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href="#Traffic"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Traffic</span> </div> </a> <ul id="toc-Traffic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Electricity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Electricity"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Electricity</span> </div> </a> <ul id="toc-Electricity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Springs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Springs"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Springs</span> </div> </a> <ul id="toc-Springs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Biology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Biology"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Biology</span> </div> </a> <ul id="toc-Biology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Team_sports_strategy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Team_sports_strategy"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Team sports strategy</span> </div> </a> <ul id="toc-Team_sports_strategy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Blockchain_networks" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Blockchain_networks"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.7</span> <span>Blockchain networks</span> </div> </a> <ul id="toc-Blockchain_networks-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Mathematical_approach" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mathematical_approach"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Mathematical approach</span> </div> </a> <button aria-controls="toc-Mathematical_approach-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Mathematical approach subsection</span> </button> <ul id="toc-Mathematical_approach-sublist" class="vector-toc-list"> <li id="toc-Example" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Example</span> </div> </a> <ul id="toc-Example-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Existence_of_an_equilibrium" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Existence_of_an_equilibrium"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Existence of an equilibrium</span> </div> </a> <ul id="toc-Existence_of_an_equilibrium-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Finding_an_equilibrium" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Finding_an_equilibrium"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Finding an equilibrium</span> </div> </a> <ul id="toc-Finding_an_equilibrium-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-How_far_from_optimal_is_traffic_at_equilibrium?" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#How_far_from_optimal_is_traffic_at_equilibrium?"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>How far from optimal is traffic at equilibrium?</span> </div> </a> <ul id="toc-How_far_from_optimal_is_traffic_at_equilibrium?-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Effect_of_network_topology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Effect_of_network_topology"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Effect of network topology</span> </div> </a> <ul id="toc-Effect_of_network_topology-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" 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Available in 25 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-25" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">25 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D9%81%D8%A7%D8%B1%D9%82%D8%A9_%D8%A8%D8%B1%D8%A7%D9%8A%D8%B3" title="مفارقة برايس – Arabic" lang="ar" hreflang="ar" data-title="مفارقة برايس" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Brayes_paradoksu" title="Brayes paradoksu – Azerbaijani" lang="az" hreflang="az" data-title="Brayes paradoksu" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%B4%D0%BE%D0%BA%D1%81_%D0%BD%D0%B0_%D0%91%D1%80%D0%B0%D0%B0%D1%81" title="Парадокс на Браас – Bulgarian" lang="bg" hreflang="bg" data-title="Парадокс на Браас" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Braess%C5%AFv_paradox" title="Braessův paradox – Czech" lang="cs" hreflang="cs" data-title="Braessův paradox" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://de.wikipedia.org/wiki/Braess-Paradoxon" title="Braess-Paradoxon – German" lang="de" hreflang="de" data-title="Braess-Paradoxon" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Paradoja_de_Braess" title="Paradoja de Braess – Spanish" lang="es" hreflang="es" data-title="Paradoja de Braess" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%BE%D8%A7%D8%B1%D8%A7%D8%AF%D9%88%DA%A9%D8%B3_%D8%A8%D8%B1%DB%8C%D8%B3" title="پارادوکس بریس – Persian" lang="fa" hreflang="fa" data-title="پارادوکس بریس" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Paradoxe_de_Braess" title="Paradoxe de Braess – French" lang="fr" hreflang="fr" data-title="Paradoxe de Braess" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Paradosso_di_Braess" title="Paradosso di Braess – Italian" lang="it" hreflang="it" data-title="Paradosso di Braess" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%A8%D7%93%D7%95%D7%A7%D7%A1_%D7%91%D7%A8%D7%A1" title="פרדוקס ברס – Hebrew" lang="he" hreflang="he" data-title="פרדוקס ברס" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Braeso_paradoksas" title="Braeso paradoksas – Lithuanian" lang="lt" hreflang="lt" data-title="Braeso paradoksas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Braess_paradoxona" title="Braess paradoxona – Hungarian" lang="hu" hreflang="hu" data-title="Braess paradoxona" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AC%E0%B5%8D%E0%B4%B0%E0%B5%87%E0%B4%B8%E0%B5%8D_%E0%B4%AA%E0%B5%8D%E0%B4%B0%E0%B4%B9%E0%B5%87%E0%B4%B3%E0%B4%BF%E0%B4%95" title="ബ്രേസ് പ്രഹേളിക – Malayalam" lang="ml" hreflang="ml" data-title="ബ്രേസ് പ്രഹേളിക" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Braess-paradox" title="Braess-paradox – Dutch" lang="nl" hreflang="nl" data-title="Braess-paradox" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%96%E3%83%A9%E3%82%A4%E3%82%B9%E3%81%AE%E3%83%91%E3%83%A9%E3%83%89%E3%83%83%E3%82%AF%E3%82%B9" title="ブライスのパラドックス – Japanese" lang="ja" hreflang="ja" data-title="ブライスのパラドックス" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Parad%C3%B2xa_de_Braess" title="Paradòxa de Braess – Occitan" lang="oc" hreflang="oc" data-title="Paradòxa de Braess" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Paradoks_Braessa" title="Paradoks Braessa – Polish" lang="pl" hreflang="pl" data-title="Paradoks Braessa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Paradoxo_de_Braess" title="Paradoxo de Braess – Portuguese" lang="pt" hreflang="pt" data-title="Paradoxo de Braess" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Paradoxul_lui_Braess" title="Paradoxul lui Braess – Romanian" lang="ro" hreflang="ro" data-title="Paradoxul lui Braess" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%B4%D0%BE%D0%BA%D1%81_%D0%91%D1%80%D0%B0%D0%B5%D1%81%D0%B0" title="Парадокс Браеса – Russian" lang="ru" hreflang="ru" data-title="Парадокс Браеса" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Braessin_paradoksi" title="Braessin paradoksi – Finnish" lang="fi" hreflang="fi" data-title="Braessin paradoksi" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Braess_paradox" title="Braess paradox – Swedish" lang="sv" hreflang="sv" data-title="Braess paradox" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Braess_Paradoksu" title="Braess Paradoksu – Turkish" lang="tr" hreflang="tr" data-title="Braess Paradoksu" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%B4%D0%BE%D0%BA%D1%81_%D0%91%D1%80%D0%B5%D1%81%D0%B0" title="Парадокс Бреса – Ukrainian" lang="uk" hreflang="uk" data-title="Парадокс Бреса" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%B8%83%E9%9B%B7%E6%96%AF%E6%82%96%E8%AE%BA" title="布雷斯悖论 – Chinese" lang="zh" hreflang="zh" data-title="布雷斯悖论" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q897194#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div 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searchaux" style="display:none">Paradox related to increasing roadway capacity</div> <p> <b>Braess's paradox</b> is the observation that adding one or more roads to a road network can slow down overall <a href="/wiki/Road_traffic" class="mw-redirect" title="Road traffic">traffic</a> flow through it. The paradox was first discovered by <a href="/wiki/Arthur_Cecil_Pigou" title="Arthur Cecil Pigou">Arthur Pigou</a> in 1920,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and later named after the German mathematician <a href="/wiki/Dietrich_Braess" title="Dietrich Braess">Dietrich Braess</a> in 1968.