
Suspicious Discontinuities - janvdberg
https://danluu.com/discontinuities/
======
btilly
The perverse incentives around these discontinuities is one of the worst
misfeatures of programs intended to help people.

Here is a real example. My niece wound up with 3 kids, divorced, with
inconsistent child support payments. So she went on food stamps. She was
looking for a better job. She found one, interviewed, and they wanted to hire
her. But the salary that they offered was $0.50/hour too much for her to stay
on food stamps, and was less than her current job. Thanks to union
regulations, the person who wanted to hire her couldn't pay her less than the
standard salary. The result is that she did not take the job.

A common result when Congress takes note of perverse incentives like this is
that they introduce a more complex program with specific terms that address
specific problems that emerged. The problem is that by creating more brackets
and more complex rules you usually create _MORE_ cliffs with perverse
incentives (though the perverse incentive at each cliff tends to be less).

Another solution is to introduce rules that attempt to identify people who are
"abusing the system" and punish them. So, for example, you can only be on
welfare for a limited time because people got outraged that single mothers
wound up on welfare for a long time. However those single mothers were acting
that was in part because they were better off on welfare looking after their
own kids than they were in a low end job and paying for daycare.

Stereotypically the first type of solution is characteristic of Democrats and
the second of Republicans. But the truth is that both parties do a lot of both
solutions.

~~~
SilasX
Relatedly, I learned about Universal Credit, an attempt by the UK to merge all
their social benefits into one coherent system, and it seems like the cliff
problem here is worse. From the BBC description:

[https://www.bbc.com/news/uk-41487126](https://www.bbc.com/news/uk-41487126)

>Under the old system many faced a "cliff edge", where people on a low income
would lose a big chunk of their benefits in one go as soon as they started
working more than 16 hours.

>In the new system, benefit payments are reduced at a consistent rate as
income and earnings increase - for every extra £1 you earn after tax, you will
lose 63p in benefits.

In other words, _after_ fixing the cliffs of the system, you would "only" face
a 63% effective tax rate for working more. So whatever system they have no is
even more discouraging. (That understates it because it's phrased post-tax;
including a positive income tax, the total effective tax is even more!)

~~~
mantap
The purpose of universal credit was not to solve any particular problem - it
was to give conservative politicians something to say when their voters
demanded _something must be done_ [about people on benefits].

The world makes a lot more sense when you realise that laws and regulations
are primarily conceived and constructed to keep incumbents in power. Maybe we
should vote in our representatives for life then they'd actually have to focus
on making things better.

~~~
M2Ys4U
>The purpose of universal credit was not to solve any particular problem - it
was to give conservative politicians something to say when their voters
demanded something must be done [about people on benefits].

It's both.

The idea of Universal Credit is a good one - it rationalised dozens of
benefits into one programme.

Unfortunately the _implementation_ is - because Tories - a terrible system
that causes more problems than it solves.

~~~
113
But no one involved in creating universal credit did so for any other reason
than as an excuse to cut the number of people who are eligible.

------
lkbm
OKCupid has a graph of reported height for men[0]. It's both shifted 2 inches
to the right of average, but also slightly flattened at the normal peak of
5'10". It's not nearly as pronounced as the graphs in this post, but it's
still notable.

[0] [https://theblog.okcupid.com/the-big-lies-people-tell-in-
onli...](https://theblog.okcupid.com/the-big-lies-people-tell-in-online-
dating-a9e3990d6ae2)

~~~
DavidVoid
Slightly off-topic but this reminded me of what is perhaps their most
depressing pair of graphs.

Look at "A woman's age vs. the age of the men sho look best to her"
[https://i.kinja-img.com/gawker-
media/image/upload/1439100617...](https://i.kinja-img.com/gawker-
media/image/upload/1439100617445469356.jpg)

and then compare that with "A man's age vs. the age of the women who look best
to him" [https://i.kinja-img.com/gawker-
media/image/upload/1439100617...](https://i.kinja-img.com/gawker-
media/image/upload/1439100617563564204.jpg)

~~~
lostmsu
Why is it depressing? Biologically this must be tuned to the best age for
having kids, and men's age hardly matters.

~~~
meristem
Quality of sperm decays with age. Just not in the same curve as egg reserves.

