Perfection and Stability of Stationary Points with Applications in Noncooperative Games

G. van der Laan, A.J.J. Talman, Z.F. Yang

Research output: Working paperDiscussion paperOther research output

206 Downloads (Pure)

Abstract

It is well known that an upper semi-continuous compact- and convex-valued mapping ö from a nonempty compact and convex set X to the Euclidean space of which X is a subset has at least one stationary point, being a point in X at which the image ö (x)has a nonempty intersection with the normal cone at x.In many circumstances there may be more than one stationary point.In this paper we refine the concept of stationary point by perturbing simultaneously both the set X and the solution concept.In case a stationary point is the limit of a sequence of perturbed solutions on a sequence of sets converging continuously to X we say that the stationary point is stable with respect to this sequence of sets and the mapping which defines the perturbed solution.It is shown that stable stationary points exist for a large class of perturbations.A specific refinement, called robustness, is obtained if a stationary point is the limit of stationary points on a sequence of sets converging to X.It is shown that a robust stationary point always exists for any sequence of sets which starts from an interior point and converges to X in a continuous way.We also discuss several applications in noncooperative game theory.We first show that two well known refinements of the Nash equilibrium, namely, perfect Nash equilibrium and proper Nash equilibrium, are special cases of our robustness concept.Further, a third special case of robustness refines the concept of properness and a robust Nash equilibrium is shown to exist for every game.In symmetric bimatrix games, our results imply the existence of a symmetric proper equilibrium.Applying our results to the field of evolutionary game theory yields a refinement of the stationary points of the replicator dynamics.We show that the refined solution always exists, contrary to many well known refinement concepts in the field that may fail to exist under the same conditions.
Original languageEnglish
Place of PublicationTilburg
PublisherMicroeconomics
Number of pages24
Volume2002-108
Publication statusPublished - 2002

Publication series

NameCentER Discussion Paper
Volume2002-108

Fingerprint

Non-cooperative Game
Stationary point
Nash Equilibrium
Refinement
Robustness
Replicator Dynamics
Bimatrix Games
Evolutionary Game Theory
Normal Cone
Solution Concepts
Upper Semicontinuous
Interior Point
Game Theory
Compact Set
Convex Sets
Euclidean space
Intersection
Game
Perturbation
Converge

Keywords

  • noncooperative games
  • stationary point
  • stability
  • equilibrium analysis

Cite this

van der Laan, G., Talman, A. J. J., & Yang, Z. F. (2002). Perfection and Stability of Stationary Points with Applications in Noncooperative Games. (CentER Discussion Paper; Vol. 2002-108). Tilburg: Microeconomics.
van der Laan, G. ; Talman, A.J.J. ; Yang, Z.F. / Perfection and Stability of Stationary Points with Applications in Noncooperative Games. Tilburg : Microeconomics, 2002. (CentER Discussion Paper).
@techreport{fc1f47c6314f493280c6116a4e027540,
title = "Perfection and Stability of Stationary Points with Applications in Noncooperative Games",
abstract = "It is well known that an upper semi-continuous compact- and convex-valued mapping {\"o} from a nonempty compact and convex set X to the Euclidean space of which X is a subset has at least one stationary point, being a point in X at which the image {\"o} (x)has a nonempty intersection with the normal cone at x.In many circumstances there may be more than one stationary point.In this paper we refine the concept of stationary point by perturbing simultaneously both the set X and the solution concept.In case a stationary point is the limit of a sequence of perturbed solutions on a sequence of sets converging continuously to X we say that the stationary point is stable with respect to this sequence of sets and the mapping which defines the perturbed solution.It is shown that stable stationary points exist for a large class of perturbations.A specific refinement, called robustness, is obtained if a stationary point is the limit of stationary points on a sequence of sets converging to X.It is shown that a robust stationary point always exists for any sequence of sets which starts from an interior point and converges to X in a continuous way.We also discuss several applications in noncooperative game theory.We first show that two well known refinements of the Nash equilibrium, namely, perfect Nash equilibrium and proper Nash equilibrium, are special cases of our robustness concept.Further, a third special case of robustness refines the concept of properness and a robust Nash equilibrium is shown to exist for every game.In symmetric bimatrix games, our results imply the existence of a symmetric proper equilibrium.Applying our results to the field of evolutionary game theory yields a refinement of the stationary points of the replicator dynamics.We show that the refined solution always exists, contrary to many well known refinement concepts in the field that may fail to exist under the same conditions.",
keywords = "noncooperative games, stationary point, stability, equilibrium analysis",
author = "{van der Laan}, G. and A.J.J. Talman and Z.F. Yang",
note = "Pagination: 24",
year = "2002",
language = "English",
volume = "2002-108",
series = "CentER Discussion Paper",
publisher = "Microeconomics",
type = "WorkingPaper",
institution = "Microeconomics",

}

van der Laan, G, Talman, AJJ & Yang, ZF 2002 'Perfection and Stability of Stationary Points with Applications in Noncooperative Games' CentER Discussion Paper, vol. 2002-108, Microeconomics, Tilburg.

