Stochastic stability of dynamic user equilibrium in unidirectional networks: Weakly acyclic game approach
研究了单向网络中固定出发时间的动态用户均衡的随机稳定性，通过将DUE分配分解、弱非循环博弈和扰动动力学稳态分布分析相结合，证明了在适当用户顺序下均衡的存在性和随机稳定性。
The aim of this study is to analyze the stability of dynamic user equilibrium (DUE) with fixed departure times in unidirectional networks. Specifically, stochastic stability, which is the concept of stability in evolutionary dynamics subjected to stochastic effects, is herein considered. To achieve this, a new approach is developed by synthesizing the three concepts: the decomposition technique of DUE assignments, the weakly acyclic game, and the asymptotic analysis of the stationary distribution of perturbed dynamics. Specifically, we first formulate a DUE assignment as a strategic game (DUE game) that deals with atomic users. We then prove that there exists an appropriate order of assigning users for ensuring equilibrium in a unidirectional network. With this property, we establish the relationship between DUE games in unidirectional networks and weakly acyclic games. The convergence and stochastic stability of better response dynamics in the DUE games are then proved based on the theory of weakly acyclic games. Finally, we observe the properties of the convergence and stability from numerical experiments. The results show that the strict improvement of users’ travel times by the applied evolutionary dynamics is important for ensuring the existence of a stochastically stable equilibrium in DUE games.