### Escalating Matters

There were a lot of great comments on my recent post about escalators, but none better than Bennett Haselton’s, which is so good I want to highlight here in a separate post.

I’m going to strip his argument down to make it even simpler, but this is all Bennett’s idea:

A New Puzzle: You’re boarding an escalator precisely at noon. You know that on a normal day, if you walk the entire way, the ride takes exactly ten minutes. But you also know that this is not a normal day, because the escalator is scheduled to be stopped for maintenance beginning at 12:05, and will at that point turn into the equivalent of a stairway. You’re planning to take a one-minute rest from walking at some point along your journey. Should you rest before 12:05, when the escalator is moving, or after 12:05, when the escalator is stopped?

Answer One:Of course you should rest while the escalator is moving, because that way, at least you make some progress while you rest.

Answer One, Reworded: Of course you shouldn’t rest while the escalator is stopped, because then you’ll spend an entire minute not getting anywhere.

Here’s the thing about Answer One: It’s completely wrong. It doesn’t make a bit of difference whether you rest from 12:00 to 12:01 or from 12:05 to 12:06 or for any other minute in between. If you don’t believe me, try an example: Suppose the escalator travels, oh, say, 20 yards per minute and your walking speed is 10 yards per minute. Then if you rest from 12:00 to 12:01, with the elevator moving, you’ll have traveled 160 yards by 12:07, and will continue to gain ten yards per minute after that. If instead you rest from 12:05 to 12:06 with the escalator stopped, you’ll have traveled exactly the same 160 yards by 12:07, and will continue to gain exactly the same ten yards per minute after that.

The Old Puzzle: You’re going to travel on a 100 yard staircase followed by a 100 yard escalator. You’re planning to take a one minute rest somewhere along the way. Should you take it on the stairs or on the escalator?

Answer One: You should rest on the escalator, because at least that way you make some progress while you rest. Or to put this another way, you shouldn’t rest on the stairs because then you’ll spend an entire minute not getting anywhere.

This time Answer One gives the right conclusion. But the reasoning can’t be right, because it’s the exact same reasoning that we applied to the New Puzzle, whereupon that reasoning led us totally astray.

Bennett’s lovely example illustrates as starkly as possible why we must reject Answer One even though it sometimes yields the right conclusion. The reason is that it also sometimes leads to the wrong conclusion. I’ve been trying to argue in the abstract that the logic of Answer One makes no sense; Bennett has done us the awesome service of pointing to a concrete example where that logic leads you inarguably astray.

It also illustrates my other main point: The real reason to rest on the escalator in the Old Puzzle is that resting on the escalator buys you more time on the escalator. Bennett has removed that advantage by giving you exactly five minutes on the escalator regardless of where you rest. In other words, when you cook up an example (like Bennett’s) in which resting on the escalator doesn’t buy you more time on the escalator, the argument for resting on the escalator vanishes in a puff of smoke.

This, incidentally, is related to a cryptic comment of my own on that earlier post, where I replied to an inquiry from Bob Murphy about my observation in an old Slate column that the fundamental confusion arises from measuring benefits in distance instead of time. (I claim that this is, in a sense that might not be entirely obvious, an equivalent description of the problem with Answer One.) In the Old Puzzle, you’re on the escalator for a fixed distance; in Bennett’s New Puzzle, you’re on the escalator for a fixed time. That illustrates exactly the distinction I had in mind, and if I find the time, I’ll write out the details sometime soon.

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#### 25 Responses to “Escalating Matters”

1. 1 1 Bennett Haselton

It’s also interesting that there are probably many variations of the scenario which are all logically equivalent, but which *feel* different (and would lead to different percentages of real people choosing to rest on the elevator instead of the stairs, even though that makes no sense in any of these scenarios).

In my original “fixed-time” version, the rule is that the escalator and the stairs are parallel side-by-side, and you can switch between them, but the eye-in-the-sky enforces that you can only be on the escalator for a total of one minute.

I suspect that in this “switch-back-and-forth-with-one-minute-cap” version, the percentage of people who would choose to rest on the escalator rather than the stairs, would be even higher than in the “escalator-becomes-stairs” version. I’m not sure why. (And in both cases it makes no sense.)

2. 2 2 Roger Schlafly

Here is an analogous problem to your new one.

Suppose a bus has an entrance door in the rear, and an exit door in front. You are taking the bus across town. At some point during the trip, you have to walk from the rear of the bus to the front. Should you do your walking while the bus is stopped at a red light, because it is better to be going forward as much as possible?

No, of course not. Nobody on a bus thinks that. Nobody would say that you should rest while the bus is moving, because that way, at least you make some progress while you rest.

