Thought: Cinderella’s Incredibly Small Foot

3/26/15

By Kevin DeLuca

ThoughtBurner

Opportunity Cost of Reading this ThoughtBurner post: $1.58 – about 7.2 minutes

I went to see the live-action remake of the Disney classic Cinderella the other day, and as I watched the prince’s men search all across the countryside for poor old Cinderella, it occurred to me that Disney characters, and especially the Disney royalty, tend to be a bit inefficient. Cinderella’s fine prince was the perfect example of sub-optimal Disney behavior. He wanted to find Cinderella after the royal ball, so he ordered his men to take the left-behind glass slipper and try to put it on every single maiden in the kingdom. This is probably the most inefficient way to find someone that I have ever heard. Also, this plan only would have worked under very specific, and fairly unlikely, conditions, which I will explain below.

Of course, it is up to ThoughtBurner to advise the creative Disney “imagineers” on ways for them to not only provide the impressionable children of the world with lessons about love, life, and happiness, but also to teach the value of efficiency in the most economic sense. And yes, for those of you who were unaware, implicit in ThoughtBurner’s mission to improve efficiency and happiness in people’s everyday lives lies the responsibility of also improving the efficiency and happiness in imaginary people’s everyday lives.

Cinderella’s Incredibly Small Foot

The prince’s plan to find Cinderella was, as I mentioned above, to take the glass slipper around the countryside and try it on every woman in the kingdom. Whoever the slipper fit would be the woman that the prince would marry. The prince is making a huge assumption here: that no other woman in the kingdom has the same foot size as Cinderella. In the movie, it shows the prince’s men trying the glass slipper on many different women, and it never seems to fit! And, for most of the women the shoe is too small – this implies that Cinderella’s foot is very small. How small would Cinderella’s foot need to be in order for the prince’s assumption to be correct, you ask?

First, we need to know the distribution of women’s foot sizes. This is actually kind of hard to find, and I could only really find data on the distribution of Japanese feet online[i]. Because Cinderella isn’t Japanese, instead I found stats on female shoe sales [ii]– the data includes the quantity of each shoe of each size that was sold in the United State in 1998. If we assume that women buy shoes that fit their feet, then the distribution of the sizes of female shoes sold should reflect the distribution of women’s foot sizes fairly accurately. Below is the distribution of female shoes sold by size:

ShoeSizeChart1

Female shoe sales by size seems to be normally distributed, which is great for us because we know a lot about the properties of things that are normally distributed (makes using statistics really easy). As you may have noticed, all of the half-sizes are all systematically lower than their whole size counterparts, which makes the distribution less nice. I’m guessing that this has a behavioral explanation (people like to buy whole sizes more than half sizes, for some reason? Update: after I published this I heard from friends that some women’s shoe stores/brands don’t sell half sizes, which is a more likely explanation for the pattern we see). If you plot only the sale of whole sizes, it looks even more like a normal distribution:

ShoeSizeChart2

Alright so, now that we know the distribution of foot sizes, we can figure out how small (or big) Cinderella’s feet needed to be in order for the prince’s plan to work. Using the same numbers that I used to create the graphs (see data link at end), I calculated the average female foot size to be 8.076 with a standard deviation of 1.468. (These are women’s shoe sizes, not inches).

Now, before we can calculated the size of Cinderella’s foot we also need to know how many people were in the prince’s kingdom. Let’s first make a few more assumptions that will favor the prince’s plan. First, assume that the prince does not actually try the slipper on every woman in the nation; instead, he stops as soon as he finds Cinderella. Also assume that the first and only woman who fits the glass slipper is actually Cinderella. The prince doesn’t know anything about how early or late in the process of shoe-trying-on Cinderella will be fitted, so we’ll assume that every woman has an equal chance of being the next woman to try the shoe on. This means that the prince is expected to try to put the shoe on half of the women in the kingdom before he finds Cinderella.

