
Couple wins millions using lottery loophole (msn.com)
361 points by onetimemanytime 6 days ago | hide | past | web | favorite | 183 comments

To all the folks saying, "how did the State not know the odds?", a lot of gambling games come out without anyone running the odds. A great example is blackjack switch [0]. The game was invented by an ex-gambler and he never really knew the odds of the game. He just convinced some casinos to try it and would advertise what their average take was. Because most people are bad at math and wouldn't play well, it was always bigger than the actual house advantage anyway.

It was only years later that someone finally ran the numbers and figured out the actual house advantage of the game [1], but no one cared because everyone who had a table was making money on it. Even with regular blackjack, the casinos always make way more than the statistical house advantage because people play poorly.

It was probably the same deal here. The State didn't care what they actual odds were, they just cared that they were making money on it.

[0] https://en.wikipedia.org/wiki/Blackjack_Switch

[1] https://wizardofodds.com/games/blackjack/switch/

I work in the gambling industry and live in au. From my perspective this is super strange. By law vendors have to publish the odd (and on some games the EV).

Publishing the odds is common. Doing the math on those odds apparently is not.

But in order to publish the odds, you have to know the odds. And the only way to know the odds is to do the math, correct?

Unless you're making a distinction for determining the odds during optimal play (which makes a lot more sense to me).

Lotteries are required to publish the odds of matching x out of y numbers. Once the design is fixed, this figure does not change.

The expected dollar value of a winning ticket, on the other hand, changes every week depending on the number of tickets sold that week as well as the amount of any unclaimed jackpot(s) rolled over from previous weeks. I won't be surprised if state lotteries are not legally required to publish these figures every week. Nobody wants to be held responsible for a precise amount published on Tuesday when a snowstorm on Friday causes ticket sales to plummet.

Many states do publish estimates of the jackpot, of course, for advertising purposes. But they're only estimates, and I've never seen estimates of anything other than the jackpot. The Selbees took advantage of smaller winnings that usually go unnoticed.

I took his comment to mean that players doing the math isn't common. The operators know the odds and they publish those odds. Players often don't care what the odds actually mean (doing the math) and they'll play anyways.

I think these people just didn't read the article. Given the state made money, it knew what the odds were. It seemed the only reason there was an investigation was to see the real world outcome of how the game was setup. What may be more remarkable is that the state didn't have the analytics to notice this trend before-hand.

The title is misleading. There was no loop-hole. There was simply a structure of the game and the people in the article were able to exploit that structure. They were playing the same rules as everyone else. Buy tickets. They just figured out the right time and the right volume for buying tickets.

I don't know how the lottery works, but I imagine he also got a bit lucky. Apparently there wasn't enough people who caught on to the trick to significantly put a dent in his game. He was also lucky that the game wasn't popular enough to prevent triggering the roll-downs. Playing just 7 times a year even over the course of 7 years isn't a great sample size to account for long term variance. I wonder if there was a chance he could have lost money on some plays.

Casinos can refuse service to someone who wins too much anyway (e.g. counting cards), so that's not such a big problem for them.



A nice illustration of the fact that there is only one mathematics, and it's equally available to everyone!

Based on the numbers alone, that's about a 20% return for the MIT group, for an annualized return (not considering compound interest) of less than 3% per year. Does this suggest that those trying to game the lottery should just turn to investing in the market? It's amusing to see that the market, which is less structured in probability and more volatile, outperforms the lottery in this aspect.

That doesn't mean they had that amount of capital under investment.

If you bet $10 every week, and every week you win $11, at the end of the year you've bet $520 and made a profit of $52. But you never had more than $10 "invested".

If that $18 million was evenly spread out over 7 years, it would be close to $50k/week, or $215k/month. Those are probably more accurate amounts of working capital for calculating ROI.

Yeah, the way the article phrased it, I have a feeling this is a useless statistic like the sum total of all the money spent on tickets.

The money is only tied up for a few days of the year while they hold tickets though, the rest of the time it could be sitting in a suitably liquid investment. In that sense it far outperforms the market.

Um, needing millions to risk on this means it isn't "available to everyone".

And yes, it is a risk, even if your mask is solid. There's always the risk that the lottery commission could say "oh, come on, the terms and conditions prohibit this kind of thing", leading to a protracted legal battle.

If you have a 60% rate of return from a single play, it only takes some 15 rounds of compounding to pass your first million, starting with $1000.


Not sure why this is down-voted. It's spot on.

Think downvoting is on first half of the comment because as is noted already, compounding.

On the legal topic, I think it’s not a prohibitive risk. It’s completely in the game’s design and completely within the lottery’s control as they basically control distribution. Running a casino isn’t a risk free operation, miscalculations cost money.

Probably because it didn't happen; these people got away with it for years, and don't seem to be in any hot water.

They bought tickets and won due to the inherent probability of the game, fair and square. No inside information, no gaming of anything.

What supposed to make mass ticket buying non-profitable is the expected return being < 1, not any hidden terms and conditions.

On the other hand, in arbitrage, it’s winner takes all.

This is a MUCH better writeup than this recently submitted link. I remember reading it at the time, very well done.

The original Mass gaming commission report is floating around out there, if anyone is looking for additional reading on this subject, though it is of limited interest, outside of a select few MIT students and a couple from Michigan.

I wish lotteries handed out many $1M prizes instead of giving $250M to one person. People understand what to do with $1M (I think) but most people are ruined by $250M.

