# Numerical Solution of Open-Loop Nash Differential Games Based on the Legendre Tau Method

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Statement

**Definition 1.**

**Definition 2.**

## 3. The Tau Method for Nonzero-Sum Differential Games

**Definition 3.**

**Theorem 1.**

**Proof of Theorem 1.**

**Theorem 2.**

**(Weierstrass approximation theorem)**Let$f\in {L}_{w}^{2}[-1,1]$and$N\in \mathbb{N}$. Then there exists a unique${f}_{N}^{*}\in {P}_{N}$, the space of all polynomials of degree at most$N$, such that

**Proof of Theorem 2.**

## 4. Illustrative Example

**Theorem 3.**

**Proof of Theorem 3.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Dockner, E.J.; Jørgensen, S.; Van Long, N.; Sorger, G. Differential Games in Economics and Management Science; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Erickson, G.M. Dynamic Models of Advertising Competition; Kluwer: Boston, MA, USA, 2003. [Google Scholar]
- Yeung, D.W.K.; Petrosjan, L. Cooperative Stochastic Differential Games; Springer: Berlin/Heidelberg, Germany, 2005. [Google Scholar]
- Jafari, S.; Navidi, H. A game-theoretic approach for modeling competitive diffusion over social networks. Games
**2018**, 9, 8. [Google Scholar] [CrossRef] [Green Version] - Bressan, A. Bifurcation analysis of a non-cooperative differential game with one weak player. J. Differ. Equ.
**2010**, 248, 1297–1314. [Google Scholar] [CrossRef] [Green Version] - Bressan, A. Noncooperative differential games. A tutorial. Milan J. Math.
**2011**, 79, 357–427. [Google Scholar] [CrossRef] - Starr, A.; Ho, Y. Further propeties of nonzero-sum differential games. J. Optim. Theory Appl.
**1969**, 3, 207–219. [Google Scholar] [CrossRef] - Starr, A.; Ho, Y. Nonzero-sum differential games. J. Optim. Theory Appl.
**1969**, 3, 184–206. [Google Scholar] [CrossRef] - Engwerda, J.C. LQ Dynamic Optimization and Differential Games; John Wiley and Sons: Hoboken, NJ, USA, 2005. [Google Scholar]
- Engwerda, J.C. On the open-loop Nash equilibrium in LQ games. J. Econom. Dynam. Control
**1998**, 22, 729–762. [Google Scholar] [CrossRef] [Green Version] - Engwerda, J.C. Feedback Nash equilibria in the scalar infinite horizon LQ game. Automatica
**2000**, 36, 135–139. [Google Scholar] [CrossRef] [Green Version] - Kossiorisa, G.; Plexousakis, M.; Xepapadeas, A.; de Zeeuwe, A.; Maler, K.G. Feedback Nash equilibria for non-linear differential games in pollution control. J. Econom. Dynam. Control
**2008**, 32, 1312–1331. [Google Scholar] [CrossRef] [Green Version] - Jiménez-Lizárraga, M.; Basin, M.; Rodríguez, V.; Rodríguez, P. Open-loop Nash equilibrium in polynomial differential games via state-dependent Riccati equation. Automatica
**2015**, 53, 155–163. [Google Scholar] [CrossRef] - Zhang, H.; Wei, Q.; Liu, D. An iterative dynamic programming method for solving a class of nonlinear zero-sum differential games. Automatica
**2011**, 47, 207–214. [Google Scholar] [CrossRef] - Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T.A. Spectral Methods: Fundamentals in Single Domains; Springer: New York, NY, USA, 2006. [Google Scholar]
- Bhrawy, A.H.; Zaky, M.A.; Baleanu, D. New numerical approximations for space-time fractional Burgers’ equations via a Legendre spectral-collocation method. Rom. Rep. Phys.
**2015**, 67, 2. [Google Scholar] - Bhrawy, A.H. An efficient Jacobi pseudospectral approximation for nonlinear complex generalized Zakharov system. Appl. Math. Comput.
**2014**, 247, 30–46. [Google Scholar] [CrossRef] - Doha, E.H.; Bhrawy, A.H.; Hafez, R.M. A Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fifth-order differential equations. Math. Comput. Modell.
**2011**, 53, 1820–1832. [Google Scholar] [CrossRef] - Doha, E.H.; Abd-Elhameed, W.M.; Bhrawy, A.H. Efficient spectral ultraspherical-Galerkin algorithms for the direct solution of 2nth-order linear differential equations. Appl. Math. Modell.
**2009**, 33, 1982–1996. [Google Scholar] [CrossRef] - Guo, B.Y. Spectral Methods and Their Applications; World Scientific: Singapore, 1998. [Google Scholar]
- Doha, E.H.; Abd-Elhameed, W.M.; Youssri, Y.H. New algorithms for solving third- and fifth-order two- point boundary value problems based on nonsymmetric generalized Jacobi Petrov–Galerkin method. J. Adv. Res.
**2015**, 6, 673–686. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Grosset, L. A note on open loop Nash equilibrium in linear-state differential games. Appl. Math. Sci.
**2014**, 8, 7239–7248. [Google Scholar] [CrossRef] - Moosavi Mohseni, R. Mathematical Analysis of the Chaotic Behavior in Monetary Policy Games. Ph.D. Thesis, Auckland University of Technology, Auckland, New Zealand, 2019. [Google Scholar]
- Nikooeinejad, Z.; Delavarkhalafi, A.; Heydari, M. A numerical solution of open-loop Nash equilibrium in nonlinear differential games based on Chebyshev pseudospectral method. J. Comput. Appl. Math.
**2016**, 300, 369–384. [Google Scholar] [CrossRef] - Doha, E.H.; Bhrawy, A.H.; Hafez, R.M. On shifted Jacobi spectral method for high-order multi-point boundary value problems. Commun. Nonlinear Sci. Numer. Simul.
**2012**, 17, 3802–3810. [Google Scholar] [CrossRef] - Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T.A. Spectral Methods on Fluid Dynamics; Springer: Berlin/Heidelberg, Germany, 1988. [Google Scholar]
- Shen, J.; Tang, T.; Wang, L.L. Spectral Methods, in: Algorithms, Analysis and Applications; Springer: Berlin/Heidelberg, Germany, 2011. [Google Scholar]
- Sorger, G. Competitive dynamic advertising: A modification of the case game. J. Econom. Dynam. Control
**1989**, 13, 55–80. [Google Scholar] [CrossRef] - Carlson, D.A.; Leitmann, G. An extension of the coordinate transformation method for open-loop Nash equilibria. J. Optim. Theory Appl.
**2004**, 123, 27–47. [Google Scholar] [CrossRef] - Cesari, L. Optimization-Theory and Applications: Problems with Ordinary Differential Equations; Springer: New York, NY, USA, 1983. [Google Scholar]

