WARDROP EQUILIBRIA —TO BE PUBLISHED IN “ENCYCLOPEDIA OF OPERATIONS RESEARCH AND MANAGEMENT SCIENCE. EDITED BY J. J. COCHRAN. WILEY”—

Wardrop equilibria are commonly used as a solution concept of network games when modeling transportation and telecommunication networks with congestion. This concept assumes that players select a route that minimizes the time or cost incurred in its traversal. This behavioral assumption admits convenient mathematical descriptions, and efficient algorithms for the computation of equilibria are available. For that reason, planners have been making use of this concept for decades for evaluating projects, optimizing tolls, estimating demands, and a myriad of applications arising from extensions of the basic model. In this article, we introduce the basic model, explain strategies for computation of equilibria, and discuss the extent of the inefficiency arising from the self-minded behavior of the players. In addition, we provide some generalizations of the basic model.

[1]  J. Kohl Der Verkehr und die Ansiedelungen der Menschen in ihrer Abhängigkeit von der Gestaltung der Erdoberfläche , 1850 .

[2]  F. Knight Some Fallacies in the Interpretation of Social Cost , 1924 .

[3]  J. G. Wardrop,et al.  Some Theoretical Aspects of Road Traffic Research , 1952 .

[4]  William S. Vickrey,et al.  A Proposal for Revising New York's Subway Fare Structure , 1955, Oper. Res..

[5]  Philip Wolfe,et al.  An algorithm for quadratic programming , 1956 .

[6]  C. B. Mcguire,et al.  Studies in the Economics of Transportation , 1958 .

[7]  W. Vickrey Congestion Theory and Transport Investment , 1969 .

[8]  Stella C. Dafermos,et al.  Traffic assignment problem for a general network , 1969 .

[9]  Robert B. Dial,et al.  A PROBABILISTIC MULTIPATH TRAFFIC ASSIGNMENT MODEL WHICH OBVIATES PATH ENUMERATION. IN: THE AUTOMOBILE , 1971 .

[10]  R. Rosenthal A class of games possessing pure-strategy Nash equilibria , 1973 .

[11]  Michel Trahan Probabilistic Assignment: An Algorithm , 1974 .

[12]  P. Robillard,et al.  Common Bus Lines , 1975 .

[13]  Carlos F. Daganzo,et al.  On Stochastic Models of Traffic Assignment , 1977 .

[14]  Deepak K. Merchant,et al.  A Model and an Algorithm for the Dynamic Traffic Assignment Problems , 1978 .

[15]  Mike Smith,et al.  The existence, uniqueness and stability of traffic equilibria , 1979 .

[16]  C. Fisk Some developments in equilibrium traffic assignment , 1980 .

[17]  Stella Dafermos,et al.  Traffic Equilibrium and Variational Inequalities , 1980 .

[18]  Andrés Weintraub,et al.  An algorithm for the traffic assignment problem , 1980, Networks.

[19]  T. Magnanti,et al.  Equilibria on a Congested Transportation Network , 1981 .

[20]  D. Hearn,et al.  CONVEX PROGRAMMING FORMULATIONS OF THE ASYMMETRIC TRAFFIC ASSIGNMENT PROBLEM , 1984 .

[21]  M. Florian Transportation planning models , 1984 .

[22]  T. Magnanti MODELS AND ALGORITHMS FOR PREDICTING URBAN TRAFFIC EQUILIBRIA , 1984 .

[23]  Alain Haurie,et al.  On the relationship between Nash - Cournot and Wardrop equilibria , 1983, Networks.

[24]  Moshe Ben-Akiva,et al.  Dynamic network equilibrium research , 1985 .

[25]  David E. Boyce,et al.  Improved Efficiency of the Frank-Wolfe Algorithm for Convex Network Programs , 1985, Transp. Sci..

[26]  Yosef Sheffi,et al.  Urban Transportation Networks: Equilibrium Analysis With Mathematical Programming Methods , 1985 .

[27]  Michael Florian,et al.  An efficient implementation of the "partan" variant of the linear approximation method for the network equilibrium problem , 1987, Networks.

