A driver can choose a route that shortens their own journey while making the combined journeys of everyone using the roads longer. The important detail is not that someone has chosen badly. It is that the decision changes another person's travel time, and that extra delay does not appear in the chooser's comparison of their own alternatives. A small, fully specified example makes the missing minutes visible.
On January 13, 2025, Reuters reported transit officials' preliminary account of lower traffic following the January 5 congestion-pricing launch in New York, United States. That reported observation is not a causal estimate. The fictional network below does not represent Manhattan.
Three journeys on two invented routes
Imagine three anonymous vehicles travelling between the same origin and destination. Every vehicle must make one trip, and each can use either of two parallel routes. The number of trips is fixed. Nobody cancels, changes destination, shares a vehicle or leaves at a different time. We compare complete route assignments for this one set of journeys, not a sequence of vehicles diverting halfway through a real trip.
Call the routes A and B. On A, each vehicle's journey takes ten minutes multiplied by the number of vehicles using A. One vehicle takes ten minutes; two vehicles each take twenty; three vehicles each take thirty. On B, each vehicle takes twenty-five minutes, whether one, two or three use it. These time functions are invented assumptions, not measurements or engineering descriptions of particular roads.
Each driver knows those functions and seeks the shortest time for their own trip. When considering a switch, the driver includes their own vehicle in the destination route's new load. A route carrying one vehicle may currently take ten minutes, but a second vehicle joining it would experience twenty. Comparing the old ten with twenty-five would therefore answer the wrong question.
The combined measure is vehicle-minutes: add the duration of all three journeys. This gives each vehicle's minute the same weight. It is not elapsed time on one clock, a money value or a count of passengers. No route toll, fuel expense, delivery penalty or different value of time enters the calculation. Keeping the unit narrow lets us identify exactly which duration changes when a driver chooses another route.
List every possible load before choosing a favourite
There are four possible counts of vehicles on A: zero, one, two or three. Since there are always three vehicles, each count also determines the number on B. The drivers have identities, but the stipulated travel times depend only on the count using a route. We can therefore calculate total time for all load combinations before deciding which named driver occupies any position.
- No vehicle on A and three on B: three journeys of twenty-five minutes give seventy-five vehicle-minutes.
- One vehicle on A and two on B: ten plus twenty-five plus twenty-five gives sixty vehicle-minutes.
- Two vehicles on A and one on B: twenty plus twenty plus twenty-five gives sixty-five vehicle-minutes.
- Three vehicles on A and none on B: three journeys of thirty minutes give ninety vehicle-minutes.
The smallest total is sixty, obtained with one vehicle on A and two on B. That conclusion follows by inspecting every possible count, not by assuming that everyone should use the route that is fastest when empty. Sending all three to A produces the largest total here, even though A offers the fastest possible individual journey when used alone.
A's own combined travel time grows from ten to forty to ninety as its load increases from one to two to three. The second vehicle therefore adds thirty vehicle-minutes on A, although its own journey lasts twenty: the other ten belong to the existing user. Removing that vehicle's twenty-five-minute journey from B leaves the five-minute increase across both routes. Counting only the new A journey would miss part of the change.
This enumeration also distinguishes an outcome from its description. Saying that A takes twenty minutes is incomplete unless the load of two vehicles is preserved. Moving a vehicle to or from A changes the premise behind that figure. B's twenty-five minutes, by contrast, remain unchanged across the permitted loads because that is expressly stipulated. Treating both times as fixed labels would erase the mechanism being examined.
Follow the five minutes saved and the ten minutes imposed
Start with the minimum-total arrangement. Give the drivers neutral names: First uses A, while Second and Third use B. Their durations are ten, twenty-five and twenty-five minutes respectively. Now consider Second choosing A instead, while the other two keep their choices. A will then carry First and Second, so each of them takes twenty minutes. Third remains on B and still takes twenty-five.
For Second, this is an improvement of five minutes: twenty instead of twenty-five. Second has correctly anticipated the extra load their own vehicle creates. There is no mistaken navigation estimate and no assumption that the road remains as fast as when First was alone. Even with an accurate forecast of their own new duration, Second prefers the move.
First experiences the other side of the decision. Their trip lengthens from ten to twenty minutes without any change in their own route choice. The imposed delay is ten minutes. Third experiences no change. Adding the individual differences gives a five-minute saving, a ten-minute loss and zero: total time rises by five, from sixty to sixty-five vehicle-minutes.
The ten minutes imposed on First are the external effect in this example. They are not the whole twenty-minute duration of Second's new trip. Nor are they the five-minute net increase in the total. Those three quantities answer different questions: how long the chooser travels, how much extra time another traveller experiences, and what happens after all changes are added together.
This distinction matters because an accurate personal comparison can still omit a real consequence. Second compares their own twenty with their own twenty-five. Nothing in that comparison contains First's additional ten. Calling the move individually advantageous and collectively time-increasing is therefore consistent. Both statements use the same complete set of stipulated times; they simply count different people's changes.
Why nobody leaves the higher-total arrangement alone
At two vehicles on A and one on B, test a change by each kind of user. An A user currently takes twenty minutes. Moving alone to B would make that user's journey twenty-five minutes, so the change is unattractive. The remaining A user would benefit from the reduced load, but the potential mover is comparing their own journey.
