Dwight School London · Natural & Unnatural Sciences · Extension
The Braess Paradox Adding a road can make everyone slower
You have seen how parallel pipes add their conductances, always producing a lower combined resistance. But what if each pipe's resistance depends on how many people use it? Then adding a new path can make the whole system worse — for everyone simultaneously. This is the Braess paradox, and it lives at the intersection of conductance, game theory, and network design.
Part 1 — The Setup
A city, two routes, traffic-dependent travel times
All vehicles want to travel from city A to city B. Let x be the fraction of all vehicles using a particular road (so x = 1 means everyone uses it; x = ½ means half do).
There are two possible routes through intermediate cities L and R:
Variant 1 — Two routes, no shortcut
Route 1: A → L → B time = 10 + 6x₁ minutesRoute 2: A → R → B time = 6x₂ + 10 minutes
Each route has one fixed segment (motorway, always 10 min) and one congested
segment (city road, time = 6x where x is the fraction of all traffic using it).
Variant 1 — network diagram
Route 1: A → L → B
At Nash equilibrium, traffic splits evenly: x₁ = x₂ = ½
t = 10 + 6×½ = 13 min
Route 2: A → R → B
By symmetry, same time — no driver can improve by switching
t = 6×½ + 10 = 13 min
Nash equilibrium. When x₁ = x₂ = ½, both routes take 13 minutes. No individual driver can reduce their travel time by switching route — if they moved to Route 1, Route 1 would get slightly more crowded, and Route 2 slightly less, until they balanced again. The system is stable at 13 minutes.
Part 2 — The Shortcut
A new road is built between R and L — and things get worse
Engineers build a new expressway connecting R directly to L. It is very fast — always just 2 minutes, regardless of traffic. A new route becomes available:
New route — the tempting shortcut
Route 3: A → R → L → B
Time = (A→R congested) + 2 min (R→L fixed) + (L→B congested)
= 6x_R + 2 + 6x_L minutes
Variant 2 — new R→L shortcut added
Why Route 3 looks tempting at first
When the shortcut is first opened and almost nobody uses it yet, Route 3 seems very attractive. Suppose half the traffic still uses Route 1 and half Route 2, so x_R ≈ ½ and x_L ≈ ½ at first:
Initial apparent time on Route 3
t₃ = 6×½ + 2 + 6×½ = 3 + 2 + 3 = 8 min
8 minutes vs 13 minutes on the old routes — of course everyone switches!
As more and more drivers switch to Route 3, both congested links (A→R and L→B) fill up. Eventually everyone uses Route 3: x_R = 1 and x_L = 1.
The new equilibrium — everybody on Route 3:
t₃ = 6×1 + 2 + 6×1 = 6 + 2 + 6 = 14 minutes
That is worse than the original 13 minutes. And no single driver can improve
by switching away: if one driver moves back to Route 1 (A→L→B), they face
x_L ≈ 1 on the A→L segment too, giving t₁ = 6×1 + 10 = 16 minutes — even worse.
The system is stuck at 14 minutes.
Before shortcut — Nash equilibrium
Traffic splits ½ / ½ on Routes 1 and 2
13 min for everyone
After shortcut — new Nash equilibrium
Everyone uses Route 3: A→R→L→B
14 min for everyone
The paradox. Adding a new high-capacity road made every single driver one minute slower — not through bad luck, but through the rational self-interest of every individual. The best collective solution is to ignore the new road entirely, but no individual driver has an incentive to do so.
Part 3 — Interactive
Adjust the shortcut time — watch the paradox switch on and off
The paradox only appears because the shortcut is fast enough to tempt everyone onto Route 3. Use the slider to change the shortcut travel time c (the time on the R→L link) and see whether the new Nash equilibrium is better or worse than the original 13 minutes.
Original equilibrium
13 min
t = 10 + α/2
New equilibrium (Route 3)
14 min
t = 2α + c
Verdict
Paradox active
New route used only when 2α + c < 10 + α/2
The threshold. The paradox is active only when the shortcut is so fast that Route 3 (time = 2α + c) beats the original equilibrium (time = 10 + α/2) even when everyone crowds onto it. Rearranging: paradox appears when c < 10 − (3α/2). For α = 6: paradox when c < 1 minute. Try the slider at c = 0 to see it most clearly, and c = 10 to see it disappear.
Part 4 — The Connection
What this means for conductance and parallel networks
Linking back to hydraulic conductance and harmonic means
The pipe and resistor examples on this page assumed fixed conductances — each pipe's resistance is a physical property of its geometry, independent of how much water flows through it. Adding a parallel pipe always increases total conductance and lowers combined resistance.
The Braess paradox shows what happens when conductance is flow-dependent: when a road (or network link) gets slower as more agents use it, adding a new link can — through the self-interested choices of individual agents — route everyone onto the congested links simultaneously, destroying the beneficial sharing that parallel paths normally provide.
System
Conductance type
Adding a parallel path...
Why
Water pipe Electric circuit
Fixed (geometry only, independent of flow)
Always helps — lowers combined resistance
Conductances add; no strategic interaction
Traffic network (selfish routing)
Congestion-dependent (worse as more use it)
Can hurt everyone if it creates a dominant but collectively bad strategy
Individual optimisation ≠ collective optimisation (Nash ≠ social optimum)
The Braess paradox has been observed in real systems: Seoul removed a motorway through the city centre and traffic improved; New York City has identified Braess-like effects in parts of its road network. In telecommunications, adding bandwidth to a congested link can sometimes slow all users if routing protocols shift everyone onto the new link.
The deeper lesson. The harmonic-mean formula for parallel resistors assumes the network is passive — each element's conductance is fixed by physics. The Braess paradox is what happens when the "network" contains agents with preferences. The mathematics of conductance tells us what happens to current or water. Game theory tells us what happens when individual decision-makers route themselves through the network. The two frameworks agree precisely when conductances are fixed and disagree precisely when they are not.