On the relationship between dynamic Nash and instantaneous user equilibria

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 discretetime version of the differential game and is used to numerically show the asymptotic behavior of openloop Nash equilibrium solutions on a simple network. A Nash equilibrium solution is also analyzed on the 18-arc network. q 1998 John Wiley & Sons, Inc. Networks 32: 141–163, 1998