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Comment by bryanlarsen | Hacker News Reader

AFAICT, many jackpot style lotteries get an expected value (EV) above 1 when the jackpots get big. They're set up so their average EV is 0.4 - 0.5, but when a jackpot isn't won, the payout is under 0.1. String a bunch of those together in a row, and the EV creeps above 1.0.

The calculations are easy to do. You do need to estimate the number of people who buy the tickets so you can calculate the odds of sharing the jackpot. The real number isn't publicly available, but newspapers often have numbers for similar jackpots in the past.

As the article mentions, lotteries protect themselves by making it logistically impossible for anybody to buy every ticket.

Here's the calculation for Canada's Lotto 6/49 in 2007: https://docs.google.com/spreadsheets/d/1vHOOdmobb3729ezH1BnO...

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While true, expected value isn't necessarily a useful metric at that point.

If I have a one in a googol chance of winning a googolplex dollars, it doesn't really matter that the expected value is going to be huge, I still won't win anything because I'll be insolvent before the requisite number of tries to have a reasonable chance of winning.

thats why the people who do this work as a company and pool their resources

It's more that the people who do this do more than just buying a single ticket with positive expected value - as described in the article.

Yeah, the Kelly Criterion is a good thing to explore in situations like this. If there's (truly) positive EV, you still should only spend a certain percentage of your bankroll on the bet. And if the bet has high payoff and low odds, your bankroll likely has to be astronomical in order for the bet to be a good decision.

True, +EV is useless if the variance is huge, as is the case with lotteries.

y'all really need to read this

https://en.wikipedia.org/wiki/St._Petersburg_paradox

Very interesting.

The funny thing is you don't need any stats to win smaller amounts of money. Just the simple question "hey what's the newest 10 dollar ticket" was enough to result in a solid net positive and a lot of beer money in college. They tended to front load the winning tickets for the newest ones.

As the article mentions, lotteries protect themselves by making it logistically impossible for anybody to buy every ticket.

I wonder why they care. In fact, they need someone to win big, so as to entice future players...

But someone will win big, sooner or later. It is more enticing that a random lucky person wins rather than a syndicate.

Because knowing that you're competing against large, well funded organizations on a regular basis makes people suddenly start to realize (to a small degree) how statistics work.

If you assume that everyone else has the same basic odds of winning that you do (1 in a Million for example) you might play. But if you find out that you have a 1:1,000,000 chance and another well funded group has a 1:10 chance, you're going to quickly realize you're just throwing away money.

If an unfair game is equally unfair for everyone, it's fair. If it's only unfair for some/most people to a greater degree than some other people, then it's unfair.

Your EV numbers are high.

If the EV was really significantly above 1.0 (say 1.5) then everyone would buy a lot of lotto tickets. It would be simple arbitrage. In reality all of those people buying extra tickets (because of the high perceived EV) would mean a higher chance of splitting the pot. Splitting the pot would reduce the real return by 1/2 for every person that purchased a ticket.

Sometimes the EV goes above 1.0 and there are several groups of people who closely monitor lotteries all over the US to spot this event. When it does happen, which is very rare, they do calculations on the expected impact, viability and if their conclusions are good, swoop in and buy hundreds of thousands of tickets.

They have people printing tickets for an entire day. They pay stores to stay open late. It's a tactical operation.

Sometimes the jackpot is so anomalously big that the general public can't buy enough tickets to bring it back down below 1.0.

The post you replied to seem to imply a "naive EV", because they already discussed odds of sharing the jackpot.

The odds of splitting the pot are calculated in my spreadsheet. In the first scenario, that decreases the EV from 2.15 to 1.11.

Problem with this calculation it it assumes that exactly one person/group undertakes the buy-every-ticket strategy. As soon as there are two or more groups the chances of splitting the pot goes to 100%.

Actually, the spreadsheet assumes that nobody is undertaking the buy-every-ticket strategy, it is calculating the EV for a single ticket.

In scenario #1, there are 1.93 tickets sold for every possible combination, meaning that on average, 1.93 people win the jackpot. If you go ahead and buy every ticket, that means 2.93 tickets per combination, reducing EV from 1.11 to 0.78.

Because of the payout model (all and only winning tickets split a fixed prize) the EV for buying multiple randomly distributed tickets and uniformly distributed tickets aren't the same. Consider if you were the only one playing a lottery with 100 combinations. If a ticket costs $1 and the jackpot is $100 one random ticket has an EV of $1. However if you buy 100 random tickets on average you'll only cover about 64 numbers, so your EV per ticket is just $.64, whereas if you buy uniformly distributed tickets the EV is constant up to the number of combinations. If you imagine raising the prize to $200 and that two people are playing this lottery and one buys 100 random tickets and the other buys every combination the random player has about a 36% of getting nothing, and will share the jackpot the remainder of the time. Overall the uniform player has an expectation that's almost twice as high as the random player, and makes a pretty strong case for buying every ticket if you're the only one that does it. It falls apart if you have to compete against another uniform player.

Who in their right mind would buy 100 tickets chosen at random _with replacement_ in your scenario?

100 people

Exactly. In fact in practice they do an even worse than random job since many people pick non-random numbers like birthdates or the number of kids they have. Any uniform purchasing strategy would probably have better EV from not buying any of the most popular combinations.

I track the EV of 5 different lotteries available in Minnesota. At this moment only one of them is over 0.5, at 0.55. I don't even think about buying a ticket until it gets over 0.8.

I'm not sure that is really logical. The odds of winning are the same regardless of if it $400m you can win or $20m. However, the effect on your life would be basically the same in either case.

To be fair, I do much the same thing. But, it doesn't really have a basis in logic. It's all psychological.

One significant difference compared to the U.S. jackpots - you need to pay taxes, unlike in Canada.

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