1 | The removal of variable $t$ in the remaining parts of the paper has also been done for simplification matters. |

**Table 1.**Optimal payoff functionals ${J}_{1}$ and ${J}_{2}$ for illustrative example with LTM as compared with RK4.

$\mathit{N}$ | ${\mathit{J}}_{1\mathit{L}\mathit{T}\mathit{M}}$ | ${\mathit{J}}_{2\mathit{L}\mathit{T}\mathit{M}}$ |
---|---|---|

8 | 0.016380209069964074615873141557194 | 0.0092479570969023022143164502992464 |

10 | 0.016380209069964074615873178289759 | 0.0092479570969023022143164746205906 |

12 | 0.016380209069964074615873178289820 | 0.0092479570969023022143164746206322 |

14 | 0.016380209069964074615873178289819 | 0.0092479570969023022143164746206318 |

$\Delta t$ | ${J}_{1RK4}$ | ${J}_{2RK4}$ |

${10}^{-4}$ | 0.016380209069970129334132078913143 | 0.0092479570969076035409551440677725 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Dehghan Banadaki, M.; Navidi, H.
Numerical Solution of Open-Loop Nash Differential Games Based on the Legendre Tau Method. *Games* **2020**, *11*, 28.
https://doi.org/10.3390/g11030028

**AMA Style**

Dehghan Banadaki M, Navidi H.
Numerical Solution of Open-Loop Nash Differential Games Based on the Legendre Tau Method. *Games*. 2020; 11(3):28.
https://doi.org/10.3390/g11030028

**Chicago/Turabian Style**

Dehghan Banadaki, Mojtaba, and Hamidreza Navidi.
2020. "Numerical Solution of Open-Loop Nash Differential Games Based on the Legendre Tau Method" *Games* 11, no. 3: 28.
https://doi.org/10.3390/g11030028