PERTURBATION STABILITY OF THE ASYMMETRIC STOCHASTIC EQUILIBRIUM ASSIGNMENT MODEL

A number of results are established regarding the stability of the asy mmetric stochastic equilibrium assignment model for general networks. En particular, we consider the marginal effect of any swap of flow fro m one route to an alternative, under various behavioural rules describ ing the way in which drivers integrate their perceived costs at equili brium with those in disequilibrium. These pairwise route how swaps are referred to as 'perturbations', since they assume that other route fl ows are held at their equilibrium levels (even though it is a non-loca l property, in the sense that the size of the flow swap is only limite d by the demand-feasibility constraints). Loosely speaking, an equilib rium is then said to be stable if following any such perturbation, the re is an incentive for drivers to re-route in the direction or the sam e equilibrium. It is seen that if the route choice is described by a r andom utility model and the cost functions are monotone, then equilibr ium is stable with respect to some behavioural adjustment assumptions, and unstable with respect to others. If, alternatively, the cost func tions satisfy a certain derivative condition, we show that stability m ay be established with respect to a greater range of behavioural adjus tments. By studying the limiting case as the perception error variance matrix tends to zero, it is shown that these concepts of stability ma y be regarded as generalisations of the 'equilibration' concept of sta bility proposed for deterministic equilibrium. Finally, simple example s are used to explore the application of the theorems above to study l ocal properties of problems with multiple stochastic equilibria. (C) 1 998 Elsevier Science Ltd. All rights reserved.