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>The paradox may have analogies in <a href="/wiki/Electrical_power_grid" class="mw-redirect" title="Electrical power grid">electrical power grids</a> and biological systems. It has been suggested that, in theory, the improvement of a malfunctioning network could be accomplished by removing certain parts of it. The paradox has been used to explain instances of improved <a href="/wiki/Traffic_flow" title="Traffic flow">traffic flow</a> when existing major roads are closed. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Discovery_and_definition">Discovery and definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Braess%27s_paradox&action=edit&section=1" title="Edit section: Discovery and definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dietrich Braess, a mathematician at <a href="/wiki/Ruhr_University" class="mw-redirect" title="Ruhr University">Ruhr University</a>, <a href="/wiki/Germany" title="Germany">Germany</a>, noticed the flow in a road network could be impeded by adding a new road, when he was working on <a href="/wiki/Traffic_modelling" class="mw-redirect" title="Traffic modelling">traffic modelling</a>. His idea was that if each driver is making the <a href="/wiki/Optimization" class="mw-redirect" title="Optimization">optimal</a> self-interested decision as to which route is quickest, a shortcut could be chosen too often for drivers to have the shortest travel times possible. More formally, the idea behind Braess's discovery is that the <a href="/wiki/Nash_equilibrium" title="Nash equilibrium">Nash equilibrium</a> may not equate with the best overall flow through a network.<sup id="cite_ref-ReferenceA_3-0" class="reference"><a href="#cite_note-ReferenceA-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p> The paradox is stated as follows:</p><blockquote><p>"For each point of a road network, let there be given the number of cars starting from it and the destination of the cars. Under these conditions, one wishes to estimate the distribution of traffic flow. Whether one street is preferable to another depends not only on the quality of the road, but also on the <a href="/wiki/Flow_density" class="mw-redirect" title="Flow density">density of the flow</a>. If every driver takes the path that looks most favourable to them, the resultant running times need not be minimal. Furthermore, it is indicated by an example that an extension of the road network may cause a redistribution of the traffic that results in longer individual running times."</p></blockquote> <p>Adding extra capacity to a <a href="/wiki/Network_(mathematics)" class="mw-redirect" title="Network (mathematics)">network</a> when the moving entities selfishly choose their route can in some cases reduce overall performance. That is because the <a href="/wiki/Nash_equilibrium" title="Nash equilibrium">Nash equilibrium</a> of such a system is not necessarily optimal. The network change induces a new game structure which leads to a (multiplayer) <a href="/wiki/Prisoner%27s_dilemma" title="Prisoner's dilemma">prisoner's dilemma</a>. In a Nash equilibrium, drivers have no incentive to change their routes. While the system is not in a Nash equilibrium, individual drivers are able to improve their respective travel times by changing the routes they take. In the case of Braess's paradox, drivers will continue to switch until they reach Nash equilibrium despite the reduction in overall performance. </p><p>If the latency functions are linear, adding an edge can never make total travel time at equilibrium worse by a factor of more than 4/3.<sup id="cite_ref-RoughgardenTardos_4-0" class="reference"><a href="#cite_note-RoughgardenTardos-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Possible_instances_of_the_paradox_in_action">Possible instances of the paradox in action</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Braess%27s_paradox&action=edit&section=2" title="Edit section: Possible instances of the paradox in action"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Prevalence">Prevalence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Braess%27s_paradox&action=edit&section=3" title="Edit section: Prevalence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 1983, Steinberg and Zangwill provided, under reasonable<sup class="noprint Inline-Template noprint Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Independent_sources" title="Wikipedia:Independent sources"><span title="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. (August 2024)">third-party source needed</span></a></i>]</sup> 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 <i>any</i> 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.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Traffic">Traffic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Braess%27s_paradox&action=edit&section=4" title="Edit section: Traffic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Induced_demand" title="Induced demand">Induced demand</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Cheonggyecheon_stream_1.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Cheonggyecheon_stream_1.jpg/220px-Cheonggyecheon_stream_1.jpg" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Cheonggyecheon_stream_1.jpg/330px-Cheonggyecheon_stream_1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/da/Cheonggyecheon_stream_1.jpg/440px-Cheonggyecheon_stream_1.jpg 2x" data-file-width="4032" data-file-height="3024" /></a><figcaption>When an expressway in Seoul was removed so a creek could be restored, traffic flow in the area improved</figcaption></figure> <p>Braess's paradox has a counterpart in case of a reduction of the road network, which may cause a reduction of individual commuting time.<sup id="cite_ref-Razemon_6-0" class="reference"><a href="#cite_note-Razemon-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>In <a href="/wiki/Seoul" title="Seoul">Seoul</a>, <a href="/wiki/South_Korea" title="South Korea">South Korea</a>, traffic around the city sped up when the Cheonggye Expressway was removed as part of the <a href="/wiki/Cheonggyecheon" title="Cheonggyecheon">Cheonggyecheon</a> restoration project.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> In <a href="/wiki/Stuttgart" title="Stuttgart">Stuttgart</a>, <a href="/wiki/Germany" title="Germany">Germany</a>, 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.<sup id="cite_ref-Knödel1969_8-0" class="reference"><a href="#cite_note-Knödel1969-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> In 1990 the temporary closing of <a href="/wiki/42nd_Street_(Manhattan)" title="42nd Street (Manhattan)">42nd Street</a> in <a href="/wiki/Manhattan" title="Manhattan">Manhattan</a>, <a href="/wiki/New_York_City" title="New York City">New York City</a>, for <a href="/wiki/Earth_Day" title="Earth Day">Earth Day</a> reduced the amount of congestion in the area.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> 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.<sup id="cite_ref-YounGastner2008_10-0" class="reference"><a href="#cite_note-YounGastner2008-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> In 2009, New York experimented with closures of <a href="/wiki/Broadway_(Manhattan)" title="Broadway (Manhattan)">Broadway</a> at <a href="/wiki/Times_Square" title="Times Square">Times Square</a> and <a href="/wiki/Herald_Square" title="Herald Square">Herald Square</a>, which resulted in improved traffic flow and permanent pedestrian plazas.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>In 2012, Paul Lecroart, of the institute of planning and development of the <a href="/wiki/%C3%8Ele-de-France" title="Île-de-France">Île-de-France</a>, 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".<sup id="cite_ref-Razemon_6-1" class="reference"><a href="#cite_note-Razemon-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> He also notes that some private vehicle trips (and related economic activity) are not transferred to public transport and simply disappear ("evaporate").<sup id="cite_ref-Razemon_6-2" class="reference"><a href="#cite_note-Razemon-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>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 <a href="/wiki/Rouen" title="Rouen">Rouen</a>, 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.