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Braess's paradox
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== Possible instances of the paradox in action == === Prevalence === In 1983, Steinberg and Zangwill provided, under reasonable{{third-party inline|date=August 2024|reason=An old paper proving theorems about a model doesn't seem like a great source for the reasonableness of that model. This also seems like an extraordinary claim.}} assumptions, the necessary and sufficient conditions for Braess's paradox to occur in a general transportation network when a new route is added. (Note that their result applies to the addition of ''any'' new route, not just to the case of adding a single link.) As a corollary, they obtain that Braess's paradox is about as likely to occur as not occur when a random new route is added.<ref>{{Cite journal | doi = 10.1287/trsc.17.3.301| title = The Prevalence of Braess' Paradox| journal = Transportation Science| volume = 17| issue = 3| pages = 301| year = 1983| last1 = Steinberg | first1 = R. | last2 = Zangwill | first2 = W. I. }}</ref> === Traffic === {{see also|Induced demand}} [[File:Cheonggyecheon stream 1.jpg|thumb|When an expressway in Seoul was removed so a creek could be restored, traffic flow in the area improved]] Braess's paradox has a counterpart in case of a reduction of the road network, which may cause a reduction of individual commuting time.<ref name=Razemon/> In [[Seoul]], [[South Korea]], traffic around the city sped up when the Cheonggye Expressway was removed as part of the [[Cheonggyecheon]] restoration project.<ref>{{cite book | last1 = Easley | first1 = D. | last2 = Kleinberg | first2 = J. | title = Networks | page = 71 | publisher = Cornell Store Press | date = 2008 }}</ref> In [[Stuttgart]], [[Germany]], after investments into the road network in 1969, the traffic situation did not improve until a section of newly built road was closed for traffic again.<ref name="Knödel1969">{{cite book|last=Knödel|first=W.|title=Graphentheoretische Methoden Und Ihre Anwendungen|url=https://books.google.com/books?id=bJ22pwAACAAJ|date=31 January 1969|publisher=[[Springer-Verlag]]|isbn=978-3-540-04668-4|pages=57–59}}</ref> In 1990 the temporary closing of [[42nd Street (Manhattan)|42nd Street]] in [[Manhattan]], [[New York City]], for [[Earth Day]] reduced the amount of congestion in the area.<ref>{{cite news | last = Kolata | first = Gina |author-link=Gina Kolata | date=1990-12-25 | work=New York Times | url=https://www.nytimes.com/1990/12/25/health/what-if-they-closed-42d-street-and-nobody-noticed.html |title=What if They Closed 42d Street and Nobody Noticed? |access-date=2008-11-16}}</ref> In 2008 Youn, Gastner and Jeong demonstrated specific routes in Boston, New York City and London where that might actually occur and pointed out roads that could be closed to reduce predicted travel times.<ref name="YounGastner2008">{{cite journal | last1 = Youn | first1 = Hyejin | last2 = Gastner | first2 = Michael | last3 = Jeong | first3 = Hawoong | title = Price of Anarchy in Transportation Networks: Efficiency and Optimality Control| journal = [[Physical Review Letters]] | volume = 101 | issue = 12 | year = 2008 | issn = 0031-9007 | pmid = 18851419 | doi = 10.1103/PhysRevLett.101.128701 | arxiv = 0712.1598 | bibcode = 2008PhRvL.101l8701Y | pages=128701 | s2cid = 20779255 }}</ref> In 2009, New York experimented with closures of [[Broadway (Manhattan)|Broadway]] at [[Times Square]] and [[Herald Square]], which resulted in improved traffic flow and permanent pedestrian plazas.<ref>{{cite episode |title=Braess' Paradox |first=Andrew |last=Boyd |series=Engines of Our Ingenuity |number=2814 |url=http://www.uh.edu/engines/epi2814.htm}}</ref> In 2012, Paul Lecroart, of the institute of planning and development of the [[Île-de-France]], wrote that "Despite initial fears, the removal of main roads does not cause deterioration of traffic conditions beyond the starting adjustments. The traffic transfer are limited and below expectations".<ref name=Razemon/> He also notes that some private vehicle trips (and related economic activity) are not transferred to public transport and simply disappear ("evaporate").<ref name=Razemon/> The same phenomenon was also observed when road closing was not part of an urban project but the consequence of an accident. In 2012 in [[Rouen]], a bridge was destroyed by fire. Over the next two years, other bridges were used more, but the total number of cars crossing bridges was reduced.<ref name=Razemon>{{in lang|fr}} Olivier Razemon, "Le paradoxde de l'« évaporation » du trafic automobile", ''[[Le Monde]]'', Thursday 25 August 2016, page 5. Published on-line as [https://www.lemonde.fr/blog/transports/2016/08/23/voitures-evaporees/ "Quand les voitures s’évaporent"] on 24 August 2016 and updated on 25 August 2016 (page visited on 3 August 2023).</ref> === Electricity === In 2012, scientists at the [[Max Planck Institute for Dynamics and Self-Organization]] demonstrated, through [[computational modelling]], the potential for the phenomenon to occur in [[Electrical grid|power transmission networks]] where [[Electricity generation|power generation]] is decentralized.