Showing posts with label new york effing city. Show all posts
Showing posts with label new york effing city. Show all posts

Tuesday, March 31, 2009

Elevator Action

Dear Dr. Math,
How do I figure out whether to take the elevator or the stairs? The elevator is faster, when it comes, but sometimes I think I end up waiting longer than it would have taken me to just walk.
Regards,
StairMaster


Excellent question, SM, but we need to clarify what criteria we're using to decide between the two options. Some people might generally prefer the stairs because they enjoy the exercise, or maybe they're worried about the possibility of being stuck alone in a metal box for 41 hours. Some people like the elevator because it's more social, and you can do that thing where you jump at the very end and it feels like you're floating. But from the way you asked, I'm assuming you're just trying to minimize time, and that's all you care about. By the way, what's the big hurry? Take time to enjoy the little things, SM; they're all we have.

To actually answer the question, we need a model for all of the relevant quantities that we care about. To keep it general, I'll use variable names instead of hard numbers, and then we can take a look at some specific examples, and you can apply the theory to your own needs.

First, there's the stair option. Let's use S to denote the time it takes to walk. Since S doesn't really change much from trip to trip, we'll treat it as a constant. If you wanted to get more sophisticated, you could account for things like how many other people were trying to take the stairs, whether you were carrying something heavy, whether you could slide down the banister, etc.

The elevator option is the more interesting one. Let's let e be the shortest possible time the elevator could take, say, if it were already there waiting for you and didn't make any other stops. Similarly, there's a maximum time the elevator could take, if it was the greatest possible number of floors away from you and someone had pushed all the buttons, or something. Denote that time by E. (Upper case for the bigger time; lower case for the smaller one.) Again, we're treating e and E as known quantities and as constants; I encourage you to measure them sometime. If S < e, it's always faster to take the stairs, no matter what. If E < S, it's faster to take the elevator, even in the worst case. The remaining possibility is that e < S < E , so sometimes one is better, sometimes the other. It sounds like that's the situation with your elevator, SM, so we'll take it as a given.

Since we don't know the actual length of time the elevator would take, we have to treat it as a random quantity with a value somewhere between e and E. Let's call the actual time T. Here again, we need to consider what information we might have about T that could tell us what kind of probability distribution is reasonable to associate to it. For example, should we expect it to usually be closer to its minimum possible value, e? That would make sense if not many other people used that elevator and it usually hung out on the correct floor--say, if we were trying to go up from the ground floor to the 7th floor in an apartment building. On the other hand, if we were trying to go down from the top floor of a busy office building, it might be more reasonable to expect T to be closer to its maximum value, E, more of the time.

In the absence of any other information, we'll assume the distribution of T is uniform on the set of times between e and E, meaning it's just as likely to be any value as any other. Another way of saying this is to say that the probability that T is less than any given value, say x , is proportional to the difference between x and e. For x = e, the probability is 0, since e is the smallest possible time; for x = E, the greatest possible time, the probability is 1; for times in the middle, the probability is between 0 and 1. In pictures, the distribution looks like this (a = e, b = E):



As a result, the expected value, or average, of T is the number halfway in between its minimum and maximum possible values, that is, (e + E)/2. So, in the sense of minimizing expected values, you should take the stairs only if S < (e + E)/2; otherwise, you're better off waiting for the elevator, on average. As in my discussion of the lottery, though, the expected value may not be the only consideration. Since the time to take the stairs is a known quantity, it has a variance of 0, and that security may be worth some trade off in expected value. On the other hand, maybe you like to gamble, SM (I don't know you that well), in which case you might prefer the thrill of betting on the higher-risk, higher-reward elevator, even if the average time is slightly greater. Over many trials, though, you'd save time choosing the option with the smallest expected value.

Just to see how this would play out with actual numbers, I'll consider a scenario that I frequently encounter when taking the subway. (And you thought this was just about elevators!) Here, the role of "stairs" will be played by the #1 downtown local train, which makes frequent stops every few blocks, and the "elevator" corresponds to the #2 downtown express train, making stops only every few stations. Let's assume that I'm already on the local train at the 96th Street station, and I'm trying to get to Times Square as quickly as possible. My options are 1) stay on the local train, which will get to Times Square after some fixed amount of time, or 2) gamble on the express train, which might get there earlier or might not.

