A couple of weeks ago, I was in Las Vegas for the annual meeting of the Association for Private Enterprise Education, where I was honored to give an invited plenary address.
From there, I went directly to Atlanta, where I gave a short talk at the Gathering for Gardner, honoring the legacy of Martin Gardner. There were a lot of other really cool talks too.
I am sorry that the Las Vegas talk was not recorded and that the recording from the Atlanta talk won’t be available for a few months. Therefore I sat down in front of my webcam and repeated both talks, sticking as close to the original words as my memory would allow. Unfortunately there is no way to recreate the audience reaction or the question and answer periods.
Click below to view either or both of those re-creations.
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The second talk is essentially a six-minute excerpt from the first. It surely benefited from the discussions here, here and here, and most especially from the comments of Bennett Haselton.
A few more words about escalators for those who care about this kind of thing:
In the past, when I’ve written about escalators, I’ve sometimes framed the question this way:
Escalator Question, Version 1: You’re going to travel from Floor One to Floor Two on the stairs and then from Floor Two to Floor Three on the escalator (or vice versa). You want to take one break. Where should you take it?
The discussion linked above, and Bennett’s comments in particular, made me realize that the issues are much starker and easier to explain if one frames the question this way:
Escalator Question, Version 2: Alice is going to travel from Floor One to Floor Two on an escalator. Bob is going to travel from Floor One to Floor Two on a staircase. Which one of them should be more willing to take a break?
The videos above reflect my belated realization that when I’m trying to explain this stuff, it’s better to start with Version 2.
Absorbed your April 4 lecture.
Clearly, comprehensibly, yes, beautifully put, especially prices and distribution, the first of political significance and the latter and later a tour de force.
All best wishes, continued.
Strictly speaking, the question as stated is under-specified, leaving us to guess the incentives. E.g. Q: Alice (on an escalator) and Bob are racing to the 2nd floor of a shopping mall. The prize for the winner is a valuable coupon. Who’s more willing to take a break?
A: Bob, of course. After a few seconds he’ll realize he’s falling behind and will slow down or take a break, settling on merely reaching the store eventually. Alice’s actions will depend on whether she sees Bob. If not, she’ll keep moving unwilling to jeopardize a sure prize.
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Enjoyed the talk!
Here is what I got out of the escalator explanation. If you are climbing stairs that might magically transform into an escalator, then you might stop and wait for the transformation. The real reason you don’t stop on the stairs is that you do not believe that the stairs are going to transform into an escalator.
As for whether the stopped man on the stairs is sacrificing a chance to make progress, you show that it depends on the definitions. It is possible to give a definition of cost that makes the alternatives appear similar.
Roger (#3): Unless I’ve misunderstood you, you appear to have missed the following simple point: If I can construct a scenario in which your reasoning leads to the wrong conclusion, then your reasoning is flawed. It does not make a bit of difference whether my scenario is terribly likely ever to occur. You seem to be saying that the usual reasoning leads to the right conclusion in all likely scenarios, and therefore the usual reasoning is correct. But that’s of course entirely wrong.
Indeed, the fact that the usual reasoning leads to the right conclusion (namely “take your break on the stairs”) 99.9% of the time is exactly why people get fooled into thinking that the usual reasoning must be correct. It takes only one counterexample (e.g. an escalator and a staircase that switch roles at a given moment) to debunk that claim.
I think a purist would say that Escalator Question, Version 2 is not strictly answerable.
Taking all your original assumptions (100 yard escalator/staircase, the escalator moves at 10 yards/minute, both Alice & Bob walk at 10 yards/minute, the rest lasts for 1 minute).
With no break, Alice walks a total of 50 yards over 5 minutes; with a break, she walks a total of 45 yards over 5.5 minutes.
With no break, Bob walks a total of 100 yards in 10 minutes; with a break, he walks a total of 100 yards in 11 minutes.
Even assuming that Bob and Alice have identical preferences, it seems like there would be no way to know how to compare their *marginal* utility of the value of arriving in 5 minutes instead of 5.5 minutes, versus the marginal utility of arriving in 10 minutes instead of 11 minutes. (Or, if you are measuring their walking effort instead of total time traveled, you have the same problem.)
I’m not sure what you mean by the “clearest” version of the question. I think the *original* original question — you take the stairs and then you take the escalator — is certainly “clear”, but most people get the right answer for the wrong reason. I think the clearest version of the question where the fallacious reasoning will lead you to the wrong answer (“If you stand still on the escalator, at least you’re making progress”) is the one from your second post (which you modestly credited to me, but I think is mostly due to you :) ) —
https://www.thebigquestions.com/2018/12/25/escalating-matters/
i.e., if you know the escalator is going to temporarily halt for a full minute some time while you’re on it, is it better to take a break while the escalator is moving, or not.
The answer that always seems to work is: maximize your time on the escalator, because that’s the only time the escalator is contributing forward-motion, so it saves you both time and energy.
Bennett (#5): Re your last point — My take on this all along has been the same as yours — the only thing that matters is how much time you spend on the escalator. But I am coming around to the point of view that although that’s absolutely correct, it tends to go over people’s heads, whereas the presentation in the current video is more likely to make them realize they’re missing something.
It is hard to believe that you guys are serious.
I guess the usual reasoning is that stopping on the stairs is sacrificing a chance to make progress. This reasoning works in all scenarios. I don’t see any flaw in it.
I define progress as advancing towards the destination without doing unnecessary work.
In Question v.1 above, you take the break on the escalator, as that gives steady progress and no extra work. In v.2, Alice takes a break on the escalator.
You might present the scenario that someone arrives at an escalator when it is not working. Does he treat it like a staircase and start climbing? Or does he stop at the bottom in the hopes that it starts working? Or does he hope that someone magically transports him to the top? It depends on his beliefs.
Continuing in the spirit of, “If I can construct a scenario in which your reasoning leads to the wrong conclusion, then your reasoning is flawed”:
In your last slide you wrote, “Why don’t people stand still on staircases? Because on a staircase, there’s usually a staircase ahead of you! But if we switched that up, people *would* take their breaks on the stairs!”
But it seems that reasoning leads to wrong answers as well. We already established that if you’re going on a normal escalator and then going onto a staircase afterwards, it’s perfectly rational to take your break on the escalator — even if, for the sake of argument, you take your break at the last portion of the escalator ride, so that there’s nothing but stairs ahead of you after your break.
I think Roger #3 is onto something when he says that the reason you rest on the stairs and then wait for them to turn into an escalator at 12:15, is that now you get to ride on an escalator for a longer portion of the ride.
I wonder if there is a simple Unified Elevator Theory that would tie together all of this, and handle all the variations of the problem that have come up:
– Does the escalator cover a fixed *distance* (the original stairs-then-elevator question), or are you on the escalator for a fixed *amount of time* (the escalator that stops moving for a short interval while you’re on it)?
– Are you trying to arrive as early as possible, or minimize walking? In the cases we considered, both goals lead to the same answer, but do they always?
And then when we have worked out the complete answer we have to read the whole thing in Mitch Hedberg’s voice: https://www.youtube.com/watch?v=yHopAo_Ohy0
Here is what I think is the complete answer that covers all cases. (And my conclusion differs from the one in the video — I argue people don’t take breaks on stairs because, well, they’re standing on stairs.)
If the problem specifies that you spend a fixed amount of *time* on an escalator or a fixed amount of *time* on stairs (and also specifies that you spend a certain amount of time resting), then:
(a) If you do take a break, it doesn’t matter where you take it. That’s because at time t, your distance traveled is always
(t – breaktime)*(walking speed) + (time spent on escalator)*(escalator speed)
and these are invariant regardless of when you took the break, as long as your break is over. (This seems intuitive.)