[28]  Pitu Mirchandani,et al.  Generalized Traffic Equilibrium with Probabilistic Travel Times and Perceptions , 1987, Transp. Sci..

[29]  Stefano Pallottino,et al.  Equilibrium traffic assignment for large scale transit networks , 1988 .

[30]  Patrick T. Harker,et al.  Multiple Equilibrium Behaviors on Networks , 1988, Transp. Sci..

[31]  Michael Florian,et al.  Optimal strategies: A new assignment model for transit networks , 1989 .

[32]  D. Van Vliet,et al.  A Full Analytical Implementation of the PARTAN/Frank-Wolfe Algorithm for Equilibrium Assignment , 1990, Transp. Sci..

[33]  A. Nagurney Network Economics: A Variational Inequality Approach , 1992 .

[34]  Yasunori Iida,et al.  RISK ASSIGNMENT: A NEW TRAFFIC ASSIGNMENT MODEL CONSIDERING THE RISK OF TRAVEL TIME VARIATION. , 1993 .

[35]  Jia Hao Wu,et al.  Transit Equilibrium Assignment: A Model and Solution Algorithms , 1994, Transp. Sci..

[36]  David K. Smith Network Flows: Theory, Algorithms, and Applications , 1994 .

[37]  L. Shapley,et al.  Potential Games , 1994 .

[38]  David Bernstein,et al.  The traffic equilibrium problem with nonadditive path costs , 1995 .

[39]  M. Bell Alternatives to Dial's logit assignment algorithm , 1995 .

[40]  Bin Ran,et al.  MODELING DYNAMIC TRANSPORTATION NETWORKS : AN INTELLIGENT TRANSPORTATION SYSTEM ORIENTED APPROACH. 2ND REV. ED. , 1996 .

[41]  Takashi Akamatsu,et al.  Cyclic flows, Markov process and stochastic traffic assignment , 1996 .

[42]  I. Milchtaich,et al.  Congestion Games with Player-Specific Payoff Functions , 1996 .

[43]  Torbjörn Larsson,et al.  A dual scheme for traffic assignment problems , 1997 .

[44]  A. Palma,et al.  Optimization formulations and static equilibrium in congested transportation networks , 1998 .

[45]  Mike Maher,et al.  Algorithms for logit-based stochastic user equilibrium assignment , 1998 .

[46]  Christos H. Papadimitriou,et al.  Worst-case equilibria , 1999 .

[47]  Torbjörn Larsson,et al.  Side constrained traffic equilibrium models: analysis, computation and applications , 1999 .

[48]  José R. Correa,et al.  Common-Lines and Passenger Assignment in Congested Transit Networks , 2001, Transp. Sci..

[49]  Christos H. Papadimitriou,et al.  Algorithms, Games, and the Internet , 2001, ICALP.

[50]  Michel Gendreau,et al.  Modeling Bus Stops in Transit Networks: A Survey and New Formulations , 2001, Transp. Sci..

[51]  Athanasios K. Ziliaskopoulos,et al.  Foundations of Dynamic Traffic Assignment: The Past, the Present and the Future , 2001 .

[52]  William H. Sandholm,et al.  Potential Games with Continuous Player Sets , 2001, J. Econ. Theory.

[53]  Henry X. Liu,et al.  An Analytical Dynamic Traffic Assignment Model with Probabilistic Travel Times and Travelers' Perceptions , 2002 .

[54]  Hillel Bar-Gera,et al.  Origin-Based Algorithm for the Traffic Assignment Problem , 2002, Transp. Sci..

[55]  Michael G.H. Bell,et al.  Risk-averse user equilibrium traffic assignment: an application of game theory , 2002 .

[56]  Tim Roughgarden,et al.  How bad is selfish routing? , 2002, JACM.

[57]  Kwang Mong Sim,et al.  The price of anarchy for non-atomic congestion games with symmetric cost maps and elastic demands , 2003, Oper. Res. Lett..