~~~
lostmsu
Does the quality of sperm have any negative effect on the child or mother?

~~~
leftyted
Older fathers are associated with all kinds of negative childbirth outcomes.

Just not as much as older mothers.

------
kauffj
At least to me, the discontinuities in things like TANF, Medicaid, CHIP, etc.
are some of the greatest signs of political dysfunction.

There's nothing remotely partisan about these being a bad thing. _Everyone_
should agree that these make no sense and mis-incentivize behavior. Yet these
flaws remain for decades despite being known and discussed for approximately
the entirety of their existence.

~~~
thedance
It’s the stated goal of the party in power to throw as many kids off food
assistance as possible. If the program has quirks that is by design.

~~~
Proziam
Both parties have had multiple turns at the wheel since many of these policies
have existed. Whichever party _isn 't_ in power blasts the other party for not
fixing it. In essence, both parties use social benefits like political
bargaining chips for buying votes because they _know_ people will show up for
those issues.

This issue is eerily similar to the 'border crisis' we've been facing. The
issues surrounding the treatment of immigrants that have been decried as
inhumane and unethical were fully in force long before 2016. Of course, it
wasn't an issue for the Democratic party until we got closer to the census and
the _other team_ was at the helm.

I don't think there's any real evidence that either party has the best
interests of the people at heart, or that they have any _genuine_ intent to
fix the issues at all.

~~~
abstractbarista
Exactly, anyone deriding a single party is in fact a pawn. Zoom out, and you
realize you're trapped in a matrix of control by all parties.

------
maxmcd
People playing speed chess (presumably) get fixated on improving their Elo
rating by the next 100 point span, leading to spikes every 100 rating points
in the distribution:
[https://lichess.org/stat/rating/distribution/bullet](https://lichess.org/stat/rating/distribution/bullet)

------
knzhou
There's a possibility that a discontinuity in data analysis might have an
impact on fundamental physics.

Many dark matter detection experiments rely on detecting an annual modulation,
i.e. an excess of detection events later in the year, when the Earth would be
going into the 'dark matter wind'. This helps one distinguish the desired
signal from backgrounds that don't change throughout the year, like cosmic
rays or ambient radioactivity.

One particular experiment has been claiming a positive detection for two
decades, leading to a lot of confusion in the field. It was recently argued
[0] that this is due to a discontinuity in their data analysis. The team might
be analyzing their annual modulation by subtracting out the average rate _in
each year_ , leading to a discontinuity at each year boundary. So if their
detection rate is slowly going up, e.g. due to detector aging, it would look
like an annual modulation, where the rate is lower in the beginning of the
year and higher at the end!

0: [https://arxiv.org/abs/2002.00459](https://arxiv.org/abs/2002.00459)

------
mgraczyk
I have firsthand experience of the frustration that these sort of
discontinuities can cause. I lost all financial aid and nearly all
"scholarship" money my senior year of undergrad after my 6 month coop at Intel
put me over some income limits (probably around $33k per year). All told, I
was probably ~$5k ahead compared to where I would have been with the
scholarships and without the income.

Still, it would have been nice to do something more productive than spend six
months at Intel for ~5k after taxes.

~~~
vxNsr
Presumably the experience itself helped you get other jobs that would’ve been
unattainable had you not had a prestigious internship at intel.

~~~
mgraczyk
Intel is not prestigious in the US software job market. Literally nobody I
have interviewed with has ever asked me about my time at Intel.

------
ceejayoz
> One reason people were looking for ways to lose money was that, in the U.S.,
> there's a hard income cutoff for a health insurance subsidy at $48,560 for
> individuals (higher for larger households; $100,400 for a family of four).
> There are a number of factors that can cause the details to vary (age,
> location, household size, type of plan), but across all circumstances, it
> wouldn't have been uncommon for an individual going from one side of the
> cut-off to the other to have their health insurance cost increase by roughly
> $7200/yr. That means if an individual buying ACA insurance was going to earn
> $55k, they'd be better off reducing their income by $6440 and getting under
> the $48,560 subsidy ceiling than they are earning $55k.