(b) The cost of the break does depend, as Landsburg said, on “what’s ahead of you”, but more precisely it depends on the speed that the system is moving at the *end* of the journey. That’s because your ghost is always a fixed *distance* ahead of you (break time multiplied by walking speed), and the speed of the system at the end (escalator speed, if any, plus your walking speed) determines the time delta between your ghost getting to the end and you getting to the end. If the system will be stairs for the first five minutes and then a 10yd/minute escalator the next five minutes and a 7yd/minute escalator for the next five minutes and a 5yd/minute escalator for the next five minutes, then assuming the distance is such that you will get off during the 5yd/minute time period whether you take a break or not, the 5yd/minute speed determines the cost of the break.
On the other hand, if the problem specifies that the stairs cover a fixed *distance* and the escalator covers a fixed *distance* (as in the original problem), then “ghosts” don’t work here because they don’t stay a fixed distance *or* time ahead of you. Instead my reasoning is as follows:
(a) If you do take a break, it does matter where you take it, because you want to maximize time spent on the escalator so that the escalator does more work for you. (If it’s a multi-escalator system, take your break on the fastest one.) I think this is intuitive as well.
(b) The cost of the break depends on the speed of the portion of the system where you take it (the faster the speed, the lower the cost, because you’re making up for it by spending more time on a fast escalator), and I also think this is intuitive from (a). The formula is not “intuitive” but: if you take a break for k minutes and the escalator speed where you take the break is e and your walking speed is w, then the delay caused by your break is kw/(e+w). (Consider a ghost for *that portion of the journey only*. While you’re taking a break, the ghost gets kw distance ahead of you. Then once you and your ghost both start walking, the time between your ghost getting to the end of that escalator segment is kw divided by the combined escalator and walking speed, e+w.) Of course if you take a break for k minutes on the stairs (e=0) this simplifies to your break costing you exactly k minutes.
So, I would modify Landsburg’s statement at the end of the video:
“The cost of your break depends not on what’s beneath your feet when you take the break, but on what’s ahead of you.”
If the problem specifies fixed *times* for stairs/escalator, then the cost of the break depends on what’s ahead of you (more precisely, the speed of the system at the end of the journey). If the problem specifies fixed *distances* covered by stairs/escalator, then the cost of the break *does* depend on what is beneath your feet when you take the break.
As for the final point, “Why don’t people stand still on staircases? Because usually, when you’re on a staircase, there’s a staircase ahead of you.”
Per the logic above, I would argue that in the real world, we’re usually in a fixed-distance problem, not a fixed-time problem (stairs from floor 1 to floor 2, escalator from floor 2 to floor 3), which means the cost of the break does depend on where you’re standing, and the reason people don’t stand still on staircases is because they’re on staircases.
Bennett (#9): I believe we’re saying exactly the same thing in different words, and I still like my words better than yours.
I claim the cost of your break depends entirely on how you’ll eventually make up for that break.
In the video, with fixed times, I point to the red distance, and say that you will *eventually* have to make up that red distance. Of course you start making up that distance at the moment your ghost exits the escalator.
If the problem specifies fixed distances, it is still the case that everything depends on where you make up for your break. With fixed distances, a break on the stairs is made up for with more time on the stairs, and a break on the escalator is made up for with more time on the escalator. That makes the break on the stairs more expensive.
I agree, I’m just saying your maxim stated in the video:
“The cost of your break depends on not on what’s beneath your feet when you take the break, but on what’s ahead of you.”
is only true under the fixed-time rules, not under the fixed-distance rules. As presented, that’s not obvious, since there are many variations of the problem that are fixed-distance.
On the other hand, I think this is incorrect:
“So, in the real world, why don’t people stand still on staircases? Because usually, when you’re on a staircase, there’s a staircase ahead of you.”
To me, the phrase “in the real world” implies the fixed-distance condition. And we agree that means the price of your break depends on where you’re standing when you take it. So that means that in the real world, people don’t stand still on stairs because they’re on stairs.
Bennett:
You wrote:
And we agree that means the price of your break depends on where you’re standing when you take it.
No.
Let’s make everything fixed distances; you go from Floor 1 to Floor 3, where each of the two legs of the journey might be either a staircase or an escalator. You take a break either shortly after leaving Floor 1 or shortly after passing Floor 2. The break is two minutes long. Your walking speed and the escalator speed are both 10 yards per minute. (Nothing depends on these speeds being equal.)
I claim that:
A. The cost of a break on stairs with stairs ahead of you is that you have to make up 20 yards on a staircase.
B. The cost of a break on an escalator with an escalator ahead of you is that you have make up 20 yards on an escalator.
C. The cost of a break on a staircase with an escalator ahead of you is that you have to make up 40 yards on an escalator.
D. The cost of a break on an escalator with a staircase ahead of you is that you have to make up 10 yards on a staircase.
(Reason for C and D: During your break, your ghost gets 20 yards ahead of you, and remains so until it reaches Floor 2. Then it switches from stairs to escalator or vice versa, so the distance between you and it either increases or decreases until you reach Floor 2, at which point the ghost is either 40 or 10 yards ahead of you.)
Comparing A with B, I think it is perfectly fair to say that when we compare the all-stairs and all-escalator situations, the break on the stairs is costlier than the break on the escalator **precisely because** the stairs have stairs in front of them and the escalator has escalator in front of it.
The cost in C happens to equal the cost in A. The cost in D happens to equal the cost in B. But again, I think it is perfectly reasonable to describe these costs by saying that the **reason** for the cost in C is that you’re going to be making up 40 yards on an escalator and the **reason** for the cost in D is that you’re going to be making up 10 yards on a staircase.
I understand that there are other ways to describe this (I’ve used those descriptions in print many times). But I make the following claims:
1. This description is also perfectly legitimate.
2. This description is probably a clunkier choice than some others when you’re comparing scenarios C and D, but:
3. This description is simpler and clearer than others when you’re comparing scenarios A and B, as well as being easier for audiences to understand.
I think the “if you stand still on the stairs you are sacrificing an opportunity to make progress” view is correct.
If you want to reach your destination in the shortest time and you have a choice between resting while moving and resting while remaining stationary then if you choose to rest while stationary your journey time will be longer as you have “sacrificed an opportunity to make progress”.
Adding a staircase to this story changes nothing. No matter where you rest you still have to go up the stairs at some point. I don’t see how it is relevant and may as well be left out.
Its true that if you rest while not moving then the rest will cost you less if you have an escalator ahead of you than if you have to walk the whole journey but your journey would still have been shorter if you had waited and taken the rest on the escalator. By choosing to rest while not on the escalator you have “sacrificed an opportunity to make progress”!
If you can walk 200M in 2 minutes then a 2 minute rest (moving or stationary) will always leave you 200M behind where you would have been with no rest. But if you rest at any speed above 0MPH then your journey time will increase by less than the 2 minutes you rested for. It is this discount on resting time that you sacrifice by resting while stationary.
Rob Rawlings (#13):
If you can walk 200M in 2 minutes then a 2 minute rest (moving or stationary) will always leave you 200M behind where you would have been with no rest. But if you rest at any speed above 0MPH then your journey time will increase by less than the 2 minutes you rested for.
But this is not true in general; in particular it is not true in the example where the stairs and escalator switch roles at (say) 12:15PM. So it seems to me that your story (if I understand it) proves too much, and therefore can’t be exactly right.
Rob (#13):
What Steve said, but more generally, do you agree or disagree with the following statements:
(1) In all of the fixed-time variations of the problem (you are on an escalator that will stop for maintenance for exactly 1 minute in the middle of your journey; or, stairs change to escalator at 12:15, etc.), if you are going to take a break then it doesn’t matter whether you take a break while the escalator is moving or standing still.
(2) While it’s true that “If you stand still on the stairs you are sacrificing an opportunity to make progress”, it’s also true that if you stand still on the escalator you are sacrificing an opportunity to make double progress (your combined walking speed plus the escalator speed).
