Some observations on stochastic user equilibrium and system optimum of trac assignment Joseph N. Prashker * , Shlomo Bekhor Faculty of Civil Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel Received 17 August 1998; received in revised form 18 May 1999; accepted 26 May 1999 Abstract Trac assignment models can be classi®ed according to the behavioral assumption governing route choice. The deterministic user equilibrium (UE), stochastic user equilibrium (SUE) and system optimum (SO) models have been studied extensively in the literature. The relationship between the UE solution and the SO solution for a given network is well known, as is the relationship between UE and SUE. The question that arises concerns the relationship between SUE and (deterministic) SO. The ¯ow pattern ob- tained from the SO solution serves as a yardstick for comparison with the ¯ow patterns obtained from the UE and SUE solutions. The investigation examines whether the stochastic equilibrium is ``closer'' than the deterministic user equilibrium to the system optimum. This paper compares the performance of the dif- ferent solutions for simple networks. The comparison is made by evaluating the relative dierence in total system times for UE and SUE solutions with respect to the SO solution. This paper also presents an ex- tension of previous results to show that the BraessÕ paradox can occur for certain ranges of demand vol- umes in the case of stochastic equilibrium and non-linear cost functions. Ó 2000 Elsevier Science Ltd. All rights reserved. Keywords: User equilibrium; System optimum; Trac assignment; Route choice 1. Introduction Trac assignment models can be classi®ed according to the behavioral assumption governing route choice. If we assume that drivers have perfect knowledge about travel costs on a network and choose the best route according to WardropÕs ®rst principle, this behavioral assumption leads to deterministic user equilibrium. The system optimum (WardropÕs second principle) is achieved Transportation Research Part B 34 (2000) 277±291 www.elsevier.com/locate/trb * Corresponding author. Tel.: +972-482-92-901; fax: +972-482-25-716. E-mail address: prashker@netvision.net.il (J.N. Prashker). 0191-2615/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII:S0191-2615(99)00025-9