Stability of the stochastic equilibrium assignment problem: a dynamical systems approach

The question of whether plausible dynamical adjustment processes converge to equilibrium was brought to attention by the compelling analysis of Horowitz (1984). The stability of stochastic equilibrium in a two link transportation network. Transportation Research 18B (1), 13-28. His analysis of discrete time processes, and the question of their convergence to stochastic equilibrium, had the significant restriction of applying only to two-link networks. In spite of a number of significant works on this 'stability' issue since, the extension of Horowitz's results to general networks has still not been achieved, and this forms the motivation for the present paper. Previous analyses of traffic assignment stability are first reviewed and classified. The key, and often misunderstood, distinctions are clarified: between stability in discrete time and continuous time, and between stability with respect to deterministic and stochastic processes. It is discussed how analyses since Horowitz's characterise much milder notions of stability. A simple dynamical adjustment process is then proposed for studying the stability of the general asymmetric stochastic equilibrium assignment problem in discrete time. Classical techniques from the dynamical systems literature are then applied in three ways, resulting in: a sufficient condition for stability, applicable to a significant subset of practical problems; a widely-applicable sufficient condition for instability; and a method for estimating domains of attraction for problems with multiple equilibria. The tests are illustrated in relation to a number of simple examples. In principle, they are applicable to networks of an arbitrary size, although further tests would be required to determine the computational feasibility of these techniques in large networks.