(3) In the fixed-distance version of the problem (stairs from floor 1 to floor 2, escalator from floor 2 to floor 3), you arrive earlier if you take your break on the escalator than if you took a break for the same amount of time on the stairs.
(4) At the end you wrote: “But if you rest at any speed above 0MPH then your journey time will increase by less than the 2 minutes you rested for.” However, even if you rest at a speed of 0MPH, then your journey time will ALSO increase by less than the 2 minutes that you rested for, IF the escalator is moving at the end of your journey. Visualize it: If you rested for 2 minutes while the escalator is stopped and you are 200M behind your ghost, and both you and your ghost are walking at 100M/minute and the escalator is moving at 100M/minute, then at the moment your ghost reaches the end point, you are moving at a combined speed of 200M/minute and you will close the 200M gap in 1 minute.
Ah, I had missed the significance of the change happening at a set time rather than a set distance. If I arrive at a fast moving escalator that doesn’t start for 2 mins I can see that the cost of my waiting for those 2 minutes (in terms of increased journey time) is lower than either if the escalator was already running or if it was never going to start but won’t my total journey time still be less if I spent those 2 minutes climbing the not-yet-started escalator and then rested for 2 mins on the escalator once it started ?
Rob is correct that not standing on the stairs is fully explained by sacrificing an opportunity to make progress.
“it’s also true that if you stand still on the escalator you are sacrificing an opportunity to make double progress”
I don’t know why you say stuff like this. The object is to explain why people stand on escalators. Sure, someone can make double progress by walking up the escalator. If the object were to explain why people walk on escalators, that would do it.
Then there are the contrived scenarios where the escalator starts and stops intermittently, or magically transforms to and from stairs. Maybe the traveler has full knowledge, and maybe not.
In those scenarios, I am not sure what we are trying to explain. What do people do? I am sure some people follow the strategy of not sacrificing an opportunity to make progress, and that is still a sensible strategy. Others will want to make that double progress.
I don’t see how the difficulty in catching up with a hypothetical ghost has anything to do with it.
I am trying to remember if I was ever on an elevator that stopped, and what people did. My guess is that most people would wait a minute to see if the escalator restarts, and start walking if it does not. If so, is that supposed to be exception to the principle of not sacrificing an opportunity to make progress?
I do not think it is. The traveler is only stopped because he is waiting for the escalator to restart. As soon as he loses confidence in the escalator, he starts walking.
Steve (#12):
I agree with everything you’re saying, but it all seems to support the conclusion that I stated earlier, not the conclusion that you stated earlier. Which of the following statements, if any, do you disagree with:
(1) Cost of break on stairs, when there’s an escalator ahead of you = Cost of break on stairs, when there are stairs ahead of you. (i.e., cost(A) = cost(C))
(2) Cost of break on escalator, when there are stairs ahead of you = Cost of break on escalator, when there’s an escalator in front of you. (i.e., cost(B) = cost(D))
(3) The costs in #1 and #2 are different, i.e. cost(A) != cost(B)
(4) Points 1-3 can be summarized as: In the fixed-distance problem, “The cost of your break depends on where you’re standing when you take it” (my words, changing “price” to “cost”).
(5) Points 1-3 *not* summarize to your statement in the video, “The cost of your break depends not on what’s beneath your feet when you take the break, but on what’s ahead of you.” (Although to be fair, this statement in the video was right after a fixed-time version of the problem, so maybe you meant it only to apply to fixed-time, not fixed-distance.)
Bennett: On your point 1), I agree that cost(A)=cost(C). I disagree that (in general) Cost of break on stairs, when there’s an escalator ahead of you = Cost of break on stairs, when there are stairs ahead of you. This is true ONLY when those costs are the same as the costs of A and C, which is ONLY true under some additional assumption (e.g. fixed distances).
I believe that most people answering this question, when they say that the cost of a break on the stairs with an escalator ahead of you is the same as the cost of a break on the stairs with stairs ahead of you, would deny that they are making any additional assumptions, and hence would be missing the key reason WHY those costs are (often but not always) the same.
In other words, it happens to be the case that often, in the real world, the necessary additional assumptions do hold. But if your explanation does not account for those extra assumptions (and why they are relevant) then your explanation is wrong.
Thank you for recording the talks and making them available! I enjoyed these. I don’t have a position on the escalator vs stairs issue, other than observe that all the discussion seems to confirm it’s an interesting question.
I do have a question about the charity selection logic. What if I am not certain of my own estimation of “most deserving charity?” To go with the starving kittens vs. puppies example: If I have a slight preference for favoring the kittens, but I’m only 55% sure it’s the correct choice, then I would probably choose to feed 11 kittens and then 9 puppies. If I think it’s a toss-up, I would pseudo-randomly select a kitten or a puppy each time I wanted to feed one. I can’t speak for other donors, but I do a fair amount of research when I donate, and I’m still not at all 100% confident of my own ranking. You could say I should continue to do research until I’m 100% certain, but I would counter that then I would pretty much never donate.
Clarifying my last statement in #20: I would never donate because I would always be stuck in research mode. Say I researched for 10 years, and got pretty confident after that – but for the situation in year 1! In those 10 years, both the world’s problems and the state of the charity industry would have changed.
I agree that the cost of a rest may be measured by the time added to the total journey by the rest. A rest on stairs with an escalator ahead costs less than a rest on an escalator with stairs ahead.
However I do not see how these additional influences on cost affect the underlying benefit of resting while moving over resting while stationary. At any point in time given the choice between resting on a fixed length escalator and resting before getting on the escalator it will always be lower cost to rest while moving on that part of the journey because one is avoiding ‘sacrificing an opportunity to make progress’.
The fact that the benefits of this progress may be outweighed by other factors (stairs ahead, the escalator becoming stairs at some point in time, a gunmen making you wait for 10mins if he sees you resting on the escalator, etc) does not remove the benefits but merely means they are not worth taking advantage of in all situations.
In short: Other things equal its better to rest while moving in order to take advantage of an opportunity to make progress’. It is however easy to come up with scenarios where other things are not equal and cause one to question the underlying intuition.
Steve (#19),
OK, for round two, which of the following statements, if any, do you disagree with:
(6) “In the real world” (the phrase you used in the video), you are far more likely to encounter a fixed-distance stairs and elevator problem, than a fixed-time stairs and elevator problem.
(7) “In the real world”, all of the additional assumptions also hold which are necessary to assert: “Cost of break on stairs, when there’s an escalator ahead of you = Cost of break on stairs, when there are stairs ahead of you”. (I actually don’t know what those additional assumptions would be. But if you disagree with this statement, I would like to know what those assumptions are that you think do not hold.)
Intuitively, of course, a person is less likely to take a break on the stairs if they see an escalator ahead of them. However, I assert that is *not* because it means the cost of the break on the stairs is any different than if there were more stairs up ahead. It’s because they can see a *lower-cost alternative* ahead of them.
In the real world, the cost of resting on the escalator is usually zero. You went to the airport 2 hours before flight time, and running up the escalator is just going to annoy others and force you to make up the lost calories by eating more pretzels on the plane. Getting to the gate sooner is not doing you any good.
If you are in danger of missing your flight then you are going to run, but it still will not matter what is ahead of you.
A better measure of cost is the work expended. Walking up the escalator is costly because it requires extra work. Taking a break on the stairs is neutral because the work is unchanged. So stopping on the escalator is normal behavior, and people only stop on the stairs if they need a rest.
Henri Hein (#20): Bob and Alice are trapped in a well and you can only save one of them. If you are 55% sure that you’d rather save Alice, why would you ever save Bob?
Bennett: Your point 7) is problematic. I claim that:
A) You’re always going to have to make up for your break eventually. Making up for that break on a staircase-ahead-of-you is always costlier than making up for that break on an escalator-ahead-of-you.
