I cracked up when I got to the marathon example. When I ran a half marathon I realized about 80% of the way through that I was on track to finish under 2:30:00 and pushed myself to make it happen. I should have guessed that sort of behavior would show up in the statistics!
After this year's Paris marathon, I ran the same per-minute graphs, and they match perfectly the "overall study" graphs with more than 9 million finishes in the article. I also added graphing by age category and gender. I don't want to deduce too much but I think it showed that young men are the most "competitive" (what I mean by that is targeting a specific time) since there are the clearest "goal time" peaks in the graphs.
IIRC There is also a discontinuity when it comes to the histogram of ages of marathon runners, because runners are binned into age groups and there is a more runners at the youngest ages of each group, I guess because it’s younger runners in each group that are more likely to run if they feel more likely to place well in their group
The UK tax system also has a bunch of unfortunate cliffs, and tapers that create >60% marginal tax rates and worse. There's a calculator here that illustrates it well https://tax-cliffs.britishprogress.org/calculator
The childcare cliff edge is probably the worst, but the personal allowance taper isn't ideal either as it's compressed over a relatively short income range
And of course all the thresholds remain frozen, creating plenty of fiscal drag on top.
The Economist has this crazy graph of UK VAT thresholds and how they are causing companies to stay small, to keep their turnover under the threshold which avoids a lot of paperwork. It's the most infuriating thing, we are literally throttling our companies, the engines of growth, because of some accounting stupidity.
I remember when my grandfather died, and my grandmother got part of his pension.
It put her pension above a max threshold for old age / low pension (cheaper phone bill, no tv taxes, ...) by, ... hold your hats... 2 Euro.
Father called the service ... "sorry, we can not do anything about that".
She lost over 100 Euro in benefits because technically... she was not low pension anymore but was financially WAY more poor then when my grandfather was alive.
So when my grandfather was alive, their combined pension was higher, they qualitied as having a low pension. When only *part* of my grandfather pension got combined into my grandmothers. Well, now your not poor...
I do not remember the exact pensions but it was a massive haircut, with the insulting on top, losing those benefits. The lost benefits made it from threading water, to drowning.
Without my dad financial help, she will have lost the house. Ironically, if she did not get my grandfathers pension part, she will have qualitied for more social assistance, and will have been better off.
> Sometimes, those lines really screw people over.
Another angle is how they affect the living. Plenty of young adults delay moving out from their parents, or starting a career, because changes in combined household income would cross one of those magic threshold, and ending up screwing the parents and younger siblings for life.
Such delaying may not be as immediately critical to the person, but it alters the course of their lives in significant ways.
I sincerely believe that a better idea in my opinion is to eliminate all means testing in government benefits altogether. No, higher taxes is not means testing. They are not the same thing at all. Removing means testing means we want the service to be used by all, which means the service should be at least usable by people who have the option to use something else. On a similar note, medicare clawback rules are cruel and unusual punishment. To punish the stupid and ill prepared while rewarding those who did tax planning five years or more before retirement is cruel.
The marathon one has a simple - and fun - explanation: it’s great to run with people!
Many (most?) marathons have pace runners who run the course, hewing to each 30 minute and 15 minute finish. So there’ll be a 3:30:00 pacer, a 3:45:00 pacer and so on. Your local pacer might even have their pace on their shirt and may even have a flag so they’re super hard to miss as you’re running.
One friend of mine runs with a speaker, to play music and keep everyone’s spirits up.
By the end of the race they’d acquired a small army of marathoners! You could see their smiles for miles, and when they all finished together they had a huge party in the recovery area with the speaker :)
I’m surprised the marathon time discontinuity isn’t bigger :)
> One of the most common comments I've seen online about this graph and/or this paper is that this is due to pace runners provided by the marathon. Section 4.4 of the paper gives multiple explanations for why this cannot be the case, once again indicating that people tend to comment without reading the paper.
Gosh that Polish language scores graph is good. Clean bell curve, slight skew, classic little bump at 100 because the measure is truncated, and then this giant mess at 30 that looks more like something out of a heart rate monitor than a normal distribution.
> this giant mess at 30 that looks more like something out of a heart rate monitor than a normal distribution.
Right, it's as if someone shot the heart with a defibrillator - because that's a good analogy of what is actually happening, given the "arbitrary negative effects for people" failing the exam are basically determining the course of one's future (or at least feel like this to the people and their parents at the time).
In particular, having passed "matura" is the lowest level kind of threshold used (both formally and informally) by employers considering job applicants; much like some higher-skill jobs still require a specific university degree, not having "matura" will cut you out of a large chunk of low-skill and entry-level jobs.
