A lot of people are focusing on the courier aspect of all this which is fine, but I didn't know there were any lotteries you could just outright win, with a profit, if you had enough money to buy enough tickets. I assumed people would design lotteries where the cost/reward ratio was such that this would never make sense.
There's a couple ways this works. Progressive jackpot games (Powerball, Mega Millions) allocate some amount of every ticket sold for winnings, but if the jackpot isn't won on a particular drawing, excess winnings are added to the prize pool for the next drawing. After a certain point, the jackpot prize for a single winner is more than the cost of buying all possible tickets. There's a chance of sharing a jackpot, which is hard to model, but makes the payout worse.
A similar game feature is "roll down", again excess prize money accumulates over several drawings, and when a certain criteria is met, the excess prize money is distributed over some set of tickets (possibly all winners). Again, this sets up the possibility of a positive expected value, and you have to consider other ticket buyers as well.
A trickier one is for scratch off games. Many lotteries share the number of tickets sold and the prizes left. If you assume all (big?) prizes are redeemed shortly after their ticket is sold, you can estimate the expected value of purchasing the remaining tickets. When the game opens, the expected value of a ticket is less than the purchase price, but depending on the observations of tickets sold and prizes redeemed, you might estimate that the expected value of the remainder of tickets has improved.
Ex: if there were 1 million scratchers printed, the cost per scratcher was $1, and there was only one prize $500,000on open the expected value of a $1 ticket would be $0.50. If the winning ticket was redeemed, the expected value of remaining tickets would be $0. If it was reported that 999,999 tickets were sold and the winner had not yet been claimed, it might be reasonable to assume a higher expected value for the last ticket --- although there's no rigorous proof there, someone may have purchased the winning ticket already and not redeemed it for whatever reason.
I'd imagine scratchers would be almost impossible unless you could somehow get all the tickets from every place they are sold. I'm not into gambling, but I guess it might work if there is X number of prizes/money per roll of tickets.
Even if you can't buy every ticket, there is well-established math about how to optimize profit from a venture with known risk and reward, and the math does not require you to exhaust the statistical universe.
There is clearly a limit on practicality though. Unless you are suggesting someone should sell their 401k and buy Powerball tickets any time the jackpot exceeds the odds of winning simply because it has positive EV?
This has happened. IIRC it was a Stanford statistitian who watched the distribution pattern for winning tickets for a certain scratch off game in Texas. She bought all of that ticket from that particular store and won $10M. I think that was her fourth scratch off win.
The way scratcher play works is that the states are required to report wins over a certain threshold. So if you keep a keen eye on the state website, it’s possible to determine how many of the big awards (say $10k+), which is where the bulk of the value is. Using that you can estimate what the outstanding prize pool is. You won’t know with any precision… but basically the idea is they look for games that have been on the market a long time, and thus sold through a large portion of their inventory, but where the big prizes are still mostly in play.
They absolutely aren’t trying to buy them all, that would just be a guaranteed loss, since they only return about 40 cents on the dollar.
> I guess it might work if there is X number of prizes/money per roll of tickets.
In most cases, there is, which is part of why a huge percentage of scratchoff prizes are won by workers at the place that sells them. Most players will scratch and redeem their prizes right in front of you, so if you watch a certain number of scratches occur in a roll and you know the prize structure of the particular card, you can calculate how many non-winning scratches you need to see for the odds to be in your favor.
I looked into this a few years ago and considered starting one of those stands that sells scratchoffs to do just this, but decided a) it wasn't quite lucrative enough to be worth it, and b) I wasn't sure of the ethics of skewing the odds against your customers like this anyway.
> I wasn't sure of the ethics of skewing the odds against your customers like this anyway.
This is interesting because I don’t think anyone would view the store as unethical for continuing to sell tickets from a roll when they know there have already been X winners from that role and therefore customer odds have gone down.
There's similar for "pack hits" and trading cards, and the regulars learn which hobby shops are reputable and which ones to avoid. Most that remain in business are not scum.
IIRC the remaining risk lies in multiple people winning or attempting arbitrage simultaneously, thus dividing the expected revenue by the number of winners. So not a free lunch.
IIRC, there was at least one case where the lottery got wise that this was happening and refused to sell the parties involved any more tickets. They had enough tickets for better than even odds, but not a guarantee. IIRC, they won.
I remember a 1990s lottery event in which a "buy all combinations" was attempted, but their physical machines they acquired were partially deficient, and they simply couldn't physically acquire enough tickets in time (as the procedure was relatively time intensive), but they still won with something like a 75% probability of success
On its face this sounds like it makes sense but why would the lottery care, at all? They get money per ticket, a story that buying lots of tickets increases your chances of winning, it's win win win for them.
I believe it's a marketing/psychology thing - people like to hear stories about how people like them won (busy person wins big from inexpensive last minute purchase) because they can relate and are more likely to buy a ticket in the next draw. Hearing that some international syndicate with big money and clever mathematicians won makes the everyman think that a big win is out of reach for them.
This scenario happens in basically any form of gambling with a progressive (grows over time until hit) jackpot. For instance there are literally pro video poker players. For some imaginary numbers let's say there's a video poker game where you play for $1 and the house has a 2% edge against perfect play. This means each time you play (in the longrun) you lose $0.02. But now let's say there's a jackpot that you have a 1 in a million chance of hitting, that has $1 million in it. This means that your expected value from the jackpot is $1 per play.
So the net result in our game is that each hand you play, you win $0.98. A skilled video poker player can get around 1000 hands per hour, so you'd be earning around $980 per hour in the longrun. Casino comps make this even more profitable. Depending on the game/casino casinos will generally comp around ~20% of their expected profit against you, and that excludes jackpots. For our imaginary $1 game with a 2% margin that means you'd also be getting $0.004 per hand back in comps. It becomes quite significant at high stakes.
It's about expected value. The assumption is more people buy in at higher payouts causing the pot to split. Usually the EV is negative. The real insight here was that this state lottery was not so popular as evidenced by long spans of time between payouts.
Net EV=cost to buy in - probability of winning * (jackpot size / number of people you split it with)
If you have a 1% chance of winning $100 your EV is $1. If you pay $1 to play you breakeven. If the pot is $200 then your EV is $2. You would pay $1 all day for that. But again the risk is more people want to play. If 2 people win then your EV drops back to even.
The lottery doesn't work the same way as most other forms of gambling. For the house, it is not zero-sum. Their profit is a fixed percentage of sales, and the bigger the prize, the more tickets people buy. The money won from the lottery was set aside at the time of purchase, it's already gone to the people running the lottery.
So the lottery makes more the bigger the prize gets. They don't really care who wins or how much they get.
If I did the math correctly, a positive-EV lottery is still guaranteed to make money for the state, because the total value of tickets purchased always exceeds the jackpot.
The lottery is always negative-EV for the average ticket-buyer, but it can sometimes be positive-EV for the marginal ticket-buyer.
