Tensions in Practice: A shortcut can slow flow and preserve a backup¶
A four-node route network
Imagine one unit of flow from S to T through intermediate nodes U and V. S→U and V→T each cost their own edge load x; U→T and S→V each cost 1. An optional U→V connector costs 0. Travelers are infinitesimally small and independently choose cheapest routes, with full information. Compare closing the connector with leaving it open. Then inspect a separate failure scenario in which both constant-cost edges disappear. The connector changes both the ordinary routing equilibrium and which paths survive that failure.
Reduce ordinary routing cost
Remove the option that draws self-routed flow onto both load-sensitive edges.
Retain the failure path
Keep the connector so a route remains if both constant-cost edges fail.
Why these aims pull against each other
The connector is harmful to the modeled ordinary equilibrium but useful in the specified failure state. An equilibrium comparison alone cannot price that reserve.
Choose an arrangement to see what changes and what remains difficult.
Compare the same rows across alternatives. Cells state explicit toy quantities, membership or permissions; colors do not supply additional meaning.
What this choice protects
What it costs
When it fits
Compare the arrangements
Close the connector
Normal flow splits equally between S–U–T and S–V–T. Each route costs 0.5 + 1 = 1.5. Removing both constant-cost edges leaves no complete path.
| Normal flow | Route cost | After failures | |
|---|---|---|---|
| S–U–T | 0.5 | 1.5 | Unavailable |
| S–V–T | 0.5 | 1.5 | Unavailable |
| S–U–V–T | Not a route | Not a route | Unavailable |
- What it protects
- All normal flow experiences cost 1.5 rather than 2 under the stipulated self-routing model.
- What it costs
- The specified simultaneous failure disconnects S from T.
- When it fits
- Plausible if lower ordinary cost warrants accepting this particular loss of backup reachability or another failure plan exists.
Illustration note: This is an editorial, deliberately bounded illustration. Its stated rules and any numbers are invented, not observations, recommended settings, or predictions.
Keep the connector
Normal flow uses S–U–V–T, putting load 1 on both load-sensitive edges and giving cost 2. Either unused side route also costs 2, so no individual has a cheaper alternative. After the two constant-cost edges fail, the connector route still exists.
| Normal flow | Route cost | After failures | |
|---|---|---|---|
| S–U–T | 0 | 2 | Unavailable |
| S–V–T | 0 | 2 | Unavailable |
| S–U–V–T | 1 | 2 | Available |
- What it protects
- A path survives the explicitly named simultaneous failure.
- What it costs
- Normal equilibrium cost rises from 1.5 to 2; keeping an extra route does not force self-routing to use it well.
- When it fits
- Plausible if this failure path is sufficiently valuable to justify the normal-time penalty under the stated routing rule.
Illustration note: This is an editorial, deliberately bounded illustration. Its stated rules and any numbers are invented, not observations, recommended settings, or predictions.
What this illustration does—and does not—establish
The source establishes the structural tension; the concrete alternatives and their conditional costs are editorial synthesis. No arrangement is a universal recommendation.
- This is an invented nonatomic equilibrium model. Costs are arbitrary units; no travel-time observations, failure probabilities or policy recommendation are supplied.
- Normal route costs are evaluated at each arrangement’s displayed normal flow. The failure column shows reachability only, not an additional equilibrium calculation.
- A coordinated dispatcher could leave the open connector unused and retain the cheaper split. Convergence, implementation cost and mixed routing behavior are outside this comparison.
Source entries
Braess's Paradox
Braess paradox Addition versus removal symmetry (sign/direction) supplies the local tension. The setting, alternative arrangements, and stipulated consequences are editorial applications.
Addition versus removal symmetry (sign/direction)
Diagnostic: before cutting, ask what the element does off-equilibrium under failure or surge; an edge useless at equilibrium may be essential under disturbance.
Self-routing is a load-bearing assumption
The paradox lives entirely in the gap between the decentralized equilibrium and the achievable optimum; it presupposes routing is selfish.