Everywhere, people say they prefer lotteries where there are more prizes with smaller wins, instead of one winner with huge jackpot. But also everywhere, their actual bahaviour is that when there is one big jackpot, they bet more.

And those who arrange lotteries do know their business.

That's probably true. They should try to educate people that if there are 250 times as many winners that your chances of winning a significant amount of money is much higher.

People aren't betting that way because they're uneducated. People generally don't play the lottery because it's statistically wise. They play because dropping $1-3 on a ticket is no big deal, and it provides a little excitement and lets them fantasize about the winnings. Fantasizing about what you'd do with $250M is a lot more exciting than fantasizing about what you'd do with $1M, and once the odds are low enough, any emotional investment in the math is going to be lost in the rounding error.

> and it provides a little excitement and lets them fantasize about the winnings

I've heard this before, but why do you need a ticket to fantasize about what you'd do with that much money? It can be a conversation starter and I can see the excitement aspect in watching your numbers come up (if you don't have a hundred combos to watch at once), but fantasizing is free.

As someone who's bought a few lottery tickets in his life, I can confirm that the fantasizing is in fact better when it's about something that could happen to you.

you could also just happen to find a winning lottery ticket on the street.

the likelihoods of the two events are similar, and one fantasy is free.

Have you tried buying a lottery ticket? It sure feels a lot different than the "free" variety you're proposing.

And props to you if you're a true rationalist who derive nearly equal amounts of satisfaction from both fantasy events, but most people don't behave like that.

The likelihood of the two events isn't at all similar. How often do you find discarded tickets for future drawings on the street?? You're much more likely to find one (or ten, or a hundred) if you buy them. Then once you're in possession of said tickets, now the odds of winning are the same.

Have you ever bought lottery tickets?

> The likelihood of the two events isn't at all similar.

1e-9 and 1e-100 should both round to 0.

and since people spend all sorts of time imagining things with likelihood _0_ (or downright counterfactual!), 1e-100 should be plenty to get you rolling.

> How often do you find discarded tickets for future drawings on the street??

you're already fantasizing a wildly improbable chains of events (buy ticket, win max amount, and then, unlike a huge number of winners avoid having your entire life ruined by it), but imagining finding a piece of paper on the street is a step too far?

and if "found it on the street" is _really_ the problem, you could always choose to fantasize that someone gives you a ticket, or even just pretend that you purchased a ticket. (that's hardly improbable, after all.)

It sounds like you fundamentally just don't understand the psychology of people who buy lottery tickets.

That's fine, then don't buy them yourself, but it makes no sense to argue against other people's actual experienced feelings and thoughts. You can't use logic to "win" this.

> You can't use logic to "win" this.

and yet here you are, producing justifications about how your fantasy has to have particular levels of believability, instead of just saying "i like doing it".

The probabilities differ by less than 0.0000001.

You could have a rich uncle in Scandinavia about to die and leave you $200M

I think if you could educate people on their expected return from a lottery ticket purchase, they wouldn't play at all.

Telling people that they have a 95% chance of winning nothing at all is not a good way of selling lottery tickets.

I think it's naive to value a lottery based solely on your monetary expected value. The value of money doesn't increase linearly, and isn't the same for everyone. $20 could have an inconsequential effect on my quality of life, while winning say $2 million could be a total life changer.

Likewise, the difference in impact between winning $0 and winning $2 million is likely a lot larger than the difference between winning $2 million and winning $4 million. So a straight EV calculation doesn't really tell the whole story.

There's also the entertainment value of participating in the lottery to consider, even when you win nothing.

Personally I like to think of lotteries as a binary thing - either I win enough money to retire now, or I don't.

I think there's more nuance than that. Winning enough to pay most of your house off wouldn't necessarily let you retire but it'd make you a lot more comfortable, for instance.

I have a graduate degree in maths and have played the lottery. The expected return isn't the point.

Expected return may be helpful but imo not sufficient without variance. Some cumulative jackpots are large enough that the expected return is actually profitable but the probability of actually hitting the numbers is vanishingly small.

However, even with positive expectation value, you should not always play. One should first check the Kelly criterion, which is well explained Russel O’Connor’s blog post:

“Do Not Play the Lottery Unless You Are a Millionaire” http://r6.ca/blog/20090522T015739Z.html

If the lottery was in the business of educating people, they would lose their customer base.

I know that this is true, but if you have a good blog post explaining these statistics i'd appreciate it!

The odds are low enough that multiplying by 250 doesn't make much of a difference.

When I play (I don't know why but somehow I find it fun) I actually play SuperLotto which has a chance of 1 in 41,416,353 vs 1 in 292,201,338 with PowerBall. Better to make a few million with better odds than a few hundred million with worse odds. Never mind that it's much more likely that I will get struck by lightning :-(

That final 7x increase to the odds is the one that makes it not worth your while?

I didn't say it's well thought out...

If you want to win life-changing money, why not take the better odds?

Exactly. Don't listen to what people say they will do, watch what they actually do.

marcus Aurelius

this forum is becoming very low in quality. someone downvoted my information about who said that

I think the issue might be that it's a common enough thought that attribution isn't important as multiple people could have derived that independently and that simply attributing where a paraphrased quote comes from without any other commentary is in itself low quality.

That actually sounds like a good idea. Though there is the old saying "the lottery is just a tax on the poor".

My aunt actually won 2 million back in 2007. She had just divorced my uncle (so I guess she is my ex-aunt) and bought a ticket and used the birthdays of me and a few others as the numbers. She was smart in that she chose to set up a trust fund with yearly pay outs instead of a lump sum. She told her sister that she was going to give her a million, but then after realizing that the tax was going to be around a million, she then changed her mind.