This is an incredibly dumb way to structure things.

~~~
myself248
What's eye-opening is the author's point about _how_ many people go about
ditching some income at the last minute -- by buying a particular type of
financial product whose value is likely to go to zero.

Which has the effect of transferring their wealth to someone else (the vendor
of the financial product) who just sold something that was effectively worth
nothing, for a great deal of money.

It's the most bizarre sort of wealth redistribution I've ever heard of.

~~~
junar
You can't necessarily do it last minute. You have to realize the loss by the
end of the tax year (Dec 31), which may be months away from the day you file
taxes.

On the other hand, if you can deduct traditional IRA contributions, you can
make prior-year contributions up until the tax filing deadline, and keep your
money. If you're eligible for an HSA or solo 401k, you can also use those to
similar effect.

~~~
PhantomGremlin
The devil is in the details. (When isn't it?)

For example, regarding health insurance subsidies (mentioned in the article),
deductible IRA contributions must be added back when calculating your income.
As are other sources of income. E.g. tax-exempt municipal bond income doesn't
count for Federal taxes but it does count for income limits for the health
care subsidy.

Offhand, I'm not familiar with income limits for an HSA. It's quite likely
that allowed income sources and deductions vary for these various programs.

~~~
junar
> deductible IRA contributions must be added back when calculating your income

Not true. IRA/401k/HSA reduce MAGI for healthcare purposes. Do not confuse
MAGI for the healthcare marketplace with MAGI on your tax return; those are
two different figures.

[https://www.healthcare.gov/income-and-household-
information/...](https://www.healthcare.gov/income-and-household-
information/income/#magi)

[https://www.healthcare.gov/reporting-
deductions/](https://www.healthcare.gov/reporting-deductions/)

~~~
PhantomGremlin
Ha. You're right about IRA deductions with respect to healthcare.

But I knew that I saw IRA being added back somewhere! It turns out I was
thinking of FAFSA. IRA deductions are added back for computing income for
financial aid purposes.

Tax exempt interest is added back for both healthcare and FAFSA.

That's why tax programs and tax accountants are essential. It's difficult for
people who don't do this for a living to remember all of this.

------
smoyer
The youth sports discontinuity was a problem for me when I became old enough
to play little league, but was exploited by me when I should have moved on to
Teener League.

I was born on the cut-off date for Little League, so I was the youngest player
on the team my first year (and the youngest for my grade the other years.) Do
to ambiguity in how the cut-off date wording was written, I ended up (legally)
playing a fourth year when I was 13. I was never a big kid but at this point I
was certainly bigger than all the 10 year-olds and more importantly had gained
coordination and experience. It was my favorite year playing baseball.

(As an aside, I paid karma for it the next year ... I spent eighth grade in a
city Teener League where I was once again the smallest, least experienced kid.
And the city league was way more cut-throat than what I'd experienced in the
suburban setting my previous four years.)

~~~
eloff
I was always small until about age 16, and I always sucked at sports and never
really got into them - partly as a result.

I wonder if we shouldn't group kids by size for sporting activities instead of
by age.

There's a famous statistic in Canada that NHL players are much more likely to
be born in January and February than September-December. The reason being is
they were nearly a year older than other kids they were paired with, so they
did better at the sport and thus kept it up as they grew.

Seems silly.

~~~
EpicEng
>I wonder if we shouldn't group kids by size for sporting activities instead
of by age.

Depending on how small your buckets are, you may end up with a scarcity
problem (players and admins/coaches). Also, age matters a lot in regards to
coordination. A 13 year old who is roughly the size as a ten year old will, on
average, have a large advantage.

Some people just aren't cut out for athletics genetically. Some grow later on
and may or may not be interested. Age seems like a fair sorting criteria and
general enough to not introduce too much complexity.

~~~
bradlys
> Some people just aren't cut out for athletics genetically. Some grow later
> on and may or may not be interested. Age seems like a fair sorting criteria
> and general enough to not introduce too much complexity.

Age isn't particularly fair but it isn't the worst either. As a kid, being
around kids younger than you can be really displeasing. When you're a kid, the
only thing you want is to be treated like you're older/better than you are. If
you keep getting put on teams for people who are small - it's likely you're
being put on teams for people who are smaller than you.

I had to play baseball with 7-yr olds for about 3 years. It wore me down as a
kid. I gave up on baseball at that point and said I wasn't doing it anymore.
The reason I had to play with them is because no coach wanted me on their team
because they saw me as too small. (Small town, they didn't group by age as
much as they did by size) I'd see kids go up to bigger-kid teams and get
bigger but I stayed the same size.

