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.