She then bought a not too fancy house on the beach for probably too much money (this was in 2007 after all). Then bought a non running yacht for $40k and a new car. Her first pay out check for the year was spent in a matter of weeks and she still had monthly payments on the new toys so she couldn't quit work either. Then she realized that the house payments were for 30 years, yet her pay out plan was only for 25 years and it would not be easy to make payments those last 5 years without more refinancing (she is still trying to figure that one out).

She got scammed many times on house repairs. When she first moved in, for what ever reason she said she needed to get the windows replaced. She paid a local contractor $9k to replace the windows. He showed up with a trailer full of windows and unloaded them all (and I guess that is when she paid him). He said he'd be back later in the week to do the install. Months went by with no contact so she found another local laborer to install them. But he quickly realized that all the windows were the wrong size and basically just junk left over from other jobs.

The $40k yacht never ran. Spent several years paying people to work on it, no telling how much. Then she finally sold it for $20k.

Then a couple of years ago, a hurricane hit the coast. Her house stood, but windows were blown out and the house got completely soaked inside. It was gutted and rebuilt for $80k. Right before this happened, she had actually fallen in a parking lot of a pet supply store after tripping over a piece of rebar that was left over from a curb demolition. She caught herself with both hands but it tore both shoulder cuffs. I don't think lawyers even got involved. The store cut her a check for $80k. It was that $80k that she used to repair the house with. The insurance still hasn't paid her back yet. Something about they wanted every contractor preapproved or something and they reject every receipt or invoice that she sends in (still an ongoing issue).

Been a wild ride...

Why take out a mortgage and lockup money at the same time?

How do you know such detailed financial numbers? It sounds like she gave her financial bookkeeping records.

Some families talk about money. I know mine does. Read old Jane Austen novels and you'll see it was common over a hundred years ago too.

Not to the detail where you could recount exact numbers from various transactions over the course of a decade. Maybe I'd memorize my own finances to that degree, but not someone else's.

I personally like the saying "the lottery is a tax on the innumerate"

Although it's a bit cruel because the "innumerate" are disproportionately poor. Lottery tickets may be driven more by despair and hope for an improbable escape than by innumeracy.

Considering how much people spend on lottery, and how much value they could get by spending that money on real products, it's innumeracy.

Innumeracy isn't about the poor. I know a lot of people who play the lottery especially when the prize is big.

My concern with the poor and the lottery is exploitative. It believe it is akin to a gambling addiction and a false hope. It is a government run institution that exacerbates a mental illness or exploiting a mental disability.

I actually watched a video on CNBC's YouTube recently about the lottery and one thing they mentioned is that they've made the odds of winning worse to let bigger jackpots build up, so they're actually going in the opposite direction of what you'd like.


The reason they don't do that is marketing; more people play for big jackpot. I think even saying there might 250 $1 million winners this week isn't as big of an enticement.

Exactly, they would never sell anywhere close to the amount of tickets as they do for the single-winner lump sums that swell to ridiculous amounts. So they wouldn't have the necessary incoming ticket sales to be able to do the 250 $1 million winners.

Around here when the jackpots get high enough you actually find really long lines just to buy tickets. No one lines up to buy the scratch offs with much better odds of winning $1 million.

When the jackpot reached a certain amount, say 500 million, they could switch from a winner takes all to essentially a raffle with 500 winners. I think it would greatly increase the number of tickets purchased when the threshold was met.

Isn't that kinda what the Windfall game did more or less?

This is what the windfall game in the article did. They were smart enough to buy as many tickets as possible during this raffle.

Taking this logic to its conclusion, the best kind of lottery would be one which paid out $1 to everyone who bought a ticket.

I understand how numbers work, so I don't usually play the lottery. But I do play when the jackpot cracks $1B. Not because I expect to win, but for the thrill of playing.

That thrill is only there when the jackpot is at $1B.

Both the major us lotteries already do this. Matching all but the last ball wins you a million.

Fiscal Refeeding Syndrome (FRS).

When I read this kind of stories about US lotteries (and there seem to be several of this kind of stories going around), I always wonder how on earth it is possible for so many cases of failed lottery rule design to exist in the first place, and for those clear failures to persist for so long.

Did no one notice that the lottery always lost huge sums of money during the Rolldown draws, while it gained money on normal draws (as it should always be the case)? No one noticed that certain convenience stores suddenly sold hundreds of thousands worth of tickets in these draws, while most other stores sold a normal amount?

I think the irony is that from the organizer's point of view, it's essentially working as intended. The "windfall" mechanic was supposed to make rollovers more enticing by increasing the payout amounts for everyone. The only reason to make a lottery more enticing is to sell more tickets.

So they sold more tickets, and made more payouts on lesser tickets. Exactly as they planned. The lottery doesn't actually lose out - it's just emptying the pot (which was filled in previous games) in return for increased sales.

It's the old adage "the house never loses" - if 50% goes in the pot and 50% goes in their pockets, it doesn't matter who wins the pot - their 50% doesn't change. If a mechanic creates an incentive for more sales, their take increases. Fantastic. But if popular perception becomes that the game is rigged, less people buy tickets - and their take decreases.

(We had an office draw that'd run for months at a time, until someone eventually won. But we wouldn't take new players until the pot had been emptied, as someone joining for the last 2 rounds and winning everything, lost us more players than we gained. That is essentially the long-term risk here too. 10 people paying in until one of them wins, feels fair. 10 people paying in until the 11th wins, doesn't. And when the game stops feeling fair, you start to lose the feeders that fill the pot in the first place.)