But why do people sit down on the bus, when if they miss their bus, they don’t just sit at the bus stop all day?

Then it does make sense to say that the guy sitting on the bus is making progress toward his destination.

3. 3 3 Rob Rawlings

Brilliant! It certainly has convinced me to change camps.

4. 4 4 Neil

Uh, yeah. If you have a limited time on an escalator (which allows you to double your speed) and that time isn’t enough for you to get to the top, you aren’t going to waste any it resting on the escalator if your goal is to get to the top as soon as you can. If you have to rest, step off. But isn’t that obvious?

5. 5 5 Neil

Oh, okay, I see. You are using it to explain why the “you’re getting somewhere” logic is wrong.

6. 6 6 yelo

Resting is in effect making your journey longer. If you walk at 10yards/minute then resting for a minute is the same as adding 10 yards to the leg of the journey where you rest.

If the leg leg of the journey where you rest is bounded by time and not distance then you cannot make up that 10 yards on that leg. If the second leg (as in the new scenario) is total-distance minus distance-covered-on first-leg then any distance not covered in the first leg is simply pushed onto the second leg, so not surprisingly it doesn’t matter which leg you rest on. The same would be true if both the escalator leg and the walking leg were time (and not distance) bound. I guess this is pretty definitive proof that the “better to move while resting” argument is spurious.

I think the legs bounded by pure distance are more interesting. You make the journey longer by resting then it turns out that the non-resting speed you travel at during this added journey time determines how much the rest time affects total journey time.

7. 7 7 Bennett Haselton

Neil, actually that leads to the wrong answer — there is no compelling argument to step *off* the escalator either. It doesn’t matter whether you rest on or off the escalator, as long as you are *on* the escalator for the maximum allotted time.

You had posited “If you have a limited time on an escalator (which allows you to double your speed)”, but the escalator doesn’t actually *double* your speed, it just adds a constant value to your speed. Since it adds the same constant value whether you’re standing or walking, it makes no difference where you rest. The only things that matter are the total time you spend on the escalator and the total time spent resting.

(If there were such a thing as a magical escalator that actually did double your speed, then it *would* make sense to rest on the stairs rather than the escalator.)

8. 8 8 Rob Rawlings

I was thinking about this some more and (probably foolishly) have decided to fight a rearguard action.

The issue at hand is to decide if Alice’s journey time is faster because:

1. She travels while resting
or
2. Is rather due to resting on the escalator buying more travel time on the escalator.

Firstly I think the stairs can be taken out of the initial version of the puzzle as they add nothing to the results. Both Bob and Alice spend 10 minutes walking up the stairs so this can be ignored.

The question then becomes: why is it better to rest on the escalator (scenario A) , rather than off it(scenario B) if you want to minimize journey time.

The number are as follows:
Scenario A:
Time resting: 1 min
Distance while resting: 10 yards
Time not resting: 4.5 mins
Distance not resting: 90 yards
TOTAL TIME: 5.5

Scenario B:
Time resting: 1 min
Distance while resting: 0 yards
Time not resting: 5 mina
Distance not resting: 100 yards
TOTAL TIME: 6

Now, to me it seems obvious that the time spent resting in A explains all the difference in time. The distance while resting would take 30 seconds travel time while not resting, which is exactly the time saved.

Its also true that resting on the escalator increases escalator time, but I struggle to see it as “buying” more time. Resting tautologically adds journey time at the non-resting travel speed. The “buying” time argument would seem to me a more complex and therefore slightly inferior way of explaining things, but its advocates seem to want to claim it refutes the “at least she’s traveling” argument!

Which brings me to the version of the puzzle described in this post that seems to disprove the “at least she’s traveling” argument and caused me to temporarily switch camps.

Ignore the stair piece and this puzzle comes down to is it better to rest on or off the escalator given a 5 minute journey time.

Scenario A (using same speeds as before):
Time resting: 1 min
Distance while resting: 10yards
Time not resting: 4mina
Distance not resting: 80
TOTAL DISTANCE: 90 yards

Scenario B:
Time resting: 1 min
Distance while resting: 0yards
Time not resting: 4mina
Distance not resting: 80 yards
TOTAL DISTANCE: 80 yards

Again better to rest on the escalator given a 5 minute journey simply because “at least you’re traveling”. Again you could say it because resting on the escalator buys you more time on the escalator. But in this case (I contend) its even more clearly tautological to claim that.

OK, so that leaves the slightly awkward fact that when you add in the stairs piece to the second version the benefits of walking while resting seem to be wiped out. I admit I do not have a great answer to this (WIP). But my provisional answer is that this scenario is constructed so that all optimization from “resting while moving” are guaranteed to be lost in the second phase, which makes it a different puzzle than the one original stated with a more complex analysis needed.