Couple failed "matura" with any kind of criminal record - and the two, I imagine, are probably correlated - and you can be screwed for life, economically speaking. It is a big deal, and teachers know it.
IDK what the current system is, but for my cohort, it was possible to repeat the exam in subsequent years - but like with any kind of degree, chances of someone doing that drop sharply with temporal distance; at some point, people just find a path for their lives and are more interested in present-day challenges than going back to finish some degree.
Indian laws have the same too. For taxation 'surcharge' applied to high income groups, a patchwork called "marginal relief" fixes it [1], however, leaves ranges in income where 100% of incremental income goes away in taxes.
Another place where thresholding is misplaced is minors (below the age of 18 years) being subject to different laws (this is for India at least), e.g., seriously different punishments for crimes like homicide or rape. I have heard that crime groups then purposefully involve minors, push the blames on them if caught, and effectively get away for much less.
> Another place where thresholding is misplaced is minors (below the age of 18 years) being subject to different laws (this is for India at least), e.g., seriously different punishments for crimes like homicide or rape. I have heard that crime groups then purposefully involve minors, push the blames on them if caught, and effectively get away for much less.
This also used to be common in the US, but nowadays for (violent?) crimes minors can be charged as adults, so this kind of thing is only done for drugs and organized stealing rather than murders and violent crimes.
I am not sure. What I have heard is that the minor pleads guilty for the entire crime (possibly in exchange for some money behind the scenes).
The broader point is that being a couple days or months younger or older than eighteen years should not result in a drastically different outcomes to begin with.
> The broader point is that being a couple days or months younger or older than eighteen years should not result in a drastically different outcomes to begin with.
The problem is that anytime you create a division there are always some cases just on the border, and if the outcome difference is sized to the difference between the centers of the classes the jump from on size of the border to another gets large.
But on the other hand if the jump isn't large you end up saying that 13 year olds and 30 year olds should be treated the same, which is clearly just worse.
You can try smoothing out the jump at the edges by adding more cuts and doing less changes from one step to the next, but I don't think you gain much in terms of recruitment of youths compared to just pushing the breakpoint fore lenient sentences to larger ages than 18.
I was on a jury for a case like that. The charges included committing the other crimes in the presence of a minor. The minor in reference was a few days away from his 18th birthday, clearly acting on his own volition, and my understanding is the minor bad already been given essentially a slap on the wrist and his record cleared when he turned 18. The judge explained after the trial was over that his sentencing has significant flexibility to take those kinds of factors into account.
Jim Roskind had a really nice talk [https://www.youtube.com/watch?v=_uaaCiyJCFA] at AWS Reinvent 2022 which highlighted the fence post problem. Basically, AWS was targeting P50 and P90 latencies in many services, and sure enough there were big clumps of latencies right under those target fence posts as engineers were basically gaming the metrics and nerfing the >P90 latencies to make it happen.
I don't think that counts as gaming the metric. It's an example of a metric that even when targeted in good faith results in consequences management didn't intend.
Good faith starts with understanding that "90" in P90 is an arbitrary number, and that the exact sampling threshold is not important to why anyone cares about that metric.
The more interesting part of the talk is how using what he calls "fenceposts" naturally results in bunching up against the fenceposts because even small regressions that cross the fenceposts are blocked but regressions of any size that don't cross a fencepost go unnoticed.
It's just a plain bad metric. Useful due to brevity though.
A decent metric for testing against might consist of a target mean, the sigma specifying a gaussian about that mean, a minimum sample size, and a hard upper cutoff.
> A simple fix for the problems mentioned above would be to have slow phase-outs instead of sharp thresholds.
How about the even simpler fix of not having phase-outs? This generally works: the cost of a subsidy generally does not increase as the recipient pays more taxes, so the higher total tax paid by a higher-income person will easily pay for the cost of a subsidy. And giving higher-income/wealthier people the same subsidies as poor people may help them appreciate the ways in which the subsidies are helpful and the ways in which they suck, which can help the whole system improve.