B) If we compare Bob, who does all his traveling on a staircase, with Alice, who does all her traveling on an escalator, then point A) is the entire story.
C) Now let’s consider Carol, who does her traveling on a fixed-length staircase followed by a fixed-length escalator, and who also plans to take exactly one break. Compared with Bob, a break on the stairs (or on the escalator for that matter) is CHEAPER for Carol because she gets to make up for it on an escalator. At the same time, a break on the stairs is COSTLIER for Carol than for Bob because it means she’ll be spending less time on the escalator. Those two things cancel out and she ends up paying the same price as Bob.
I claim it is always true (under any assumptions, fixed distance, fixed time, or anything else) that *being able to make up for your break on an escalator instead of a staircase reduces your costs*. It is also true that with fixed distances, there are other offsetting considerations, but the statement between the asterisks is still true.
This is exactly why I now think it is much much cleaner to focus on Bob versus Alice, where there are no offsetting considerations, than to talk about Bob versus Carol, where there are.
I also think that when people hear the question “Why do people stand still on escalators but not stairs”, they tend to imagine Bob versus Alice, not Bob versus Carol. Therefore focusing on Bob and Alice is quite responsive to the question people have in mind, and fortunately that’s the case where the issue is cleanest — the cost of your break depends entirely on where you’re going to be making up for that break.
Henri Hein (#20):
Suppose there are two “charity slot machines” in a casino. One of them, when you put in a dollar, wires $0.50 to a charity. The other, when you put in a dollar, wires $2.00 to the same charity. But you don’t know which machine is which, and the machine never tells you what happened after you put in a dollar. Would you agree this is a reasonable model of the problem where you don’t know which mode of giving is more effective?
But, suppose you are 55% sure that the machine which wires $2 to charity is the one on the right. Would you put 55% of your money into the machine on the right? No, the way to maximize the expected value of your donation is to put all of your money into the machine on the right. For each dollar you put in, the expected benefit to the charity is 0.45*$0.50 + $0.55*$2 if you put it in the machine on the right, and 0.55*$0.50 + $0.45*$2 if you put it in the machine on the left. It doesn’t matter how many dollars you’ve put in already.
Steve (#27):
I confess I don’t understand what you mean by “making up for your break” (on stairs or on an escalator). I have been assuming that in the standard definition of the problem, if you take a break, you never “make up for the break”. You just arrive at the endpoint later. Assuming the “cost” of the break is the delay, then in the fixed-distance version of the problem, the cost (delay) depends on where you are standing when you take the break, and nothing else.
What do you mean by “making up for the break”? Do you actually mean that at some later point, the person is going to walk faster than their usual speed, until they catch up with where they would have been if they had walked normal speed but never taken the break at all?
Bennett: “Making up for your break” means spending time traveling to your destination after your ghost has already arrived there.
You keep referring to the “fixed distance version” of the problem but I have no idea what you mean by that. Alice travels from the fixed point A to the fixed point C on an escalator. Bob travels from the fixed point A to the fixed point C on the stairs. Carol travels from the fixed point A to the fixed point B on stairs and then from the fixed point B to the fixed point C on an escalator. When you talk about “the fixed distance version of the problem” are you talking about a comparison between Alice and Bob, or between Bob and Carol, or something else?
If you mean Alice and Bob (which is the case I addressed in the video) then the question is “WHY is a break costlier for Bob than for Alice?”. You observe that Bob rests on the stairs and Alice on the escalator, and that the break is costlier for Bob. That doesn’t mean it’s costlier for him BECAUSE he rests on the stairs. It is costlier for him because he MAKES UP for the break on stairs, as can be seen by the fact that we can CHANGE THE COST OF HIS BREAK AFTER THE FACT by switching the stairs to an escalator. If the break on the stairs were the CAUSE of the higher cost, then there would be nothing we could do to change that cost once it’s been incurred.
In #26 Steve wrote:
“Now let’s consider Carol, who does her traveling on a fixed-length staircase followed by a fixed-length escalator, and who also plans to take exactly one break. Compared with Bob [who travels the whole distance on stairs], a break on the stairs… is CHEAPER for Carol because she gets to make up for it on an escalator.”
I maintain this is incorrect. If Bob or Alice take a 1 minute break on the stairs, they will both arrive at the endpoint 1 minute later than they would have if they had never taken the break.
To answer your question in #29:
A “fixed distance” version of the problem is where the stairs and elevator are each located over a fixed distance, but neither of them change speeds over the whole duration of the problem. (e.g. stairs from Floor 1 to Floor 2 and elevator from Floor 2 to Floor 3.) A “fixed time” version of the problem is where you spend the whole time on a single escalator, but the problem specifies the times that it is stopped and the times that it is moving. (e.g. your version in the video, “The stairs turn into an escalator at exactly 12:15 PM”, or from your earlier blog post, “You are riding an escalator except some time while you are riding the escalator will shut down for maintenance for exactly 1 minute”.) (I thought this was all clear after post #9 but hope it’s clear now.)
As I said in post #9, I maintain:
(a) In a fixed-time problem:
(i) if you do take a break, it doesn’t matter where you take it
(ii) the cost of the break is determined by the speed the escalator is moving at the end of the journey — the delay is kw/(e+w) where k is the break duration, w is your walking speed, and e is the escalator speed at the end point of the journey.
(b) In a fixed-distance problem:
(i) if you do take a break, the cost depends on where you are standing when you take the break – the faster the escalator, the lower the cost of the break.
(ii) the cost of the break is kw/(e+w) where k is break duration, w is walking speed and e is the escalator speed at the point where you are standing when you take the break.
Bennett (#30) wrote:
“Compared with Bob [who travels the whole distance on stairs], a break on the stairs… is CHEAPER for Carol because she gets to make up for it on an escalator.”
I maintain this is incorrect.
Well, yes, it certainly becomes incorrect if you choose to quote only part of it. The full quote says that the break on the stairs is cheaper for Carol for one reason, costlier for Carol for a different reason, and that the two effects cancel out.
Bennett (#31):
To answer your question in #29:
A “fixed distance” version of the problem is where the stairs and elevator are each located over a fixed distance, but neither of them change speeds over the whole duration of the problem. (e.g. stairs from Floor 1 to Floor 2 and elevator from Floor 2 to Floor 3.)
This does not answer my question, because it contains the undefined term “the problem”. Problem A (the problem in the video) compares two people, each of whom is exogenously confined to an staircase or an escalator, and asks who pays more for taking a break. Problem B concerns one person, traveling first on stairs and then on an escalator, and asks where that person should take a break. There are also problems C, D and E. I have trouble following you when you talk about “the problem” and don’t tell me what problem you’re addressing.
Bennett:
Let’s clearly specify the problem and see where our alternative theories lead us:
Alice travels on an escalator from point A to point B. Bob travels on a staircase from point A to point B. If Alice takes a two-minute break, she adds one minute to her travel time. If Bob takes a two-minute break, he adds two minutes to his travel time. The quesion is: What accounts for that difference?
Theory One: It’s because Bob is male and Alice is female. Breaks are always costlier for males.
We can debunk Theory One by changing the setup so that Alice takes the stairs and Bob takes the escalator. Then Theory One makes the wrong prediction.
Theory Two: It’s because Bob took a break on the stairs and Alice took a break on the escalator.
We can debunk Theory Two by changing the setup so that the stairs are scheduled to start moving (and/or the escalator is scheduled to stop) at a particular time. Now Theory Two makes the wrong prediction.
Theory Three: It’s because, during the time when Bob is making up for his break (i.e. after he would have reached the destination if not for that break), he’s on a staircase, while during the time Alice is making up for her break, she’s on an escalator.
I claim there is no debunking of Theory Three analogous to the debunkings of Theories One and Two. If you disagree, please tell me what that debunking would consist of.
My debunking of Theory Three is any fixed-distance version of the model where you take a break on anything other than the final segment of the journey.