Perhaps not surprisingly, there doesn’t seem to be much data on the population size or demographic characteristics of magical fairytale kingdoms. I will use the next best substitute, the imaginary medieval worlds of gamers, on which there apparently has been research done on the populations of towns, villages, and cities or kingdoms. Thanks to S. John Ross and his page with some guesses of the population density of kingdoms (which seem to be based on real cities[iii]), we can provide a range of estimates of the size of the kingdom that Cinderella lived in, and according to that we can see how small her foot needed to be in order for the prince’s plan to work.

Remember that about half of the population will be women, and that the prince only has to try the slipper on half of these women before he is expected to find Cinderella. In the chart below, I calculated how many women need to go through the slipper test before Cinderella would be found. Next, I found which percentile Cinderella’s shoe size would need to be in order for her to be the only person with that shoe size in that kingdom. Next, I converted the percentile to a z-score, and last I calculated how small (or big) her shoe size would have needed to be by subtracting (or adding) z-score-many standard deviations to the average shoe size.

CinderellaFootChart

As you can see from the chart, even with a ‘kingdom’ as small as 300 people, Cinderella’s shoe size would have to be in the bottom 1.3% of the population, and she would be a size 4.82. To put this into perspective, a woman’s size 4 is barely over 8 inches[iv], and most vendors don’t sell below a woman’s shoe size 4. If the kingdom had a population of around 50,000 (about one-sixteenth the size of Austin[v], maybe not unreasonable for a “kingdom”), Cinderella would have to have a shoe size in the bottom 0.007% of the population, which corresponds to a size 2.5 (which I think means she would have to buy kid shoes). If Cinderella had lived in New York City, with a population of 8.4 million[vi], she would need a shoe size of 0.88. While possible that Cinderella just had really really small feet and the prince just had a really really small kingdom, it seems highly unlikely that the prince’s plan would have worked. On top of that, the prince’s men are also going to be tied up traveling the kingdom. Assuming that these men are compensated for their work by the prince, I’m sure that the people of… wherever… would not be happy to know their taxpayer money was being wasted on such an inefficient manhunt.

I will now prescribe a much simpler way to find Cinderella, and show you that he could have found Cinderella much more quickly for only a fraction of the cost of his original method. Obviously, the prince should not try the shoe on every woman. All I suggest to the prince is that he test only blonde females in his kingdom.

(Note: In the new movie, Cinderella always has blonde hair. I was looking through pictures of the old Disney film online, and Cinderella’s hair color seems to change between blonde, dirty blonde, orange, and brown, and I can’t tell if this is representative of Cinderella’s true hair color or if we have another black-blue white-gold dress thing going on. All of the “modern” pictures of Cinderella[vii] go with a blonde-haired version. And according to this Cinderella wiki[viii], Cinderella’s hair color is “Strawberry-Blonde.” I will therefore continue with the assumption that Cinderella had blonde or a darkish blonde hair color, though I admit that if this assumption fails then my strategy would need to be slightly modified.)

How many women in the kingdom have blonde hair? This would normally be impossible to know, except for the fact that the prince actually invited every woman in the kingdom to his ball. This is perfect for us, because we have the entire population of interest trapped inside the palace for random sampling. Also, conveniently, Cinderella was the last one to arrive to the ball so we know that there is no selection bias (if you were worried that hair color was correlated to promptness or something). So, let us take a random sample of women from the royal ball and see how many are blonde. Here is a snapshot of the ball from the new film:

CINDERELLA

Notice anything? Including Cinderella, I count only 4 (possibly) blonde women out of the 19 pictured. Since this is a random sample of all of the women in our population of interest, we expect the mean from the sample to equal the mean of the population. So, in this kingdom only about 4 out of 19 women, or 21% of women, should have blonde hair. (If you use pictures of the ball from the old animated film, even less of the women are blonde.)

