Design a lottery you would never play

The four numbers in a format, and what each one buys

The odds. With two drums the top prize needs every main number and every bonus ball, so

$\frac{1}{P}=\binom{N}{k}\binom{N_b}{k_b},$

and 5 from 69 with 1 from 26 gives 11,238,513 × 26 = 292,201,338. With one drum the bonus balls come out of what the main draw left behind, a line holding all $k$ main numbers cannot also hold one, and the top prize is simply $\binom{N}{k}$. Every tier below is the multivariate hypergeometric

$P(m,j)=\frac{\binom{k}{m}\binom{k_b}{j}\binom{N-k-k_b}{\,k-m-j\,}}{\binom{N}{k}}$

for one drum, and the product of two independent hypergeometrics for two.

The prize ladder. A tier taking share $s$ of the stake returns exactly $s$ per unit staked to every line, whatever its odds: rarer tier, larger cheque, same expectation. Only the total moves expected value — the split between tiers decides who gets it, not how much comes back. A fixed cash prize $v$ at probability $p$ contributes $pv$ instead, and is the only kind whose contribution the odds can change.

The default ladder pays every outcome at or rarer than 1 in 30. Applied to 5-from-69 plus 1-from-26 that reproduces Powerball's real nine tiers exactly — it admits the bonus ball alone at 1 in 38 and refuses two main numbers at 1 in 28 — and the any-prize figure comes out at the operator's advertised 1 in 24.9 without being given it anywhere.

The break-even, and why it usually does not exist. Sales follow a power law $L=L_0 (J/J_0)^{\beta}$. Expected co-winners are $\lambda \approx L/T$, and the expected share is the identity , which for large is . The jackpot's contribution to one line is then

$\frac{J\,c\,(1-t)}{T}\cdot\frac{1-e^{-\lambda}}{\lambda}\;\longrightarrow\;\frac{J\,c\,(1-t)}{L_0}\left(\frac{J_0}{J}\right)^{\beta}\;\propto\;J^{\,1-\beta}.$

It grows without limit only when $\beta<1$, is flat at $\beta=1$, and falls when $\beta>1$. A game whose players respond more than proportionally to a bigger prize can never be worth playing at any prize, however large. That is a fact about the players, not the rules, and it is the one thing in this studio a designer cannot fix. Because the curve is not monotone, the search scans a logarithmic grid for the peak first and bisects only inside the rising part.

When the odds stop mattering. There is a second regime hiding in the same identity, and it is the most surprising thing this studio produces. Once sales are large enough that $\lambda \gg 1$, the share is $\approx 1/\lambda = T/L$ and the jackpot's contribution to one line becomes

$\frac{J\,c\,(1-t)}{T}\cdot\frac{T}{L} \;=\; \frac{J\,c\,(1-t)}{L},$

with the odds $T$ cancelling out entirely. Set the studio to three hundred million lines and move the pool from 50 to 80: the break-even jackpot does not move at all, to twelve significant figures. Set it to one million lines and the same change multiplies the break-even by the odds ratio exactly, which is the intuition everyone arrives with. In a crowded game the prize is being divided among the same crowd either way, and how hard it was to reach makes no difference to what a line is worth.

Everything else is closed form. The sum curve is the exact count of $k$-subsets by total; the neighbours are $1-\binom{N-k+1}{k}\big/\binom{N}{k}$; the draws to see every number are the coupon collector's exact expectation, not the harmonic approximation.

Sit in the designer's chair for ten minutes

A lottery is four numbers and a price list. How many balls you draw, how many there are to draw from, whether there is a second machine, and what the ladder pays. Everything anyone has ever said about a lottery — that the odds are absurd, that the jackpot is life-changing, that somebody always wins eventually — is downstream of those four choices, and the choices are not arbitrary. They solve a problem.

The problem is that the two things a lottery has to sell pull against each other. People buy a ticket for the jackpot, and the jackpot is only enormous if almost nobody wins it. But a game nobody ever wins is a game people stop playing. So the designer needs long odds at the top and frequent wins at the bottom, and has to pay for both out of the same ticket.

Watch how it is done. Raise the pool by one and the jackpot odds lengthen while nothing else on the page moves much — that is the cheapest lengthening there is, and it is why every large lottery's pool has grown over the decades. Then add the second drum. A bonus ball from its own machine multiplies the odds by the size of that machine without changing anything a player recognises: the ticket still shows five numbers, and the game is twenty-six times harder.

Longer odds mean fewer winners, fewer winners mean the prize rolls over, and a rolling jackpot sells more tickets than any advertisement ever written. That is the engine.

The ladder does the other half of the job. Set it to pay only the jackpot and watch how little the return rate changes — the small prizes barely cost anything, because a tier at one in thirty pays out a few pence a line. What they buy is the sentence "I win something quite often", which is true, and which keeps the habit alive between the rollovers. The small prizes are not where the money is. They are where the routine is.

There is one more thing worth doing before you leave the controls, because it is the only result here that surprised the people who built the page. Set the sales very high — three hundred million lines — and then move the pool. The odds change enormously and the break-even jackpot does not move at all. In a game that crowded the prize is being shared among the same people whatever the odds are, so how hard it was to reach stops mattering to what a line is worth. Drop the sales to a million and the same change moves the break-even in exact proportion to the odds, which is what everyone expects. Both are the same formula; they are just different ends of it.

Then there is the number the designer does not control. Sales grow with the jackpot, and every extra buyer is another person to share it with. Push the elasticity above one — players responding more than proportionally to a bigger prize — and the break-even jackpot does not merely get large. It stops existing. However big the prize gets, the crowd arriving to chase it carries the extra money away as fast as the operator adds it, and a line is never worth its price at any headline that could ever be printed. Powerball's own record puts its elasticity below one, which is why its break-even is a real, finite, occasionally-reached number rather than an impossibility.

You cannot design your way out of the last line of this page. Whatever you do to the inputs, the return rate stays below the price — a game that broke that rule would be a charity with a drum. The point of the exercise is not to invent a lottery worth playing. It is to see, in your own hands, that the game you already play was built to hit exactly the numbers it hits.