I don't understand why this is an issue since each drawing's odds are independent. Perhaps I don't understand how these pools work.

That's the disconnect. Each draw's odds are independent, but each draw's winnings aren't.

So there's a collectivism that goes into a growing pot. This is the only thing that really makes a "rollover" enticing to players. If each draw was an isolated incident, the "windfall" mechanic the article describes would be in place for every draw. So a collective pitches in, and the pot is distributed amongst the collective's members (the players) according to how successful each ticket is. So if there's no 6-number winners, there's more left in the pot for the 5-number winners. If there's 5-number winners, there's more left in the pot for the 4-number winners, etc.

(Either that, or the house makes out like bandits. State lotteries are usually regulated to keep a distinction between the pot and the profit, hence such pot-emptying mechanisms.)

But if the pot rolls over - it's not distributed, but added to the pot for the next draw - you now have more than one collective. One collective that's contributed to the pot (over n draws), and one collective that participates in the winning game (over 1 draw). And if there's a significant disconnect between the two, then yes - as one commenter put it, sour grapes. It's the difference between feeling like you've lost a fair game, and feeling like you've been hustled.

Technically it makes zero difference. But if it makes people less inclined to play in future, then it's bad for the long-term health of the game.

It becomes an issue because you introduce humans and human emotion into something that is only viable because of the emotional draw that gets people to play. That emotion feels betrayed... feels so the actual facts aren't relevant... then they don't keep playing and the game dries up.

It's an issue because of "sour grapes". The people that had been in the game longer feel the newcomers didn't deserve to win. Feeling that way led them to no longer play - and that is the issue. Fewer players = smaller pot = fewer players = smaller port = fewer player.... you get the idea.


The state never "loses." In this case, for example, the MIT group "17 and $18 million ... over a seven-year period, earning at least $3.5 million in profits," yet the state made $120 million. Even if the state noticed, why would they care? Pay out $3.5 million to make $120 million? Why not. The problem is that it :"sounds" rigged.


Someone noticed, because they tipped off the Boston Globe that "in certain Massachussetts locations, Cash Winfall tickets were being sold at an extraordinary volume." Again, it "sounds" rigged.

The average person is not going to think that a lottery has a "loophole" in it. The average person thinks something like that can't happen.

The lottery’s and retailers’ profits are each a fixed amount of the ticket price, and the rest goes into the prize pool. They don’t lose any money unless they commit to paying more prize money than they allocated to the pool. Jackpots are explicitly a (large) fraction of the unallocated award funds and so cannot make the lottery operation insolvent— only an excess of fixed-value awards can do that. In fact, the increased sales on rolldown draws just mean more profit for the operators and reduce their biggest risk, which is something happening to the stored funds used to pay out prizes.

Long before that point, lotteries that decide to innovate in game design don't have a professional actuary on the staff to run the ruler over the designs?

That's the weird part for me. Surely inventing a whole new game mechanic, must be their industry's version of rolling your own encryption?

Well, that also seems like something state governments would do.

I suspect the designers were aware of this shortcoming, but were not concerned. As explained by others in this thread, the state still made money.

The house always wins, the lottery didn't lose money during the rolldown, nor any other time. Quite the opposite.

They also did notice.


>And, there was also the fact that lottery officials in Massachusetts had started to figure out that the Selbees and the MIT students had identified an advantage, but had done very little to combat it.

>"How do I become a member of the [Selbees'] club when I retire?" one lottery official joked in an email that later became public.

These lotteries are usually run by state commissions, staffed by career bureaucrats and sometimes led by appointees. I don't think there's a deep well of talent that is familiar with how to implement robust monitoring, security, and optimization systems.

Lotteries are run by a couple of companies, and the multi-state systems are run by fairly secretive organizations that have their share of issues.

End of the day, the providers don’t have much interest in finding problems, and the states have limited ability to detect them. Protecting the integrity of the system can be largely addressed with PR.

They were featured on 60 minutes twice, including last night. The rendering on Microsoft News is off!

You can see the entire video here.


Love hearing stories like this where people gained because of some loophole. Does anyone know if there is a compilation somewhere of these?

See also:


Irish Lottery, 1992, brute force - you could purchase all ~2M combinations for < IR£1M . The company played dirty when they realised what was happening, then later changed the game when they'd finished the forehead slapping: https://en.wikipedia.org/wiki/National_Lottery_%28Ireland%29...

There is a story similar to the dollar coin gig to earn reward miles. A long time ago when plastic gift cards were first a thing, stores didn't have a way to update the balance. The scheme would be to buy a $100 gift card from Walmart using your credit card (getting the miles). You would then use the gift card to buy a $0.35 pack of gum. Since the store couldn't update the balance, they would give you cash as change. You could then just deposit the change to your bank and pay off the credit card. Rinse, repeat.

So for $0.35, you could get $100 in equivalent reward miles. Now stores have changed, they can now update balances on the card and therefore won't issue cash as change.

To expand, this still goes on, it's just a lot more convoluted and has become quite the cat and mouse game. It's been very interesting watching how manufactured spending has evolved.

I know people used to do this by buying the $1 coins from the US treasury, and then just take the coins to the bank and depositing them in order to manufacture spending.

I think because the original gift cards just effectively said "I'm a $100 card" in the magstripe.

Now, gift cards operate more like limited credit cards.

