Prof. Dr. Dietrich Braess

      Faculty of Mathematics
      Ruhr University Bochum
      44780 Bochum
      Germany

     Tel. (+49.234) 322 3405 (new)
     FAX. (+49.234) 3214 750
 
     If you want to see a photo,
     we offer you one from the homepage of Constructive Approximation.

Areas of actual research

Numerical Treatment of Elliptic Differential Equations

 - Multigrid methods
 - cg-algorithms with nonstandard preconditioners
 - Finite elements in solid mechanics - Locking phenomena
 - Stokes-solvers - Mortar elements in domain decomposition
 - Multigrid methods for saddle point problems

Nonlinear Approximation Theorie

 - rational approximation
 - approximation problems und methods from complex analysis
 - learning theory

Optimization

Coeditor of the Journals

• Constructive Approximation until 1999
• SIAM Journal on Numerical Analysis (SINUM) until 2000
• Numerical Linear Algebra

• Electronic Mail

  braess(at)num.ruhr-uni-bochum.de
  
  
  For good reasons here (at) was not replaced by the known symbol.
  

• A Paradox on Traffic Networks

 References

D. Braess, Über ein Paradoxon aus der Verkehrsplanung.
   Unternehmensforschung 12, 258 - 268 (1968)
   [On a paradox of traffic planning]
   PDF-file
   An English version is in preparation
 
Some earlier definitions are found in the following:

J.G. Wardrop, Some theoretical aspects of road traffic Research.
in Proc. of the Inst. of Civil Engineers, Part II, 325-378 (1952)

M.J. Beckmann, C.B. McGuire, C.B. Winston,
Studies in the Economics of Transportation.
Yale Univ. Press, New Haven 1956
W. Knödel, Graphentheoretische Methoden und ihre Anwendungen.
   Springer-Verlag 1969, pp 57 - 59

J.D. Murchland, Braess's paradox of traffic flow.
   Transpn. Res. 4, 391 - 394 (1970)

L.J. Leblanc, An algorithm for the discrete network design problem.
   Transpn. Sci. 9, 183 - 199 (1975)

M.J. Smith, In a road network, increasing delay locally
   can reduce delay globally.
   Transpn. Res. 12, 419 - 422 (1978) 

M.J. Smith, Traffic control and route choice; a simple example.
   Transpn. Res. 13 B, 289 - 295 (1979)
   
C. Fisk, More paradoxes in the equilibrium assignment problem.
   Transpn. Res. 13 B, 305 - 309 (1979)

N.F. Stewart, Equilibrium vs system-optimal flow: some examples.
   Transpn. Res. 14 A, 81 - 84 (1980)

M. Frank, The Braess paradox.
   Math. Programming 20, 283 - 302 (1981)

W.I. Zangwill and C.B.Garcia.
   Pathways to Solutions. Fixed Points and Equilibria, p. 176.
   Prentice-Hall, Englewood Cliffs 1981

A. Taguchi, Braess' paradox in a two-terminal transportation network.
   J. Oper. Res. Soc. of Japan 25, 376 - 388 (1982)

R. Steinberg and W. Zangwill, The prevalent of Braess' paradox.
   Transpn. Sci. 17, 301 - 318 (1983)

S. Dafermos and A. Nagurney, On some traffic equilibrium theory paradoxes.
   Transpn. Res. B 18, 101 - 110 (1984)

J. Liebman, L. Lasdon, L. Schrage and A. Waren,
   Modeling and Optimization with GINO. p. 53.
   Boyd & Fraser/The Scientific Press, Danvers, MA (1986). (Out of sale)

R. Steinberg and R. Stone, The prevalence of paradoxes
   in transportation equilibrium problems.
   Transpn. Sci. 22, 231 - 141 (1988)

J.E. Cohen and F.P. Kelly, A paradox of congestion in a queuing network.
   J. Appl. Prob. 27, 730 - 734 (1990)

New York Times. What if they closed 42nd Street and nobody noticed?
   NYT 25 December 1990, p.38. 

J.E. Cohen and P. Horowitz, Paradoxial behaviour of mechanical and
   electrical networks. Nature 352, 699 - 701 (August 1991)

F.P. Kelly, Network routing.
   Phil. Trans. R. Soc. London A 337, 343 - 367 (1991)

I. Peterson, Strings and springs net mechanical surprise.
   Science News 140, 118 (August 1991)

S. Catoni and S Pallottino. Traffic equilibrium paradoxes.
   Transpn. Sci. 25, 240 - 244 (1991)

M. De Luca and A. Maugeri, Variational inequalities applied to the study
   of paradoxes in equilibrium problems.
   Optimization 25, 249 - 259 (1992)

Laurence R. Rilett and M. Van Aerde,
   Modelling distributed real-time route guidance strategies in a traffic
   network that exhibits the Braess paradox.
   Proc. 2nd International Vehicle Navigation and Information Systems
   Conference (Dearborn, Mich., Oct. 1991) pp. 577-587.