According to the schedule, it takes the local train about 10 minutes to get from 96th Street to Times Square. And the express train takes 6 minutes but only runs every 12 minutes (in the middle of the day). Therefore, the least possible time it could take is 6 minutes, and the greatest possible time is 18 minutes. Assuming the distribution of times to be uniform between these extremes gives us an average travel time of (6 + 18)/2 = 12 minutes, which is 2 minutes more than the local train. Hence, to minimize expected value, I should always stay on the #1. Here, the randomness of the travel time has more to do with how long I have to wait for the train than how long the trip will actually take, and it's somewhat reasonable to treat this as being uniformly distributed, since the train runs on a regular schedule (every 12 minutes) but I don't know what time it is currently relative to that schedule.

Of course, in practice the situation is more complicated than that. For example, there are actually two possible express trains I can take (the #2 or the #3), just as there might be two possible elevators you could take, and I'll take whichever comes first. If we treated these both as uniformly random variables, independent of each other, then the distribution of the time until the next train would not be uniform and would, in fact, have a smaller expected value.

For the numbers I gave above, it actually works out that the #2/#3 express option has the same expected value, 10 minutes, as the local train. So it's back to personal preference to break the tie. Another interesting variant is to consider how many people you see waiting for the train, elevator, bus, etc. and use that as a way to estimate the amount of time they've already been waiting. For example, if you see 3 people waiting and you know people tend to show up at the rate of 2 per minute, then you could estimate the time already waited at 1.5 minutes, reducing the maximum possible travel time by that same amount and perhaps tipping the balance.

Hope that helps some, and let me know whether the time you save turns out to be greater than the time spent thinking about the problem. I've got to go now to catch a (local) train.

-DrM

Monday, February 9, 2009

In the Big Apple, I prefer Honeycrisp.

I've been called out by Short Round over at alt85 again, concerning a recent article in The New York Times:

The article included one piece of information with direct relevance to the little people: "a new study from the Center for an Urban Future, a nonprofit research group in Manhattan, estmates that it takes $123,322 to enjoy the same middle-class life as someone earning $50,000 in Houston." [Tugs nervously at collar.] And since the average median* per-capita income in Houston in 1999 (according to houstontx.gov) was $20,101, and since the Urban Future people's figures would suggest that $20,101 in Houston is worth less than $49,578 in New York (for reasons that the newly returned Dr. Math could surely explain better than I,** unless he disagrees, in which case I challenge him to a duel)... Well, New York is f**kin' expensive. Not news.
Short Round


Sir, I accept!

So, I'm not generally opposed to the conclusion that New York ¢ity is an expensive place to live. (God knows I could use an extra $500K a year to spend on all those things that I've heard the city is supposedly famous for but that I'm too poor to experience.) The authors of this article seem to be basically assuming that conclusion from the beginning. In a sense, all this "news" piece is even claiming to do is put some quantitative weight behind a stereotype that we've all pretty much agreed on already. But since it involves numbers, I can't resist picking apart their methodology a little. The Devil, as always, is in the details:

First off, I had to do some considerable digging to even get to the original source of this email-forward-ready statement that $50,000 Houston dollars is equivalent to $123,322 Dollars New York ($NY). The Times article cites a report from the oddly-named Center for an Urban Future, which used a cost-of-living calculator from the CNN (yes, CNN) website, which had as its source material a survey done by the Council for Community and Economic Research (C2ER), in which they hired surveyors to sample prices from various cities they wanted to compare (more on that later). The (Center for an Urban Future) report is a 52 page document entitled "Reviving the City of Aspiration" about ongoing trends in the middle class of America, particularly in New York. One problem right off the bat is that the authors never precisely define what they mean by "middle class". They write, "In this study, we use ['middle class'] to indicate those who own homes or who have the prospect of becoming homeowners, earn at least in the middle quintile of wages and enjoy a modicum of economic stability." They then go on to wax poetic for a while about the important contributions middle class Americans make to society (including "providing the customer base for a wide mix of businesses across the city," adding to New York's "street life" and, somewhat circularly, owning homes). But setting aside the logical hiccup for a minute, it's still not clear from the definition who exactly qualifies as middle class. Rather, it's somewhat clear what the minimum standards are for membership--you have to own a home or have "the prospect" of doing so, earn "at least in the middle quintile of wages," which is sloppily phrased but I'm guessing means you have to earn more than at least 40% of people in the area, and have "a modicum" of economic stability, which they explain as being able to consistently pay your bills--but there seems to be no clear maximum standards. For example, would someone earning $250K per year in the 98th percentile be considered middle class, assuming he owned a house and could pay his bills (for monocle cleaning and storage)? Maybe, by the authors' definition, but certainly not by mine.