(Hope I didn’t screw up my timings!)

9. 9 9 Neil

Bennett,

It is obvious—that I was wrong. Thanks.

One more puzzle.

There are no stairs! Just two consecutive escalators of equal length. One escalator is faster than the other. You can’t get to the top without resting. Should you rest on the fast escalator or the slow one?

10. 10 10 Sprobert

@Rob Rawlings

The mistake in your argument for Bennett’s example is that you are assuming the rest comes before the escalator. In this case, you are worse off, because (like the original problem if you rested on the stairs) you are denying yourself time on the escalator.

If you compare resting on the escalator to resting AFTER the escalator, then you’ll the result really is the same, AFTER 6 minutes:
Resting on escalator scenario:
1 min resting (on escalator): 10 yards
4 min walking on escalator: 80 yards
1 min walking after escalator: 10 yards
TOTAL: 100 yards in 6 minutes

Resting after escalator scenario:
5 min walking on escalator: 100 yards
1 min resting after escalator: 0 yards
TOTAL: 100 yards in 6 minutes

Nothing has been gained by resting on the escalator, as long as both individuals spend the full five minutes on the functioning escalator.

11. 11 11 Rob Rawlings

@#10

Yes, I see that if you take the first 6 minutes it makes no difference if you rest on the escalator or off as the rules of this version of the puzzle specifies that if you rest on the escalator you have to walk the next minute, and if you walk the whole 5 mins on the escalator you have to rest the 6th minute, and this wipes out all the benefits form “travelling while resting” in the first 5 minutes.

But does that really show that the “at least you’re traveling” argument loses out to the “buy more time on the escalator” one or merely (as I suspect) that its possible to construct examples where “at least you’re traveling” is obscured by the rules of the construct?

In the simplest case of a single escalator journey its so obvious that its best (if you wish to optimize journey time) to move while resting rather than not moving while resting that I fail to see how this cannot still (at some level) apply to more complex scenarios.

12. 12 12 Rob Rawlings

And I think in Steve’s original example it really does boil down to the simplest case of a single escalator journey that Bob or Alice can rest either on or off (since the 10 minutes walking up – and not resting on – the stairs cannot be avoided so is irrelevant).

Curious if people think I am an off base here ?

13. 13 13 Eric

While this formulation seems to disprove the “moving while resting” argument, it also, perhaps, suggests that our intuition isn’t as bad as the original post suggests.

Our intuitions are honed in the real world where staircases and escalators do not change in length. Here, we effectively have a variable length escalator, and this is not a situation we encounter often enough to get any intuition about.

14. 14 14 Rob Rawlings

I get that these puzzles are set up so there is trade-off between the 2 legs of the journey.

In the original version where the distance on the escalator is fixed if you choose to rest on the escalator you get in return a matching speedup on the stairs AND some additional time on the escalator making this a worthwhile exchange to improve journey time.

In the later version where the time on the escalator is fixed if you choose to rest on the escalator you get in return a matching speedup on the stairs BUT no additional time on the escalator making this a neutral exchange.

My point is that in Steve’s original version you don’t actually need this trade-off story to explain things. The extra distance covered while resting on the escalator is enough to explain why this is the best option. Any “trade-off” this choice makes with the walking stage just takes back some of the gains.

15. 15 15 Neil

The point is that resting on the escalator shortens the distance you have to walk, whereas resting on the stairs does not. Since you walk at constant speed, the time to destination is less. The “moving while resting argument” is correct, except in Bennett’s problem where your time on the escalator is fixed.

16. 16 16 Bennett Haselton

@Neil #9 yes you should rest on the faster escalator, and similarly the reason is that if your total resting time is taken as a given, you want to maximize *time* spent on the fastest escalator possible, in order to get the maximum distance-adding benefit from it.

17. 17 17 Rob Rawlings

lets restate the 2 puzzles to see what the real choices are.

Original version:
(Separating out mandatory steps so we can see the real differences between Bob and Alice’s journey)

Both Bob and Alice have to
1. Walk up stairs for 10 minutes
2. Walk on escalator for 4.5 minutes

But Alice
3. Rests on escalator for a minute
While Bob
4. Rests on the stairs for a minute
5. Walks on the escalator for 30 secs

Alice finishes the whole journey at the same time that Bob finishes step 4 and still has step 5 to go. This can only be because option 3 (“resting while moving”) is better than option 4 (“resting while not moving”) as its the only difference.

New Scenario: (As specified by Steve: 300 yards long, escalator moves at 20 yards per minute and your walking speed is 10 yards). I have broken out mandatory steps so we can see the real choice.