In particular: Bob travels from floor 1 to floor 3 on a staircase. Alice travels from floor 1 to floor 2 on a staircase and floor 2 to floor 3 on an escalator. Both Alice and Bob take a break between floor 1 and floor 2. I can quote directly from Theory Three: “During the time when Bob is making up for his break (i.e. after he would have reached the destination if not for that break), he’s on a staircase, while during the time Alice is making up for her break, she’s on an escalator,” which implies that the break should be more expensive for Bob than for Alice. But it’s not; it costs the same for both of them.
My Theory Four as stated is:
– In all fixed-time variants of the problem, if you take a break, the cost is the same no matter where you take it, and the cost is kw/(e+w) where k is break time, w is walking speed, and e is the escalator speed *at the exit point of the system*.
– In all fixed-distance variants of the problem, if you take a break, the cost depends on the moving speed of the system where you are standing when you take the break, and the cost is kw/(e+w) where the e is the escalator speed where you take the break.
Bennett: I do not buy your “debunking” of Theory Three.
Your Alice and your Bob travel from Floor 1 to Floor 2 on staircases. Absent breaks, they would reach Floor 2 at time 0.Instead, they reach floor 2 at (say) time 10. At time 10, Bob is still 200 yards behind his ghost. Alice, however, is MORE than 200 yards behind her ghost, because her ghost boarded the escalator at time 0.
So for Bob, “making up for his break” means traveling 200 yards on a staircase. For Alice, “making up for her break” means traveling MORE than 200 yards on an escalator. The fact that Alice ends up paying the same price as Bob EVEN THOUGH SHE HAS MORE DISTANCE TO MAKE UP FOR is due entirely to the fact that she gets to make up that distance on an escalator.
Indeed, if the escalator stops moving as soon as Alice boards it, then her break costs her MORE than Bob’s break costs him, because she has sacrificed progress while breaking AND sacrificed the chance to board the escalator while it was still moving.
A break sets you back a certain distance. The cost of the break is the time it takes you to cover that distance. That time will be shorter if you cover the distance on an escalator.
Your example is misleading because you’ve created an environment where Alice’s and Bob’s breaks set them back by different distances. But that misses the whole point of the puzzle.
The puzzle is: If Alice takes the escalator and Bob takes the stairs, then their breaks set them back by the same distance, so how can one break be costlier than the other?
It is easiest to grasp this puzzle in an environment where Alice’s and Bob’s breaks do indeed set them back by the same distance.
Your experiment poses pretty much the same puzzle in a less clear way; you are essentially asking In such-and-such a setup, Alice’s break and Bob’s break set them back by different distances, so how can both breaks be equally costly?
This has the same answer as the puzzle I prefer to pose, but I think it is a little harder to explain, to no good additional purpose.
Steve (#25) and Bennett (#27):
I was not clear. What I mean is that I have an unsolvable uncertainty in my rankings of preferred charities. I will speak for myself here, but I suspect the same will be true for lots of other individual donors.
To start somewhere, let’s say I am trying to evaluate protecting free speech in the US vs saving starving children in Africa. Even if I had some unrealistic belief that quelling free speech in the US will stop all progress on all problems all over the world, it would be impossible for me to accurately assess the actual threat it is facing. Since humans have a tendency to inflate problems we have recently thought about, even at my most rational, I could not trust my own assessment of that threat.
Then I have the problem that even if I had a good reason to put a problem on top of the list, I don’t know if the charity I have selected to fight that problem is effective. If I had perfect knowledge about how they spend their money, I might prefer to donate to a charity fighting problem #2 on my list if they would make much better use of my donation. Even with perfect knowledge, I would have some difficult comparisons to make. How many mosquito nets equal one new well?
Even though my confidence in a particular problem-charity selection is low, I do have reason to believe that at least some of the problems on my short-list is worth troubling with, and at least one of the charities on my short-list for each problem is effective at fighting it. I still say a mixed strategy is better when faced with this amount of uncertainty.
I gave the 55% as an example where given my superficial* analysis, I might want to use a weighted strategy instead of an even one, but most of the time I suspect donating to each charity on my short-list evenly is the best I can do.
*For us amateurs – I’m neither a full-time donor working for a fund or a humanities major – the analysis will always be superficial.
Henri Hein (#37): Of course there is always uncertainty in anything you do. If you are comparing mosquito nets to wells, you cannot know for sure whether a dollar’s worth of netting is more valuable (by whatever standards you apply) than a dollar’s worth of wells.
But at some point you have to say, “okay, given my uncertainty, and given how I see the probabilities, is it better to buy the netting or to buy the well?”. You answer that question. You act accordingly. Now you decide to give a little more, so you ask exactly the same question again, with exactly the same uncertainty you had yesterday. Why should the answer change?
Steve (#38):
Two problems.
1. If I estimate the relative value of nets vs. wells at 50/50 (or 1:1), I don’t see it matters which one I pick. I could choose nets or wells the first day and nets or wells the second day.
2. Sure, the very next day, things would not have changed. It’s likely 6 months or a year later that I donate. In that timeframe, I *will* expect the world to have changed. I have no reason to believe it will have changed in favor of my previous choice, neither do I have a reason to believe it will have changed in favor of the choice I passed over. I don’t want to spend 20 hours on research every time I write a check. So I’m back to point #1.
Any journey of a fixed or variable distance that consists of a series of steps of fixed or variable length escalators and/or fixed or variable length stairs can be described by a series of steps each of which has one of the following formats (Walking pace and escalator speed assumed to be constant for simplicity):
– rest on stairs for n minutes
– rest on escalator for n minutes
– walk up stairs for n minutes
– walk up escalator for n minutes
For example Bob’s journey could be described as:
1: rest on stairs for two minutes
2 : walk up escalator for one minutes
For any walk that contains a step of the format ‘rest on escalator for n minutes’ if one substitutes for that step ‘rest on stairs for the same number of minutes’ then in all cases total journey time will increase. Similarly for any journey that contains a step of the format ‘rest on stairs for n minutes’ then in all cases if one substitutes ‘rest on escalator for the same number of minutes’ then the total journey time will decrease. The conclusion can only be that motion while resting gives an opportunity to make progress compared to remaining still while resting.
The issue I see with the examples of a fixed length journey where an escalator turns to stairs at a fixed point in time (or vice versa) is that the steps that describes the journey where one rests on the escalator and the steps that describe the the journey where one rests on the stairs are so different from each other that it becomes very confusing to do a meaningful comparison (as the discussion between Bennett and Steve possibly shows). However If one takes the step by step description of each of these journeys and does the substitutions described above then one gets the same results and I contend this shows the benefits of moving well resting is a universal rule.
Rob (#40) – do you agree or disagree with the following statement: In any fixed-time version of the problem (that is, the entire system is an escalator, which is moving for some fixed time period, and stationary for some time period), then assuming you take a break for 1 minute, you arrive at your destination at the same time regardless of whether you take the break while the escalator is moving or whether it’s stationary. This is why I’m saying that the rule of “You might as well take a break on the escalator because at least you’re still getting somewhere”, is faulty — because in the fixed-time variants of the problem, it leads to the wrong answer.
yes, I agree that in what you call the fixed-time version of the problem one arrives at the destination at the same time regardless of whether you take the break while the escalator is moving or whether it’s stationary.
But I disagree that it disproves the “You might as well take a break on the escalator because at least you’re still getting somewhere” for the following reason. Taking a situation where the stairs become an escalator after 3 mins and I take a break on the stairs my steps might be:
1. Break on the stair for 2 mins
2. Walk on the stairs for 1 mins
3. Ride on the escalator for 2 mins
If for step 1 rather than resting on the stairs I had rested on an escalator and if the other steps had stayed the same then its clear that I will able to reach my destination sooner (I will not need to ride the escalator for the full 2 mins in step 3 to reach the end of the journey).