Now, if we trace this comparison-of-cities data all the way back to its source, the C2ER survey, we find an interesting disparity. The basic idea of the survey was to follow some sample of people around and make a log of the prices of all the things they paid for--clothes, food, entertainment, travel, etc.--to get a measurement of the relative cost of living in different places. However, in the guidelines for the survey participants, it says specifically that the authors are not looking for middle class consumers to follow around (they changed their original survey language because "it was too easily confused with 'middle class,' which isn't the same thing at all"); rather, they focus on a population they call "moderately affluent professional and managerial households", who are characterized as "a household consisting of both spouses and one child (for pricing apartments, it is assumed that the couple is childless or the individual is single)" with the criteria that "both spouses hold college degrees; at least one has an established professional or managerial career," and, most significantly, "household income is in the top quintile for the area" (emphasis mine). For most cities, they say that the household annual income should be "between $70,000 and $100,000;" however, as they say, "the appropriate income range will be higher in traditionally high-cost places like New York..." So our monocle-polishing Uncle Moneybags the hedge fund manager would be included in the survey.

What's the real problem with this? Apart from the fact that we've gotten, explicitly, pretty far away even from the ill-defined "middle class" of the Urban Future report, upon whose homeowning backs the street life of the city rests, we've also gotten into some shaky statistical territory, where I believe we're not even comparing apples to apples anymore, but rather something like apples to different kinds of apples (Fuji to Jonagold), to learn all about oranges. And also the middle class. I don't have any hard data to back me up here, but my sense from having lived in New York for a little while now is that, due to the presence of so many ultra rich celebrities and financiers, the shape of the distribution of incomes here is more heavily slanted towards the top ("fat-tailed," as they say), meaning not only is the average income higher, but the relative difference between the top 20% and those of us way down in the middle is considerably greater than in other U.S. cities. In pictures, the graph of incomes in New York is more like this:















than this:


















In the latter case (Houston), it doesn't take much more income to put you in the top 20%, but in New York, it takes considerably more. So, the potential gap in luxury lifestyles is exaggerated, and as a result, more especially luxurious opportunities open up for those who can afford them. There really just isn't a Houston equivalent of buying a $150 truffle and foie gras burger at Bistro Moderne or paying $75K per year for a personal driver or all the other outrageous things the Times article mentions.

Which all brings me all the way around to the point: that measuring what it costs to uphold a "standard of living" is an extremely difficult and subtle problem, one which requires a great deal of precision and care. And it may not really be possible when the markers of that standard vary so greatly from place to place. New York is a pretty special town with no real equivalent anywhere else in the U.S., and in fact, based on the ways we live our lives, renting instead of owning, riding the subway instead of driving, eating fancy burgers made out of goose liver... it may not even make sense to think of it as part of the U.S., despite its importance as a cultural hub.

Like the old song goes, "New York, New York, it's a pretty special town with no real equivalent anywhere else in the U.S., and in fact, based on the ways we live our lives, renting instead of owning, riding the subway instead of driving, eating fancy burgers made out of goose liver... it may not even make sense to think of it as part of the U.S., despite its importance as a cultural hub."

-DrM


P.S.--To Short Round: oddly enough, it seems that "average median" is correct there. In the report, they averaged together the median incomes of the various ethnic groups in the suburbs of Houston, presumably with some weighting. Hence, average median income. Weird.