Look up Joan Ginther - ex-Stanford statistician and she won scratchcard lotteries multiple times (I seem to recall by arbing non-uniform geographical distribution of winning cards). Unfortunately I can't find a technically satisfying article and don't have ability to find one right now...

“A nationwide reporting adventure tracks improbably frequent lottery winners”:




$80 BILLION a year in lottery spending, that was shocking to me. Only 30 some countries have bigger yearly budgets. https://en.wikipedia.org/wiki/List_of_countries_by_governmen...

Remember that you can look at a lot of data on the US like this and come up with seemingly shocking figures, simply because of the huge population of the USA and their higher than average disposable income. I bet if you looked at other things like total amount spent on other optional/nonessential items - chewing gum or fabric softener, for example - you'd end up with a dollar value in excess of what many governments spend :-)

The chewing gum market is a bit smaller (but impressive nonetheless), 110-ish countries have revenues comparable to sales in North America:



Oh, I know. Our GDP does it. If US can spend $80 Billion on lottery tickets imagine what we can spend to win a war. Makes you think more than twice before poking the bear.

It's very common for people who win small prizes to 'reinvest' their winnings though. Someone who spends $5 and wins $25 might then just buy $25 of tickets with that win. If they're playing online that money never really exists - it's just an account balance in a database going up and down.

Slightly more than spent on Food Stamps I believe? Is there a correlation?

The title of an article I would never have clicked on if it were not referenced via Hacker News...

A wonderful account of this is given in Ellenberg's "How Not to Be Wrong": https://en.wikipedia.org/wiki/How_Not_to_Be_Wrong

I highly recommend the book; it is a popular mathematics book, written by a real mathematician, discussing some real mathematics, with an engaging style.

Can anyone explain the strategy like I'm 5? I'm confused by his answer.

It might make a little more sense with the numbers. WinFall was a typical lottery game where you choose 6 of 46 numbers[1]. As with most lottery games, there is a jackpot payout but there are also payouts for matching 3, 4, and 5 of the numbers. The odds breakdown goes like this[2]:

  Result | Odds
  6 of 6 | 1 in 9,366,819
  5 of 6 | 1 in 39028.41
  4 of 6 | 1 in 800.58
  3 of 6 | 1 in 47.40
  2 of 6 | 1 in 6.83
The issue with WinFall is that when the jackpot increased past $5MM, the expected value of the non-jackpot winnings becomes favorable to the player. I.e. a 5-of-6 matching ticket might normally yield $4,000 but may yield >$20,000 for a "Roll-down" game. So if you go out and buy 200,000 tickets for the roll-down game (assuming all tickets are different and numbers are selected at random), you should expect to have 5 5-of-6 tickets, 250 4-of-6 tickets, 4,219 3-of-6 tickets, and 29,283 2-of-6 tickets. The combined winnings of all of those tickets makes it profitable.

[1]: https://www.mass.gov/files/documents/2016/08/vv/lottery-cash...

[2]: https://en.wikipedia.org/wiki/Lottery_mathematics

This is what I was looking for! Thanks.

so in each game the money that is not won is collected in the pool until the pool is bigger than the overall cost for all tickets?

and in other games this doesn't work because they are 'winner-takes-all'?

If nobody wins the big jackpot ("guess all the numbers") for some time, parts of it are given to the smaller winners ("guess a few of the numbers"). In this case, which happened once in a while, it became more likely that you'll win more than you have bet. Usually this means you'll win some $50; they have bet many, many times, and won $50 times many.

Doesn't work any more, rules have changed so this way of winning was removed.

But which set of tickets do you buy?

From the article, it seems that it doesn't matter, they're random. (Or were random, apparently it's over now, doesn't matter.)

To make up an example:

Let's say that betting $1 in some way has a 4% chance of winning, and the payout is 20x bet, so if you win on one ticket, you get $20. So, you buy a lot of tickets (say 1000), and you pay $1000. Most of them don't win anything, but 4% of them win $20; $1000 x 4% x 20 = $800; you have won some, but you have $200 less than you started with. Not a good deal.

However, if nobody wins the jackpot, it is distributed amongst smaller payouts, which changes the equation: still 4% chance, but payout is now $30. Again, you pay $1000, buy 1000 tickets, most don't win anything, but the 4% that do, you get $1000 x 4% x 30 = $1200. You have $200 more than you started with. Profit! Of course, this will only work if you buy many tickets; buying one or two, you're unlikely to win anything at all.

The actual numbers were different in these cases, but the general principle is the same.


That's the important part here. You can always win the lottery if you can buy enough tickets. To match all (n) numbers, you need to buy one ticket for every possible combination. To match (n-1), you need to buy a lot less.

But in a normal game, the cost of doing so is more than the payout. So if I have a dollar-game "guess a number between one and ten", and you can win $6 - it's not worth buying ten tickets, even if it guarantees you a win. But if I say "one day only, double the winnings!" - now $10 guarantees you $12 back.

The odds don't change, just the profitability of them.

In a normal lottery, if the pot gets large enough, even with the normal rules the "expected payout" of a ticket becomes higher than its list price. It's just that the chances of actually winning are so astronomically low that you'd have to be Jeff Bezos to stand a good chance of winning your bet (i.e. being able to buy enough tickets such that your chances of winning are worth the play, and then especially being able to absorb the loss when you lose anyway because it's still random).

What's interesting with the rolldown mechanism is not just that the "expected payout" of a ticket becomes worth more than its price, but that the number of winners is such that you can actually reasonably expect to turn a profit by buying a modest number of tickets.