As a kid - it was super disheartening to be stuck with the same 7 to 8 year
olds for all that time and seeing people you used to play with in the bigger
kid league. It sucked.

That said - baseball is probably the ONLY sport I can imagine being where you
could move kids up by age and not by size as much. Football, soccer,
basketball, etc. all involve a serious size component to them whereas baseball
is less so - in my mind.

For reference - I grew up to be a normal sized adult. I was just super late on
the curve compared to my peers.

~~~
SamReidHughes
When I was a kid, our club soccer travel team of 4th graders beat an
intramural team of 6th graders by 3 goals. Our coach lost $100, because he'd
bet we'd win by at least 4. I think soccer and basketball much moreso involve
a coordination and fitness component than size. In youth soccer, you have to
outrun the other kids and handle the ball well. Messi is 5'7". In basketball,
of course, height does help a lot there, but amongst local kids, better to be
able to make shots, dribble, and get around the court.

Football was the only sport that segregated by size in my school system's
leagues, in middle school. Well, besides wrestling, of course. They're contact
sports.

------
aeturnum
This is a lovely post and good general demonstration on this phenomenon. The
drug charging graph is particularly stark.

I think the most interesting graph is the final one, for marathon finishing
times. The idea that people will fudge numbers to meet (or avoid) official
definitions is commonly believed (and shown repeatedly here). It's more
interesting to observe that it's not just people using motivated reasoning,
but also that people will alter their performance to hit highly visible
targets.

~~~
Wehrdo
I think another factor for the marathon finishing time discontinuities is the
use of pacers --- people who run carrying signs at a pre-defined speed
(usually targeting a specific finishing time, e.g. 3:15). Lots of people who
have finishing goals will run in a pack with this person so they know they're
going fast enough, then will all finish just under the target time.

------
BurningFrog
The Swedish system solves much of this, at a price.

In Sweden, the principle is that _everyone_ pay into the system, and
_everyone_ gets benefits from it.

So everyone, rich or poor, gets a $130/month payment per child under 18. There
are many other examples.

The price is that while the US has a very progressive tax system, where high
income earners pay a lot of tax, while the low & middle pay very little, in
Sweden _everyone_ pays a lot of tax. It (kinda) works because _everyone_ also
get a lot back from the system.

So Sweden has very high taxes, but most people get similar value back. You can
think of it as a forced savings system in some ways.

------
stared
Ad Polish end of high school exams: quite a few years ago I created an
interactive widget to explore it [https://github.com/stared/delab-
matury](https://github.com/stared/delab-matury).

There are a few jokes about the Matural Distribution (Matura is the name of
this exam) or the Polish Gaussian.

It is the Polish language, which is the most discontinuous. However:

\- There are quite a few scores for the essay which are subjective (e.g. on
the style).

\- It is an official procedure that in the case of results on the edge, the
person marking should mark an ambiguous score in the favour of the student.

~~~
gwd
> It is an official procedure that in the case of results on the edge, the
> person marking should mark an ambiguous score in the favour of the student.

The funny thing about the "almost there" argument is that what it means is the
_actual_ line is more fuzzy, but a bit lower. That is, if the "official" cut-
of is 30, and teachers are willing to "find an extra point" for people with
scores of 28 and 29 but not 27, then the _actual_ cut-off is 27.

I had a prof at uni who for programming projects would say: "The deadline is
6pm on Friday. But there's a 6-hour grace period where you're allowed to be
late." Inevitably someone in the class would say, "Doesn't that mean that the
_actual_ deadline is at midnight?" To which he would respond, "Yes, of course.
But if someone submits a project at 12:01 and it's rejected, when they say
'But I was only 1 minute late!', I can say, 'No, you were 6 hours late.'"

~~~
jcranmer
When I was a TA, the rule for homework assignments was that they were due at
midnight, with a grace period chosen per-assignment of 1 to 6 hours,
undisclosed to students. Alas, I could only choose the grace period on 5
minute increments, so I couldn't pick a time like 3:13.

