Stability, sensitivity analysis, and computation of competitive network equilibria : Anna Nagurney, 1983 (Dissertation in the division of Applied Mathematics, Brown University, Providence, RI 02912)

Abstract

Competitive network equilibrium problems originate in situations where each user of a network acting independently seeks to minimize his personal disutility until, finally, an equilibrium is established characterized by the property that no user has any incentive to change his strategy, given that all other users retain their present strategies. Such competitive equilibrium problems, that arise in economics, operations research, and control theory can be formulated as a variational inequality. Although a plethora of algorithms is available for determining the equilibrium, relatively little effort has been extended towards the computational testing of these algorithms. In addition, the important for the applications questions of stability and sensitivity of the equilibrium have not been addressed yet in any general context. In this thesis we study the stability and sensitivity of the equilibrium for two applications, namely, the traffic equilibrium problem and the spatial economic equilibrium problem. We consider the general asymmetric traffic equilibrium problem with fixed demands. We show that, under certain assumptions, the equilibrium traffic pattern depends continuously upon the assigned travel demands and travel cost functions. We then focus on the delicate question of predicting the direction of the change in the traffic pattern and the incurred travel costs resulting from changes in the travel cost functions and travel demands and attempt to elucidate certain counter-intuitive phenomena such as Braess paradox. Next we consider the general spatial economic equilibrium problem. Under certain conditions we show that the multicommodity equilibrium price pattern depends continuously upon the market supply and demand functions and the multicommodity equilibrium shipment pattern depends continuously upon the transportation cost functions. We then address the important question of predicting the direction of the change in the price and shipment pattern and the incurred supplies, demands, and transportation costs resulting from changes in the supply, demand, and transportation cost functions. Finally, we proceed to test the relative efficiency of the main computational techniques for computing the solution of the variational inequality, the relaxation and projection methods. Our results indicate that the form of certain functions affects the performance of the two basic techniques.