That requires a lot of faith in the fairness of the particular lottery.

Official lotteries are usually heavily regulated and audited to ensure they comply with their published rules. These are state lotteries we're talking here, not random gambling websites.

A. Any lotto requires faith. B. Selbee didnt put all his money in initially. C. The parent comment is just explaining a concept w a hypothetical example

All of them. They bought 1100 to ensure that every combination of some size was in their set of tickets.

I don’t think 1100 tickets is enough to cover all combinations of 4 numbers, but I don’t know the original lottery number range so I’m not sure.

They apparently announced when this was going to happen, so it was more about timing than buying specific tickets.

The jackpot grows off of people losing money. Once people have lost enough money ($5M), the chances of winning the money with a single ticket are raised, however you can no longer win the whole $5M with a single ticket - the money is now distributed over a larger pool of tickets.

After that it becomes a good bet, because you can bet smaller amounts, but still win more than you put in initially - you're profiting off of people who lost money before you.

Someone can always win the whole 5M. It is just that if the prize for large enough and no one win, then the amount of the small prizes increased. It was a way to keep the total payout low.

Even weirder is the statement that the State that organized the lottery made more money thanks to them buying massive amount of tickets ( or maybe it was the publicity around the first newspaper article that triggered more interest, and so more participant in general ?)

I understand the opposite: the State didn't lose any money, (but didn't make any money from these people winning, either); it made money from lottery as usual, but not from these people. (There's an implication that more people in general bought tickets, didn't win, and so the organizer made money from those - but it's not entirely clear.)

The state makes money from every ticket purchased. The lottery simply pays out a fixed percentage of ticket revenue. More ticket sales means more money to the state, regardless of whether those tickets, when purchased, had an expected positive or negative return.

If the ticket has an expected positive return for the buyer, then the state must be losing money.

The ticket only has an expected positive value return under certain limited circumstances, not all the time (i.e. when jackpot has not been won for a time). State is not losing money in total - just moving around the distribution of prizes, but the sum total remains constant.

If anyone would be losing money here, it would be a hypothetical winner of the jackpot - but as the jackpot is only redistributed when there is no such winner for some time...

The prize pool losses the money, not the state. This was a mechanism to keep the overall prize pool from getting too large. Of one person won the main prize, then they would get a car majority of the prize pool. Otherwise the prize pool was essentially distribute among all the players. Yes technically if enough people played, they could have spent more than the prize pool, but, if enough people played, then the chance that one person won goes up as well.

It has an expected positive return only on the days there was a Rolldown.

Money came from people who played on days there wasn't one.

No, because the payout for the state (the amount they planned on losing) was always the same. The amount was "lost" regardless, it just depended on if it went to one big winning ticket or hundreds/thousands of smaller winning tickets.

For this to be true, every additional winning ticket purchased must result in a net loss to the state. But it doesn't, every additional winning ticket purchased is money going to the state. The losers in this case are the other people with winning tickets, because their payouts become smaller¹.

¹Well, given one assumption about how the game is run. The other assumption is that the 3- and 4-number winning tickets have a fixed payout calibrated such that this doesn't fully exhaust the pot, and in this case the "loser" for each additional winning ticket is whoever wins the full pot next time as the pot will be lower than it would otherwise have been.

I think in this case the winnings were fixed and the surge in sales in Rolldown days would cause the pot to be drawn down much more, meaning a longer delay before the next Rolldown day.

So Rolldowns probably became less frequent, but otherwise the lottery made their money and everyone took home exactly the winnings they would have in that particular game.

I feel like mathematically, you must be right. One of the sides must be calculating their expected value wrong.

For example I can offer to pay a $3 jackpot (split across winners) for a $1 game for guessing heads or tails. If more than 3 people play the game, I make money. For each player, the naive way to calculate expected return is $3 * 0.5 + $0 * 0.5 - $1 = $0.5. So the players also think they are making money. But once you account for jackpot splitting, that "expected value" isn't positive anymore. To calculate this correctly, the player (who has less information) need some way to model the distribution of the number of other players.


He isn't right. The state doesn't work with expected values in any way. Money comes in via ticket sales, some of it goes into the state budget, the rest goes out as lottery payouts. There is no element of chance in any of that.

As you point out here and I point out elsewhere, the expected value of a ticket for the purchaser depends on the total number of tickets sold, which has been mostly elided from the discussion. In fact, it's mostly irrelevant as to this past event; at the actual ticket volumes, returns really were positive on the days in question.

The way the game is presented in the article is that the low payout numbers are fixed (so a 3 match on a given rolldown game pays out $50 regardless of how many winners there are). This is consistent with the state lotteries I know. In this scenario every ticket sold causes the state to lose money, on average.

No. Every ticket sold causes the prize pool to lose money. The amount the state makes increases.

It's a zero sum game. If one side is positive, the other side must be negative.

If jackpot isn't paid out, state gets (ticket_sales), players in total gets (-ticket_sales).

If jackpot is paid out, state gets (ticket_sales - jackpot), players gets (jackpot - ticket_sales). The winning players get ((jackpot - winning_ticket_sales) / num_winners) each, the rest gets (losing_ticket_sales).

Either way the expected value for an individual player is always the opposite sign of the expected value of the state.

The interpretation that people might be using is that: when the tickets sold is low, it's positive expected value for the players. When sales cross the jackpot, it's positive expected value for the state. But then it becomes negative value for the player! There's no point in time where both sides are winning.

