Why Jackpot Size Drives Sales When It Cannot Change Your Odds

A rollover jackpot changes exactly one thing about your ticket: the amount printed on the prize. It does not change the number of combinations, so it does not change your probability of holding the winning one.

Sales respond anyway, and they respond hard. This article covers why, and then the part that gets left out — that the bigger jackpot comes with a sharing penalty that eats much of the apparent gain.

The demand evidence

Philip J. Cook and Charles T. Clotfelter, The Peculiar Scale Economies of Lotto, American Economic Review, volume 83, issue 3 (1993), pages 634–643.

Their result is the foundation of lotto game design. Because a larger player base supports larger jackpots, and larger jackpots draw disproportionately more play, lotto exhibits a scale economy in demand: bigger populations produce higher sales per capita, not merely higher sales. This is why small jurisdictions join multi-state and multi-country games — the jackpot is the product.

David Forrest, Robert Simmons and Neil Chesters, Buying a Dream: Alternative Models of Demand for Lotto, Economic Inquiry, volume 40, issue 3 (2002), pages 485–496, tested this directly on UK National Lottery data. They found that jackpot size exerts an influence over and above variations in effective price — that is, players are not simply responding to the improved expected value that a rollover delivers. The headline number itself does work that the arithmetic does not account for.

That is the empirical anomaly. Now the mechanism.

Probability neglect and scope insensitivity

Two documented patterns are relevant, and both are about the same underlying failure: very large and very small numbers are not felt in proportion.

Probability neglect. Cass R. Sunstein, Probability Neglect: Emotions, Worst Cases, and Law, Yale Law Journal, volume 112 (2002), page 61. Sunstein's argument is that when an outcome is emotionally vivid, people attend to the outcome and largely disregard its probability. Attention goes to what it would be like, not how likely it is. A $1.5 billion jackpot is a more vivid outcome than a $200 million one, and that vividness does work regardless of the probability attached.

Scope insensitivity. In a well-known contingent-valuation experiment, respondents were asked what they would pay to prevent migratory birds drowning in oil ponds. Willingness to pay was essentially flat across scenarios of 2,000, 20,000 and 200,000 birds — roughly $80, $78 and $88 respectively. A hundredfold change in scope produced no change in valuation. The result is discussed in Carson and Mitchell's review of scope tests.

The lottery analogy is not a study, and it is offered here as an analogy rather than a finding: below some threshold, a probability stops being felt as a magnitude at all. One in 45 million and one in 292 million are both simply "basically never" — so the probability side of the calculation goes flat, and only the prize side varies. When one input to a mental multiplication stops responding, the product tracks the other input alone.

The part that is left out: the sharing penalty

Here is the arithmetic almost no jackpot coverage includes.

Suppose N tickets are sold for a draw with C total combinations. Treat the lines as chosen independently and uniformly (an approximation — real players cluster, which makes the following worse, not better). Then the number of other jackpot winners follows a Poisson distribution with mean:

lambda = N / C

Given that you hold a winning combination, the expected fraction of the jackpot you receive is:

E[1 / (1 + K)] = (1 - e^(-lambda)) / lambda

Working that out:

Now apply it. A rollover that triples the advertised jackpot typically does so alongside a large sales increase, which raises lambda. Take an illustrative pair:

The advertised prize went up by 200%. The amount you would actually expect to collect went up by about 21%. Your probability of winning did not move at all.

This is not hypothetical. On 13 January 2016 the US Powerball jackpot of $1.586 billion — the largest to that date — was won by three tickets, in California, Florida and Tennessee. Each received roughly $528.8 million as an annuity, or $327.8 million as cash (Forbes). The billion-and-a-half-dollar headline paid out at about a third of that per winner, before tax.

The full mechanics of how pools are divided are in how prize pools are split, and the number sharing risk and syndicate split tools will run your own figures.

The one genuine effect of a rollover

To be fair to rollovers: they do improve expected value, because prize money carried forward from a previous draw is money you are not paying for this time. In rare configurations this is enough to push a draw positive — the conditions are set out in when a lottery is genuinely +EV, and the break-even jackpot calculator finds the threshold for any game.

But the same sales surge that makes the jackpot exciting is what erodes the improvement. The rollover raises the numerator; the crowd raises lambda; the sharing penalty claws a large share back. Any honest expected-value calculation on a record jackpot has to include the split, and most published ones do not.

The summary

• Odds: unchanged. A rollover alters the prize, not the combination count.
• Sales: sharply up, by more than the change in effective price justifies (Forrest, Simmons & Chesters 2002).
• Expected share: down, because lambda rises with sales.
• Expected value: up, but by less than the headline suggests.

If the jackpot number is what makes a draw worth entering for you, that is a preference, not an error. It is worth knowing that the number on the billboard and the number you would receive are not the same quantity, and that the gap widens exactly when the billboard is at its most impressive.

The marketing built around that moment is covered in draw-night marketing psychology.