It was only years later that someone finally ran the numbers and figured out the actual house advantage of the game , 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.
Unless you're making a distinction for determining the odds during optimal play (which makes a lot more sense to me).
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.
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.
Incredibly, the MIT group bet between $17 and $18 million on Cash Winfall over a seven-year period, earning at least $3.5 million in profits. Almost the exact same rate of return as the Selbees.
A nice illustration of the fact that there is only one mathematics, and it's equally available to everyone!
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.
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.
Not sure why this is down-voted. It's spot on.
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.
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 this submission:
And those who arrange lotteries do know their business.
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.
the likelihoods of the two events are similar, and one fantasy is free.
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.
Have you ever bought lottery tickets?
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.)
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.
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".
Telling people that they have a 95% chance of winning nothing at all is not a good way of selling lottery tickets.
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.
“Do Not Play the Lottery Unless You Are a Millionaire”
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...
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.
That thrill is only there when the jackpot is at $1B.
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?
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.)
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.
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.
>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?
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.
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.
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.
You can see the entire video here.
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.
Now, gift cards operate more like limited credit cards.
I highly recommend the book; it is a popular mathematics book, written by a real mathematician, discussing some real mathematics, with an engaging style.
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
and in other games this doesn't work because they are 'winner-takes-all'?
Doesn't work any more, rules have changed so this way of winning was removed.
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.
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.
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.
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...
Money came from people who played on days there wasn't one.
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.
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.
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.
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)
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.
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.
And is anybody calculating the taxes on the prize, which also goes to fed/state?
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.
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.
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...
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.
* 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.
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
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.
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.
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.
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.
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.
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.