### Abstract

This paper, having a tutorial character, is intended to provide an introduction to the theory of noncooperative differential games. Section 2 reviews the theory of static games. Different concepts of solution are discussed, including Pareto optima, Nash and Stackelberg equilibria, and the co-co (cooperative-competitive) solutions. Section 3 introduces the basic framework of differential games for two players. Open-loop solutions, where the controls implemented by the players depend only on time, are considered in Section 4. These solutions can be computed by solving a two-point boundary value problem for a system of ODEs, derived from the Pontryagin maximum principle. Section 5 deals with solutions in feedback form, where the controls are allowed to depend on time and also on the current state of the system. In this case, the search for Nash equilibrium solutions leads to a highly nonlinear system of Hamilton-Jacobi PDEs. In dimension higher than one, we show that this system is generically not hyperbolic and the Cauchy problem is thus ill posed. Due to this instability, feedback solutions are mainly considered only in the special case with linear dynamics and quadratic costs. In Section 6, a game in continuous time is approximated by a finite sequence of static games, by a time discretization. Depending of the type of solution adopted in each static game, one obtains different concepts of solutions for the original differential game. Section 7 deals with differential games in infinite time horizon, with exponentially discounted payoffs. Section 8 contains a simple example of a game with infinitely many players. This is intended to convey a flavor of the newly emerging theory of mean field games. Modeling issues, and directions of current research, are briefly discussed in Section 9. Finally, the Appendix collects background material on multivalued functions, selections and fixed point theorems, optimal control theory, and hyperbolic PDEs.

Original language | English (US) |
---|---|

Pages (from-to) | 357-427 |

Number of pages | 71 |

Journal | Milan Journal of Mathematics |

Volume | 79 |

Issue number | 2 |

DOIs | |

State | Published - Dec 1 2011 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Milan Journal of Mathematics*,

*79*(2), 357-427. https://doi.org/10.1007/s00032-011-0163-6

}

*Milan Journal of Mathematics*, vol. 79, no. 2, pp. 357-427. https://doi.org/10.1007/s00032-011-0163-6

**Noncooperative Differential Games.** / Bressan, Alberto.

Research output: Contribution to journal › Review article

TY - JOUR

T1 - Noncooperative Differential Games

AU - Bressan, Alberto

PY - 2011/12/1

Y1 - 2011/12/1

N2 - This paper, having a tutorial character, is intended to provide an introduction to the theory of noncooperative differential games. Section 2 reviews the theory of static games. Different concepts of solution are discussed, including Pareto optima, Nash and Stackelberg equilibria, and the co-co (cooperative-competitive) solutions. Section 3 introduces the basic framework of differential games for two players. Open-loop solutions, where the controls implemented by the players depend only on time, are considered in Section 4. These solutions can be computed by solving a two-point boundary value problem for a system of ODEs, derived from the Pontryagin maximum principle. Section 5 deals with solutions in feedback form, where the controls are allowed to depend on time and also on the current state of the system. In this case, the search for Nash equilibrium solutions leads to a highly nonlinear system of Hamilton-Jacobi PDEs. In dimension higher than one, we show that this system is generically not hyperbolic and the Cauchy problem is thus ill posed. Due to this instability, feedback solutions are mainly considered only in the special case with linear dynamics and quadratic costs. In Section 6, a game in continuous time is approximated by a finite sequence of static games, by a time discretization. Depending of the type of solution adopted in each static game, one obtains different concepts of solutions for the original differential game. Section 7 deals with differential games in infinite time horizon, with exponentially discounted payoffs. Section 8 contains a simple example of a game with infinitely many players. This is intended to convey a flavor of the newly emerging theory of mean field games. Modeling issues, and directions of current research, are briefly discussed in Section 9. Finally, the Appendix collects background material on multivalued functions, selections and fixed point theorems, optimal control theory, and hyperbolic PDEs.

AB - This paper, having a tutorial character, is intended to provide an introduction to the theory of noncooperative differential games. Section 2 reviews the theory of static games. Different concepts of solution are discussed, including Pareto optima, Nash and Stackelberg equilibria, and the co-co (cooperative-competitive) solutions. Section 3 introduces the basic framework of differential games for two players. Open-loop solutions, where the controls implemented by the players depend only on time, are considered in Section 4. These solutions can be computed by solving a two-point boundary value problem for a system of ODEs, derived from the Pontryagin maximum principle. Section 5 deals with solutions in feedback form, where the controls are allowed to depend on time and also on the current state of the system. In this case, the search for Nash equilibrium solutions leads to a highly nonlinear system of Hamilton-Jacobi PDEs. In dimension higher than one, we show that this system is generically not hyperbolic and the Cauchy problem is thus ill posed. Due to this instability, feedback solutions are mainly considered only in the special case with linear dynamics and quadratic costs. In Section 6, a game in continuous time is approximated by a finite sequence of static games, by a time discretization. Depending of the type of solution adopted in each static game, one obtains different concepts of solutions for the original differential game. Section 7 deals with differential games in infinite time horizon, with exponentially discounted payoffs. Section 8 contains a simple example of a game with infinitely many players. This is intended to convey a flavor of the newly emerging theory of mean field games. Modeling issues, and directions of current research, are briefly discussed in Section 9. Finally, the Appendix collects background material on multivalued functions, selections and fixed point theorems, optimal control theory, and hyperbolic PDEs.

UR - http://www.scopus.com/inward/record.url?scp=81955161122&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=81955161122&partnerID=8YFLogxK

U2 - 10.1007/s00032-011-0163-6

DO - 10.1007/s00032-011-0163-6

M3 - Review article

AN - SCOPUS:81955161122

VL - 79

SP - 357

EP - 427

JO - Milan Journal of Mathematics

JF - Milan Journal of Mathematics

SN - 1424-9286

IS - 2

ER -