<sup id="cite_ref-Razemon_6-3" class="reference"><a href="#cite_note-Razemon-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Electricity">Electricity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Braess%27s_paradox&action=edit&section=5" title="Edit section: Electricity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 2012, scientists at the <a href="/wiki/Max_Planck_Institute_for_Dynamics_and_Self-Organization" title="Max Planck Institute for Dynamics and Self-Organization">Max Planck Institute for Dynamics and Self-Organization</a> demonstrated, through <a href="/wiki/Computational_modelling" class="mw-redirect" title="Computational modelling">computational modelling</a>, the potential for the phenomenon to occur in <a href="/wiki/Electrical_grid" title="Electrical grid">power transmission networks</a> where <a href="/wiki/Electricity_generation" title="Electricity generation">power generation</a> is decentralized.<sup id="cite_ref-rdmag_mpi_12-0" class="reference"><a href="#cite_note-rdmag_mpi-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>In 2012, an international team of researchers from Institut Néel (CNRS, France), INP (France), IEMN (CNRS, France) and UCL (Belgium) published in <i><a href="/wiki/Physical_Review_Letters" title="Physical Review Letters">Physical Review Letters</a></i><sup id="cite_ref-PalaBaltazar2012_13-0" class="reference"><a href="#cite_note-PalaBaltazar2012-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> a paper showing that Braess's paradox may occur in <a href="/wiki/Mesoscopic_physics" title="Mesoscopic physics">mesoscopic</a> 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 <a href="/wiki/Scanning_gate_microscopy" title="Scanning gate microscopy">scanning gate microscopy</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Springs">Springs</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Braess%27s_paradox&action=edit&section=6" title="Edit section: Springs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Braess-Paradoxon_der_Mechanik.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Braess-Paradoxon_der_Mechanik.svg/220px-Braess-Paradoxon_der_Mechanik.svg.png" decoding="async" width="220" height="241" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Braess-Paradoxon_der_Mechanik.svg/330px-Braess-Paradoxon_der_Mechanik.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Braess-Paradoxon_der_Mechanik.svg/440px-Braess-Paradoxon_der_Mechanik.svg.png 2x" data-file-width="1901" data-file-height="2085" /></a><figcaption>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.</figcaption></figure> <p>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.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>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 <a href="/wiki/Series_and_parallel_springs" title="Series and parallel springs">mechanical sense</a>) 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. </p><p>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 <a href="/wiki/Hooke%27s_law" title="Hooke's law">Hooke's law</a> and the way springs work in series and in parallel. </p> <div class="mw-heading mw-heading3"><h3 id="Biology">Biology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Braess%27s_paradox&action=edit&section=7" title="Edit section: Biology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Adilson_E._Motter" title="Adilson E. Motter">Adilson E. Motter</a> and collaborators demonstrated that Braess's paradox outcomes may often occur in biological and ecological systems.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> Motter suggests removing part of a perturbed network could rescue it. For resource management of endangered species <a href="/wiki/Food_webs" class="mw-redirect" title="Food webs">food webs</a>, 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.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Team_sports_strategy">Team sports strategy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Braess%27s_paradox&action=edit&section=8" title="Edit section: Team sports strategy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>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.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Blockchain_networks">Blockchain networks</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Braess%27s_paradox&action=edit&section=9" title="Edit section: Blockchain networks"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Braess's paradox has been shown to appear in blockchain payment channel networks, also known as layer-2 networks.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> 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. </p><p>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. </p> <div class="mw-heading mw-heading2"><h2 id="Mathematical_approach">Mathematical approach</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Braess%27s_paradox&action=edit&section=10" title="Edit section: Mathematical approach"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Example">Example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Braess%27s_paradox&action=edit&section=11" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/File:Braess_paradox_road_example.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/Braess_paradox_road_example.svg/500px-Braess_paradox_road_example.svg.png" decoding="async" width="500" height="129" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/Braess_paradox_road_example.svg/750px-Braess_paradox_road_example.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/01/Braess_paradox_road_example.svg/1000px-Braess_paradox_road_example.svg.png 2x" data-file-width="750" data-file-height="194" /></a><figcaption></figcaption></figure> <p>Consider a road network as shown in the adjacent diagram on which 4000 drivers wish to travel from point Start to End. The travel time in minutes on the Start–A road is the number of travellers (T) divided by 100, and on Start–B is a constant 45 minutes (likewise with the roads across from them). If the dashed road does not exist (so the traffic network has 4 roads in total), the time needed to drive Start–A–End route with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> drivers would be <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {a}{100}}+45}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>a</mi> <mn>100</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mn>45</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {a}{100}}+45}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1fc3b7c99fd600572c02423650b8ced3a364ef8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.467ex; height:3.343ex;" alt="{\displaystyle {\tfrac {a}{100}}+45}"></span>. The time needed to drive the Start–B–End route with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> drivers would be <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {b}{100}}+45}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>b</mi> <mn>100</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mn>45</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {b}{100}}+45}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39bb989ca15cd918c60092b46142b0995a02d38e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.467ex; height:3.843ex;" alt="{\displaystyle {\tfrac {b}{100}}+45}"></span>. As there are 4000 drivers, the fact that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b=4000}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>=</mo> <mn>4000</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b=4000}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46ddf0ae3526d4dabbc6d52693387fc78a785952" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.816ex; height:2.343ex;" alt="{\displaystyle a+b=4000}"></span> can be used to derive the fact that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=b=2000}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> <mo>=</mo> <mn>2000</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=b=2000}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a9f2fe136cb1a503b3977ae732dcf01b49ff676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.074ex; height:2.176ex;" alt="{\displaystyle a=b=2000}"></span> when the system is at equilibrium. Therefore, each route