Abstract
Bi-level optimization problems arise in hierarchical decision making, where players of different ranks are involved. The situation is described by the so-called Stackelberg game. The players of lower rank, called followers, react to decisions made by the first rank player, also called the leader. Situations similar to this arise for instance in mixed economies, land-use, traffic signal setting and in the particularly well-known network design problem. Previously, solution techniques based on the replacement of the bi-level problem by a bi-criteria one were proposed. However, it was shown subsequently, by the means of counter examples, that the bi-level problem is not equivalent to the bi-criteria formulation. In this note we demonstrate that the bi-criteria approach is not necessarily obsolete. We derive conditions under which we show that a Stackelberg equilibrium is indeed a Pareto optimum.
I thank Dr. D.T. Luc for the discussions. This work has been partially supported by the Swedish Communication Research Board (KFB).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J.P. Aubin (1979) Mathematical Methods of Game and Economic Theory. North-Holland, Amsterdam.
F.A. Al-Khayal, R. Horst and P.M. Pardalos (1992) Global Optimization of Concave Functions subject to Separable Quadratic Constraints–An Application to Bi-Level Programming, Annals of Operations Research, vol. 34, pp. 125–147
J. F. Bard (1983) An Efficient Point Algorithm for a Linear Two-Stage Optimization Problem, Operations Research, vol. 31, pp. 670–684
Ben-Ayed and C.E. Blair (1990) Computational Difficulties of Bilevel Linear Programming, Operations Research, vol. 38, pp. 556–560
Ben-Ayed, C.E. Blair, D.E. Boyce and L.J. LeBlanc (1992) Construction of Real-World Bilevel Linear Programming Model of the Highway Network Design Problem, Annals of Operations Research, vol. 34, pp. 219–254
C. Blair (1992) The Computational Complexity of Multi-Level Linear Programs, Annals of Operations Research, vol. 34, pp. 13–19
W.R. Blundet and J.A. Black (1984) The Land-Use/Transport System. 2nd edition. Pergamon Press, Australia.
G.E. Cantarella and A. Sforza (1987) Methods for Equilibrium Network Traffic Signal Setting, in “Flow Control of Congested Networks”, A.R. Odoni et al. (ed.$), NATO ASI Series, vol. F38, Springer-Verlag, Berlin, pp. 69–89
G. Improta (1987) Mathematical Programming Methods for Urban Network Control, in “Flow Control of Congested Networks”, A.R. Odoni et al. (ed.$), NATO ASI Series, vol. F38, Springer-Verlag, Berlin, pp. 35–68
R.G. Jeroslow (1985) The Polynomial Hierarchy and Simple Model for Competitive Analysis, Mathematical Programming, vol. 32, pp. 131–153
L.J. Leblanc and D.E. Boyce (1986) A Bilevel Programming Algorithm for Exact Solution of the Network Design Problem with User Optimal Flows, Transportations Research, vol. 20B, pp. 259–265
D.T. Luc (1989) Theory of Vector Optimization. Lecture Notes in Economics and Mathematical Systems 319, Springer-Verlag, Berlin.
P. Marcotte (1983) Network optimization with continuous control parameters, Transportation Science, vol. 17, pp. 181–197
P. Marcotte (1988) A Note on the Bilevel Programming Algorithm by Leblanc and Boyce, Transportations Research, vol. 22B, pp. 233–237
M. Simaan and J.B. Cruz, Jr (1973) On the Stackelberg Strategy in Nonzero-Sum Games, JOTA, vol 11, pp. 533–555
H. von Stackelberg (1952) The Theory of the Market Economy. Oxford University Press.
S. Suh and T. Kim (1992) Solving Nonlinear Bilevel Programming of the Equilibrium Network Design Problem - A Comparative Review, Annals of Operations Research, vol. 34, pp. 203218
H. Thy, A. Migdalas and P. Värbrand (1993) A Global Optimization Approach for the Linear Two-Level Program, Journal of Global Optimization, vol. 13, pp. 1–23
H. Thy, A. Migdalas and P. Värbrand (1994) A Quasiconcave Minimization Method for Solving Linear Two-Level Programs, Journal of Global Optimization, vol. 4, pp. 243–263
G. Unlu (1987) A Linear Bilevel Programming Algorithm Based on Bicreteria Programming, Computers and Operations research, vol. 14, pp. 173–179
J.G. Wardrop (1952) Some Theoretical Aspects of Road Traffic Research, Proceedings of the Institution of Civil Engineering, pp. 325–362
U.-P. Wen and S.-T. Hsu (1989) A Note on a Linear Bilevel Programming Algorithm Based on Bicreteria Programming, Computers and Operations Research, vol. 16, pp. 79–83
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Migdalas, A. (1995). When is a Stackelberg Equilibrium Pareto Optimum?. In: Pardalos, P.M., Siskos, Y., Zopounidis, C. (eds) Advances in Multicriteria Analysis. Nonconvex Optimization and Its Applications, vol 5. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-2383-0_11
Download citation
DOI: https://doi.org/10.1007/978-1-4757-2383-0_11
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-4748-2
Online ISBN: 978-1-4757-2383-0
eBook Packages: Springer Book Archive