<ref name=rdmag_mpi>{{Citation |author=Staff (Max Planck Institute) |date=September 14, 2012 |title=Study: Solar and wind energy may stabilize the power grid |magazine=[[R&D Magazine]] |at=rdmag.com |url=https://www.rdworldonline.com/study-solar-and-wind-energy-may-stabilize-the-power-grid/ |access-date=September 14, 2012 }}</ref> In 2012, an international team of researchers from Institut Néel (CNRS, France), INP (France), IEMN (CNRS, France) and UCL (Belgium) published in ''[[Physical Review Letters]]''<ref name="PalaBaltazar2012">{{cite journal | last1 = Pala | first1 = M. G. | last2 = Baltazar | first2 = S. | last3 = Liu | first3 = P. | last4 = Sellier | first4 = H. | last5 = Hackens | first5 = B. | last6 = Martins | first6 = F. | last7 = Bayot | first7 = V. | last8 = Wallart | first8 = X. | last9 = Desplanque | first9 = L. | last10 = Huant | first10 = S. | title = Transport Inefficiency in Branched-Out Mesoscopic Networks: An Analog of the Braess Paradox | journal = Physical Review Letters | volume = 108 | issue = 7 | pages = 076802 | year = 2012 | issn = 0031-9007 | doi = 10.1103/PhysRevLett.108.076802 | arxiv = 1112.1170 | orig-year = 6 Dec 2011 (v1) | bibcode=2012PhRvL.108g6802P | pmid=22401236| s2cid = 23243934 }}</ref> a paper showing that Braess's paradox may occur in [[Mesoscopic physics|mesoscopic]] electron systems. In particular, they showed that adding a path for electrons in a nanoscopic network paradoxically reduced its conductance. That was shown both by simulations as well as experiments at low temperature using [[scanning gate microscopy]]. === Springs === [[File:Braess-Paradoxon der Mechanik.svg|thumb|On the right are two springs joined in series by a short rope. When the short rope connecting B and C is removed (left), the weight hangs higher.]] A model with springs and ropes can show that a hung weight can rise in height despite a taut rope in the hanging system being cut, and follows from the same mathematical structure as the original Braess's paradox.<ref>{{cite web |last1=Mould |first1=Steve |title=The Spring Paradox |url=https://www.youtube.com/watch?v=Cg73j3QYRJc |website=YouTube |date=29 July 2021 |access-date=2 December 2022 |language=en}}</ref> For two identical springs joined in series by a short rope, their total spring constant is half of each individual spring, resulting in a long stretch when a certain weight is hung. This remains the case as we add two longer ropes in slack to connect the lower end of the upper spring to the hung weight (lower end of the lower spring), and the upper end of the lower spring to the hanging point (upper end of the upper spring). However, when the short rope is cut, the longer ropes become taut, and the two springs become parallel (in the [[Series and parallel springs|mechanical sense]]) to each other. The total spring constant is twice that of each individual spring, and when the length of the long ropes is not too long, the hung weight will actually be higher compared to before the short rope was cut. The fact that the hung weight rises despite cutting a taut rope (the short rope) in the hanging system is counter-intuitive, but it does follow from [[Hooke's law]] and the way springs work in series and in parallel. === Biology === [[Adilson E. Motter]] and collaborators demonstrated that Braess's paradox outcomes may often occur in biological and ecological systems.<ref>{{Cite journal | doi=10.1002/bies.200900128| pmid=20127700| pmc=2841822|title = Improved network performance via antagonism: From synthetic rescues to multi-drug combinations| journal=BioEssays| volume=32| issue=3| pages=236–245|year = 2010|last1 = Motter|first1 = Adilson E.| arxiv=1003.3391}}</ref> Motter suggests removing part of a perturbed network could rescue it. For resource management of endangered species [[food webs]], in which extinction of many species might follow sequentially, selective removal of a doomed species from the network could in principle bring about the positive outcome of preventing a series of further extinctions.<ref>Sahasrabudhe S., Motter A. E., [http://www.nature.com/articles/ncomms1163 Rescuing ecosystems from extinction cascades through compensatory perturbations], Nature Communications 2, 170 (2011)</ref> === Team sports strategy === It has been suggested that in basketball, a team can be seen as a network of possibilities for a route to scoring a basket, with a different efficiency for each pathway, and a star player could reduce the overall efficiency of the team, analogous to a shortcut that is overused increasing the overall times for a journey through a road network. A proposed solution for maximum efficiency in scoring is for a star player to shoot about the same number of shots as teammates. However, this approach is not supported by hard statistical evidence, as noted in the original paper.<ref>{{Cite journal |bibcode = 2009arXiv0908.1801S |title = The price of anarchy in basketball |journal = Journal of Quantitative Analysis in Sports|volume = 6 |issue = 1 |last1 = Skinner|first1 = Brian|last2 = Gastner|first2 = Michael T|last3 = Jeong|first3 = Hawoong|year = 2009|arxiv = 0908.1801|doi = 10.2202/1559-0410.1217|citeseerx = 10.1.1.215.1658|s2cid = 119275142 }}</ref> === Blockchain networks === Braess's paradox has been shown to appear in blockchain payment channel networks, also known as layer-2 networks.<ref>{{Cite book |last1=Kotzer |first1=Arad |last2=Rottenstreich |first2=Ori |chapter=Braess Paradox in Layer-2 Blockchain Payment Networks |year=2023 |title=2023 IEEE International Conference on Blockchain and Cryptocurrency (ICBC) |chapter-url=https://ieeexplore.ieee.org/document/10174986 |pages=1–9 |doi=10.1109/ICBC56567.2023.10174986 |isbn=979-8-3503-1019-1 }}</ref> Payment channel networks implement a solution to the scalability problem of blockchain networks, allowing transactions of high rates without recording them on the blockchain. In such a network, users can establish a channel by locking funds on each side of the channel. Transactions are executed either through a channel connecting directly the payer and payee or through a path of channels with intermediate users that ask for some fees. While intuitively, opening new channels allows higher routing flexibility, adding a new channel might cause higher fees, and similarly closing existing channels might decrease fees. The paper presented a theoretical analysis with conditions for the paradox, methods for mitigating the paradox as well as an empirical analysis, showing the appearance in practice of the paradox and its effects on Bitcoin's Lightning network.
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