Simply by testing only blonde women, the prince would have spent only 21% of the original total cost if the rest of his plan had been exactly the same. In other words, I would have reduced his costs by about four-fifths with a super obvious modification to his strategy. To save even more on costs, he could have put out a royal decree saying something like: “I will compensate travel costs for any blonde woman who comes to my palace if she fits into this glass slipper.” Then, only Cinderella and other blonde women who think they might fit into the glass slipper would have any incentive to go the palace to try on the slipper. If Cinderella is the only one who fits the slipper, then the prince will just have to pay whoever takes Cinderella to the palace for the travel costs. If other women fit the slipper, then the prince will have to use something other than shoe size to identify Cinderella and may have to pay additional costs of transporting false positives. In either case though, I suspect it would be less expensive than having his men travel the countryside testing every blonde woman one at a time. (I know that this wouldn’t have actually worked in the movie since Cinderella is locked in the attic or whatever, but the prince’s original plan wouldn’t have worked either for the same reason. My method still has much lower expected costs.)

Also, by using my plan, the prince would have found Cinderella in about one-fifth the time it took him in his original plan, assuming no fixed time costs. If he wanted to find Cinderella as quickly as possible, he could have sent out a single man to each household with instructions to bring any blonde women in the household to the prince’s castle for a shoe fitting. Rather than taking the glass slipper to each house, just bring all the possible Cinderellas to the palace in one fell swoop and test the slipper on them. I’m not sure if this would be cost effective, but compared to his original plan he could probably use all the money I saved him to justify any additional costs of getting his Cinderella sooner. And isn’t that more romantic too? What princess wouldn’t love a prince who in addition to doing everything he could to find her as quickly as possible also did it while minimizing expected costs?! I can see the headline now: “Kingdom’s Tiniest-Footed Woman Marries Cost Minimizing Prince In Record Time!”

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[i] http://www.dh.aist.go.jp/en/research/centered/foot/

[ii] Footwear Impression Evidence – Page 191: http://books.google.com/books?id=xLVUjzkK3rgC&pg=PA191&lpg=PA191&dq=191&prev=http://print.google.com/print%3Fie%3DUTF-8%26q%3DThe%2BProfessional%2B%2522Shoe%2BFitting%2522&sig=asPShS7HyeGpZ-jWulrJSJIZh8E&hl=en%23v=onepage&q=191&f=false#v=snippet&q=191&f=false

[iii] http://www222.pair.com/sjohn/blueroom/demog.htm

[iv] http://www.shoesizingcharts.com/

[v] http://en.wikipedia.org/wiki/Austin,_Texas

[vi] http://en.wikipedia.org/wiki/New_York_City

[vii] http://www.disneystore.com/animation-movies-entertainment-cinderella-blu-ray-and-dvd-digitial-copy-combo-pack-diamond-edition/mp/1320018/1000316/

[viii] http://disney.wikia.com/wiki/Cinderella_(character)

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Data

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Thought: The “YOLO” Effect – Inappropriately Discounting The Future

3/12/15

By Kevin DeLuca

ThoughtBurner

Opportunity Cost of Reading this ThoughtBurner post: $1.46

You can hear the call coming from inside the gated frat house parties across college campuses every Friday night. The Lonely Island called it “the battle cry of a generation.”[i] Even in everyday conversation, hesitation, prudence, and caution are all greeted with a new, four-letter challenge: YOLO.

The expression “you only live once” (YOLO) is often meant to serve as a justification for risky or unwise behavior. The idea behind the message, however, is stronger than that. Because we only have one life to live – you only live once – we should not give up current opportunities. We should do what we want to do, when we want it, because you might not have the same options later as you do now (since we only live once). Similar phrases have captured this idea throughout history; “carpe diem”, “live life to the fullest”, and “live like there’s no tomorrow.”

Before I speak of the YOLO effect, I’ll introduce the common idea (in economics) of “discounting the future” in choice models. The basic concept behind discounting is that people value the present more than the future. Easy example: would you rather have $10 right now or $10 in 5 years? Most people say $10 right now, since $10 right now is more valuable than $10 in 5 years; you might need that $10 now and it’s not clear that the $10 will be as useful in 5 years. (see here[ii] for more). In order to account for this phenomenon, economists often place some discount factor on future payoffs in optimization problems.

In a very broad and abstract sense then, our total life happiness can be modeled as:

Happy1

Where each subscript (1, 2, …, T) indicates our happiness in some time period (until we die at period T), and where β is a discount factor between 0 and 1.

