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.
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.