized, thus when c is minimized, which occurs at \(a=L/4\). In which case, \(c=L^2/16\), \(\max p^\bullet =\alpha /2-L/16= 2.812\), \(\max Q^\bullet =2.812L/2=4.219\), \(\max u^\bullet =2.812\cdot 4.219=11.865\).

  8. \(\underline{\alpha \le \alpha _1}~ Q{=}2(\alpha -p) , u=2(\alpha -p)p, u^\prime /2 =(\alpha -p) -p \rightarrow p^\bullet {=}\frac{1}{2}\alpha \rightarrow Q^\bullet =2(\alpha -\frac{\alpha }{2})=\alpha \).

     \(\alpha {-}p^\bullet {=}a \rightarrow \alpha _1{=}2a.\)       \(\underline{\alpha _1\le \alpha \le \alpha _2}~ Q{=}2a,~p^\bullet {=}\alpha -a\).       

    \(\underline{\alpha _2\le \alpha \le \alpha _3}~ Q{=}a+\alpha -p, u^\prime /2 =a+\alpha -p-p \rightarrow p^\bullet {=}(\alpha +a)/2 \rightarrow Q^\bullet =a+\alpha -(\alpha +a)/2 =(\alpha +a)/2. ~\alpha {-}p^\bullet {=}a \rightarrow \alpha _2{=}3a\).

    \(Q^\bullet =(\alpha +a)/2 = L/2 \rightarrow \alpha _3{=}L-a\).       \(\underline{\alpha \ge \alpha _3}~Q{=}L/2,~p^\bullet {=}\alpha -(L/2-a)\). If \(a>L/4\), a is to be replaced by \(L/2-a\) in Eq.(19).

  9. \(u_1=p_1^cQ_1, Q_1=(\alpha -p_1^c){\overline{s}}+t(a{\overline{s}}-{\overline{s}}^2/2-a^2)\).       \(\frac{\partial {\overline{s}}}{\partial p_1}=-\frac{1}{2t}(\cos \gamma -\sin \gamma )\equiv -\frac{1}{2t}\Delta (\gamma ). \)       \(\frac{\partial u_1}{\partial p_1}=\cos \gamma [(\alpha -p_1^c){\overline{s}}+t(a{\overline{s}}-{\overline{s}}^2/2-a^2)]+p_1^c[-\cos \gamma {\overline{s}}-\frac{\Delta }{2t}(\alpha -p_1^c)+t(-a\frac{\Delta }{2t}+{\overline{s}}\frac{\Delta }{2t})]\)

    \(a=b \rightarrow p_1=p_2=p \rightarrow p_1^c=p_2^c=p(\sin \gamma +\cos \gamma )=p\Sigma \rightarrow {\overline{s}}=\frac{L}{2} \rightarrow \frac{\partial u_1}{\partial p_1}=0 \rightarrow \)

    \(\cos \gamma [(\alpha -p\Sigma )\frac{L}{2}+t(a\frac{L}{2}-(\frac{L}{2})^2/2-a^2)]+p\Sigma [-\cos \gamma \frac{L}{2}-\frac{\Delta }{2t}\big (\alpha -p\Sigma +t(a-\frac{L}{2}\big )\big )]=0\rightarrow \)

    \(p^2\Sigma ^2\Delta +\Sigma \big [-Lt\cos \gamma -Lt\cos \gamma -\Delta \big (\alpha +t(a-\frac{L}{2})\big ]p + t\cos \gamma \big [L\alpha +t(aL-\frac{L^2}{4}-2a^2)\big ]=0\,,\)  \(\Sigma \Delta =\cos ^2\gamma -\sin ^2\gamma =\cos 2\gamma \).