takes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2000}{100}}+45=65}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2000</mn> <mn>100</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mn>45</mn> <mo>=</mo> <mn>65</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2000}{100}}+45=65}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9592cae8f8941e5d7c8fb3c70e96fac646230a66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:14.713ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2000}{100}}+45=65}"></span> minutes. If either route took less time, it would not be a Nash equilibrium: a rational driver would switch from the longer route to the shorter route. </p><p>Now suppose the dashed line A–B is a road with an extremely short travel time of approximately 0 minutes. Suppose that the road is opened and one driver tries Start–A–B–End. To his surprise he finds that his time is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2000}{100}}+{\tfrac {2001}{100}}=40.01}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2000</mn> <mn>100</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2001</mn> <mn>100</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mn>40.01</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2000}{100}}+{\tfrac {2001}{100}}=40.01}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/812f411662f6c53091db5c8492e187de43ea295b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:19.484ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2000}{100}}+{\tfrac {2001}{100}}=40.01}"></span> minutes, a saving of almost 25 minutes. Soon, more of the 4000 drivers are trying this new route. The time taken rises from 40.01 and keeps climbing. When the number of drivers trying the new route reaches 2500, with 1500 still in the Start–B–End route, their time will be <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2500}{100}}+{\tfrac {4000}{100}}=65}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2500</mn> <mn>100</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4000</mn> <mn>100</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mn>65</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2500}{100}}+{\tfrac {4000}{100}}=65}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c1c7e28c586c3ee06022249a44ef355ef4aedb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:16.512ex; height:3.843ex;" alt="{\displaystyle {\tfrac {2500}{100}}+{\tfrac {4000}{100}}=65}"></span> minutes, which is no improvement over the original route. Meanwhile, those 1500 drivers have been slowed to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 45+{\tfrac {4000}{100}}=85}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>45</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4000</mn> <mn>100</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mn>85</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 45+{\tfrac {4000}{100}}=85}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e76e0c104d5feea985a01a7a14decdb253547ac6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:14.713ex; height:3.843ex;" alt="{\displaystyle 45+{\tfrac {4000}{100}}=85}"></span> minutes, a 20-minute increase. They are obliged to switch to the new route via A too, so it now takes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {4000}{100}}+{\tfrac {4000}{100}}=80}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4000</mn> <mn>100</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4000</mn> <mn>100</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mn>80</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {4000}{100}}+{\tfrac {4000}{100}}=80}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2e330fe9a197cf12a6b04e64edd2ac0a26d9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:16.512ex; height:3.843ex;" alt="{\displaystyle {\tfrac {4000}{100}}+{\tfrac {4000}{100}}=80}"></span> minutes. Nobody has any incentive to travel A-End or Start-B because any driver trying them will take 85 minutes. Thus, the opening of the cross route triggers an irreversible change to it by everyone, costing everyone 80 minutes instead of the original 65. If every driver were to agree not to use the A–B path, or if that route were closed, every driver would benefit by a 15-minute reduction in travel time. </p> <div class="mw-heading mw-heading3"><h3 id="Existence_of_an_equilibrium">Existence of an equilibrium</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Braess%27s_paradox&action=edit&section=12" title="Edit section: Existence of an equilibrium"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If one assumes the travel time for each person driving on an edge to be equal, an equilibrium will always exist. </p><p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{e}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{e}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04fd7bebe685606dbda35996daf31ca85fad38e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.72ex; height:2.843ex;" alt="{\displaystyle L_{e}(x)}"></span> be the formula for the travel time of each person traveling along edge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span> when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> people take that edge. Suppose there is a traffic graph with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bffc1d36b6465edf0ff6f9950096c46f85888c44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.328ex; height:2.009ex;" alt="{\displaystyle x_{e}}"></span> people driving along edge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span>. Let the energy of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(e)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>e</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(e)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8c3c60376e68f0651f4f927633006aadb64601e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.668ex; height:2.843ex;" alt="{\displaystyle E(e)}"></span>, be </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{x_{e}}L_{e}(i)=L_{e}(1)+L_{e}(2)+\cdots +L_{e}(x_{e})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mrow> </munderover> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{x_{e}}L_{e}(i)=L_{e}(1)+L_{e}(2)+\cdots +L_{e}(x_{e})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fe63b9920ee6661365bb8df181e0279ea9da0a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:41.103ex; height:7.009ex;" alt="{\displaystyle \sum _{i=1}^{x_{e}}L_{e}(i)=L_{e}(1)+L_{e}(2)+\cdots +L_{e}(x_{e})}"></span></dd></dl> <p>(If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{e}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{e}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d506b6e0be0a0ff76b0b6e9ae14a0e778c131b58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.589ex; height:2.509ex;" alt="{\displaystyle x_{e}=0}"></span> let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(e)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>e</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(e)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fc75077bb6c7b5d4dfde2cc4dd18d69541205d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.929ex; height:2.843ex;" alt="{\displaystyle E(e)=0}"></span>). Let the total energy of the traffic graph be the sum of the energies of every edge in the graph. </p><p>Take a choice of routes that minimizes the total energy. Such a choice must exist because there are finitely many choices of routes. That will be an equilibrium. </p><p>Assume, for contradiction, this is not the case. Then, there is at least one driver who can switch the route and improve the travel time. Suppose the original route is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{0},e_{1},\ldots ,e_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{0},e_{1},\ldots ,e_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e75c8463f07654dd116b216072cd3008fc892b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.79ex; height:2.009ex;" alt="{\displaystyle e_{0},e_{1},\ldots ,e_{n}}"></span> while the new route is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e'_{0},e'_{1},\ldots ,e'_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> <mo>,</mo> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e'_{0},e'_{1},\ldots ,e'_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/267f948c3fd96bb4775ff3a746ff8d3c2e402b33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.246ex; height:2.843ex;" alt="{\displaystyle e'_{0},e'_{1},\ldots ,e'_{m}}"></span>. Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> be total energy of the traffic graph, and consider what happens when the route <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{0},e_{1},...e_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{0},e_{1},...e_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe1e18f72037b9c855e3db398d4b365b5122971e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.747ex; height:2.009ex;" alt="{\displaystyle e_{0},e_{1},...e_{n}}"></span> is removed. The energy of each edge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebdc3a9cb1583d3204eff8918b558c293e0d2cf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.883ex; height:2.009ex;" alt="{\displaystyle e_{i}}"></span> will be reduced by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{e_{i}}(x_{e_{i}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{e_{i}}(x_{e_{i}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7b374fbb17ec42810b27f0cf8d1d8a4e9fd3225" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.968ex; height:3.009ex;" alt="{\displaystyle L_{e_{i}}(x_{e_{i}})}"></span> and so the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> will be reduced by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{i=0}^{n}L_{e_{i}}(x_{e_{i}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i=0}^{n}L_{e_{i}}(x_{e_{i}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fc738fe6353bfc61a62d33a7eeb5161591b1651" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.71ex; height:3.176ex;" alt="{\textstyle \sum _{i=0}^{n}L_{e_{i}}(x_{e_{i}})}"></span>. That is simply the total travel time needed to take the original route. If the new route is then added, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e'_{0},e'_{1},\ldots ,e'_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> <mo>,</mo> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e'_{0},e'_{1},\ldots ,e'_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/267f948c3fd96bb4775ff3a746ff8d3c2e402b33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.246ex; height:2.843ex;" alt="{\displaystyle e'_{0},e'_{1},\ldots ,e'_{m}}"></span>, the total energy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> will be increased by the total travel time needed to take the new route. Because the new route is shorter than the original route, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> must decrease relative to the original configuration, contradicting the assumption that the original set of routes minimized the total energy. </p><p>Therefore, the choice of routes minimizing total energy is an equilibrium. </p> <div class="mw-heading mw-heading3"><h3 id="Finding_an_equilibrium">Finding an equilibrium</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Braess%27s_paradox&action=edit&section=13" title="Edit section: Finding an equilibrium"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The above proof outlines a procedure known as <a href="/wiki/Best_response" title="Best response">best response</a> dynamics, which finds an equilibrium for a linear traffic graph and terminates in a finite number of steps. The algorithm is termed "best response" because at each step of the algorithm, if the graph is not at equilibrium then some driver has a best response to the strategies of all other drivers and switches to that response. </p><p>Pseudocode for Best Response Dynamics: </p> <pre>Let <i>P</i> be some traffic pattern. <b>while</b> <i>P</i> is not at equilibrium: compute the potential energy <i>e</i> of <i>P</i> <b>for each</b> driver <i>d</i> in <i>P</i>: <b>for each</b> alternate path <i>p</i> available to <i>d</i>: compute the potential energy <i>n</i> of the pattern when <i>d</i> takes path <i>p</i> <b>if</b> <i>n</i> < <i>e</i>: modify <i>P</i> so that <i>d</i> takes path <i>p</i> <b>continue</b> the topmost <b>while</b> </pre> <p>At each step, if some particular driver could do better by taking an alternate path (a "best response"), doing so strictly decreases the energy of the graph. If no driver has a best response, the graph is at equilibrium. Since the energy of the graph strictly decreases with each step, the best response dynamics algorithm must eventually halt. </p> <div class="mw-heading mw-heading3"><h3 id="How_far_from_optimal_is_traffic_at_equilibrium?"><span id="How_far_from_optimal_is_traffic_at_equilibrium.3F"></span>How far from optimal is traffic at equilibrium?</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Braess%27s_paradox&action=edit&section=14" title="Edit section: How far from optimal is traffic at equilibrium?"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the travel time functions are linear, that is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{e}(x)=a_{e}x+b_{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{e}(x)=a_{e}x+b_{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33683c24070975ddaa336fa8421f952e74ee778c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.213ex; height:2.843ex;" alt="{\displaystyle L_{e}(x)=a_{e}x+b_{e}}"></span> for some <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{e},b_{e}\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{e},b_{e}\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4d9b51796770d47a189257f726fd2107e76ee53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.519ex; height:2.509ex;" alt="{\displaystyle a_{e},b_{e}\geq 0}"></span>, then at worst, traffic in the energy-minimizing equilibrium is twice as bad as socially optimal.<sup id="cite_ref-EasleyKleinberg_19-0" class="reference"><a href="#cite_note-EasleyKleinberg-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>Proof: Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span> be some traffic configuration, with associated energy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(Z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>Z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(Z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bad908841c19ebb23401d8bb42ff6f8d9345dbe1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.265ex; height:2.843ex;" alt="{\displaystyle E(Z)}"></span> and total travel time <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(Z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>Z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(Z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbe19adbc728bd141dc419327e261d8236370989" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.126ex; height:2.843ex;" alt="{\displaystyle T(Z)}"></span>. For each edge, the energy is the sum of an <a href="/wiki/Arithmetic_progression" title="Arithmetic progression">arithmetic progression</a>, and using the formula for the sum of an arithmetic progression, one can show that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(Z)\leq T(Z)\leq 2E(Z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>Z</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>Z</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mn>2</mn> <mi>E</mi> <mo stretchy="false">(</mo> <mi>Z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(Z)\leq T(Z)\leq 2E(Z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bee3c181eced290c6d227d1c34fdd045165bd90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.016ex; height:2.843ex;" alt="{\displaystyle E(Z)\leq T(Z)\leq 2E(Z)}"></span>. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{o}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{o}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efac161f49e493f2d423a3aac9a084e5ab2cd5e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.617ex; height:2.509ex;" alt="{\displaystyle Z_{o}}"></span> is the socially-optimal traffic flow and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d83bcb46e7db16904749df9b09463d0db4bfc197" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.586ex; height:2.509ex;" alt="{\displaystyle Z_{e}}"></span> is the energy-minimizing traffic flow, the inequality implies that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(Z_{e})\leq 2E(Z_{e})\leq 2E(Z_{o})\leq 2T(Z_{o})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≤</mo> <mn>2</mn> <mi>E</mi> <mo stretchy="false">(</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≤</mo> <mn>2</mn> <mi>E</mi> <mo stretchy="false">(</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≤</mo> <mn>2</mn> <mi>T</mi> <mo stretchy="false">(</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(Z_{e})\leq 2E(Z_{e})\leq 2E(Z_{o})\leq 2T(Z_{o})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82f0cc902be59a28fb3ec9a5269196af843d54ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.25ex; height:2.843ex;" alt="{\displaystyle T(Z_{e})\leq 2E(Z_{e})\leq 2E(Z_{o})\leq 2T(Z_{o})}"></span>. </p><p>Thus, the total travel time for the