I know that switching step 1 in this way means that the description ‘the stairs become an escalator after 3 mins’ is no longer true but I don’t think that matters. The fact that switching the rest in step 1 from stairs to escalators leads to a faster journey must be (I claim) because “If you take a break on the escalator at least you’re still getting somewhere” is correct.
Your claim seems to be based on the fact that the fixed-time version of the problem might lead to the following alternatives:
1. Rest on the stairs for 2 mins
2. Walk on the stairs for 1 mins
3. Ride on the escalator for 2 mins
or
4. walk on the stair for 3 mins
5. rest on the escalator for 2 mins
give the same journey time.
But (I contend) these 2 alternative journey cannot be compared in a way that sheds any light on the issue at hand as they share little in common. If however one switches the rest in step 1 from steps to escalator then the journey time falls and if switches the rest in step 5 from escalator to stairs the journey time increases. I believe this demonstrates the truth of the “taking a break on the escalator is better because at least you’re still getting somewhere” view.
Steve (#29, #36, etc.):
Perhaps the reason we’re talking past each other is that I’ve been trying to state the simplest *formula* for what the cost of the break is, and you’re trying to state an *explanation* for why the cost is what it is.
(1) By my understanding, you agree with me that in the generalized version of the model (including both fixed-time and fixed-distance variants): (a) in a fixed-time model, the delay caused by the break is exactly kw/(e+w) where e is the escalator speed *at the end point of the entire system*; and (b) in a fixed-distance model, the delay caused by the break is kw/(e+w) where e is the escalator speed where you are standing when you take the break (which may be 0 if you are on a “stairs” segment). Is that correct? I would assume you agree with that, because that’s just algebra.
(2) When you say: “The cost of your break depends not on what’s beneath your feet when you take the break, but on what’s ahead of you” — in a generalized multi-segment version of the model (e.g. stairs from floor 1 to floor 2 and escalator from floor 2 to floor 3), does your statement become “The cost of your break depends on what’s ahead of you IN THE CURRENT SEGMENT”. In other words, the reason people don’t take breaks on the stairs segment is because there are more stairs ahead of them in the current segment. (And in *other* other words, I haven’t “disproven” your model by the fact that there is an escalator even further ahead of the person, because you’re using “ahead of you” to mean only in the current segment.)
If that’s the case, then we’re saying the same thing. Because of course “there are stairs ahead of you in the current segment” is the same thing as “you are standing on stairs”.
To express my point about charity selection more clearly, I want to illustrate it with a model.
Let’s say we have 26*26 charities. Organize them with an alphabetical designator ordered by my preferred criteria: ‘AA’ is my most preferred charity, ‘ZZ’ the least. I’m also going to give them a cardinal effectiveness rating between 0 and 1. 1 would mean they work towards the world-problem that worries me the most as efficient as humanly possible. 0 would be something like ‘send overpriced flowers from sister’s floral boutique to children in hospitals that may or may not be sick and/or allergic to pollen.’ The effectiveness rating is not crucial to the point, but will help illustrate.
I select 4 charities that I consider donating to. If I had perfect knowledge, I would select the first 4 charities according to my criteria. They will look something like this (effectiveness rating in parenthesis):
AA (0.9982), AB (0.9972), AC (0.9962), AD (0.9949)
I don’t have perfect knowledge. Even spending many hours researching these 26*26 charities, I doubt I have as much as 1% of the datapoints required to have this perfect knowledge. In reality, my list will look more like this:
GQ (0.7497), BC (0.9530), PO (0.4110), FQ (0.7856)
They are unordered on purpose. I just selected them as my shortlist. Now I have to pick one to donate to. I could use the strategy that Steve recommends: pick one based on available data to me at the time and send all my donations to it. I could use a mixed strategy: either pick one at random every time I donate, or round-robin between them. If I’m only concerned about optimizing for the probability that my money will be put to good use (again, according to my own criteria), it doesn’t matter which of these strategies I pick.
If I am also concerned that my donations all go to the near-dud that is PO, a mixed strategy is better. I think this is a reasonable concern to have. Since the mixed strategy doesn’t reduce expected return, I claim it’s the better one.
You could say I should continue to do research until I have eliminated all the worst charities from my short-list. As an amateur, I respond this is an unreasonable expectation. Not only do I not have that much time to spend on researching, I’m not trained and do not necessarily have the tools to find the required data. Speaking and listening to other people about how they donate, I am also pretty confident in saying that a lot of people spend a miniscule amount of research before donating.
Henri Hein (#44):
If I am also concerned that my donations all go to the near-dud that is PO, a mixed strategy is better. I think this is a reasonable concern to have. Since the mixed strategy doesn’t reduce expected return, I claim it’s the better one.
This depends entirely on what you mean by “better”. If the better strategy is the one with the best chance of making the world a better place (according to your own criteria) then the best strategy is certainly to bullet everything on the charity with the highest rating (according to your own current knowledge).
If you’re concerned about something other than the good that you’re doing for the world — in other words, if your motivations are not strictly charitable — then (and only then) some other strategy might be better.
Bennett (#43):
does your statement become “The cost of your break depends on what’s ahead of you IN THE CURRENT SEGMENT”.
No. My statement is exactly what it’s always been: The cost of your break depends on what’s ahead of you WHEN YOU MAKE UP FOR THE BREAK.
You keep insisting on talking about “the problem” without specifying what that means. What **I** mean by “the problem” (in this instance) is “Why is a break costlier for Bob, who takes the stairs from point A to point B than it is for Alice, who takes the escalator from point A to point B?”
One way to state the answer is that Alice gets more time on the escalator than Bob does. Another way to say the same thing is that Alice makes up for her break on an escalator while Bob makes up for his break on the stairs. I’ve learned from experience that most people understand this better when you say it the second way rather than the first. Your experience might differ, but otherwise I cannot figure out what your objection is.
Edited to add: My comment 49 below is an extended version of this comment; if you read that one, you can skip this one.
Henri Hein (#39):
It’s likely 6 months or a year later that I donate. In that timeframe, I *will* expect the world to have changed.
Agreed completely. But it doesn’t address the question of why you’d contribute to both projects on the same day.
Henri Hein:
The point you seem to keep overlooking is this:
Jack and Jill both agree that Kitten #1 is more deserving than Puppy #1. They both agree that there’s some uncertainty about this. THey also both agree that Kitten #2 is exactly as deserving as Kitten #1.
Jack says: I want to do the most good I can for the world. I believe my best shot at that is to save Kitten #1 instead of Puppy #1. I might be wrong about this, but it’s my best shot. So I will save Kitten #1 instead of Puppy #1, and for the exact same reason I will save Kitten #2 instead of Puppy #1.
Jill says: I want to feel good about myself. I believe my best shot at this is to save Kitten #1 instead of Puppy #1. I might be wrong about this, but it’s my best shot. But if I also save Puppy #1 instead of Kitten #1, then I increase the chance that at least one of my contributions was the best contribution possible, and that makes me feel better about myself. So I will save Puppy #1 instead of Kitten #2.
If your goal is to maximize the good you’re doing for Kittens and Puppies, you bullet. If your goal is to maximize the probability that one of your contributions was optimal, you diversify. The charitable impulse is the former; the selfish impulse is the latter.
I do not object to your selfish impulse if it gets you to save some kittens and/or puppies who you might not have saved at all. But that impulse is surely not consistent with pure charity.
Bennett: Let me spell this out in excruciating detail. Feel free to pinpoint whatever you disagree with.
1) If you take a break, your “ghost” reaches the destination before you do.
2) Set D = the distance between you and your ghost at the moment your ghost reaches the destination.
3) Set T = time difference between your ghost’s arrival and your arrival.
4) Set R = your average rate of travel between your ghost’s arrival and your arrival.
5) Note that T=D/R.