The B user currently takes twenty-five minutes. Moving alone to A would raise A's load from two to three, making the mover's journey thirty minutes. This change is also unattractive. Thus no individual can shorten their trip by changing routes while everybody else keeps their choice. The arrangement is stable against an individual deviation under the model's objective.
The other counts fail that test. With nobody on A, a B user can obtain ten instead of twenty-five by moving. With one on A, a B user can obtain twenty instead of twenty-five. With all three on A, one user can obtain twenty-five instead of thirty by moving to B. Only the count of two on A passes every individual comparison.
There is one stable count, not one unique assignment of named people. Any of the three drivers can be the person on B, giving three identity-specific arrangements with the same times. Likewise, the minimum total can be achieved with any one driver alone on A. The model does not select which person receives which position or predict how an initial arrangement develops over time.
Swapping positions preserves the load
A simultaneous exchange of positions is different from a unilateral move. If one A user and the B user swap routes, the load remains two on A and one on B. The former B user saves five minutes and the former A user loses five; total time stays sixty-five. Such a swap changes who experiences the longer trip without removing the external-delay gap. Reaching sixty requires changing the load count, not simply rotating the occupants of the existing positions.
The lower total is not a gain for every driver
Return to the stable arrangement with First and Second on A, Third on B. To reach the minimum total while keeping First on A, move Second back to B. First's duration falls from twenty to ten. Second's rises from twenty to twenty-five. Third remains at twenty-five. The ten-minute improvement for First exceeds the five-minute deterioration for Second, producing the five-minute total reduction.
That is not an improvement for everyone. The person asked to use B loses five minutes relative to their stable position. A lower sum does not remove that disadvantage, and an analyst should not describe it away as a misunderstanding. It is precisely why the minimum-total arrangement is not sustained by each person independently choosing their own shortest trip.
The example also shows why the identity of the affected person should remain attached to the calculation. Two lists of sorted durations reveal the totals, but not who moves from twenty to twenty-five. A decision about an assignment concerns people occupying positions, not only the positions themselves. The same numerical optimum admits several assignments whose individual beneficiaries differ.
No compensation, agreement or enforcement arrangement has been specified. The arithmetic therefore does not establish how a lower-total pattern could be implemented or whether a particular implementation would be acceptable. It identifies the conflict that such a discussion would have to address: somebody's privately attractive move adds more time elsewhere than it saves for that person.
Two different thresholds explain the gap
The effect is not tied to the single value twenty-five. Keep A's time function unchanged and temporarily call B's constant duration b. Consider values strictly between twenty and thirty minutes. This is a comparison within the invented model, not an estimate of a range for a real road. The purpose is to see which inequalities produce the difference.
When one vehicle already uses A, a B user who joins it will take twenty minutes. The move is individually attractive whenever b is greater than twenty. But the total with one on A is ten plus twice b; with two on A it is forty plus b. The second total exceeds the first whenever b is less than thirty.
Between twenty and thirty, both statements hold. The driver saves b minus twenty minutes, while the existing A user loses ten. The saving is positive but smaller than ten. Subtracting the saving from the imposed delay leaves thirty minus b additional vehicle-minutes overall. Substituting twenty-five recovers the five-minute total increase already traced person by person.
The stability and minimum comparisons also persist throughout this open interval. Two A users prefer their twenty to b, and the B user prefers b to thirty. One on A has a lower total than none because b exceeds ten, and a lower total than three because b is below forty. Its total is also below the two-on-A total because b is below thirty. At the excluded endpoints, some comparisons become ties and require separate wording.
An observed journey does not reveal the imposed delay
Suppose a record shows that Second travelled on A in twenty minutes. That is enough to describe the journey experienced under the actual assignment in the model. It is not enough, on its own, to establish how much delay Second imposed on First. To calculate that effect, we also need First's duration under the alternative assignment without Second on A.
Here the alternative is known because we stipulated the entire function: ten minutes with one vehicle and twenty with two. In an observed setting, two journeys on different days would not automatically supply the same comparison. Other traffic, departure times and conditions could differ. The invented calculation avoids those issues by defining them away; it does not provide an empirical method that has already resolved them.
A route estimate can therefore be accurate for its intended purpose and still be insufficient for this wider question. Predicting the chooser's own duration is one task. Describing how the choice changes other users' durations is another. The second requires a relationship between load and time, together with a clear alternative assignment, rather than simply a more detailed record of the chooser's completed trip.
Keep the business question as precise as the arithmetic
For an organisation considering several journeys together, the example distinguishes three questions: which route is shortest for this vehicle given the others' choices, which complete assignment has the lowest total time, and which people gain or lose when the assignment changes. Our answers differ because the route's duration responds to its users. No faster vehicle or shorter underlying task is required for that difference.
Those questions do not settle a wider transport policy. The model has no passengers, changing trip demand, delivery priorities, costs of coordination or alternative departure times. Adding any of those features would change what should be counted or which decisions are available. Nor does the model supply a tariff or prove the effects of a real charging programme. Its finding concerns one clearly defined interaction.
The decisive calculation remains small: one driver saves five minutes, another loses ten, and the third is unaffected. The chooser's benefit is genuine, yet it is not the complete change. Keeping the imposed delay visible explains why a personally quicker route can increase combined travel time without requiring anybody to be confused about their own journey.