Stochastic stability of dynamic user equilibrium in unidirectional networks: Weakly acyclic game approach

Dynamic system optimal traffic assignment with atomic users: Convergence and stability

“…Kuwahara and Akamatsu, 1993;Akamatsu et al, 2015;Wada et al, 2019). Especially, with this property and the theory of weakly acyclic games, Satsukawa et al (2019) established the convergence and stability of dynamic user equilibrium in a unidirectional network. In a similar manner, we prove the convergence and stochastic stability of evolutionary dynamics in a DUE-FCP game in an SBPR-1 network.…”

“…Specifically, under the best response dynamics, a route profile can shift from an equilibrium state to another state with an equal total cost since users can switch their routes among their best response routes, as mentioned above. Such a switch could cause the following ripple effect (Satsukawa et al, 2019): route change of a user affects utilities of other users and makes their routes into non-best response routes. This means that these users can find routes whose marginal costs are lower than those they currently select after the shift.…”

“…We then obtain the following theorem and propositions regarding theoretical properties of a DUE-FCP game in an SBPR-1 network in the same way as in Satsukawa et al (2019):…”

“…Under the fixed implementation scheme (or in a DUE-FCP game), owing to the ordering property shown in Proposition 3, a route change of a user affects the utility of users who depart later from the same origin, but not vice versa. By this temporal asymmetry of the interaction, users having earlier departure times become likely to choose their ex-post best response routes earlier during the convergence process (Satsukawa et al, 2019). By contrast, under the evolutionary implementation scheme (or in a DSO game), as externalities are perfectly internalised, the better/best response of any user leads the traffic state toward an equilibrium state, regardless of the order of his/her departure time.…”

“…We then propose the following algorithm: This algorithm derives an equilibrium state by assigning users to their best response routes in the order of their departure times (please refer to Satsukawa et al (2019) for the details of the role of each step). Thanks to Proposition 3, it is guaranteed that the utility of the user in Group A is not changed by route change of any user in Group B.…”