You have to
1. Walk up moving escalator for 4 minutes (120 yards)
2. Walk up stationary escalator for 15 minutes (150 yards)

But you can choose between
3. Rest on moving escalator for a minute (20 yards)
4. Walk up stopped escalator for 1 minute (10 yards)
or
5. Rest on stopped escalator for a minute (0 yards)
6. Walk on moving escalator for a minute (30 yards)

The benefit of 3 (resting while moving)over 5 (resting while not moving) is then wiped out by the penalty of 4 compared to 6 that comes as an unavoidable part of your choice. So , I contend, this example does not refute the “resting while moving” theory after all as this scenario does not sufficiently isolate the reason for the (lack of) difference in journey time.

18. 18 18 Bennett Haselton

Well yes, resting on a moving escalator does cover more ground than resting on a stopped escalator :) it’s just that as long as you take total resting time as a given, it will have to be offset somewhere else.

Look at it this way: Over any time period, your total distance traveled equals:
- the time you spend walking, times your walking speed
plus
- the time you spend on a (moving) escalator (whether standing or walking), times the speed of the moving escalator

If everything is fixed except the time you spend on the moving escalator, that’s what you maximize. If that’s fixed too, then it doesn’t matter what you do. I don’t think this reasoning will ever lead you astray.

19. 19 19 Rob Rawlings

@18.

I agree with what you say and if I had to write a maxim to provide guidance on these kinds of puzzles then “if possible maximize the time spent on the fastest leg” would be it – (Though “always rest on the fastest leg” would never actually cause you to have a slower journey as far as I can see).

My point really was just to try and show that the “If you rest on the escalator, at least you’re making progress” argument ((that Steve was aiming to debunk) has some merit in explaining why Alice has a faster journey time than Bob if you break down the problem to its core.

20. 20 20 Neil

Resting on the fastest leg isn’t the right logic. Suppose I have to bike two miles to a destination, one mile flat and one mile up hill. I can pedal 20 mph on the flat, but only 10 mph uphill. I have to rest somewhere along the trip. It doesn’t matter whether I rest on the slow leg or the fast leg. In either case I am stopped while I rest and my total time is the same. The difference in the escalator case is that the escalator moves me along while I rest.

21. 21 21 Sprobert

@Rob 17,

You have the right math but you’re drawing the wrong conclusions.

In the original problem, Alice gets 5.5 minutes on the escalator and Bob only gets 5 minutes. So the proper question is, what matters? The fact that Alice was moving when resting, or the fact that Alice rested on the escalator gave her more time on the escalator?

Then you did the math right for Bennett’s example, but refused to treat it as the counter-example that it is. If what matters is “moving when resting”, there MUST be a difference in Bennett’s example, and your math proves there isn’t. Why is there no difference? Because the TIME on the escalator was the same regardless of when you’ve rested.

This counterexample makes it abundantly clear: resting when being moved only matters if it allows you more time to be moved the external force.

22. 22 22 Rob Rawlings

@#21

Yes, I get that the more time you spend on thee escalator the lower your total travel time, and that when scenario is setup so that escalator time is fixed then so is travel time fixed.

But that is irrelevant to my point. I am claiming that Steve’s attempted debunking of the “You should rest on the escalator, because at least that way you make some progress while you rest.” argument fails to work.

In my comment #17 above I identify that when you remove all the mandatory steps in the original scenario you really are left with a choice between resting while moving and resting while not moving, and the most obvious reason why the first option is better is because it means “at least you are moving while resting”.

23. 23 23 Steve Landsburg

Sprobert: Yes, you’ve said this perfectly. Thanks.

24. 24 24 Neil

But it is *because* the escalator is moving that the extra time spent on the escalator shortens the time to your destination. If, for example, the escalator stops moving when you rest, there is no advantage. (This is actually equivalent to Bennett’s example.) So the moving while resting logic is perfectly okay.

25. 25 25 Richard D.

SL: “This time Answer One gives the right conclusion. But the reasoning can’t be right, because it’s the exact same reasoning that we applied to the New Puzzle, whereupon that reasoning led us totally astray.
Bennett’s lovely example illustrates as starkly as possible why we must reject Answer One even though it sometimes yields the right conclusion.”

No, this deduction is unwarranted, because you compare apples to oranges; one example features fixed time, the other, fixed distance. Thus, a single line of reasoning may be invalid in one case, but valid in the other.

Consider:
I) Adam proposed marriage to Eve, in order to form a long term relationship, for the purpose of making babies and raising a family.

II) Adam proposed marriage to Steve, in order to form a long term relationship, for the purpose of making babies and raising a family.

The identical reasoning is correct in one case, dubious in the other.