The grace period I set for the hardest homework assignment was 1:05--probably
cut off a few people who were hoping to extend a two-week assignment by 3 or 4
hours.

~~~
gwd
After reading about human beings' natural tendency to "anchor" on things like
the next 100-point ELO rating, that probably a better way of doing things
would be to say it was due at midnight, but there was a 10-minute grace
period. people naturally will tend to aim for midnight, regardless of whether
it's a hard deadline or not.

IOW, my professor's insight that there had to be a hard deadline _somewhere_
was correct. But he missed the fact that not all hard deadlines are equal: a
hard deadline that is 10 minutes past a natural "anchor" is much easier for
anchor-prone humans to meet than a hard deadline right on a natural "anchor".

Which is a bit strange, because the official university policy with regard to
scheduling actually already took into account this natural anchoring: It was
official policy that if a schedule said 10:00-11:00, the class was _actually_
10:10-11:00. The idea being that it's much easer for the "hard end" of a class
to be at 11:00 than 10:50.

------
cs702
Any rules that can be gamed, sooner or later, inevitably, _will_ be gamed.

The "suspicious discontinuities" shown in this fantastic post are evidence for
it.

It seems to be an Iron Law of Human Nature.

~~~
hn_throwaway_99
This 'Iron Law of Human Nature' is a corollary of Goodhart's Law:
[https://en.wikipedia.org/wiki/Goodhart%27s_law](https://en.wikipedia.org/wiki/Goodhart%27s_law)

------
brilee
The Polish test distribution is an interesting counterexample: it demonstrates
that there is _not_ a discontinuity. Instead of failing immediately at a score
of 29, the graph suggests that you start failing probabilistically, with
greater probability as your score drops. The bump at 30 is merely the
mechanism by which this happens. Thus, this is in fact an example of a
continuous threshold of the type Dan is recommending.

------
leto_ii
The Economist recently had some more detailed plots on the crack cocaine
discontinuities: [https://www.economist.com/graphic-
detail/2020/01/18/smoking-...](https://www.economist.com/graphic-
detail/2020/01/18/smoking-gun-evidence-emerges-for-racial-bias-in-american-
courts)

------
fyp
Another real world example I saw recently was in programming competitions.

There's a tradition of displaying a color for your username based on your ELO
rating (for example people would brag about being Red or Yellow on topcoder).

So people tend to stop competing forever after hitting the next level just to
preserve their color!

Graph distribution of ratings:
[https://codeforces.com/blog/entry/52470](https://codeforces.com/blog/entry/52470)

------
petethepig
Another example is Russian 2011 elections:
[https://akarlin.com/2011/12/measuring-churovs-
beard/](https://akarlin.com/2011/12/measuring-churovs-beard/)

~~~
totetsu
Here is a study applying the same integer percentages analysis to the 2018
elections in Russia
[https://rss.onlinelibrary.wiley.com/doi/full/10.1111/j.1740-...](https://rss.onlinelibrary.wiley.com/doi/full/10.1111/j.1740-9713.2018.01141.x)

------
gowld
The youth-sports thing has spread to academics, where parents "redshirt" their
kids in order to get them higher GPAs and test scores which are curved against
their grade-level peers not their age. College admissions committees are
easily duped by this.

------
fizixer
Off-topic: I'm looking at turnout histograms and I'm having the hardest time
making sense of it. Why is registered voter count on the y-axis instead of
x-axis, since that should be the independent variable?

Then I realized/recalled that is what histograms do. They flip the plot
between the two axis. The original, non-histogram, plot is really turnout
(y-axis) vs registered-voters. But that would look very different (generally
such plots are very noisy, with a running-average that mimics a horizontal
line). Then, if you wish to know what's the distribution of registered-voters
with respect to turnout, you take a histogram.

edit: on second thought, the plot still doesn't make sense. The y-axis should
be # of polling stations, not # of registered-voters.

edit 2: This article [0] has a plot that confirms my suspicion (y-axis is
supposed to be number/frequency of polling stations, not number of registered
voters).

[0]
[https://doi.org/10.1111/j.1740-9713.2018.01141.x](https://doi.org/10.1111/j.1740-9713.2018.01141.x)

------
gowld
Sickening of these "unintended" consequences are that programs designed to
help the poor end up being used by the rich and well-connected -- parents who
"disown" their children" to make them look poor during college application, or
who lie about or overemphasize their racial background to claim affirmative
action rewards.