energy-minimizing equilibrium is at most twice as bad as for the optimal flow. </p> <div class="mw-heading mw-heading3"><h3 id="Effect_of_network_topology">Effect of network topology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Braess%27s_paradox&action=edit&section=15" title="Edit section: Effect of network topology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mlichtaich<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> proved that Braess's paradox may occur if and only if the network is not a <a href="/wiki/Series%E2%80%93parallel_graph" title="Series–parallel graph">series-parallel graph</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Braess%27s_paradox&action=edit&section=16" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Downs%E2%80%93Thomson_paradox" title="Downs–Thomson paradox">Downs–Thomson paradox</a> – Paradox in traffic engineering related to improvements in the road network</li> <li><a href="/wiki/Jevons_paradox" title="Jevons paradox">Jevons paradox</a> – Efficiency leads to increased demand</li> <li><a href="/wiki/Marchetti%27s_constant" title="Marchetti's constant">Marchetti's constant</a> – Average commuting time</li> <li><a href="/wiki/Lewis%E2%80%93Mogridge_position" title="Lewis–Mogridge position">Lewis–Mogridge position</a> – Theory of road traffic</li> <li><a href="/wiki/Price_of_anarchy_in_congestion_games" title="Price of anarchy in congestion games">Price of anarchy in congestion games</a> - a quantitative analysis of the loss in efficiency caused by congestion externalities.</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Braess%27s_paradox&action=edit&section=17" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFPigou2017" class="citation cs2">Pigou, Arthur Cecil (24 October 2017), <a rel="nofollow" class="external text" href="https://dx.doi.org/10.4324/9781351304368-1">"Welfare and Economic Welfare"</a>, <i>The Economics of Welfare</i>, Routledge, pp. 3–22, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4324%2F9781351304368-1">10.4324/9781351304368-1</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-351-30436-8" title="Special:BookSources/978-1-351-30436-8"><bdi>978-1-351-30436-8</bdi></a><span class="reference-accessdate">, retrieved <span class="nowrap">24 March</span> 2023</span></cite><span 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(2010). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2841822">"Improved network performance via antagonism: From synthetic rescues to multi-drug combinations"</a>. <i>BioEssays</i>. <b>32</b> (3): 236–245. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1003.3391">1003.3391</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fbies.200900128">10.1002/bies.200900128</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2841822">2841822</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/20127700">20127700</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=BioEssays&rft.atitle=Improved+network+performance+via+antagonism%3A+From+synthetic+rescues+to+multi-drug+combinations&rft.volume=32&rft.issue=3&rft.pages=236-245&rft.date=2010&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC2841822%23id-name%3DPMC&rft_id=info%3Apmid%2F20127700&rft_id=info%3Aarxiv%2F1003.3391&rft_id=info%3Adoi%2F10.1002%2Fbies.200900128&rft.aulast=Motter&rft.aufirst=Adilson+E.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC2841822&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABraess%27s+paradox" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">Sahasrabudhe S., Motter A. E., <a rel="nofollow" class="external text" href="http://www.nature.com/articles/ncomms1163">Rescuing ecosystems from extinction cascades through compensatory perturbations</a>, Nature Communications 2, 170 (2011)</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSkinnerGastnerJeong2009" class="citation journal cs1">Skinner, Brian; Gastner, Michael T; Jeong, Hawoong (2009). "The price of anarchy in basketball". <i>Journal of Quantitative Analysis in Sports</i>. <b>6</b> (1). <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0908.1801">0908.1801</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009arXiv0908.1801S">2009arXiv0908.1801S</a>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.215.1658">10.1.1.215.1658</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2202%2F1559-0410.1217">10.2202/1559-0410.1217</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119275142">119275142</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Quantitative+Analysis+in+Sports&rft.atitle=The+price+of+anarchy+in+basketball&rft.volume=6&rft.issue=1&rft.date=2009&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119275142%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2009arXiv0908.1801S&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.215.1658%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.2202%2F1559-0410.1217&rft_id=info%3Aarxiv%2F0908.1801&rft.aulast=Skinner&rft.aufirst=Brian&rft.au=Gastner%2C+Michael+T&rft.au=Jeong%2C+Hawoong&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABraess%27s+paradox" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKotzerRottenstreich2023" class="citation book cs1">Kotzer, Arad; Rottenstreich, Ori (2023). <a rel="nofollow" class="external text" href="https://ieeexplore.ieee.org/document/10174986">"Braess Paradox in Layer-2 Blockchain Payment Networks"</a>. <i>2023 IEEE International Conference on Blockchain and Cryptocurrency (ICBC)</i>. pp. 1–9. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FICBC56567.2023.10174986">10.1109/ICBC56567.2023.10174986</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/979-8-3503-1019-1" title="Special:BookSources/979-8-3503-1019-1"><bdi>979-8-3503-1019-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Braess+Paradox+in+Layer-2+Blockchain+Payment+Networks&rft.btitle=2023+IEEE+International+Conference+on+Blockchain+and+Cryptocurrency+%28ICBC%29&rft.pages=1-9&rft.date=2023&rft_id=info%3Adoi%2F10.1109%2FICBC56567.2023.10174986&rft.isbn=979-8-3503-1019-1&rft.aulast=Kotzer&rft.aufirst=Arad&rft.au=Rottenstreich%2C+Ori&rft_id=https%3A%2F%2Fieeexplore.ieee.org%2Fdocument%2F10174986&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABraess%27s+paradox" class="Z3988"></span></span> </li> <li id="cite_note-EasleyKleinberg-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-EasleyKleinberg_19-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEasleyKleinberg" class="citation web cs1">Easley, David; Kleinberg, Jon. <a rel="nofollow" class="external text" href="http://www.cs.cornell.edu/home/kleinber/networks-book/networks-book.pdf">"Networks, Crowds, and Markets: Reasoning about a Highly Connected World (8.3 Advanced Material: The Social Cost of Traffic at Equilibrium)"</a> <span class="cs1-format">(PDF)</span>. <i>Jon Kleinberg's Homepage</i>. Jon Kleinberg. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150316015111/http://www.cs.cornell.edu/home/kleinber/networks-book/networks-book.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 16 March 2015<span class="reference-accessdate">. Retrieved <span class="nowrap">30 May</span> 2015</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Jon+Kleinberg%27s+Homepage&rft.atitle=Networks%2C+Crowds%2C+and+Markets%3A+Reasoning+about+a+Highly+Connected+World+%288.3+Advanced+Material%3A+The+Social+Cost+of+Traffic+at+Equilibrium%29&rft.aulast=Easley&rft.aufirst=David&rft.au=Kleinberg%2C+Jon&rft_id=http%3A%2F%2Fwww.cs.cornell.edu%2Fhome%2Fkleinber%2Fnetworks-book%2Fnetworks-book.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABraess%27s+paradox" class="Z3988"></span> – This is the preprint of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780521195331" title="Special:BookSources/9780521195331">9780521195331</a></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMilchtaich2006" class="citation journal cs1">Milchtaich, Igal (1 November 2006). <a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/pii/S0899825605001284">"Network topology and the efficiency of equilibrium"</a>. <i>Games and Economic Behavior</i>. <b>57</b> (2): 321–346. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.geb.2005.09.005">10.1016/j.geb.2005.09.005</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/10419%2F259308">10419/259308</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0899-8256">0899-8256</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Games+and+Economic+Behavior&rft.atitle=Network+topology+and+the+efficiency+of+equilibrium&rft.volume=57&rft.issue=2&rft.pages=321-346&rft.date=2006-11-01&rft_id=info%3Ahdl%2F10419%2F259308&rft.issn=0899-8256&rft_id=info%3Adoi%2F10.1016%2Fj.geb.2005.09.005&rft.aulast=Milchtaich&rft.aufirst=Igal&rft_id=https%3A%2F%2Fwww.sciencedirect.com%2Fscience%2Farticle%2Fpii%2FS0899825605001284&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABraess%27s+paradox" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Braess%27s_paradox&action=edit&section=18" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBraess1969" class="citation journal cs1 cs1-prop-foreign-lang-source">Braess, D. (1969). <a rel="nofollow" class="external text" href="https://homepage.ruhr-uni-bochum.de/Dietrich.Braess/paradox.pdf">"Über ein Paradoxon aus der Verkehrsplanung"</a> [On a Paradox of Traffic Planning] <span class="cs1-format">(PDF)</span>. <i>Unternehmensforschung</i> (in German). <b>12</b>: 258–268.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Unternehmensforschung&rft.atitle=%C3%9Cber+ein+Paradoxon+aus+der+Verkehrsplanung&rft.volume=12&rft.pages=258-268&rft.date=1969&rft.aulast=Braess&rft.aufirst=D.&rft_id=https%3A%2F%2Fhomepage.ruhr-uni-bochum.de%2FDietrich.Braess%2Fparadox.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABraess%27s+paradox" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://homepage.rub.de/Dietrich.Braess/Paradox-BNW.pdf">(translation by Nagurney & Wakolbinger)</a></li> <li>Katharina Belaga-Werbitzky: „Das Paradoxon von Braess in erweiterten Wheatstone-Netzen mit M/M/1-Bedienern“ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-89959-123-2" title="Special:BookSources/3-89959-123-2">3-89959-123-2</a></li> <li>Translation of the Braess 1968 article from German to English appears as the article "On a paradox of traffic planning," by D. Braess, A. Nagurney, and T. Wakolbinger in the journal Transportation Science, volume 39, 2005, pp. 446–450. <a rel="nofollow" class="external text" href="http://supernet.som.umass.edu/cfoto/braess-visit/braessvisit.html">More information</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110927012722/http://supernet.som.umass.edu/cfoto/braess-visit/braessvisit.html">Archived</a> 27 September 2011 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIrvine1993" class="citation journal cs1">Irvine, A. D. (1993). "How Braess' paradox solves Newcomb's problem". <i>International Studies in the Philosophy of Science</i>. <b>7</b> (2): 141–160. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F02698599308573460">10.1080/02698599308573460</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Studies+in+the+Philosophy+of+Science&rft.atitle=How+Braess%27+paradox+solves+Newcomb%27s+problem&rft.volume=7&rft.issue=2&rft.pages=141-160&rft.date=1993&rft_id=info%3Adoi%2F10.1080%2F02698599308573460&rft.aulast=Irvine&rft.aufirst=A.+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABraess%27s+paradox" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteinbergZangwill1983" class="citation journal cs1">Steinberg, R.; Zangwill, W. I. (1983). "The Prevalence of Braess' Paradox". <i>Transportation Science</i>. <b>17</b> (3): 301. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1287%2Ftrsc.17.3.301">10.1287/trsc.17.3.301</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transportation+Science&rft.atitle=The+Prevalence+of+Braess%27+Paradox&rft.volume=17&rft.issue=3&rft.pages=301&rft.date=1983&rft_id=info%3Adoi%2F10.1287%2Ftrsc.17.3.301&rft.aulast=Steinberg&rft.aufirst=R.&rft.au=Zangwill%2C+W.+I.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABraess%27s+paradox" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRapoportKuglerDugarGisches2009" class="citation journal cs1">Rapoport, A.; Kugler, T.; Dugar, S.; Gisches, E. J. (2009). <a rel="nofollow" class="external text" href="https://www.parisschoolofeconomics.eu/IMG/pdf/Choices_of_routes.pdf">"Choice of routes in congested traffic networks: Experimental tests of the Braess Paradox"</a> <span class="cs1-format">(PDF)</span>. <i>Games and Economic Behavior</i>. <b>65</b> (2): 538–571. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.geb.2008.02.007">10.1016/j.geb.2008.02.007</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Games+and+Economic+Behavior&rft.atitle=Choice+of+routes+in+congested+traffic+networks%3A+Experimental+tests+of+the+Braess+Paradox&rft.volume=65&rft.issue=2&rft.pages=538-571&rft.date=2009&rft_id=info%3Adoi%2F10.1016%2Fj.geb.2008.02.007&rft.aulast=Rapoport&rft.aufirst=A.&rft.au=Kugler%2C+T.&rft.au=Dugar%2C+S.&rft.au=Gisches%2C+E.+J.&rft_id=https%3A%2F%2Fwww.parisschoolofeconomics.eu%2FIMG%2Fpdf%2FChoices_of_routes.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABraess%27s+paradox" class="Z3988"></span></li> <li>T. Roughgarden. "The Price of Anarchy." MIT Press, Cambridge, MA, 2005.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Braess%27s_paradox&action=edit&section=19" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output 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Paradoxes (December 2005)</a> <a rel="nofollow" class="external text" href="http://download.microsoft.com/download/3/a/7/3a7fa450-1f33-41f7-9e6d-3aa95b5a6aea/MSDNMagazineDecember2005en-us.chm">Direct Link</a></li> <li><a rel="nofollow" class="external text" href="http://homepage.ruhr-uni-bochum.de/Dietrich.Braess/#paradox">Dietrich Braess's homepage</a></li> <li><a rel="nofollow" class="external text" href="http://www.davros.org/science/roadparadox.html">The Road Network Paradox</a></li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=Cg73j3QYRJc">The Spring Paradox</a>, <a href="/wiki/YouTube" title="YouTube">YouTube</a> video by <a href="/wiki/Steve_Mould" title="Steve Mould">Steve Mould</a> explaining Braess's Paradox as well as a closely related <a href="/wiki/Spring_(device)" title="Spring (device)">spring</a> paradox</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist 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abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Paradoxes" title="Template:Paradoxes"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Paradoxes" title="Template talk:Paradoxes"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Paradoxes" title="Special:EditPage/Template:Paradoxes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Notable_paradoxes" style="font-size:114%;margin:0 4em">Notable <a href="/wiki/Paradox" title="Paradox">paradoxes</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Philosophical</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Paradox_of_analysis" title="Paradox of analysis">Analysis</a></li> <li><a href="/wiki/Buridan%27s_bridge" title="Buridan's bridge">Buridan's bridge</a></li> <li><a href="/wiki/Dream_argument" title="Dream argument">Dream argument</a></li> <li><a href="/wiki/Epicurean_paradox" title="Epicurean paradox">Epicurean</a></li> <li><a href="/wiki/Paradox_of_fiction" title="Paradox of fiction">Fiction</a></li> <li><a href="/wiki/Fitch%27s_paradox_of_knowability" title="Fitch's paradox of knowability">Fitch's knowability</a></li> <li><a href="/wiki/Argument_from_free_will" title="Argument from free will">Free will</a></li> <li><a href="/wiki/New_riddle_of_induction" title="New riddle of induction">Goodman's</a></li> <li><a href="/wiki/Paradox_of_hedonism" title="Paradox of hedonism">Hedonism</a></li> <li><a href="/wiki/Liberal_paradox" title="Liberal paradox">Liberal</a></li> <li><a href="/wiki/Meno" title="Meno">Meno's</a></li> <li><a href="/wiki/Mere_addition_paradox" title="Mere addition paradox">Mere addition</a></li> <li><a href="/wiki/Moore%27s_paradox" title="Moore's paradox">Moore's</a></li> <li><a href="/wiki/Newcomb%27s_paradox" title="Newcomb's paradox">Newcomb's</a></li> <li><a href="/wiki/Paradox_of_nihilism" title="Paradox of nihilism">Nihilism</a></li> <li><a href="/wiki/Omnipotence_paradox" title="Omnipotence paradox">Omnipotence</a></li> <li><a href="/wiki/Preface_paradox" title="Preface paradox">Preface</a></li> <li><a href="/wiki/Wittgenstein_on_Rules_and_Private_Language" title="Wittgenstein on Rules and Private Language">Rule-following</a></li> <li><a href="/wiki/Sorites_paradox" title="Sorites paradox">Sorites</a></li> <li><a href="/wiki/Ship_of_Theseus" title="Ship of Theseus">Theseus' ship</a></li> <li><a href="/wiki/White_Horse_Dialogue" title="White Horse Dialogue">White horse</a></li> <li><a href="/wiki/Zeno%27s_paradoxes" title="Zeno's paradoxes">Zeno's</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Logical</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Barber_paradox" title="Barber