Equilibrium strategies in dynamic games with multi-levels of hierarchy

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)749-754
Number of pages6
JournalAutomatica
Volume17
Issue number5
DOIs
StatePublished - Sep 1981

Fingerprint

Cost functions
Data storage equipment
Decision making

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.

In: Automatica, Vol. 17, No. 5, 09.1981, p. 749-754.

Research output: Contribution to journalArticle

@article{71c381553ff54e8a84c0228270f4a720,
title = "Equilibrium strategies in dynamic games with multi-levels of hierarchy",
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.",
keywords = "Game theory, Stackelberg strategies, dynamic games, hierarchical decision making, hierarchical systems, large-scale systems, optimal systems",
author = "Basar, {M Tamer}",
year = "1981",
month = "9",
doi = "10.1016/0005-1098(81)90022-4",
language = "English (US)",
volume = "17",
pages = "749--754",
journal = "Automatica",
issn = "0005-1098",
publisher = "Elsevier Limited",
number = "5",

}

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 -