### Abstract

This paper considers noncooperative equilibria of three-player dynamic games with three levels of hierarchy in decision making. In this context, first a general definition of a hierarchical equilibrium solution is given, which also accounts for nonunique responses of the players who are not at the top of the hierarchy. Then, a general theorem is proven which provides a set of sufficient conditions for a triple of strategies to be in hierarchical equilibrium. When applied to linear-quadratic games, this theorem provides conditions under which there exists a linear one-step memory strategy for the player (say, J1) at the top of the hierarchy, which forces the other two players to act in such a way so as to jointly minimize the cost function of J1. Furthermore, there exists a linear one-step memory strategy for the second-level player (say, J2), which forces the remaining player to jointly minimize the cost function of J2 under the declared equilibrium strategy of J1. A numerical example included in the paper illustrates the results and the convergence property of the equilibrium strategies, as the number of stages in the game becomes arbitrarily large.

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

Pages (from-to) | 749-754 |

Number of pages | 6 |

Journal | Automatica |

Volume | 17 |

Issue number | 5 |

DOIs | |

State | Published - Sep 1981 |

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### Keywords

- Game theory
- Stackelberg strategies
- dynamic games
- hierarchical decision making
- hierarchical systems
- large-scale systems
- optimal systems

### ASJC Scopus subject areas

- Control and Systems Engineering
- Electrical and Electronic Engineering

### Cite this

**Equilibrium strategies in dynamic games with multi-levels of hierarchy.** / Basar, M Tamer.

Research output: Contribution to journal › Article

*Automatica*, vol. 17, no. 5, pp. 749-754. https://doi.org/10.1016/0005-1098(81)90022-4

}

TY - JOUR

T1 - Equilibrium strategies in dynamic games with multi-levels of hierarchy

AU - Basar, M Tamer

PY - 1981/9

Y1 - 1981/9

N2 - This paper considers noncooperative equilibria of three-player dynamic games with three levels of hierarchy in decision making. In this context, first a general definition of a hierarchical equilibrium solution is given, which also accounts for nonunique responses of the players who are not at the top of the hierarchy. Then, a general theorem is proven which provides a set of sufficient conditions for a triple of strategies to be in hierarchical equilibrium. When applied to linear-quadratic games, this theorem provides conditions under which there exists a linear one-step memory strategy for the player (say, J1) at the top of the hierarchy, which forces the other two players to act in such a way so as to jointly minimize the cost function of J1. Furthermore, there exists a linear one-step memory strategy for the second-level player (say, J2), which forces the remaining player to jointly minimize the cost function of J2 under the declared equilibrium strategy of J1. A numerical example included in the paper illustrates the results and the convergence property of the equilibrium strategies, as the number of stages in the game becomes arbitrarily large.

AB - This paper considers noncooperative equilibria of three-player dynamic games with three levels of hierarchy in decision making. In this context, first a general definition of a hierarchical equilibrium solution is given, which also accounts for nonunique responses of the players who are not at the top of the hierarchy. Then, a general theorem is proven which provides a set of sufficient conditions for a triple of strategies to be in hierarchical equilibrium. When applied to linear-quadratic games, this theorem provides conditions under which there exists a linear one-step memory strategy for the player (say, J1) at the top of the hierarchy, which forces the other two players to act in such a way so as to jointly minimize the cost function of J1. Furthermore, there exists a linear one-step memory strategy for the second-level player (say, J2), which forces the remaining player to jointly minimize the cost function of J2 under the declared equilibrium strategy of J1. A numerical example included in the paper illustrates the results and the convergence property of the equilibrium strategies, as the number of stages in the game becomes arbitrarily large.

KW - Game theory

KW - Stackelberg strategies

KW - dynamic games

KW - hierarchical decision making

KW - hierarchical systems

KW - large-scale systems

KW - optimal systems

UR - http://www.scopus.com/inward/record.url?scp=0019609737&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0019609737&partnerID=8YFLogxK

U2 - 10.1016/0005-1098(81)90022-4

DO - 10.1016/0005-1098(81)90022-4

M3 - Article

AN - SCOPUS:0019609737

VL - 17

SP - 749

EP - 754

JO - Automatica

JF - Automatica

SN - 0005-1098

IS - 5

ER -