Perfection and Stability of Stationary Points with Applications in Noncooperative Games. / van der Laan, G.; Talman, A.J.J.; Yang, Z.F.

Tilburg : Microeconomics, 2002. (CentER Discussion Paper; Vol. 2002-108).

Research output: Working paperDiscussion paperOther research output

TY - UNPB

T1 - Perfection and Stability of Stationary Points with Applications in Noncooperative Games

AU - van der Laan, G.

AU - Talman, A.J.J.

AU - Yang, Z.F.

N1 - Pagination: 24

PY - 2002

Y1 - 2002

N2 - It is well known that an upper semi-continuous compact- and convex-valued mapping ö from a nonempty compact and convex set X to the Euclidean space of which X is a subset has at least one stationary point, being a point in X at which the image ö (x)has a nonempty intersection with the normal cone at x.In many circumstances there may be more than one stationary point.In this paper we refine the concept of stationary point by perturbing simultaneously both the set X and the solution concept.In case a stationary point is the limit of a sequence of perturbed solutions on a sequence of sets converging continuously to X we say that the stationary point is stable with respect to this sequence of sets and the mapping which defines the perturbed solution.It is shown that stable stationary points exist for a large class of perturbations.A specific refinement, called robustness, is obtained if a stationary point is the limit of stationary points on a sequence of sets converging to X.It is shown that a robust stationary point always exists for any sequence of sets which starts from an interior point and converges to X in a continuous way.We also discuss several applications in noncooperative game theory.We first show that two well known refinements of the Nash equilibrium, namely, perfect Nash equilibrium and proper Nash equilibrium, are special cases of our robustness concept.Further, a third special case of robustness refines the concept of properness and a robust Nash equilibrium is shown to exist for every game.In symmetric bimatrix games, our results imply the existence of a symmetric proper equilibrium.Applying our results to the field of evolutionary game theory yields a refinement of the stationary points of the replicator dynamics.We show that the refined solution always exists, contrary to many well known refinement concepts in the field that may fail to exist under the same conditions.

AB - It is well known that an upper semi-continuous compact- and convex-valued mapping ö from a nonempty compact and convex set X to the Euclidean space of which X is a subset has at least one stationary point, being a point in X at which the image ö (x)has a nonempty intersection with the normal cone at x.In many circumstances there may be more than one stationary point.In this paper we refine the concept of stationary point by perturbing simultaneously both the set X and the solution concept.In case a stationary point is the limit of a sequence of perturbed solutions on a sequence of sets converging continuously to X we say that the stationary point is stable with respect to this sequence of sets and the mapping which defines the perturbed solution.It is shown that stable stationary points exist for a large class of perturbations.A specific refinement, called robustness, is obtained if a stationary point is the limit of stationary points on a sequence of sets converging to X.It is shown that a robust stationary point always exists for any sequence of sets which starts from an interior point and converges to X in a continuous way.We also discuss several applications in noncooperative game theory.We first show that two well known refinements of the Nash equilibrium, namely, perfect Nash equilibrium and proper Nash equilibrium, are special cases of our robustness concept.Further, a third special case of robustness refines the concept of properness and a robust Nash equilibrium is shown to exist for every game.In symmetric bimatrix games, our results imply the existence of a symmetric proper equilibrium.Applying our results to the field of evolutionary game theory yields a refinement of the stationary points of the replicator dynamics.We show that the refined solution always exists, contrary to many well known refinement concepts in the field that may fail to exist under the same conditions.

KW - noncooperative games

KW - stationary point

KW - stability

KW - equilibrium analysis

M3 - Discussion paper

VL - 2002-108

T3 - CentER Discussion Paper

BT - Perfection and Stability of Stationary Points with Applications in Noncooperative Games

PB - Microeconomics

CY - Tilburg

ER -

van der Laan G, Talman AJJ, Yang ZF. Perfection and Stability of Stationary Points with Applications in Noncooperative Games. Tilburg: Microeconomics. 2002. (CentER Discussion Paper).