## Friday, October 01, 2004

### Lifting Waits

I work on the fourth floor of my building. Usually I walk up for the exercise, but if I’m feeling lazy I’ll take the annoyingly slow elevator. This morning I was feeling lazy. But one of my fellow riders must have felt even lazier: she exited at the second floor. As I continued my ascent, I recalled that the elevators of the main building at NYU (my grad school) only stopped on alternate floors – one elevator bank for evens, another for odds – and that got me thinking about the economics of floor skipping.

Say you have an 8-story building. And suppose, for simplicity, that the same number of people wish to get off at every floor; we can treat this as one person per floor. In order to maximize the satisfaction of potential passengers, should the elevator stop at every floor? And if not, how far should it rise before stopping?

Set aside, for the moment, unusual strategies like NYU’s; assume our elevator stops at every floor above some threshold. The marginal benefit (MB) of skipping a floor is proportionate to the number of people who would be inconvenienced by waiting. Skipping the 2nd floor benefits all six people destined for floors 3-8. Skipping the 3rd floor benefits the five destined for floors 4-8. Skipping the 4th floor benefits the four destined for floors 5-8. And so on. Thus, the MB of skipping floors is decreasing, but at a constant rate; see the table below. More importantly, the MB for every floor increases as a function of the number of stories in the building. E.g., if we had a 10-story building, then skipping the 2nd floor would benefit eight people instead of six, skipping the 3rd floor would benefit seven people instead of five, etc.

Now consider the marginal cost (MC) of floor-skipping. Assume that each person whose floor is skipped will walk up (this assumption will be relaxed in my next post). And assume there is constant disutility to walking up flights, so that walking up two flights is exactly twice as annoying as walking up one (another assumption to be relaxed in my next post). Then the MC of skipping the 2nd floor is 1 (a single person walks up one flight); the MC of skipping the 3rd floor is 3 (one person walks up one flight, one person walks up two); the MC of skipping the 4th floor is 6 (one person walks up one, one person walks up two, one person walks up three). The MC therefore rises at an increasing rate. And note that, unlike the MB, the MC of skipping a given floor is not a function of the number of stories above. Skipping the 2nd floor causes one person to have to walk up one flight, regardless of how tall the building is.
```
skipped fl.		MB (time)		MC (effort)
2nd			6			1
3rd			5			3
4th			4			6
5th			3			10
6th			2			15
7th			1			21

```
.
Since MB is falling while MC is rising, it’s reasonable to conclude there will be a turning point where we switch from MB > MC (skip the floor) to MB < MC (stop at the floor). However, in order to find the optimal policy, the MB and MC need to be comparable. In the table above, MB is in terms of time saved, MC in terms of climbing effort. Let’s say we think the disutility of waiting for one elevator stop is equal to the disutility of climbing one flight. Then in this case, the optimal policy would skip the 2nd and 3rd floors, and stop on the 4th through 8th.

The key result here is that elevators should skip more floors in taller buildings. More floors means a higher benefit of skipping any given floor, without a corresponding increase in the cost of skipping it. For example, if this were an 11-story building, the MB of skipping any floor would rise by 3, and then it would make sense to skip the 4th floor.

Strangely, most buildings have elevators that stop on all floors. One possible explanation is that some people, notably the elderly and disabled, have a prohibitively high cost of walking up even one floor. To accommodate them, the elevator must stop at any floor (if the button is pushed). This problem could be obviated by a smart elevator capable of identifying passengers with special needs. But where technology fails us, social norms could fill the void. The choice, by an able-bodied person, to exit the elevator on a floor that should be skipped could be punished via public ridicule and/or beatings.

More later.

FB said...

Well, isn't this the logic by which, on parallel routes, bus lines have more stops than metros, and trains have even fewer?

Anonymous said...

While this is true in heavy traffic, when there is light traffic the elevator can afford to stop at every floor, since it makes no one wait. So rather than only stopping at some floors, the elevator should run an alternate "rush-hour" algorithm. This could be triggered by a large number of floor presses, or just time of day.

-Patri Friedman