6) Alice is on an escalator. Bob is on a staircase. THE PROBLEM is defined to be the question “How can it be that T_Alice differs from T_Bob?”.
7) In view of 5), the answer must be one of the following (or both):
THEORY ONE: D_Alice differs from D_Bob or
THEORY TWO: R_Alice differs from R_Bob.
8) Most people’s intuition favors THEORY ONE.
9) THEORY ONE is false. In fact D_Alice=D_Bob.
10) Therefore the correct answer must be THEORY TWO.
11) R = M + W where M is the average speed of the medium you’re traveling on (after your ghost has arrived) and W is your average walking speed during that time.
12) We have assumed that Alice and Bob have the same walking speed when they walk, and we have assumed that neither takes a rest after their ghosts depart. Therefore W_Alice=W_Bob.
13) In view of 11) and 12), the difference between R_Alice and R_Bob must be due a difference between M_Alice and M_Bob.
14) In other words, the correct answer to “How can T_Alice differ from T_Bob” is THEORY TWO, which comes down to “M_Alice differs from M_Bob”, which can be reworded as: AFTER THE GHOSTS DEPART (i.e. during the time you are making up for your break), Alice is on an escalator and Bob is on stairs. And that’s all that matters.
15) QED.
16) You have posed some other problems (all of which, confusingly, you refer to as “THE PROBLEM”). In some of those problems, D_Alice and D_Bob are different. In those problems, the difference between T_Alice and T_Bob can be partly attributable to a difference in D_Alice and D_Bob and partly attributable to a difference in R_Alice and R_Bob, to different extents. That’s because different problems have different answers. None of those problems is the problem I addressed in the video.
17) Yes, it is quite true in all these problems that all that matters is the amount of time you spend on the escalator. For many purposes it’s a useful observation. I do not think it’s nearly as useful as the presentation above for explaining why THEORY ONE is wrong and THEORY TWO is therefore right in the problem I was actually addressing.
My understanding is that the question under discussion is “Why do people stand still on escalator but not on stairs”.
In scenarios where D_Alice=D_Bob you have proven that the reason must be down to the fact that this distance can be made up faster on escalators than stairs and so the cost is lower for escalator journeys.
But what of those scenarios where D_Alice!=D_Bob? In one of these cases of a fixed length staircase followed by a fixed length escalator and where Alice rests on the escalator and Bob on the stairs D_Alice is lower by the exact distance that she travels during her rest on the escalator. Her rest is lower cost than Bob’s
I must therefor conclude that at least in some cases the answer “Why do people stand still on escalator but not on stairs” is partially down to the fact that on escalators they are taking advantage of an opportunity to make progress.
Rob Rawlings (#50):
In scenarios where D_Alice=D_Bob you have proven that the reason must be down to the fact that this distance can be made up faster on escalators than stairs and so the cost is lower for escalator journeys.
Cool! We agree on what I think is the most interesting point.
But what of those scenarios where D_Alice!=D_Bob? In one of these cases of a fixed length staircase followed by a fixed length escalator and where Alice rests on the escalator and Bob on the stairs D_Alice is lower by the exact distance that she travels during her rest on the escalator. Her rest is lower cost than Bob’s
Agreed also.
I must therefor conclude that at least in some cases the answer “Why do people stand still on escalator but not on stairs” is partially down to the fact that on escalators they are taking advantage of an opportunity to make progress.
You have to be very careful about what you mean by this. Suppose Carol takes a fixed-length staircase followed by a fixed-length escalator. And suppose Carol takes a break on the stairs. Then at some point, her ghost is going to board the escalator before she does, and get farther ahead of Carol because it is on an escalator while she’s still on a staircase.
You want to say (I think) that that’s an example of Carol sacrificing an opportunity to make progress.
I want to say that’s NOT an example of Carol sacrificing an opportunity to make progress WHILE SHE IS RESTING. It is instead Carol sacrificing an opportunity to make FUTURE progress (during the time that here ghost is already on the escalator and she is not).
So I maintain two things:
1) Even for Carol, her rest on the stairs is not costlier than a break on the escalator because of the missed opportunity to make *current* progress. It is costlier because of the missed opportunity to make FUTURE progress. This missed opportunity shows up partly as a change in D and partly as a change in R. But it’s still all about what’s AHEAD of her at the time of the break, not what’s beneath her feet at the time of the break.
2) For an audience that is trying to wrap its head around this for the first time, it is best to focus on Alice and Bob while ignoring Carol, whose situation is somewhat more complicated. But the bottom line — what matters is what’s AHEAD of you — still applies to Carol as well, though in a somewhat more complicated way than it applies to Alice and Bob.
Henri Hein (#44): Let me say this one more way.
Alice cares about how many kittens and puppies are saved.
Bob cares about how many kittens and puppies *his money* saves.
They both believe there’s a 55% chance that it’s better to save a kitten than a puppy.
When they save a kitten, they both risk making a mistake.
For Alice, the cost of that mistake is that she’s saved a puppy when she could have saved a kitten. But she minimizes that sort of cost by saving the kitten, and for Alice that’s all that matters.
For Bob, the cost of the mistake includes the possibility that HIS MONEY has not saved any puppies, even when it should have.
Notice that Alice CAN’T avoid that cost by diversifying, because she doesn’t care where HER money goes; she cares only about where the total of ALL money goes — and given the vast number of other contributors, there is no risk of no puppies being saved.
So Alice bullets, Bob diversifies. Alice cares about the help the kittens and puppies get; Bob cares about the help the kittens and puppies get FROM HIM. I have no difficulty calling Alice’s motives charitable and Bob’s at least partly egoistic.
This is all clearer if you do the math—-the expected value that comes from small contributions to a large charity is linear in those contributions, and therefore maximized when you bullet. The expected value that comes from YOUR contributions is non-linear in those contributions and therefore maximized when you diversify.
@Steve 51
I’m not quite grasping the intuition but am trying to understand the logic which I take to be the following follows:
– D_Carol when Carol rests on the stairs is greater than D_Carol when Carol rests on the escalator.
– When Carol rests on the stairs her ghost carries on walking so when Carol resumes walking her ghost is already walking up the escalator and this (Carol walking on the stairs while her ghost walks up the escalator) is what creates the greater D_Carol.
– This opportunity for a greater D_Carol would not exist for a stairs only journey so therefore it must be the existence of the future escalator journey that increases the costs of a stairs rest in this scenario.
Am I in the right ballpark here ?
rob: yes! this is not exactly how i’d have chosen to word it, but i believe your words are capturing the way i want to think about this.
Steve (#47):
> But it doesn’t address the question of why you’d contribute to both projects on the same day.
I never suggested contributing to both projects the same day.
Steve (#48):
> If your goal is to maximize the good you’re doing for Kittens and Puppies, you bullet. If your goal is to maximize the probability that one of your contributions was optimal, you diversify. The charitable impulse is the former; the selfish impulse is the latter.
What if the goals are not in conflict? If I have perfect information, they are not in conflict. I would select the ‘best’ one, which meets both goals. If I have no information, they are not in conflict, since my choice is going to be arbitrary anyway.
Steve (#45):
> If the better strategy is the one with the best chance of making the world a better place (according to your own criteria) then the best strategy is certainly to bullet everything on the charity with the highest rating (according to your own current knowledge).
But I don’t know which one that is. I don’t know how to say it more clearly. Not only don’t I know which one that is, I cannot know which one it is.
Steve (#52):
> They both believe there’s a 55% chance that it’s better to save a kitten than a puppy.
I don’t have any problem with the logic when there is a clear indication one is better than the other. My question is what to do when there is no information, such that the choice is effectively 50/50. (Or more realistically: not enough information to decide between kittens and puppies).
Henri (#58):
My question is what to do when there is no information, such that the choice is effectively 50/50.
Then you might as well flip a coin and bullet accordingly. If your motives are purely charitable, there is still absolutely no reason to diversify.