A β of 1 would mean that we value each period’s happiness exactly the same ($10 in 5 years is just as good as $10 right now), whereas a β of 0 would indicate that we don’t care at all about future happiness ($10 in 5 years doesn’t make you happy in the slightest).

Everyone is trying to maximize their total happiness – we are people and everyone wants to be happy. How do we do it? It looks like a straightforward problem, but the tricky part is that happiness in one period often affects happiness in future periods. Binge watching Netflix might make you extremely happy in period 1, but in period 2 (the next day, right before your exam) your happiness will could be very low (as you cram for your test). Maybe you can’t pull it off either, and you end up failing your test during period 3, and maybe eventually even failing out of school months later (period 100 or something). Your total happiness would have been much higher if you had suffered in period 1 and 2 – studying instead of watching Sons of Anarchy – and then receiving much higher payoffs in future periods.

How does this relate to only living once? In a sense, the phrase YOLO is an attempt to alter the value of discounting. It suggests that we make the present value of opportunities and payoffs much greater than any future payoffs. Another way to say this is that it places an inappropriate (as in non-optimal) discount factor on future payoffs (and consequences). To an extreme, it suggests using a β of 0 (in other words, discount the future completely). This is “The YOLO Effect.” Rather than basing you decisions on some objective function that takes into account present payoffs and future payoffs, The YOLO Effect changes your optimization problem to the form:

Happy2

Since β = 0, which simply becomes:

Happy3

This makes the maximization problem very easy; in order to maximize happiness, simply maximize happiness in the current period, period 1 (H1). Though it may be true that in certain periods, the solution to your lifetime happiness problem is the same as the solution to the current-period maximization problem, it seems unlikely that this would happen very often. The more likely scenario is that you end up getting high payoffs in the current period (high H1) and get lower payoffs in the future (low H2 or even low H100). While it is true that, if you are lucky, the YOLO mentality can lead to extremely high payoffs ( could end up being really, really high), strictly speaking the resulting cumulative amount of happiness from YOLO-based decisions will not be higher than the cumulative lifetime happiness derived from decisions based on appropriate future discounting (which is not the case if β = 0). In other words, the YOLO effect is causing people all over the world to be cumulatively less happy over their lifetime.

In order to counteract the YOLO effect, I propose a new catch phrase: “Live like you’ll live an average lifespan conditional on your particular demographic characteristics!” Not as catchy, I admit, and unfortunately the acronym is a bit complicated (“LLYLAALCOYPDC” doesn’t exactly roll off the tongue). But, if you keep this phrase in mind whenever you make choices, and if you practice saying it enough to yell it quickly at parties and then proceed to do something responsible, you can ensure that you and your friends are maximizing your expected cumulative lifetime happiness – even when the YOLO effect is at its strongest.

(A bit of an aside – The Lonely Island song referenced at the beginning of the article actually takes YOLO to mean something like “Be extra careful with your life, because YOLO.” They meant this sarcastically of course – that is not how the term is used colloquially. Their interpretation, however, implies that you should instead maximize your expected life time. Notice that this is not the same as maximizing your happiness and is, in fact, also a suboptimal strategy (even though you may live longer). A longer life is not always a better life. It does not necessarily lead to inappropriate discounting of future happiness as I’ve described above, but it does cause suboptimal decision making, if your intention is to be as cumulatively happy as possible.)

But maybe a phrase meant to counteract the YOLO effect doesn’t actually help anyone to make decisions (other than to, potentially, stop them from doing something that is obviously suboptimal). The information you need, you might be thinking, is really 1) how long you are expected to live and 2) how much you should discount the future – the ‘appropriate’ discount rate.

Let’s start with the first bit. Using the Social Security website life expectancy calculator[iii], you can enter some information and it will predict how long you’ll live (it predicted that I would live an additional 59.2 years, not bad I guess). The U.S. Census Bureau has some interesting tables[iv] as well to help you figure out how much longer you can expect to live. This[v] website also has a bunch of different life expectancy stats, though I’m not sure how reputable it is based on its underground-esque appearance. Of course there is a lot of uncertainty, so I won’t claim that the ‘average age’ or ‘life expectancy’ is at all close to your expected lifetime – especially if you say “YOLO” a lot. In terms of estimating your expected lifespan, it might be best to ‘start’ at the average life expectancy, which is 78.8 years (for the United States[vi]), and then adjust from there based on your healthy/unhealthy habits, your family history, medical conditions, your ability to make wise decisions and so on.

