alt.hn

6/27/2026 at 1:32:20 PM

Suspicious Discontinuities (2020)

 https://danluu.com/discontinuities/

by tosh

6/27/2026 at 3:31:56 PM

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!

by fwipsy

6/27/2026 at 4:23:59 PM

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.

by chantepierre

6/27/2026 at 8:14:58 PM

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

by brightbeige

6/27/2026 at 3:17:31 PM

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.

by mnahkies

6/27/2026 at 4:56:13 PM

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.

Graph: https://www.economist.com/cdn-cgi/image/width=600,quality=10... from this article: https://www.economist.com/britain/2024/04/22/how-to-fix-brit...

by rwmj

6/27/2026 at 5:05:25 PM

Same in slovenia for eg. kindergarten prices, 1eur over the line, 90eur more per month in fees.

by ajsnigrutin

6/27/2026 at 7:24:06 PM

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.

by dadkins

6/27/2026 at 6:08:42 PM

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

by ziofill

6/27/2026 at 6:06:43 PM

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.

[1] https://cleartax.in/s/marginal-relief-surcharge

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.

by alok-g

6/27/2026 at 5:47:17 PM

> 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.

by amluto

6/27/2026 at 3:13:31 PM

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.

by christianbryant

6/27/2026 at 3:45:42 PM

Would test score problem be solved if teachers graded individual questions, not entire test?

by dvh

6/27/2026 at 4:02:55 PM

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.

by irishcoffee

6/27/2026 at 5:14:44 PM

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.

by jameshart

6/27/2026 at 6:32:44 PM

Isn't that equivalent to just setting the passing threshold to 25, with the same incentives?

by aesthesia

6/27/2026 at 4:50:12 PM

If you grade individual questions, you don't know the total score.

by dvh

6/27/2026 at 5:15:02 PM

The incentive to find the extra point would partially disappear.

by rvba

6/27/2026 at 4:58:22 PM

yeah, slow phase-outs seems ideal- I have often wondered about that with gross and and taxable income.

by initramfs

6/27/2026 at 7:30:37 PM

Haven’t seen danluu stuff here for a while!

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.

by ant6n

6/27/2026 at 4:01:38 PM

[ EDIT: Since the undated article is actually from Feb 2020, predating the expansion of federal health insurance subsidies, I withdraw everything I said below. ]

----------------------------------

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.

by PaulDavisThe1st

6/27/2026 at 6:00:08 PM

Goes to show that having a creation date placed somewhere near the top of an article or blog post is a good idea.

by XRG

6/27/2026 at 5:09:28 PM

The post is from February 2020.

I can't see a date on the post (on mobile) but the archives link has month and year.

by forbiddenlake

6/27/2026 at 5:34:47 PM

Well then, I take it all back.

by PaulDavisThe1st

6/27/2026 at 4:45:10 PM

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.

by norseboar

6/27/2026 at 4:49:20 PM

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.

by PaulDavisThe1st

6/27/2026 at 3:38:34 PM

I never understood why taxes or similiar absolute points aren't gradients instead.

by hsuduebc2

6/27/2026 at 5:59:26 PM

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.

by Georgelemental

6/27/2026 at 4:24:00 PM

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.

by blharr

6/27/2026 at 4:59:03 PM

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.

by entrope

6/27/2026 at 4:37:13 PM

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.

by bostik

6/27/2026 at 5:19:56 PM

Are you talking about this tax system? <https://nordisketax.net/pages/en-GB/taxation/?country=finlan...>

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?

by degamad

6/27/2026 at 7:43:03 PM

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.

by zipy124

6/27/2026 at 5:00:32 PM

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.

by initramfs

6/27/2026 at 5:48:49 PM

They may not trust it if they can't understand it.

by mmooss

6/27/2026 at 5:11:33 PM

You might just be taking people too literally.

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.

by encoderer

6/27/2026 at 6:29:43 PM

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.

by gs17

6/27/2026 at 5:45:59 PM

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.

by tmoertel

6/27/2026 at 6:37:43 PM

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?

by bee_rider

6/27/2026 at 4:47:23 PM

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.

by marcosdumay

6/27/2026 at 3:54:28 PM

Because that's harder to write the laws for

by KylerAce

6/27/2026 at 4:02:45 PM

Not if you assume people could understand basic math, such ... oh, any continuously valued polynomial ....

by PaulDavisThe1st

6/27/2026 at 5:12:05 PM

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).

[1] https://www.pennstatelawreview.org/penn-statim/dont-be-silly... (2013) ...which argues that this is a good thing!

[2] https://www.medicalnewstoday.com/articles/what-is-the-medica...

by TimorousBestie

6/27/2026 at 5:27:29 PM

(2020)

by Sniffnoy