Abstract
We extend a general network theorem of Calvert and Keady (CK) relating to the minimum number of arcs needed to guarantee the occurrence of the Braess Paradox. We rephrase the CK theorems and express our proof in the terminology of traffic networks. CK described their theorem in relation to a two-terminal network of liquid in pipes. Approximately stated, it is: if every relationship between flow and head difference is not a power law, with the same (power) s on each arc, given at least 6 pipes, one can arrange (lengths of) them so that Braess's paradox occurs, i.e. one can increase the conductivity of an indi- vidual pipe yet require more power to maintain the same consumptions. In relation to the original Braess situation of traf- fic network flows, the relationship is between flow and link-cost on a congested link. Our extended theorem shows that 5 pipes (roads, links, arcs) arranged in a Wheatstone Bridge (WB) network (as in the original Braess network) are necessary and sufficient to produce a Braess paradox (BP) in a two-terminal network (not limited to liquid in pipes) if at least one of the five has a different conductivity law (not power s).
Highlights
Braess [1] described a “paradoxical” traffic network for which an extra road added with the express aim to relieve congestion, instead increased the travel time for all users
Because the Calvert and Keady (CK) theorem applies for any arbitrary arcs 4 and 6, in the final equivalent five arc network, arc 4 is an arbitrary arc with the power law of the combination
If either the original arc 4 or the added sixth arc has a different conductivity function, the series combination does not have a simple power law s. This series connection produces the type of sixarc network which CK have shown can be arranged to have a Braess paradox (BP) Note, that it is equivalent to a five-arc Wheatstone Bridge (WB) network in which one arc, arc 4, does not have the simple power-law s
Summary
Braess [1] described a “paradoxical” traffic network for which an extra road (arc, link) added with the express aim to relieve congestion, instead increased the travel time for all users. CK proved several theorems concerning the occurrence or non-occurrence of the Braess Paradox (hereinafter referred to as BP) Their final result: “Approximately stated, it is: if every relationship between flow and head difference is not a power law, with the same (power) s on each arc, given at least 6 pipes, one can arrange (lengths of) them so that Braess's paradsox occurs, i.e. one can increase the conductivity of an individual pipe yet require more power to maintain the same consumptions2.”. Stated, in the language of the CK abstract, (but in traffic terms): If every relationship between flow U and cost C is not a power law, with the same power s on each arc, given at least 6 arcs, one can arrange lengths of them so that Braess's paradox occurs.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