(And of course that interpretation is pretty meaningless because you don't want to calculate the expected return based on the current number of tickets sold. You want the forecast of the eventual number of tickets sold. The state never really loses, even at the start because they have a pretty good idea of what this number will be in the end)

It’s a zero-sum game, but there are 3 parties: the state general fund, the prize pool, and the players. By regulation, a fixed percentage of each ticket sale goes to the state general fund and the rest goes to the prize pool. Also by regulation, all the money that goes into the prize pool must eventually be paid out to players.

The unpaid prize pool getting too big can cause problems for the operator: there may be accusations by the regulator that it’ll never be paid out, or it could be stolen or mismanaged leaving the operator insolvent because they no longer have the funds they are required to give to players.

A rolldown event is the operator choosing to increase the payout schedule to an expected win for the purpose of reducing their liability of unpaid prizes; tickets having a positive expected value is the entire point.

Swarming the rolldown day would just have the effect of diminishing the prize pool faster, and therefore a longer delay before the next rolldown day.

Actually, the state doesn't seem to enter into the equation at all, that's a red herring. State gets (ticket_sales - prize_money), sure. This amount is predetermined by the rules (usually "X% goes to state, 100-X% goes to prize pool). From here, the only changes happen in the way that the prize pool is distributed to winners: part goes to jackpot, part goes to smaller prizes; if jackpot too large, recalculate with a larger part going to smaller prizes. The amount that state is paid out is never influenced by this.


In round 1, state sells tickets for 100 ($, $thousand, $billion, doesn't matter). 50% goes to state, 20% goes to jackpot, 30% goes to smaller prizes. State gets 50, jackpot is 20, prizes of 30 are paid out.

Round 2: sell 100, get 50, jackpot 20+20 (from last round), prizes of 30 are paid out.

Round 3: sell 100, get 50, jackpot 40+20, prizes of 30 are paid out.

Round 4: sell 100, get 50, jackpot 60+20, prizes of 30 are paid out.

Round 5: sell 100, get 50, jackpot 80+20=100, recalculation triggers, new jackpot 0, prizes of 30+100 are paid out.

Round 6: sell 100, get 50, jackpot 0+20, prizes of 30 are paid out.

Note that in every round, the amount that state receives is independent of the relationship between jackpot and other prizes.

minus 5% for 'administration' which is outrageous. Its a fixed cost to administer the lottery. Should not be a percentage.

And is anybody calculating the taxes on the prize, which also goes to fed/state?

I have no idea what the exact numbers are. I have tried to illustrate how the temporary increase in the smaller payouts has no bearing on how much the state makes, yet it makes the game temporarily profitable. Obviously this still needs to be profitable for the players after taxes, expenses and whatnot, otherwise it would be a complete non-story: "person does not get richer playing lottery."

From what I understand, money paid out was calculated after removing state profit from sales. So the state always profited, whether prizes were paid or not. It was effectively a per-ticket tax.

What changed was the distribution of winnings among players and operator. If everyone had cottoned on the “winning strategy”, the prize would have simply been divided in smaller and smaller parts - at some point, the winning strategy would have become a losing one - and the operator wouldn’t get to keep any “unclaimed prize”, only whatever fixed per-ticket fee they might have applied.

That’s precisely what the rules aimed for, btw. They wanted to unload prize money at higher rates over a certain threshold, for their own reasons - and they did. It just so happened that a small number of players benefited disproportionately more than others, because they played smart.

Imho the operators chickened out - as long as their profit model allowed for all prize money to be really given away, they could have continued, maximizing their overall revenue. However, if “keeping some of the prize money” was part of the expected operator profit, then yeah, it couldn’t be allowed to go on.

The simple correction is that the pot doesn't live in your pocket. And the lottery don't offer a fixed jackpot, they offer the contents of the pot.

So a realistic correction of this, would be that for each $1 game, 50 cents goes in the pot, 50 cents goes in your pocket. If 3 people play and all 3 win, their payout is 50 cents. Obviously 50/50 odds lead to a disappointing game for the players - a bigger pot needs lower odds - but either way, the house never loses.

For that particular round of the game, yes.

Overall? No, unless the game is very badly designed.

The whole situation arises from rolling over the losses between rounds, which "sweetens" the pot for intelligent players. It's also a good way to inflate your numbers if you know that your game might get cut next year (e.g. it might be better to show turnover of 100mm and payouts of 90mm than turnover of 10mm and payouts of 8mm).


...wouldn't that be better under all possible circumstances? Would you rather have $10 million or $2 million?


100mm with 99mm payout vs 10mm and 8.5mm payout then...

You're not thinking about this the right way. The ticket doesn't have a positive expected return as a structural feature of the lottery. It has a positive expected return conditional on the total number of tickets sold being less than a fixed value.

Are state lottery ticket purchases taxed? That'd also bring in a chunk of change.

That is correct: since these were not usual players who abstained from playing, but rather casual players who bought extra tickets just on rolldown, while the State got a cut from every ticket sale, they were effectively pouring extra money into the lottery, increasing the State's income.

Even if they re-invested their previous winnings, that would be twice (or more) that the State would get its cut from the same money, still increasing its income.

The money was "stolen" from the person that would have won the jackpot. The maximum jackpot was just smaller, it didn't make a difference to the lottery who actually won that money. They could say that the jackpot will be split among everyone that has exactly one number right in the next round, and still make profit.

* Expected value of the lottery tickets is normally negative. (You pay $1, you expect on average to get back less than $1).