Here is what I hope is a more elegant explanation for why (in the fixed stairs followed by fixed escalator scenario) a rest on the stairs is more costly than a rest on the escalator.
Its because a rest on the stairs leads to a greater D_Carol and its this that need to be explained.
It can’t be because when Carol rests on the the escalator ‘at least she is moving’. She and her ghost are level at the end of the stars so this scenario is now exactly the same as the escalator-only version that Steve covers in #49
It can’t be because when Carol rests on the stairs she is ‘giving up an opportunity to make progress’ because during her rest D_Carol only increases to the same level as in the stairs-only version of the problem which again is covered by Steve in #49
It’s therefore must be something else. When Carol rests on the stairs she cedes to her ghost a later opportunity to travel on the escalator while Carol is still on the stairs. This event lays in the future at the time of her rest and is what leads to the increase in D_Carol that needs to be explained.
So I now agree that Steve is correct to claim “its all about what’s AHEAD of her at the time of the break, not what’s beneath her feet at the time of the break.”!
Rob: Your latest post is great. I agree with it 100%. Thanks for the added clarity.
Rob (#60) and Bennett:
Here is a riff on Rob’s formulation which I think should be the final word on this, though maybe one or the other of you will prove me wrong.
Alice travels from location 0 to location 1 on a staircase/escalator, any part of which might or might not be moving at any rate at any moment.
For each time t, and for each location x (between 0 and 1), set r(x,t) be the rate at which this contraption is moving at location x and time t.
Note that this general formulation includes as special cases all of Bennett’s “fixed-time” and “fixed-distance” scenarios, as well as a great variety of other scenarios.
Alice has a fixed walking speed w. She takes a two minute break, which ends at time, T0, when Alice is at location X0.
I ask you the question: What was the cost of this break — by which I mean “How much time did this add to Alice’s journey from 0 to 1?”.
Which values of the function r(x,t) would be useful to you when you attempt to answer this question?
Claim I:All values with X less than X0 and T less than T0 are irrelevant. You can compute the answer to the question without knowing these values.
Claim II:A quite fair way to state Claim I is that it’s quite entirely irrelevant whether the stairs are moving at the time and place where Alice takes her break. All that matters is what’s ahead of her.
Steve @62
Wouldn’t r(x,t) for the period before X0 but after Alice has started her break be relevant to the cost of the break?
For example: If the section of the escalator Alice would have been on just after she starts her break (but which she is not on because of the break) starts to move faster than the bit Alice is on, wouldn’t this be a factor relevant to the cost of her break and that occurs before X0 ?
In other words: Alice’s ghost can be further ahead than Alice could walk in 2 minutes if the escalator can change speed during Alice’s break for the section the ghost is on but not Alice and this I think affects the cost of the break.
Note — This has been edited again. I had some less-thans and greater-thans mixed up, among other things. I think it’s right now.
Rob: Ok, I think you are saying this:
At time T0-2, Alice has reached location X0-H and begins her break. The break lasts till time T0, at which point the (possibly moving) stairs have carried Alice to location X0. So her path is indicated by the black line (which is drawn as a straight line for convenience only).
If Alice forgoes her break, then her path becomes the red line.
Take as given the values of r(x,t) outside the blue oval. Then in Scenario One, where the values are low inside the blue oval, the red line deviates some amount from the black line. In Scenario Two, where the values are high inside the blue oval, the red line deviates substantially more from the black line. This deviation affects the cost of Alice’s break, so that values inside the blue oval — where X is less than X0 — do matter.
[Though they matter in a complicated way, because it’s possible that toward the upper right the red line passes through regions of low r(X,T) and the black line through regions of high r(X,T)….]
This means that my claim as stated wrong in general (though still right in the specific case where the stairs are completely stationary throughout Alice’s break, so that X0-H=X0 and the problem area does not exist).
The right general statement should be something like this: Let Alice’s path be given by x=f(t). Then the only values of r that matter are at points where x is greater than f(t). In other words, all that matters are conditions that are ahead of Alice at the time when those conditions occur.
Bennett (if you’re still reading) (and Rob and others if interested): Maybe this will help.
We will always have you travel a fixed distance from Floor Zero to Floor One (the “first segment”), and then the same fixed distance from Floor One to Floor Two (the “second segment”). We will always have you take your break on the first segment.
Your walking speed is 100 yards per minute.
A) Suppose the first segment is a non-moving staircase. You break for two minutes.
A1) If the second segment is a non-moving staircase, your break costs you two minutes (that is, it adds two minutes to your travel time).
A2) If the second segment is a moving escalator, your break costs you two minutes.
A3) If the second segment is a moving escalator that at some point stops moving, your break can easily cost you more than two minutes, because you miss your chance to ride the escalator.
Conclusion: the cost of your break depends on what’s ahead of you. If what’s ahead of you moves at a fixed speed (be it a motionless staircase or a standard escalator) the cost of your break is two minutes. If it moves at a variable speed, the cost of your break is something else.
What matters is not the *speed* of what’s ahead of you, but the *variation* in that speed. But it’s still all a matter of what’s ahead of you. I think you might have confused yourself by comparing a staircase with a standard escalator, both of which move at constant speeds, and therefore both of which end up costing you the same amount. But that’s only because the map from (what’s ahead of you) to (cost of break) is not one-to-one; that doesn’t mean it’s a constant map.
B) Segment One is an escalator. You take a two-minute break, all on the escalator. Note that for the duration of that two-minute break, there is an escalator ahead of you, regardless of what’s on Segment Two.
B1) If Segment Two is a staircase, the cost of your break is less than two minutes. Comparing this to A1), you might say “Ah! The only difference between A1 and B1 is that in A1 I took my break on a staircase and in B1 I took it on an escalator!” And the costs of those breaks were different! Therefore it does matter where I take my break!
But I say that is wrong — there is another difference between A1 and B1. That difference is that in A1, there is always a staircase ahead of you, whereas in B1, there is an escalator ahead of you for the duration of your break, and a staircase ahead of you thereafter. And I claim it’s that difference that accounts for the different cost of the break.
B2) Segment Two is an escalator. Now the cost of your break is the same as in B1). This is again because the escalator and the staircase both move at constant speeds. If we replaced them with something that moves at a variable speed, your cost would be different. The cost of the break depends entirely on what’s ahead of you.
I hope this at least makes clear what I’ve been trying to say.
I finally grasped Steve’s point on this issue after his post #49 and now no longer think that a rest on a escalator compared to a rest on stairs gives benefits because ‘at least you are moving’ but always due to other factors.
However after my #63 and Steve’s #64 I now question whether ‘the cost of your break depends on what’s ahead of you’ is always true. Taking A3 from #65: If my 2 minute break takes me exactly to the end of segment 1 and if segment 2 is an escalator that stops moving 1 minute after the time I start my break then I have missed an opportunity to ride that escalator (as I would have reached segment 2 one minute earlier with no break). This chance occurs during my break and not ahead of it. I end my break further behind my ghost than the ghost walked during those 2 minutes.
Its true that one piece of the cost (the speed of the catch-up leg) always occurs after your break has ended but I think that other costs and/or benefits can occur during the break.
Rob (#66): But that extra cost, even if it occurs during your break, is still occurring in a location you haven’t yet reached — in other words, in a location that’s ahead of you. No?
Steve (#67)
Its ahead of you in terms of the space you cover including your rest, but not in terms of the space you would have covered with no rest – so I’m not sure how to pin it down in time. The fact that I end my break further behind my ghost than the ghost walked during the break feels relevant to the issue but I’m really not sure.
However the main point is that any gains/losses incurred as a result of the speed of the stage 2 escalator at the point you reach it are entirely unrelated to any idea of ‘at least you are moving while resting’ or ‘you’re sacrificing a chance to progress’ and entirely down to the configuration of the various stages of the journey.
It’s too bad nobody could get into this one.