What this game actually paid back

A roll-down is the one kind of edge that ever came from a rulebook rather than a broken machine: when nobody wins the top prize the money is pushed into the tiers below, and for one night the game can pay out more than it takes.

Realised return, and what makes a night exceed it

For one draw, the realised return is the money out over the money in:

$R_d=\frac{\text{prizes paid}_d}{\text{lines}_d\times\text{price}}.$

Neither term is published directly, and both can be recovered. The prizes are the sum over tiers of winners times prize, with an advertised annuity read at its cash value — a Powerball jackpot quoted over thirty years is not thirty years of money handed over on the night, and treating it as such produced returns above 2,000% before the correction. The lines come from the crowd-neutral tier where the game has one; see the Sharing Map for how and for what it costs when it does not.

The era in one figure is not the mean of the ratios. It is

$R_{\text{era}}=\frac{\sum_d \text{prizes}_d}{\sum_d \text{lines}_d\times\text{price}},$

because a draw that sold twice as many lines is twice as much of what the game paid back. Averaging the per-draw ratios would give a quiet Tuesday the same say as a record rollover, and the two differ by an order of magnitude in both terms.

Why a night goes above one. Four causes, in a hierarchy of certainty rather than of preference:

1. The jackpot was won. Money accumulated over previous draws is paid out on this one, so the night's outflow has little to do with the night's sales. This is the game working as designed and explains the overwhelming majority of nights above the price. Detected from the top base tier having winners.
2. A roll-down. The top prize was not won and a lower tier's prize rose far above what that tier usually pays, which is money arriving from above. Detected as a prize at least three times the median for that same tier across the era — a threshold fixed before reading, and deliberately blunt, because a pari-mutuel tier moves by tens of percent every week and a tighter rule would call every busy Saturday a roll-down.
3. Thin sales. Fixed prizes divided by unusually few tickets. Measured against the era's own tenth percentile of sales, not an absolute figure, because "few" means nothing across games three orders of magnitude apart in size.
4. Not explained. Kept as a category rather than folded into the others. A residual that quietly disappeared would make the classification look better than it is.

The first two are mutually exclusive by construction: a roll-down is created by nobody winning the top prize.

What this cannot do. Every quantity above is measured after the draw. The pre-draw expectation of a return is a different object requiring an advertised jackpot, a sales forecast and a sharing model, and each of those carries its own error; the page reports only what happened.

The only edge that ever came from the rulebook

There are three ways a lottery has actually been beaten, and only one of them was legal, repeatable, and available to anyone who could read.

The first was a broken machine — balls that were not identical, a drum that did not mix. The second was fraud: Eddie Tipton wrote the random number generator for the Multi-State Lottery Association and put a back door in it. The third is the interesting one, because nobody had to break anything.

Some lotteries carry a roll-down. If nobody matches all the numbers, the top prize does not roll over to next week — it is pushed down into the tiers below, so the people who matched four or five numbers share money that was never meant for them. On such a night the game can pay out more than it takes in. Massachusetts ran a game called Cash WinFall with exactly this rule, and for seven years syndicates in Michigan and Massachusetts bought tickets by the hundred thousand on the nights it triggered. The state's own inspector general concluded afterwards that no law had been broken. The rules said what they said.

This page exists to show how rare that is.

The line on the first chart is what a lottery normally does: it sits somewhere between a third and a half of what was staked, week after week, for years. The marked points are the nights it went above the price. There will be some — a jackpot gets won, and a pile of money accumulated over eleven draws is paid out on the twelfth — but a won jackpot is not an edge. It is the game working. Somebody has to win eventually, and the night they do is not a night anybody could have identified in advance.

Which is the point to hold on to. Every figure on this page is about a draw that has already happened. A night above the price is visible only afterwards, and knowing that eleven such nights occurred in fifteen years tells you nothing whatever about the next one. Cash WinFall was different precisely because its roll-down was announced — the rule was published, the trigger was arithmetic, and the syndicates knew before buying. No game on this site works that way today, and if one did, it would be here in the roll-down column rather than in an advertisement.