Operations Research & Transportation Networks
Braess’s Paradox & Wardrop Equilibrium Latency Formula
Calculate pre-bridge vs post-bridge Wardrop Nash travel time on a 4-link diamond network and prove when adding a zero-latency shortcut increases EVERY commuter’s travel time!
Prove Dietrich Braess’s (1968) paradox: adding capacity can slow EVERY user down!
In Dietrich Braess’s (1968) diamond network, total commuter flow F_tot travels from Start to End via Top (bottleneck a_1 * x + highway b_0) or Bottom (highway b_0 + bottleneck a_1 * x). Without a cross-bridge, traffic splits 50/50 for Wardrop user-equilibrium latency T_before = b_0 + 0.5 * a_1 * F_tot (matching the Social Optimum!). Now open an instant zero-latency cross-bridge connecting the first bottleneck directly to the second: selfish drivers abandon the highway whenever zig-zag latency 2 * a_1 * x_zig <= b_0, yielding post-bridge Wardrop latency T_after = MIN(2 * b_0, 2 * a_1 * F_tot) for a_1 * F_tot < 2 * b_0 (and b_0 + 0.5 * a_1 * F_tot above 2 * b_0) — making EVERY driver slower across the entire Braess window (2/3) * (b_0 / a_1) < F_tot < 2 * (b_0 / a_1)!
Test Braess’s exact classic numbers: F_tot = 4,000 cars (B2), bottleneck congestion slope a_1 = 0.01 min/car (B3, i.e., x / 100), and highway travel time b_0 = 45 min (B4): BEFORE the bridge, 2,000 cars take each route in 45 + 20 = 65.00 min. AFTER opening the free bridge, zig-zag becomes a dominant strategy for all 4,000 cars, pushing EVERY driver’s commute up to 40 + 40 = 80.00 min (+15.00 min Braess Paradox penalty!)! Test F_tot = 2,000 (light traffic) to verify the bridge genuinely saves -15.00 min (40 vs 55 min).
| Demand Flow F_tot (a_1=0.01, b_0=45) | Before Bridge T_before | With Bridge T_after & Braess Change |
|---|---|---|
| Light Demand (F = 2,000) | 55.00 min | 40.00 min (-15.00 min Faster) |
| Classic Braess Regime (F = 4,000) | 65.00 min | 80.00 min (+15.00 min SLOWER!) |
| 3-Path Mixed Regime (F = 6,000) | 75.00 min | 90.00 min (+15.00 min SLOWER!) |
| Heavy Saturation (F = 10,000) | 95.00 min | 95.00 min (0.00, Bridge Unused) |
Post-Bridge Wardrop Nash Equilibrium Travel Time T_after Formula
Piecewise-exact Wardrop latency across all three regimes: pure zig-zag (a_1*F <= b_0), 3-route indifference (b_0 < a_1*F < 2*b_0 -> T_after = 2*b_0), and unused bridge (a_1*F >= 2*b_0).
=LET(f_fl, B2, a_bt, B3, b_hw, B4, IF(OR(f_fl <= 0, a_bt <= 0, b_hw <= 0), "Invalid Network Input", LET(af_v, a_bt * f_fl, t_af, IF(af_v <= b_hw, 2 * af_v, IF(af_v < 2 * b_hw, 2 * b_hw, b_hw + 0.5 * af_v)), ROUND(t_af, 4))))Braess Paradox Delay Penalty (T_after - T_before) Formula
Positive values flag Braess’s Paradox — where closing the bridge (or charging a Pigouvian congestion toll a_1 * x on bottlenecks) strictly reduces every commuter’s travel time!
=IF(ISTEXT(E3), E3, ROUND(E3 - E2, 4))Try the example
Use the original values or change them to test your understanding.