Once you have a good idea of how long you have left to live, and have come to terms with your own mortality, you now need to figure out how much you ought to discount the future. This, as you may imagine, is not an easy question to answer. There have been studies that empirically try to estimate how people discount the future, and much of the data fits a hyperbolic discounting model (see this[vii] paper, section 4.1 and 5.1) where, generally:

Happy4

With t being time and  being some constant indicating the strength of the discounting. The implication of hyperbolic discounting is that your discounting factor becomes smaller as you consider different situations that are further away in the future. For example, suppose you are comparing the current period (say t = 0) with the next (t = 1) (also assume that  = 1 for simplicity). You would discount today’s happiness by  = 1 (so, not at all) and the next period’s happiness by  = 0.5, or half! So if you receive $100 today you value it at $100, and if you receive $100 tomorrow you value it at only $50 (in current value). In other words, $100 tomorrow would only make you $50-worth happier today. But now let’s compare t = 100 to t = 101. This gives  0.0099 for day 100 and about  0.0098 for day 101. So if you receive $100 on day 100 you value it at about $0.99 (in current value), and if you receive $100 on day 101 you value it at about $0.98. The difference between the two times is the same as the first example – just 1 day – but the difference in your valuation is only $0.01, rather than $50.

Now, we don’t know that these empirical studies necessarily reveal anything about the “best” way to discount the future – the people studied may not have been acting optimally. At this point, however, there is no consensus among economists as to which model of future discounting is optimal, or even which is most accurate. We don’t know whether these people were maximizing their lifetime happiness or whether they were following some other behavioral strategy, but if they were acting optimally then that’s good news for hyperbolic discounting models. Hyperbolic discounting seems to work well to explain the observed data in general, and may be a good, easy approximation.

I’m usually against using rules of thumb or heuristics to solve these sorts of problems, but I actually think that using my anti-YOLO prescribed saying, “Live like you’ll live an average lifespan conditional on your particular demographic characteristics!”, actually works pretty well. It essentially prompts you to consider the cumulative lifetime happiness maximization problem rather than the current-period maximizing problem (which is usually suboptimal). When you say this in your head or out loud, you should be considering:

-The potential for tradeoff in payoffs between time periods

-Your expected lifespan

-How payoffs may affect total lifespan

-How your preferences may change in the future (anticipate your future needs and try to predict how happy certain things will make you in the future as you grow older)

-How to appropriately discount the future, and

-The amount of uncertainty and risk surrounding each of your estimations (how wrong you could be and the consequences of being wrong)

Unfortunately, many of these things are person and preference dependent, so it is hard to give answers or advice that is any more specific. Ultimately, I am leaving you to figure out the solution to your own lifetime happiness maximization problem. Your success depends on how well you can accurately and optimally incorporate all of the things I mentioned above into your optimization problem formation, in addition to how well you can resist the allure of the instant gratification offered by the YOLO effect. For this problem, you are in the best position to estimate the maximizing solution, since you have the most information about yourself and how it might affect the form of the problem.

Essentially what I am saying – and I think that this creed is one that many economists would adopt in problems that rely so heavily on the individual – is one thing: “You do you.”

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[i] https://www.youtube.com/watch?v=z5Otla5157c

[ii] See http://en.wikipedia.org/wiki/Temporal_discounting for more.

[iii] http://www.socialsecurity.gov/OACT/population/longevity.html

[iv] http://www.census.gov/compendia/statab/cats/births_deaths_marriages_divorces/life_expectancy.html

[v] http://www.worldlifeexpectancy.com/usa/life-expectancy

[vi] http://www.cdc.gov/nchs/fastats/life-expectancy.htm

[vii] http://www.cmu.edu/dietrich/sds/docs/loewenstein/TimeDiscounting.pdf

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