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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.

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

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).

> 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.

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