Beautiful! I noticed this on the chess ratings distribution on Lichess: players like to cross a nice multiple of 100 and make an extra effort to avoid dropping below if they can: https://imgur.com/a/Db7fQdX
I also appreciate discontinuities and while I won't comment on the data in the paper itself without cross-referencing, I will say that a couple of these examples hold true for me from applied observation over the years. When I was old enough to start caring about insurance for health and property, or became a parent and had to begin forecasting costs for college and what loans really represented, I began looking at observable data much differently. Working in the software industry, you begin to see the complicated systems at work at the C-Level, and the seemingly odd relationships with unrelated organizations start to become clear. Being an educated voter, a discerning consumer of products, and turning a critical eye on world news all require the ability to see processes, their patterns and the discontinuities within them. While there may not always be a useful explanation behind all of them, seeing them in the first place is essential to navigating so-called reality successfully.
it depends what the purpose of grading is. if the people is to purely assess, grading question by question is better. if the purpose is to provide feedback, knowing how the student did in general can substantially help with identifying what to emphasize
The reddit explanation in the post addresses your question I believe. If someone is at a 28 or 29 a few "charity" points can be found in subjectively-graded tests.
I think that really just reflects the fact that on subjectively graded tests the score really doesn’t have that many significant figures of accuracy. That a regrading can find 3 to 5 points by being more generous - or presumably take 3 to 5 off by being harsher - says that really you could save a lot of effort by treating the final grades as bucketing into 10 point bands and treating 25-35 as the actual cutoff.
Unfortunately this is the final exam from pre-university education. You can think of it as one's first educational degree, a step below bachelor's. It's expected - particularly by employers - that everyone should achieve it.
I think because the gradient is simply too confusing for laypeople to understand.
Even for a simple system like US social security that has a gradient. For every $2 you make over the limit, you lose $1 in benefits. I've heard countless times misconceptions of people thinking they'd be losing money (as in literally having less money net) by working.
Assuming that normies cannot understand gradients or feedback loops is IMO the source of a large amount of "tactical-level" problems in society, particularly around healthcare, taxation, and politics.
The better alternative would be to assume that normies are, in fact, capable of understanding these tools, and in the process forcing them to understand them by setting an expectation. I mean, this is what people should be told as answer when they ask, "what will learning math be useful for me in my life?". No, multiplying polynomials will not be useful to 99% of the people. But having a feel for basic linear algebra and feedback loops would be, because they describe the behavior of the simple and interesting control systems in any and all areas of life - which includes the Social Security examples of yours, too.
The worst thing is, letting majority of people off the hook here doesn't just impact them individually; in democracies, it prevents systems require certain level of understanding from being created in the first place. As shown in plenty of examples brought up in the discussion thread here.
So I guess one answer to "how is any of that math going to be useful for me in the future?" is, "if you all learn it, it will allow you to stop keeping yourselves in poverty in stupid and entirely unintentional ways".
Given that normies don't geneally understand tax brackets, I can understand not wanting to do more complex stuff either.
I don't know how many people I've heard express that they didn't want to move into the next tax bracket because they didn't want to pay that rate on their whole income, but it's a lot.
A very small minority are worried about hitting a ceiling that removes eligibility for important services, and some are worried about the weird phaseouts where marginal rates are very high, but most just don't understand.
Maybe that means having a more complicated system is fine because it's already incomprehensible to most, but I dunno.
A single benefit usually has an appropriate incentive structure, but a lot of people get multiple benefits -- even from different levels of government (local, state, federal) -- and adding up phase-outs in different systems can result in marginal phase-outs rates above 100%. It's hard to avoid that entirely given that we want to have a lot of transfers to the bottom of the income distribution while phasing those out by roughly the median. It would be easier to avoid phase-outs above (say) 80% of marginal income is we only had federal and state aid as predictable money transfers, but for various reasons we provide a lot of transfers in-kind or with limited authorized uses. Those limitations aren't necessarily wrong, but they do mean that transfers aren't fungible, so there's an incentive to provide transfers for other "good" uses, and that diversity is what makes it hard to bound the marginal phase-outs for everyone.
This does happen in Finnish tax system. Your tax rate (percent with one decimal) is calculated based on your annual gross income. Rates are supposed to be calculated smoothly, and they are certainly calculated for each individual separately.
In reality they are step functions. It is surprisingly common to have people refuse promotions because if would put them above an income tax threshold, bump up their rate, and end up with less money after taxes in the end.
The UK tax system is far from fair but at least it has clear brackets: income above threshold X is taxed at rate Y.
Because that is a marginal system, (and unless they've messed up the calculations, which they haven't in this case) you should never end up with less from earning more. Can you give an example of two income amounts where the lower income ends up with more money after-taxes than the higher income?
Or is it the additional municipal, church, or health levies mentioned on that page which have the discontinuities?
I've encountered people talking about turning down a promotion to not "get above a tax bracket". They have invariably been wrong and just not understood the basic math, or done any research.