paradox">Barber</a></li> <li><a href="/wiki/Berry_paradox" title="Berry paradox">Berry</a></li> <li><a href="/wiki/Bhartrhari%27s_paradox" class="mw-redirect" title="Bhartrhari's paradox">Bhartrhari's</a></li> <li><a href="/wiki/Burali-Forti_paradox" title="Burali-Forti paradox">Burali-Forti</a></li> <li><a href="/wiki/Paradox_of_the_Court" title="Paradox of the Court">Court</a></li> <li><a href="/wiki/Crocodile_dilemma" title="Crocodile dilemma">Crocodile</a></li> <li><a href="/wiki/Curry%27s_paradox" title="Curry's paradox">Curry's</a></li> <li><a href="/wiki/Epimenides_paradox" title="Epimenides paradox">Epimenides</a></li> <li><a href="/wiki/Free_choice_inference" title="Free choice inference">Free choice paradox</a></li> <li><a href="/wiki/Grelling%E2%80%93Nelson_paradox" title="Grelling–Nelson paradox">Grelling–Nelson</a></li> <li><a href="/wiki/Kleene%E2%80%93Rosser_paradox" title="Kleene–Rosser paradox">Kleene–Rosser</a></li> <li><a href="/wiki/Liar_paradox" title="Liar paradox">Liar</a> <ul><li><a href="/wiki/Card_paradox" title="Card paradox">Card</a></li> <li><a href="/wiki/No%E2%80%93no_paradox" title="No–no paradox">No-no</a></li> <li><a href="/wiki/Pinocchio_paradox" title="Pinocchio paradox">Pinocchio</a></li> <li><a href="/wiki/Quine%27s_paradox" title="Quine's paradox">Quine's</a></li> <li><a href="/wiki/Yablo%27s_paradox" class="mw-redirect" title="Yablo's paradox">Yablo's</a></li></ul></li> <li><a href="/wiki/Opposite_Day" title="Opposite Day">Opposite Day</a></li> <li><a href="/wiki/Paradoxes_of_set_theory" title="Paradoxes of set theory">Paradoxes of set theory</a></li> <li><a href="/wiki/Richard%27s_paradox" title="Richard's paradox">Richard's</a></li> <li><a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's</a></li> <li><a href="/wiki/I_know_that_I_know_nothing" title="I know that I know nothing">Socratic</a></li> <li><a href="/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel" title="Hilbert's paradox of the Grand Hotel">Hilbert's Hotel</a></li> <li><a href="/wiki/Temperature_paradox" title="Temperature paradox">Temperature paradox</a></li> <li><a href="/wiki/Barbershop_paradox" title="Barbershop paradox">Barbershop</a></li> <li><a href="/wiki/Catch-22_(logic)" title="Catch-22 (logic)">Catch-22</a></li> <li><a href="/wiki/Chicken_or_the_egg" title="Chicken or the egg">Chicken or the egg</a></li> <li><a href="/wiki/Drinker_paradox" title="Drinker paradox">Drinker</a></li> <li><a href="/wiki/Paradoxes_of_material_implication" title="Paradoxes of material implication">Entailment</a></li> <li><a href="/wiki/Lottery_paradox" title="Lottery paradox">Lottery</a></li> <li><a href="/wiki/Plato%27s_beard" title="Plato's beard">Plato's beard</a></li> <li><a href="/wiki/Raven_paradox" title="Raven paradox">Raven</a></li> <li><a href="/wiki/Imperative_logic#Ross's_paradox" title="Imperative logic">Ross's</a></li> <li><a href="/wiki/Unexpected_hanging_paradox" title="Unexpected hanging paradox">Unexpected hanging</a></li> <li>"<a href="/wiki/What_the_Tortoise_Said_to_Achilles" title="What the Tortoise Said to Achilles">What the Tortoise Said to Achilles</a>"</li> <li><a href="/wiki/Heat_death_paradox" title="Heat death paradox">Heat death paradox</a></li> <li><a href="/wiki/Olbers%27s_paradox" title="Olbers's paradox">Olbers's paradox</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Economic</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Allais_paradox" title="Allais paradox">Allais</a></li> <li><a href="/wiki/The_Antitrust_Paradox" title="The Antitrust Paradox">Antitrust</a></li> <li><a href="/wiki/Arrow_information_paradox" title="Arrow information paradox">Arrow information</a></li> <li><a href="/wiki/Bertrand_paradox_(economics)" title="Bertrand paradox (economics)">Bertrand</a></li> <li><a class="mw-selflink selflink">Braess's</a></li> <li><a href="/wiki/Paradox_of_competition" title="Paradox of competition">Competition</a></li> <li><a href="/wiki/Income_and_fertility" title="Income and fertility">Income and fertility</a></li> <li><a href="/wiki/Downs%E2%80%93Thomson_paradox" title="Downs–Thomson paradox">Downs–Thomson</a></li> <li><a href="/wiki/Easterlin_paradox" title="Easterlin paradox">Easterlin</a></li> <li><a href="/wiki/Edgeworth_paradox" title="Edgeworth paradox">Edgeworth</a></li> <li><a href="/wiki/Ellsberg_paradox" title="Ellsberg paradox">Ellsberg</a></li> <li><a href="/wiki/European_paradox" title="European paradox">European</a></li> <li><a href="/wiki/Gibson%27s_paradox" title="Gibson's paradox">Gibson's</a></li> <li><a href="/wiki/Giffen_good" title="Giffen good">Giffen good</a></li> <li><a href="/wiki/Icarus_paradox" title="Icarus paradox">Icarus</a></li> <li><a href="/wiki/Jevons_paradox" title="Jevons paradox">Jevons</a></li> <li><a href="/wiki/Leontief_paradox" title="Leontief paradox">Leontief</a></li> <li><a href="/wiki/Lerner_paradox" title="Lerner paradox">Lerner</a></li> <li><a href="/wiki/Lucas_paradox" title="Lucas paradox">Lucas</a></li> <li><a href="/wiki/Mandeville%27s_paradox" title="Mandeville's paradox">Mandeville's</a></li> <li><a href="/wiki/Mayfield%27s_paradox" title="Mayfield's paradox">Mayfield's</a></li> <li><a href="/wiki/Metzler_paradox" title="Metzler paradox">Metzler</a></li> <li><a href="/wiki/Resource_curse" title="Resource curse">Plenty</a></li> <li><a href="/wiki/Productivity_paradox" title="Productivity paradox">Productivity</a></li> <li><a href="/wiki/Paradox_of_prosperity" title="Paradox of prosperity">Prosperity</a></li> <li><a href="/wiki/Scitovsky_paradox" title="Scitovsky paradox">Scitovsky</a></li> <li><a href="/wiki/Service_recovery_paradox" title="Service recovery paradox">Service recovery</a></li> <li><a href="/wiki/St._Petersburg_paradox" title="St. Petersburg paradox">St. Petersburg</a></li> <li><a href="/wiki/Paradox_of_thrift" title="Paradox of thrift">Thrift</a></li> <li><a href="/wiki/Paradox_of_toil" title="Paradox of toil">Toil</a></li> <li><a href="/wiki/Tullock_paradox" class="mw-redirect" title="Tullock paradox">Tullock</a></li> <li><a href="/wiki/Paradox_of_value" title="Paradox of value">Value</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Decision theory</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abilene_paradox" title="Abilene paradox">Abilene</a></li> <li><a href="/wiki/Apportionment_paradox" title="Apportionment paradox">Apportionment</a> <ul><li><a href="/wiki/House_monotonicity" title="House monotonicity">Alabama</a></li> <li><a href="/wiki/Coherence_(fairness)" title="Coherence (fairness)">New states</a></li> <li><a href="/wiki/State-population_monotonicity" class="mw-redirect" title="State-population monotonicity">Population</a></li></ul></li> <li><a href="/wiki/Arrow%27s_impossibility_theorem" title="Arrow's impossibility theorem">Arrow's</a></li> <li><a href="/wiki/Buridan%27s_ass" title="Buridan's ass">Buridan's ass</a></li> <li><a href="/wiki/Chainstore_paradox" title="Chainstore paradox">Chainstore</a></li> <li><a href="/wiki/Condorcet_paradox" title="Condorcet paradox">Condorcet's</a></li> <li><a href="/wiki/Decision-making_paradox" title="Decision-making paradox">Decision-making</a></li> <li><a href="/wiki/Paradox_of_voting" title="Paradox of voting">Downs</a></li> <li><a href="/wiki/Ellsberg_paradox" title="Ellsberg paradox">Ellsberg</a></li> <li><a href="/wiki/Fenno%27s_paradox" title="Fenno's paradox">Fenno's</a></li> <li><a href="/wiki/Fredkin%27s_paradox" title="Fredkin's paradox">Fredkin's</a></li> <li><a href="/wiki/The_Green_Paradox" title="The Green Paradox">Green</a></li> <li><a href="/wiki/Hedgehog%27s_dilemma" title="Hedgehog's dilemma">Hedgehog's</a></li> <li><a href="/wiki/Inventor%27s_paradox" title="Inventor's paradox">Inventor's</a></li> <li><a href="/wiki/Kavka%27s_toxin_puzzle" title="Kavka's toxin puzzle">Kavka's toxin puzzle</a></li> <li><a href="/wiki/Morton%27s_fork" title="Morton's fork">Morton's fork</a></li> <li><a href="/wiki/Navigation_paradox" title="Navigation paradox">Navigation</a></li> <li><a href="/wiki/Newcomb%27s_paradox" title="Newcomb's paradox">Newcomb's</a></li> <li><a href="/wiki/Parrondo%27s_paradox" title="Parrondo's paradox">Parrondo's</a></li> <li><a href="/wiki/Preparedness_paradox" title="Preparedness paradox">Preparedness</a></li> <li><a href="/wiki/Prevention_paradox" title="Prevention paradox">Prevention</a></li> <li><a href="/wiki/Prisoner%27s_dilemma" title="Prisoner's dilemma">Prisoner's dilemma</a></li> <li><a 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