Ch. Pöppe, Paradoxes Verhalten physikalischer und ökonomischer Systeme.
   Spektrum der Wissenschaft, 23 - 26, Nov. 1992

T. Bass. Road to ruin. Discover 56 - 61, May 1992

P.A. Samuelson, Tragedy of the open road:
   Avoiding paradox by use of regulated public utilities
   that charge corrected Knightian tolls.
   J. of Int. and Comparative Econ. 1, 3 - 12 (1992)

P.A. Samuelson, Tragedy of the commons:
  Efficiency rents to the rescue of the free-road inefficiencies and paradoxes.
  In "Does Economic Space Matter?"
  (Essays in honor of Mei Greenhut. H. Ohta and J.-F. Thisse, eds),
  pp. 71 - 80. St. Martin's Press, New York (1992)

B. Calvert and G. Keady, Braess's paradox and power-law nonlinearities
   in networks.  J. Australian Math. Soc. B 35, 1 - 22 (1993)

A. Knop, Warum mehr Strassen den Verkehrsfluss bremsen.
   nature 76 - 77, 3/1993

G. Alperovich. Neglected welfare aspects in Braess's paradox.
   Int. J. Transport Economics XX (2), 215 - 220 (1993)

A. D. Irvine, How Braess's paradox solves Newcomb's problem.
   Int. Studies Philosophy Science 7, 141 - 160 (1993)
   reprinted in Peter Danielson (ed.), Modeling Moral and Rational Agents,
   Oxford: Oxford University Press, 1997/98

R. Arnott and K. Small. The economics of traffic congestion.
   American Scientist 82, 446 - 455, Sept/Oct 1994

A. Hallefjord, K. Joernsten, and S. Storoey.
   Traffic equilibrium paradoxes when travel demand is elastic.
   Asia-Pacific J. Operational Res. 11, 41 - 50 (1994)

Y.A. Korilis, A.A. Lazar, and A. Orda. Architecting noncooperative networks.
   IEEE J. on Selected Areas in Communications 13, 1241 - 1251 (1995)

N. Bean. Secrets of network success. Physics World (Feb. 1996), 30 - 33

A. Maugeri. Variational and quasi-variational inequalities in network
   flow models. Recent developments in theory and algorithms. 
   In "Variational inequalities and network equilibrium problems."
   (F. Giannessi and A. Maugeri, eds.), pp 195 - 211,
   Plenum, New York 1995

N.G. Bean and P.G. Taylor. Can Braess's paradox occur in loss networks? 
   University of Adelaide, Nov 1994

B. Calvert and G. Keady, Braess's paradox and power-law nonlinearities
   in networks. II.
   In "Proceedings of the First World Congress of Nonlinear Analysts,
   WC 528", Florida Aug 92, Volume III, pp. 2223 - 2230.
   (V. Lakshmanthan, ed.), W. de Gruyter 1996

T.W. Körner. The Pleasures of Counting. pp. 268 - 275
   Cambridge University Press 1996

   deutsch: Mathematisches Denken, SS 370 - 378, 1998

C. Daganzo. Two paradoxes of traffic flow on networks
   with physical queues. II Symposium Ingeneria de los Transportes,
   Madrid, 22-24 May 1996, pp. 55 - 62

L. Marinoff. How Braess' paradox solves Newcomb's problem: Not!
   International Studies in the Philosophy of Science 10, 217 - 237 (1996)

E.I. Pas and S.L. Principio, Braess' paradox: Some new insight.
   Transpn. Res. B 31, 265 - 276 (1997)

W. Blum. Die Logik des Paradoxen. Die Zeit Nr. 52 (1997), S. 36

C.M. Penchina, Braess paradox: maximum penalty in a minimal
   critical network. Transpn. Res. A 31 (5), 379 - 388 (1997)

G. Alperovich. An economic interpretation of Braess' paradox.
   Int. J. Transport Economics 145 - 155 (1997)

Y.A. Korilis, A.A. Lazar, and A. Orda.
   Capacity allocation under noncooperative routing.
   IEEE Trans. on Automatic Control 42 (3) 309 - 325 (1997)

N. Arora, S. Sen and M. Gordin.
   Resolving social dilemmas using genetic algorithms: Initial results.
   The Seventh International Conference on Genetic Algorithms.
   July 19-23, 1997, Michigan State University

B. Calvert, W. Solomon, and I. Ziedins.
   Braess's paradox in a queuing network with state depending routing.
   J. Applied Probability 34, 134 - 154 (1997)

B. Calvert. The Downs-Thomson effect in a Markov process.
   Prob. in the Engineering and Info. Sciences, 11, 327 - 340, (1997) 

N.G. Bean, F.P. Kelly, and P.G. Taylor.  Braess's paradox in loss networks.
   J. Applied Probability 34, 155 - 159 (1997)

N.G. Bean and P.G. Taylor. Braess's paradox in communication networks? 
   EMAC '98 (E.O. Tuck and J.A.K. Stott, eds.) pp. 107 - 110.
   The Institution of Engineers, Australia, 1998

