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Unformatted text preview: SELFISH ROUTING IN CAPACITATED NETWORKS ´ ´ JOSE R. CORREA, ANDREAS S. SCHULZ, AND NICOLAS E. STIER MOSES Sloan School of Management and Operations Research Center Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge, MA 021394307 Abstract. According to Wardrop’s first principle, agents in a congested network choose their routes selfishly, a behavior that is captured by the Nash equilibrium of the underlying noncooperative game. A Nash equilibrium does not optimize any global criterion per se, and so there is no apparent reason why it should be close to a solution of minimal total travel time, i.e. the system optimum. In this paper, we offer positive results on the eﬃciency of Nash equilibria in traﬃc networks. In contrast to prior work, we present results for networks with capacities and for latency functions that are nonconvex, nondifferentiable, and even discontinuous. The inclusion of upper bounds on arc ﬂows has early been recognized as an important means to provide a more accurate description of traﬃc ﬂows. In this more general model, multiple Nash equilibria may exist and an arbitrary equilibrium does not need to be nearly eﬃcient. Nonetheless, our main result shows that the best equilibrium is as eﬃcient as in the model without capacities. Moreover, this holds true for broader classes of travel cost functions than considered hitherto. Date : June 2003; revised February 2004. 2000 Mathematics Subject Classification. Primary 90C35; 90B10, 90B20, 90C25, 90C27, 91A10, 91A13, 91A43. Key words and phrases. Selfish Routing, Price of Anarchy, Traﬃc Assignment, System Optimum, Nash Equilibrium, Performance Guarantee, Multicommodity Flow. OR/MS classification. Games: Noncooperative; Mathematics: Combinatorics; Networks/graphs: Multicom modity, Theory; Transportation: Models. 1 2 ´ ´ JOSE R. CORREA, ANDREAS S. SCHULZ, AND NICOLAS E. STIER MOSES 1. Introduction It is a common behavioral assumption in the study of traﬃc networks modeling congestion effects and therefore featuring ﬂowdependent link travel times, that travelers choose routes that they perceive as being the shortest under the prevailing traﬃc conditions. In other words, travelers minimize their individual travel times (Kohl 1841). The situation resulting from these individual decisions is one in which drivers cannot reduce their journey times by unilaterally choosing another route, which prompted Knight (1924) to call the resulting traﬃc pattern an equilibrium. Nowadays it is indeed known as the user equilibrium (Dafermos and Sparrow 1969), and it is effectively thought of as a steadystate evolving after a transient phase in which travelers successively adjust their route choices until a situation with stable route travel costs and route ﬂows has been reached (Larsson and Patriksson 1999). In a seminal contribution, Wardrop (1952) stated two principles that formalize this notion of equilibrium and the alternative postulate of the minimization...
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This note was uploaded on 12/04/2011 for the course ESD 15.094 taught by Professor Jiesun during the Spring '04 term at MIT.
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