Non-cooperative and semi-cooperative differential games

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

In this paper we review some recent results on non-cooperative and semicooperative differential games. For the n-person non-cooperative games in one-space dimension, we consider the Nash equilibrium solutions. When the system of Hamilton-Jacobi equations for the value functions is strictly hyperbolic, we show that the weak solution of a corresponding system of hyperbolic conservation laws determines an n-tuple of feedback strategies. These yield a Nash equilibrium solution to the non-cooperative differential game. However, in the multi-dimensional cases, the system of Hamilton-Jacobi equations is generically elliptic, and therefore ill posed. In an effort to obtain meaningful stable solutions, we propose an alternative “semi-cooperative” pair of strategies for the two players, seeking a Pareto optimum instead of a Nash equilibrium. In this case, the corresponding Hamiltonian system for the value functions is always weakly hyperbolic.

Original languageEnglish (US)
Title of host publicationAnnals of the International Society of Dynamic Games
PublisherBirkhauser
Pages85-104
Number of pages20
DOIs
StatePublished - Jan 1 2009

Publication series

NameAnnals of the International Society of Dynamic Games
Volume10
ISSN (Print)2474-0179
ISSN (Electronic)2474-0187

Fingerprint

Cooperative Game
Differential Games
Nash Equilibrium
Non-cooperative Game
Equilibrium Solution
Hamilton-Jacobi Equation
Hamiltonians
Value Function
Pareto Optimum
n-tuple
Conservation
Hyperbolic Systems of Conservation Laws
Stable Solution
Feedback
Hamiltonian Systems
Weak Solution
Person
Strictly
Alternatives
Differential games

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty
  • Applied Mathematics

Cite this

Shen, W. (2009). Non-cooperative and semi-cooperative differential games. In Annals of the International Society of Dynamic Games (pp. 85-104). (Annals of the International Society of Dynamic Games; Vol. 10). Birkhauser. https://doi.org/10.1007/978-0-8176-4834-3_5
Shen, Wen. / Non-cooperative and semi-cooperative differential games. Annals of the International Society of Dynamic Games. Birkhauser, 2009. pp. 85-104 (Annals of the International Society of Dynamic Games).
@inbook{7b5430a70e6e4568a9aa1c6b09a729dc,
title = "Non-cooperative and semi-cooperative differential games",
abstract = "In this paper we review some recent results on non-cooperative and semicooperative differential games. For the n-person non-cooperative games in one-space dimension, we consider the Nash equilibrium solutions. When the system of Hamilton-Jacobi equations for the value functions is strictly hyperbolic, we show that the weak solution of a corresponding system of hyperbolic conservation laws determines an n-tuple of feedback strategies. These yield a Nash equilibrium solution to the non-cooperative differential game. However, in the multi-dimensional cases, the system of Hamilton-Jacobi equations is generically elliptic, and therefore ill posed. In an effort to obtain meaningful stable solutions, we propose an alternative “semi-cooperative” pair of strategies for the two players, seeking a Pareto optimum instead of a Nash equilibrium. In this case, the corresponding Hamiltonian system for the value functions is always weakly hyperbolic.",
author = "Wen Shen",
year = "2009",
month = "1",
day = "1",
doi = "10.1007/978-0-8176-4834-3_5",
language = "English (US)",
series = "Annals of the International Society of Dynamic Games",
publisher = "Birkhauser",
pages = "85--104",
booktitle = "Annals of the International Society of Dynamic Games",

}

Shen, W 2009, Non-cooperative and semi-cooperative differential games. in Annals of the International Society of Dynamic Games. Annals of the International Society of Dynamic Games, vol. 10, Birkhauser, pp. 85-104. https://doi.org/10.1007/978-0-8176-4834-3_5

Non-cooperative and semi-cooperative differential games. / Shen, Wen.

Annals of the International Society of Dynamic Games. Birkhauser, 2009. p. 85-104 (Annals of the International Society of Dynamic Games; Vol. 10).

Research output: Chapter in Book/Report/Conference proceedingChapter

TY - CHAP

T1 - Non-cooperative and semi-cooperative differential games

AU - Shen, Wen

PY - 2009/1/1

Y1 - 2009/1/1

N2 - In this paper we review some recent results on non-cooperative and semicooperative differential games. For the n-person non-cooperative games in one-space dimension, we consider the Nash equilibrium solutions. When the system of Hamilton-Jacobi equations for the value functions is strictly hyperbolic, we show that the weak solution of a corresponding system of hyperbolic conservation laws determines an n-tuple of feedback strategies. These yield a Nash equilibrium solution to the non-cooperative differential game. However, in the multi-dimensional cases, the system of Hamilton-Jacobi equations is generically elliptic, and therefore ill posed. In an effort to obtain meaningful stable solutions, we propose an alternative “semi-cooperative” pair of strategies for the two players, seeking a Pareto optimum instead of a Nash equilibrium. In this case, the corresponding Hamiltonian system for the value functions is always weakly hyperbolic.

AB - In this paper we review some recent results on non-cooperative and semicooperative differential games. For the n-person non-cooperative games in one-space dimension, we consider the Nash equilibrium solutions. When the system of Hamilton-Jacobi equations for the value functions is strictly hyperbolic, we show that the weak solution of a corresponding system of hyperbolic conservation laws determines an n-tuple of feedback strategies. These yield a Nash equilibrium solution to the non-cooperative differential game. However, in the multi-dimensional cases, the system of Hamilton-Jacobi equations is generically elliptic, and therefore ill posed. In an effort to obtain meaningful stable solutions, we propose an alternative “semi-cooperative” pair of strategies for the two players, seeking a Pareto optimum instead of a Nash equilibrium. In this case, the corresponding Hamiltonian system for the value functions is always weakly hyperbolic.

UR - http://www.scopus.com/inward/record.url?scp=85055038364&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85055038364&partnerID=8YFLogxK

U2 - 10.1007/978-0-8176-4834-3_5

DO - 10.1007/978-0-8176-4834-3_5

M3 - Chapter

AN - SCOPUS:85055038364

T3 - Annals of the International Society of Dynamic Games

SP - 85

EP - 104

BT - Annals of the International Society of Dynamic Games

PB - Birkhauser

ER -

Shen W. Non-cooperative and semi-cooperative differential games. In Annals of the International Society of Dynamic Games. Birkhauser. 2009. p. 85-104. (Annals of the International Society of Dynamic Games). https://doi.org/10.1007/978-0-8176-4834-3_5