  10. \((p^\star _{1,2}(\gamma ))^c=\frac{2t\Sigma \frac{\cos \gamma }{\cos 2\gamma } \left( \alpha L-2c\right) }{\lambda '+\sqrt{\lambda '^2-4t\Sigma \frac{\cos \gamma }{\cos 2\gamma } \left( \alpha L-2c\right) }}= \frac{2t\Sigma \left( \alpha L-2c\right) }{(\lambda '\frac{\cos 2\gamma }{\cos \gamma })+\sqrt{(\lambda '\frac{\cos 2\gamma }{\cos \gamma })^2-4t\Sigma \frac{\cos 2\gamma }{\cos \gamma } \left( \alpha L-2c\right) }}\). It is \(\lim _{\gamma \rightarrow \pi /4}\lambda '\frac{\cos 2\gamma }{\cos \gamma }=2L\Sigma t\). Thus,\(\lim _{\gamma \rightarrow \pi /4}(p^\star _{1,2})^c= \frac{\alpha L-2c}{2L}=\frac{\alpha }{2}-\frac{c}{L}=p^\bullet \).

  11. \(u_1=p_1^cQ_1\), \(Q_1=(\alpha -p_1^c){\overline{s}}+t(a{\overline{s}}-{\overline{s}}^2/2-a^2)\).    \(\frac{\partial {\overline{s}}}{\partial p_1}=-\frac{\cos 2\gamma }{2t}.\)   \(\frac{\partial u_1}{\partial p_1}=\cos ^2\gamma [(\alpha -p_1^c){\overline{s}}+t(as-{\overline{s}}^2/2-a^2)]+p_1^c[-\cos ^2\gamma {\overline{s}}-\frac{\cos 2\gamma }{2t}(\alpha -p_1^c)+t(-a\frac{\cos 2\gamma }{2t}+{\overline{s}}\frac{\cos 2\gamma }{2t})]\).   \(a=b \rightarrow p_1=p_2=p \rightarrow p_1^c=p_2^c=p \rightarrow {\overline{s}}=\frac{L}{2} \rightarrow \frac{\partial u_1}{\partial p_1}=0 \rightarrow \)

    \(\cos ^2\gamma [(\alpha -p)L+t(aL-(\frac{L}{2})^2-2a^2)]+p\big [-\cos ^2\gamma L -\frac{\cos 2\gamma }{t}[(\alpha -p)+t(a-\frac{L}{2})]\big ]=0\rightarrow \).

    \(\cos 2\gamma p^2+\big [-2t\cos ^2\gamma L -\cos 2\gamma [\alpha +t(a-\frac{L}{2})]\big ]p+t\cos ^2\gamma [\alpha L+t(aL-(\frac{L}{2})^2-2a^2)]=0\).

  12. \((p^\star _{1,2}(\gamma ))^c=\frac{2t\frac{\cos ^2\gamma }{\cos 2\gamma } \left( \alpha L-2c\right) }{\lambda '+\sqrt{\lambda '^2-4t\frac{\cos ^2\gamma }{\cos 2\gamma } \left( \alpha L-2c\right) }}= \frac{2t\left( \alpha L-2c\right) }{(\lambda '\frac{\cos 2\gamma }{\cos ^2\gamma })+\sqrt{(\lambda '\frac{\cos 2\gamma }{\cos ^2\gamma })^2-4t\frac{\cos 2\gamma }{\cos ^2\gamma } \left( \alpha L-2c\right) }}\). It is \(\lim _{\gamma \rightarrow \pi /4}\lambda '\frac{\cos 2\gamma }{\cos \gamma }=2Lt\). Thus,\(\lim _{\gamma \rightarrow \pi /4}(p^\star _{1,2}(\gamma ))^c=\frac{2(\alpha L-2c)}{2L+2L}=\frac{\alpha }{2}-\frac{c}{L}=p^\bullet \).