[58]  Hong Kam Lo,et al.  Network with degradable links: capacity analysis and design , 2003 .

[59]  Tim Roughgarden The price of anarchy is independent of the network topology , 2003, J. Comput. Syst. Sci..

[60]  José R. Correa,et al.  Sloan School of Management Working Paper 4319-03 June 2003 Selfish Routing in Capacitated Networks , 2022 .

[61]  Patrice Marcotte,et al.  A Strategic Flow Model of Traffic Assignment in Static Capacitated Networks , 2004, Oper. Res..

[62]  Michael Patriksson,et al.  Algorithms for Computing Traffic Equilibria , 2004 .

[63]  Michael Patriksson,et al.  Sensitivity Analysis of Traffic Equilibria , 2004, Transp. Sci..

[64]  Ryuichi Kitamura,et al.  Simulation Approaches in Transportation Analysis , 2005 .

[65]  Haijun Huang,et al.  Mathematical and Economic Theory of Road Pricing , 2005 .

[66]  G. Santos Urban Congestion Charging: A Comparison between London and Singapore , 2005 .

[67]  Eitan Altman,et al.  A survey on networking games in telecommunications , 2006, Comput. Oper. Res..

[68]  Roberto Cominetti,et al.  A frequency-based assignment model for congested transit networks with strict capacity constraints: characterization and computation of equilibria , 2006 .

[69]  Robert B. Dial,et al.  A path-based user-equilibrium traffic assignment algorithm that obviates path storage and enumeration , 2006 .

[70]  Amnon Rapoport,et al.  Navigating congested networks with variable demand: Experimental evidence , 2006 .

[71]  D. Hearn,et al.  Mathematical and Computational Models for Congestion Charging , 2006 .

[72]  José R. Correa,et al.  Network Games with Atomic Players , 2006, ICALP.

[73]  G. Santos,et al.  Road Pricing: Lessons from London , 2006 .

[74]  Satish V. Ukkusuri,et al.  Dynamic Traffic Equilibrium , 2007 .

[75]  Chaitanya Swamy,et al.  The effectiveness of Stackelberg strategies and tolls for network congestion games , 2007, SODA '07.

[76]  Michael Patriksson,et al.  Chapter 10 Traffic Equilibrium , 2007, Transportation.

[77]  Georgia Perakis,et al.  The "Price of Anarchy" Under Nonlinear and Asymmetric Costs , 2007, Math. Oper. Res..

[78]  Thomas Pitz,et al.  Commuters route choice behaviour , 2007, Games Econ. Behav..

[79]  Roberto Cominetti,et al.  Markovian traffic equilibrium , 2007, Math. Program..

[80]  Adrian Vetta,et al.  A Priority-Based Model of Routing , 2008, Chic. J. Theor. Comput. Sci..

[81]  José R. Correa,et al.  A Geometric Approach to the Price of Anarchy in Nonatomic Congestion Games , 2008, Games Econ. Behav..

[82]  Guido Gentile,et al.  Linear User Cost Equilibrium: a new algorithm for traffic assignment , 2009 .

[83]  Michael Florian,et al.  A New Look at Projected Gradient Method for Equilibrium Assignment , 2009 .

[84]  Umang Bhaskar,et al.  Equilibria of atomic flow games are not unique , 2009, SODA.

[85]  John Morgan,et al.  Network architecture and traffic flows: Experiments on the Pigou-Knight-Downs and Braess Paradoxes , 2007, Games Econ. Behav..

[86]  Amnon Rapoport,et al.  Choice of Routes in Congested Traffic Networks: Experimental Tests of the Braess Paradox , 2005, Games Econ. Behav..

[87]  Tim Roughgarden,et al.  Algorithmic Game Theory , 2007 .

[88]  Hillel Bar-Gera,et al.  Traffic Assignment by Paired Alternative Segments , 2010 .

[89]  William H. Sandholm,et al.  Population Games And Evolutionary Dynamics , 2010, Economic learning and social evolution.

[90]  N. Stier-Moses,et al.  Wardrop Equilibria with Risk-Averse Users , 2010 .