G. Keady. The Colebrook-White formula for pipe networks.
   (Preprint Jan. 1995)
   J. Hydraulic Engineering (Amer.Soc.Civil Engineers) 1998

J.E. Cohen, Cooperation and self-interest: Pareto-inefficiency of
   Nash equilibria in finite random games.
   Proc. Natl. Acad. Sci. USA 95, 9724 - 9731 (1998)

C. Gawron, An iterative algorithm to determine the dynamic user
   equilibrium in a traffic simulation model.
   Int. J. Mod. Phys. C 9(3), 393 - 407 (1998)

Hai Yang and M.G.H. Bell,
   A capacity paradox in network design and how to avoid it.
   Transpn. Res. 32A, 539 - 545 (1998)

C. Daganzo. Queue spillovers in transportation networks with choices.
   Transpn. Sci. 32, 3 - 11 (1998)

Y.A. Korilis, A.A. Lazar, and A. Orda,
   Avoiding the Braess paradox in non-cooperative networks,
   J. Appl. Prob. 36, 211 - 222 (1999).

H-K. Chen, C-F. Hsueh, and C-Y. Wang,
   The dynamic system-optimal route choice problem and toll policies.
   Transp. Planning and Technol. 22, 201 - 228 (1999)

K. Tumer and D.H. Wolpert,
   Collective intelligence and Braess' paradox,
   In "Proc. Seventeenth National Conference
   on Artificial Intelligence, pp. 104 - 109, 
   Austin, TX, August 2000.

H. Kameda, E. Altman, T. Kozawa, and Y. Hosokawa,
   Braess-like paradoxes in distributed computer systems.
   IEEE Trans. on Automatic Control 45 (9) 1687 - 1690 (2000)

J. N. Hagstrom, R. A. Abrams,
   Characterizing Braess's paradox for traffic networks.
   Proc. of IEEE 2001 Conference on Intelligent Transportation Systems,
   pp. 837 - 842 (2001).

A. Nagurney, J. Dong,
   Paradoxes in networks with zero emission links:
   Implications for telecommunications versus transportation.
   Transportation Research D 6, 283-296 (2001)

T. Roughgarden,
   On the severity of Braess's paradox:
   Designing networks for selfish users is hard.
   J. of Computer and System Sciences (to appear).
   A preliminary version is found in the Proc. of the 42nd Annual
   IEEE Symposium on FOCS 2001, pp. 472-481. 

I. Milchtaich,
   Network topology and the efficiency of equilibrium. (June 2001).
   Bar-Ilan University Economics Working Paper No. 12-01.

T. Roughgarden, É. Tardos,
   How bad is selfish routing?
   Journal of the ACM, 49(2), 236 - 259 (2002)

H. Kameda, O. Pourtallier.
   Paradoxes in distributed decisions on optimal load balancing
   for networks of homogeneous computers.
   J. of the ACM 49, 407-433 (2002)

H. Kameda,
   How harmful the paradox can be in Braess/Cohen-Kelly-Jeffries networks,
   Proc. IEEE INFOCOM 2002, New York, June 2002.

C.M. Penchina, L.J. Penchina,
   The Braess paradox in mechanical, traffic, and other networks.
   Amer. J. Phys. 71, 479-482 (2003)

A. Nagurney,
   Influence of Beckmann, McGuire, and Winsten's Studies 
   in the Economis of Transportation on
   Innovation in Modeling, Methodological Developments and Applications.
   UMASS preprint 2003

H. Lin, T. Roughgarden, É. Tardos,
   A stronger bound on Braess's paradox. 
   Proceedings of the ACM-SIAM Symposium on Discrete Algorithms,
   2004, pp. 333 - 334.

H. Lin, T. Roughgarden, É. Tardos,
   A. Walkover: Braess's paradox, Fibonacci numbers,
   and exponential inapproximability (submitted)

S. Robinson,
   The price of anarchy,
   SIAM news 37/Number 5 (June 2004)


  Links to further references

       Books by Dietrich Braess

Approximationstheorie.

 Vorlesungstext für die Fernuniversität Hagen (deutsch). 480 S. 1979 

Advances in Multigrid Methods.

 D. Braess, W. Hackbusch, and U. Trottenberg, Eds.

 Vieweg, Braunschweig-Wiesbaden 1985

Nonlinear Approximation Theory.

 Springer-Verlag, Berlin-Heidelberg-New York 1986

Finite Elemente: Theorie, schnelle Löser 

 und Anwendungen in der Elastizitätstheorie.

 Springer-Verlag, Berlin-Heidelberg-New York 1992, 1997, 2003

Numerical Methods in Approximation Theory.

 D. Braess and L.L. Schumaker, Eds.

 Birkhäuser-Verlag, Basel 1992

Finite Elements: Theory, Fast Solvers

 and Applications in Solid Mechanics.

 Cambridge University Press, Cambridge, 1997, what is new in 2001

• You can obtain a list of corrections and extensions for the books on Finite Elements