References

  1. Hotelling, H.: Stability in competition. Econ. J. 39(153), 41–57 (1929)

    Article  Google Scholar 

  2. d’Aspremont, C., Gabszewicz, J.J., Thisse, J.F.: On hotelling’s stability in competition. Econometrica 47(5), 1145–1150 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  3. Lerner, A.P., Singer, H.W.: Some notes on duopoly and spatial competition. J. Polit. Econ. 45(2), 145–186 (1937)

    Article  Google Scholar 

  4. Smithies, A.: Optimum location in spatial competition. J. Pol. Econ. 49(3), 423–439 (1941)

    Article  Google Scholar 

  5. Li, H., Du, J., Massar, S.: Continuous-variable quantum games. Phys. Lett. A 306, 73–8 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Lo, C.F., Kiang, D.: Quantum Stackelberg duopoly. Phys. Lett. A 318, 333–336 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Lo, C.F., Kiang, D.: Quantum Bertrand duopoly with differentiated products. Phys. Lett. A 321(2), 94–98 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Alonso-Sanz, R., Adamatzky, A.: Cellular automaton simulation of the quantum war of attrition game. Quantum Inform. Process. 19(10), 1–20 (2020)

    Article  MathSciNet  Google Scholar 

  9. Alonso-Sanz, R., Adamatzky, A.: Spatial simulation of the quantum Bertrand duopoly game. Phys. A 557, 124867 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  10. Alonso-Sanz, R., Martin-Gutierrez, S.: The free-rider in the quantum Stackelberg duopoly game. Phys. A 554, 124271 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  11. Alonso-Sanz, R.: Simulation of the quantum Cournot duopoly game. Phys. A 534, 122116 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  12. Garcia-Perez, L., Grau-Climent, J., Losada, J.C., Alonso-Sanz, R.: Cellular automaton simulation of the quantum Hotelling game with reservation cost. Quant. Inform. Process. 20(7), 227 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  13. Alonso-Sanz, R.: Quantum Game Simulation. Springer, New York (2019)

    Book  MATH  Google Scholar 

  14. Grau-Climent, J., Garcia-Perez, L., Losada, J.C., Alonso-Sanz, R.: Simulation of the hotelling-smithies game: hotelling was not so wrong. Commun. Nonlinear. Sci. Numer. Simul. 112, 106513 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  15. Puu, T.: Hotelling’s Ice cream dealers with elastic demand. Ann. Reg. Sci. 36(1), 1–17 (2002)

    Article  ADS  Google Scholar 

  16. Frackiewicz, P.: Remarks on quantum duopoly schemes. Quant. Inform. Process. 15(1), 121–136 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Chen, Y., Qin, G., Wang, A.M.: Quantization of the location stage of Hotelling model. arXiv preprint arXiv:1410.2779 (2014)

  18. Rahaman, R., Majumdar, P., Basu, B.: Quantum Cournot equilibrium for the Hotelling-Smithies model of product choice. J. Phys. A Math. Theor 45(45), 455301 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kameshawari, A.V.S., Balakrishnan, S.: Cournot and Stackelberg duopoly games in the purview of modified EWL scheme. Quantum Inform. Process. 20(10), 1–15 (2021)

    MathSciNet  Google Scholar 

  20. Ikeda, K., Aoki, S.: Theory of quantum games and quantum economic behavior. Quantum Inform. Process. 21(1), 1–29 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  21. Aumann, R.J.: Correlated equilibrium as an expression of Bayesian rationality. Econometrica 55(1), 1–18 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  22. Du, J., Ju, C., Li, H.: Quantum entanglement helps in improving economic efficiency. J. Phys. Math. Gen. 38(7), 1559 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Iskakov, M., Iskakov, A.: Solution of the Hotelling’s game in secure strategies. Econ. Lett. 117(1), 115–118 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gabszewicz, J., et al.: Location Theory. Taylor & Francis, London (2013)

    Book  Google Scholar 

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Acknowledgements

This work has been funded by the Spanish Grant PID2021-122711NB-C21. The computations of this work were performed in FISWULF, an HPC machine of the Int. Campus of Excellence of Moncloa, funded by the UCM and Feder Funds.

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  1. Complex Systems Group, Universidad Politénica de Madrid, 28040, C. Universitaria, Madrid, Spain

    Luis Garcia-Perez, Juan Grau-Climent, Juan C. Losada & Ramon Alonso-Sanz

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Garcia-Perez, L., Grau-Climent, J., Losada, J.C. et al. The quantum Hotelling–Smithies game. Quantum Inf Process 22, 38 (2023). https://doi.org/10.1007/s11128-022-03780-7

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