~~~
PhantomGremlin
_parents who "disown" their children" to make them look poor during college
application_

I'm sure this happens. Parents can also get divorced, the kid then becomes a
dependent of a poor parent.

But it goes both ways. E.g. in a case I'm familiar with, a friend of my
daughter:

1) her father died when she was young, IIRC a junior in high school

2) now she's poor, mother doesn't have or make much money. Fortunately there
are things like life insurance and social security survivors benefits to help.

3) colleges like to give good financial aid packages to poor people.

4) mother remarries, but this is a 2nd marriage and so people are older, with
existing children

5) now mother is part of a household with a lot higher income; sucks for the
daughter because now she isn't getting jack-shit in terms of college financial
aid

I certainly wouldn't expect her new stepfather to want to pay $20,000 or more
per year tuition for an 18 year old girl whose mother he just married.
Especially when he has his own kids to think of for college.

Unfortunately for the kid, community college became the only option.

------
raviolo
Would creating, say, online marketplace to help people lose desired amount of
money _with certainty_ \- for example, via some sort of specially crafted
financial instrument - violate any applicable rules & regs? Idea for the next
YC batch?

~~~
PhantomGremlin
You don't need to do anything special for this. I don't see the need for a
startup to "help" with this. Existing options are already sufficient. It would
take a financial professional less than a day to write up a comprehensive
guide to such a strategy.

E.g. if you want to lose $5,000 just buy a bunch of options that are way out
of the money, that have a perhaps 1% chance of paying off.

99% of the time you lose $5,000, which is what you wanted.

1% of the time you win $500,000, which is a nice problem to have. That payoff
will probably dwarf the financial penalty of whatever discontinuity you were
trying to work around.

------
jennyyang
Losing money on put options won't adjust your income very much. You can only
claim up to $3000/year on capital losses against income. So the example
wouldn't work unless some of the income was also capital gains.

------
bemmu
Our post office has a shipping discount, where you get 10% off all of your
shipments you've sent during the month if you send 1000 packages or more. Once
you get near this limit, you get into the perverse situation where it makes
sense to send empty packages to nowhere.

------
nazgulnarsil
Another example is that many men who are 5'10" and 5'11" report being six
foot, which shows up in a lump on an otherwise normal distribution.

------
ww520
IRS would do well catching tax evasion by running some statistic models to
find suspicious discontinuities.

~~~
jacques_chester
They do. Tax agencies worldwide have used statistical techniques for decades.

~~~
TeMPOraL
Benford's law is probably the most known example.

[https://en.wikipedia.org/wiki/Benford%27s_law](https://en.wikipedia.org/wiki/Benford%27s_law)

------
throwaway023421
The statistics on the amount of cocaine show an appalling pattern of lying and
false evidence from the police.

It's even more appalling that people hardly acknowledge such bias.

~~~
lolc
Please do not jump to conclusions and read the footnote which addresses that
point explicitly: "drug seizure records, amounts seized do not appear to be
the cause of this change."

~~~
tqi
I do not have access to the original paper, but author's assessment that the
bunching is due to prosecutorial discretion is a little puzzling to me. If it
is true that prosecutors are choosing to prosecute cases that are above the
cutoff at a higher rate than those below, I would have expected everything
above 280g to increase, rather than a solitary spike at 280g?

~~~
wahern
From Note 1:

> prosecutors can charge people for amounts that are not the same as the
> amount seized and then notes

If someone is caught with 500g you can simply charge 280g and make your life
easier. If you charge 500g the defendant will argue it was just 450g, which
even if irrelevant makes the case seem weaker to the jury--death by a thousand
cuts. But if you charge 280g and present evidence of 500g, it makes the case
seem more like a slam-dunk to the jury.

~~~
tqi
Ah interesting/makes sense - thanks for the explanation!

------
selimthegrim
How about the US killing like 10 “Al Qaeda No 2”s in a row for a while in Iraq

~~~
mumblemumble
If you order your hit list right, you can kill a continuous streak of #2s,
until there's only one person left.

~~~
kevin_thibedeau
With only a few 100k casualties in collateral damage.

~~~
banads
And twice as many more potential #2s spring up as a result