Lacking those basic skills might be a reason not to promote someone though, so perhaps it all works out for the best
The UK system doesn't either though. The rate for 100-125k is higher than for 125k+ due to the phasing out of the personal allowance. It gets worse if you have kids and can even result in a >100% marginal rate.
I agree, but almost everyone today can use a computer or smartphone. They can type in their income, and the computer can calculate it, providing them an average number of what their percentage of actual taxable income was- I think Turbo Tax and other software might do something like this.
They don't have to understand how it works to do their own taxes.
I have a different take. I think the state wants the system to be this rigid because, paradoxically, that makes it more flexible for them.
For example, they can keep certain tax exemption thresholds low while overall incomes rise. That means more people gradually fall into higher tax brackets or lose exemptions, even though the money they earn is worth less over time.
The rigidity itself is not a bug but a feature. It is a sneaky way to raise the taxes without openly raising taxes. But it's just a theory.
I've heard the same thing -- if they take a job they will lose money. What they really mean is that if they take a job (trade time for money), they will lose some of the pension they have already earned. This is a real economic loss (even if they might have a few more bucks at the end of the week) to say nothing of their lost time.
It's especially hard when you have a combination of many government programs at all levels (federal, state, local, many different kinds of taxes, many different kinds of welfare). Even if every individual program uses a gradient, it's still possible that summing all the programs together leads to a >100% effective marginal tax rate.
Why do we have legislated cliffs instead of gradients? Because approximately nobody understands lerp. And linear interpolation is the simplest (nontrivial) gradient scheme. Consequently, we get cliffs or, if we're lucky, lookup tables that approximate gradients with stair-step successions of small cliffs.
This isn’t a very satisfying explanation to me. I mean, the idea of a cliff causing bad incentives is extremely widely understood (to the point where people under progressive taxation policies avoid taking overtime because they misunderstand the policy and believe the cliff to be there).
I wonder why we don’t define these things in terms of logistic functions or something. That’s high school or at least first semester college math, right?
I was surprised to see the "Big Beautiful Bill" tax incentive for financing a new US-made car had a gradual phase out above $100,000, although it's implemented as a step function instead of a linear one.
It's very well established internationally that income taxes are defined by gradients. I have no idea why politicians want to reinvent them so often in other kinds of taxes.
I hear this explanation a lot but I think it's a convenient fiction. Lawmakers, at least in the US, don't seem to write their own legal texts very often. Sometimes they don't even read them [1]! Congress has no problem producing monstrously complicated laws, in any case.
I think it's likely that politicians and their funding sources have found ways to profit off of these discontinuities. The infamous Medicare "donut hole" [2] was arguably a "benefit discontinuity" of the sort mentioned by the author and pharmaceutical companies profited off of it (more than a hypothetical Medicare structure without a donut hole, not relative to the spending cap that replaced it—which profits them even more).
One way to reduce income may be to buy a bunch of stocks with high variance - for any that is in the red during the year, sell and rebuy them. Any that has a gain, leave them be.
[ EDIT: Since the undated article is actually from Feb 2020, predating the expansion of federal health insurance subsidies, I withdraw everything I said below. ]
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The opening story is fabricated and/or bullshit.
Last year (2025) there was no limit on income for health insurance subsidies. That ended for this year, but last year there would have been no reason for anyone who knew what they were doing to try to lose money to drop their income (especially in the cited range of $48-55k/year).
That is the case this year, in most states (thankfully not where I live), but that's not what TFA is talking about.
Suspicious? It certainly makes me skeptical that the author has got the details of the other examples correct.
It's always a good idea, and it's rather an obvious point - not to mention, something schools try to drill into kids since they're single-digit years old.
At this point I treat lack of dating articles as being done intentionally, and as evidence of malicious intent - i.e. that the entity publishing it has a reason they don't want readers to know the creation date, and there's really no reason other than trying to get away with lies or otherwise screw the reader over.
(Above is a heuristic I apply by default; this post is an example of a case-by-case exception, because of the type of site, my impression of articles read in the past, and of the author themselves.)
This might vary state-by-state, CA MediCal for instance did limit the subsidies based on income last year. I don't think it was an all-or-nothing cutoff, but I do think there were points around the 50k mark where the delta between your subsidy and the one for the lower bracket was higher than the loss you'd take by a few thousand dollars.
The subsidies were from the federal government, managed as part of your federal income tax return. They had nothing to do with additional state subsidies.
The basic story was that until the end of 2025, nobody in the USA had any reason to pay more than (roughly) 8.3% of their AGI for health insurance.
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