* At a certain point, if the jackpot grows too large, the game switches to a new "windfall" mode where the expected value is actually _positive_ to bleed off the jackpot. (You pay $1, you expect on average to get back more than $1).

* The couple and the MIT students only played during the positive expected value times.

Overall the game still makes money for the state, it's just it has two distinct regimes, and the people playing during negative-EV essentially bankroll both the state and the positive-EV players.

There is a great lecture about it from The Royal Institution https://www.youtube.com/watch?v=kZTKuMBJP7Y

"That was satisfactory."

I love that this guy, who is probably, what, 80? 81? pulled off this huge math heist after probably a lifetime spent doing math running a convenience store.

I bet it feels great to have that sense of, like, ease with himself in old age, having made a mark on the world in a way that hurt no one and helped a lot of people.

He stuck the landing

He hurt all the lottery players he beat.

They bet money on a game that somebody else played better.

In fact not. Their winnings did not take a $1 of winnings from anyone else.

I was wondering how both these syndicates and the state we winning. I think I understand now.

Lotteries are zero sum games so no extra money is being created. All ticket purchases either go to the state or the winners. So if the players have a positive return how can the state also be winning?

The players have a positive return only for the rollover round. The players have a negative return for all other rounds, where the state makes its money, and the syndicates don't play.

So the people who lost are all the suckers who played in non-rollover rounds.

Every decade or so some "lottery loophole" like this is discovered, exploited, publicized, and rules are updated to close it. I'm curious to see if another loophole resulting in a massive rework of the lottery system will ever present itself, or if the laws and rules have finally prevented that after the last few.

I really enjoyed this story! The best part is people like to think this loop hole is easy “profit”. No one pointed out the fact that they would go on 10 day benders buying/sorting/etc. to get the profit! Even this loop hole for easy profit was incredibly arduous in manual labor.

Great story.. but extremely shitty interviewer in my view.

Great story. Now they should go make another million or two selling the movie rights!

Good for them! Thinking is rewarded.

I just wonder, do you have to think very high about yourself to play such a game and enjoy it? Or very low? I just can not imagine myself feeling good about this.

Of course, the feeling dumbs a bit: compare the feelings you had after using your first illegally copied floppy disk or mp3 track and the later ones, but still.

Why would you feel bad? What woukd you have down that was wrong? You aren't taking money from anyone else.

You are taking money from the prize pool, i.e. payouts for future winners.

You, and everyone else. That is the rule of the game. You aren't doing anything wrong. By that logic, any money winner is taking payours from future winners.

> You aren't doing anything wrong.

That's debatable. Lotteries are supposed to be completely up to chance. If by your own ingenuity you exploit a weakness then you are tipping the scales in your favor, which is unfair to less clever players.

It's basically the tragedy of the commons; everyone has the same chance of winning, but only you know you can safely risk more and win more, lessening the winning of others.

He didn't have insider knowledge though. He used the games own rules to play an advantage. Those rules were available to everyone, everyone could have done the same.

The rules are available to everyone, but having enough cash to purchase that many lottery tickets certainly isn't. Unfortunately this is yet another system that enables the already rich to get richer on the backs of the poor.

Perhaps if the poor didn’t spend so much on lottery tickets their station might improve? If they know they are playing a losing bet, why keep doing it? And these folks that were winning were hardly hedge-fund managers. If the “poor” wanted to play to win, perhaps they could get everyone on their block to chip in to buy shares just like the old couple did? They could get all of their friends to save what they would have spent in the lottery over the course of the year, put that fund into the “pot” and then win, reinvesting it and repeating the process.

It’s hard to be sympathetic to people that make bad decisions as a habit. For example, if it’s a 1/5,000,000 chance of winning and they still keep playing, that’s on them. The have very little sympathy for losing lottery players. If people choose to spend their money like that, that’s their business, but that doesn’t invite sympathy when it doesn’t work out. It isn’t like they are being defrauded or even cheated. They have access to the same rules of the game and they have access to math just like everyone else.

> Perhaps if the poor didn’t spend so much on lottery tickets their station might improve?

I agree, and indeed this is a good argument for prohibiting lotteries. They're a form of predatory gambling that relies largely upon irrationality, addiction, and desperation.

Any of the millions of poor people that buy lottery tickets every week could have spent a much smaller initial sum and worked up to the millions. It just would have taken longer.

According to the couple quoted in the article, taking advantage of a 'Rolldown' draw required an initial investment of $1,100 (equivalent to around $1,500 now). That's not a sum easily available to those who are poor.

In this case, the Rolldown was specifically designed to increase the payouts in order to encourage more people to buy tickets.

This is a feature, not a bug.

The bug was the fact that it was only two groups of people figured this out.

Nobody improperly took anything from anyone.

This really only shows how much the lottery is an innumeracy tax. There was nothing hiding the fact that you can safely risk more and win more -- everyone had full access to this knowledge -- it was published right at the place of purchase. It is not like it was some insider secret available to only to friends of some corrupt programmer or something. It took him only 3 minutes to figure it out. You or I could have done the same.

So, totally fair.

Unfair to the less clever players? How is it unfair? They can do the exact same strategy. It isn’t my fault of someone else doesn’t “get” it. It’s a game, not a civil right.

But the lottery is not a game of skill or knowledge. It's a game of chance!

In these cases the prize pool didn't roll over. The states wanted to entice people to play and they fully succeeded in that goal...

Lotteries are a regressive tax; I don't think this particular case was more or less regressive.

I'd enjoy it very much and not feel bad at all.

I first read about these people a year or so ago so CBS "news" has finally caught up.
