On the relationship between dynamic Nash and instantaneous user equilibria

Abstract

The problem of a dynamic Nash equilibrium traffic assignment with schedule delays on congested networks is formulated as an N-person nonzero-sum differential game in which each player represents an origin-destination pair. Optimality conditions are derived using a Nash equilibrium solution concept in the open-loop strategy space and given the economic interpretation as a dynamic game theoretic generalization of Wardrop's second principle. It is demonstrated that an open-loop Nash equilibrium solution converges to an instantaneous dynamic user equilibrium solution as the number of players for each origin-destination pair increases to infinity. An iterative algorithm is developed to solve a discrete-time version of the differential game and is used to numerically show the asymptotic behavior of open-loop Nash equilibrium solutions on a simple network. A Nash equilibrium solution is also analyzed on the 18-arc network. © 1998 John Wiley & Sons, Inc. Networks 32: 141–163, 1998