Talk:Braess's paradox

message: The slowing of the speed of money due to hoarding by those with much more than they spend, is less than optimal, and may be comparatively detrimental, when compared with the value of the residue when a portion of the hoarded value is pumped into circulation through those who are more likely to spend it immediately (increase consumption), and to increase productivity (employment). <BR>
Can this financial Braess' paradox be proven as an apparent choice against best interest?  By analogy, the traffic flow slows when the 'short-cut' is introduced because the immediate (selfish routing) response of some drivers reduces their travel time (until it is used by a majority), at which time it is easy for them to see that their travel time has actually increased, and many return to their previous route (ignoring the short-cut)(stop hoarding). The traffic analogy seems to break-down because it is easy for drivers to compare their daily travel time by route, as it changes with the migration of other drivers to the new route, and then back, as the new route fails.<BR>
It is not so easy for "the 1%", or mega-corporations(stakeholders), to compare the alternative to hoarding because 'speeding up money flow' is a slowly responding marginal benefit.<BR>
--Wikidity (talk) 04:39, 23 June 2015 (UTC)
thread_title:  Financial Example 

message: Isn't it Braess'? Not Braess's... Apostrophe-'S' is used when the owner of the object doesn't not end with 'S', just an apostrophe is used for owners ending with an 'S'.
For instance, George Lucas' Star Wars, not George Lucas's Star Wars.  —Preceding unsigned comment added by 24.219.164.86 (talk) 15:44, 27 September 2010 (UTC)
thread_title:  Possessive Apostrophe 

message: I think the following information should be added:
- If latency function are linear then adding an edge can never make total travel time at equilibrium worse than by a factor of 4/3
- At worst traffic in equilibrium is twice as bad as socially optimal
--Grondax (talk) 15:33 11 Novermber 2008 (EST)
: It looks like someone added the thing about traffic at equilibrium, but the proof given only implies that traffic at the energy minimizing equilibrium is at most twice as bad. Is it true in general (or is the energy-minimizing equilibrium the only one in general)? If so, please add that if you have a proof or citation. Also, please add the thing about adding an edge never making an equilibrium more than 1/3 worse if you have a citation or proof. --skeptical scientist (talk) 04:50, 14 November 2015 (UTC)
thread_title:  Add: How far from optimal is traffic at equilibrium 

message: This example is flawed..
Suppose I'm a 'rational' traveler, and the third route gets added.  I know that everyone is taking the new route, thinking it will be faster, so I take Start->B->End instead.  Start->B takes 45 minutes.  B->End takes 0 minutes, because everyone else is trying the new 'faster' route.  I get home in 45 minutes.  Everyone else eventually realizes that this new route isn't any faster, and everyone ignores it, and eventually I suppose it gets closed to reduce maintenance costs.
And this example is flawed, and I dare say the entire theory, because people will COMMUNICATE with one another.  —Preceding unsigned comment added by 71.193.207.197 (talk) 00:41, 3 May 2008 (UTC)
replies: text: I agree the example is flawed, but I don't think its necessary for people to communicate with each other to find the optimal solution. If x people take Start->A->End, y people take Start->A->B->End, and z people take Start->B->End, then (x, y, z) = (1301, 896, 1803) yields an optimal solution of 63 minutes on average and 71 minutes in the worst route (the z route). Given enough time and sensitivity, isn't it likely that uninformed drivers would naturally arrive at this equilibrium? --Beefyt (talk) 17:52, 4 September 2008 (UTC)
replies: text: Only if the z drivers were unaware that they could get from Start to B in 21.97 minutes by taking the route via A instead of going directly to B, which takes 45 minutes. Similarly the direct route from A to End takes 45 minutes but going first to B makes it 26.99 minutes. Once they realize this the drivers will start switching their routes in favour of the shorter ones, which will in turn become longer due to the increased traffic. We eventually end up in the situation described in the article. In other words, your equilibrium is unstable, given the assumption that drivers can choose their route freely and that they are minimizing their own time spent on the way. I can't see any error in the example. 81.83.2.11 (talk) 21:42, 30 November 2008 (UTC)

text: I don't think that the example is flawed. The original message here has flawed reasoning: if everyone is taking the new route (Start-A-B-End) then the proposed route (Start-B-End) will take 45 minutes for Start->B plus 4000/100=40 minutes for B->End, as everyone taking the new route must also travel on the B->End segment. So the travel time will be 85 minutes for this route, not 45 minutes. The rational traveller will take the new route with everyone else. Perhaps the explanation in the article isn't/wasn't clear on this point. This is clearly not the optimal solution in terms of average travel time, but is the result of each driver individually choosing the fastest route. 121.98.212.178 (talk) 06:50, 25 June 2009 (UTC)

text: Exactly, that's the right answer! The RedBurn (ϕ) 14:53, 11 October 2017 (UTC)
replies: text: No, the conditions are stated for 4000 travellers leaving START. The rates of travel on each section apply where number of travellers T is up to and including 4000. Each section may have greater capacity, at which the rates of travel (45 mins and T/100 mins and 0 mins) may stay the same, but that's not the problem here. The problem here is to deal with the 4000 known to be leaving START. Akld guy (talk) 19:32, 11 October 2017 (UTC)
thread_title:  The example is flawed. 

message: Shouldn't that be "Braess' paradox"? Paradoctor (talk) 08:35, 6 March 2010 (UTC)
thread_title:  Article title 

message: The term x_e has not been previously defined in the section. If it refers to total number of people on edge e then it should be e_x . 
Secondly, the variable x refers to the number of people:
 "Let L_e(x) be the formula for the cost of x people" 
as well as the route: "x_0, x_1, \ldots, x_n".  — Preceding unsigned comment added by 155.69.2.12 (talk) 09:19, 16 February 2014 (UTC)
replies: text: I noticed that this proof was using both x_e and e_x inconsistently, and fixed it. I prefer x_e, since it's a quantity x (consistent with prior uses of x as a number of travelers) dependent on an edge e. This is more naturally denoted x_e than e_x. I also changed the routes to e_i (before) and e'_j (after) so that e with subscript now consistently means an edge and x with subscript now consistently means a number of travelers. The disadvantage is there are now double-subscripts in a few places (e.g. x_{e_i}), but I think the result is easier to follow overall. I also did a few other proof-rewrites to make the proofs easier to follow and more precise. (By the way, the edits to this page from  207.198.105.22 were mine.) --skeptical scientist (talk) 04:40, 14 November 2015 (UTC)
thread_title:  Wrong notation  in "Existence of an equilibrium" 

message: Among the so-called "Possible instances of the paradox in action", only the first few traffic examples are actual examples of Braess's paradox. People moving to public transport or staying at home is not an example.
Also, for the other fields, few, if any, of the examples are analogous. Some of them are related to the prisoner's dilemma and Nash equilibria, though.
The article could definitely benefit from having a clearer definition in the introduction, accompanied by an illustration of a minimal example.2001:700:1200:5118:95BA:2753:A33A:2A52 (talk) 14:05, 1 November 2018 (UTC)
thread_title:  Inexact analogies 

message: Is there an error on the first citation in the bibliography?  It says (1969) but when I follow the link the publication says 1968 on it.
Edit: actually even the text for [1] says that it was observed in 1920, but the reference provided points to a chapter in a book from 2002 or so.  — Preceding unsigned comment added by 131.155.69.19 (talk) 19:03, 22 November 2023 (UTC)
thread_title:  Citation Error? 

message: I came across a discussion of Braess's paradox at: https://www.quora.com/What-are-examples-of-Braess-Paradox-in-real-life/answer/Sridhar-Mahadevan-6?ch=99&share=68016dc7&srid=u30T This may provide additional examples or references that can be used to improve this article. (I hope I am not being redundant, again!) Thanks! --Lbeaumont (talk) 15:29, 26 December 2020 (UTC)
thread_title:  Possible source of additional references 

message: The article states that the situation in which the paradox appears is a multiplayer prisoner dilemma. However prisoner dilemmas are always multiplayer, since they involve at least two players. The redaction should therefore be changed to something like "a prisoner's dilemma with more than 2 players". Carllacan (talk) 09:15, 9 July 2022 (UTC)
thread_title:  Prisioner dilemma games are always multiplayer. 

message: Now that a centralized (well, distributed within a central company) algorithm is suggesting optimal routes to a significant fraction of drivers, have the operators of the Google vehicle navigation service observed Braess' Paradox and attempted to improve their routing algorithms to reduce its impact?  For example, if there are three routes, they could suggest that 70% of the cars go the "fast way" and suggest 10% go another way and 20% a third way.  This seems like an interesting research area, has it been investigated somewhere that Wikipedia could cite? Gnuish (talk) 03:44, 11 October 2022 (UTC)
thread_title:  Application to Google Map navigation? 

message: Why are the travel times of only two legs of the network dependent on the current number of users while the other two are fixed regardless of users? Am I to imagine an expansive, underutilized road of many lanes, with a strictly enforced speed limit? And the other segment is an unpoliced, undersized highway that can provide instantaneous transportation? And both paths from start to end use each type, alternating type at the point where both paths tangentially contact, but only allow for a one-way transfer?
I fail to see how this is a relevant model to any actual city's traffic grid. The parameters of the model appear to be carefully designed to produce an unusual result. After giving it thought myself, I challenge someone to sketch a pipe network or electronic circuit with equivalent behavior. For the arguably cherry-picked/myopic phenomenon of traffic improving upon road closure (as referenced in the article), I would expect the chilling effect to be the primary cause. 192.33.240.46 (talk) 16:38, 2 August 2023 (UTC)
thread_title:  Assumptions of the traffic example