Mega Millions Frenzy

Mega Millions Frenzy

It’s difficult to make economic forecasts in uncharted territory but lottery officials have set a new standard for missing the mark.  Everyone realized that the Mega Millions jackpot would set a new record and that lottery sales would surge, but no one knew by how much.  Lottery officials consistently underestimated sales over the past few days.  The first announced Mega Millions jackpot was $460 million but reached $640 million by the end of the week.  Higher jackpots result from higher ticket sales, but more than two-thirds of new sales revenue goes to the states and consolation prizes.  This means that Mega Millions sales over the past three days were almost three times higher than suggested by the original advertised jackpot. 

Lottery officials seem to have badly underestimated Mega Millions frenzy.  Are lottery officials really that bad at forecasting sales?  Lottery officials are almost certainly required to make cautious forecasts so that the advertised jackpot rarely overstates the actual jackpot.  Do regulations requiring lottery officials to systematically underestimate jackpots depress sales?

Perhaps not, because conservative forecasts may promote sales as gradual increases in advertised jackpots are reported by the news media.  An unbiased forecast would be revised down as often as it is revised up, but a decrease in the advertised jackpot could provide negative publicity.  Would we have seen the same Mega Millions frenzy if the advertised jackpot had been revised down during the past few days?

The nature of Mega Millions arithmetic means that if a jackpot has accumulated over time, as we observed from January 24 until this week, increases in the size of the advertised jackpot during the week is bad news for the customer.  This is because rising sales during the week increase the jackpot by far less than they increase the chance of a winning ticket sharing the jackpot.

If the actual jackpot had been $460 million as originally “forecast” by officials, the winning ticket would have been expected to generate a payout about 20% more than Friday’s winning tickets from a much larger jackpot.  With fewer sales during the week the most likely outcomes are that a winning ticket shares the jackpot with one other contestant or receives the prize alone.  As sales grew during the past few days it became much more likely that the jackpot would be split three ways, which is what happened last night.

Unless lottery officials change the Mega Millions rules we won’t see a jackpot this large for at least five years.  The accumulated jackpot grew to record levels because over 800 million tickets didn’t match the winning numbers over 18 previous drawings.  The odds of this happening are less than one in one hundred.  Over the past four years the typical jackpot was awarded after accumulating over 8 weeks, and Friday’s jackpot was just the third that had accumulated for more than 15 drawings.

When the next record jackpot occurs players should be skeptical of the initial jackpot announcements and expect the advertised amounts to increase as the next Mega Millions frenzy appears to gain momentum.  When this happens players should also remember that jackpot increases during a week of record Mega Millions sales actually decrease the expected payout from a winning ticket.

Why a Mega Millions Ticket is a Good Bet, But… (Revised 03.30.2012)

The Mega Millions multi-state lottery is now advertising a jackpot of $640 million for tonight’s drawing. The cash value of the jackpot is $460 million, a more accurate estimate of the present value of the prize. The jackpot has accumulated because there has been no winner for 18 consecutive drawings – since January 24th. This means that those of us who buy Mega Millions tickets for the first time this week have been subsidized by the millions of players who contributed to the jackpot over the past two months.

The probability of correctly selecting all six numbers for Friday’s drawing is one in 175 million. It is likely that there will be multiple jackpot winners. Even after taking this into account, the expected value of a one dollar Mega Millions ticket is about 90 cents.

About 18 cents of every dollar collected from Mega Millions sales are used to pay consolation prizes. Another 32 cents of each dollar goes into the cash jackpot. The cash value of Friday’s drawing is estimated to be about $200 million higher than in Tuesday’s drawing. In other words lottery officials expect about 625 million lottery tickets to be sold between now and Friday.

If lottery officials are correct there is a 96% chance that someone will pick the winning numbers on Friday, and the expected number of winners, conditional on the jackpot being awarded, is about 3. This means that even after splitting the jackpot, the expected value of a Mega Millions ticket is about 90 cents (including 18 cents in expected consolation prizes).

In general state sponsored lottery games have low expected values for players, relative to other forms of gaming. Lotteries pay out about half of their proceeds in prizes, on average. The rollover nature of the jackpot in state lotteries means that a lottery ticket will have a high expected value on occasion. Until the recent surge in sales, this week would have been the highest expected value for a Mega Millions ticket in the history of the game.

Finally, because the jackpot will be taxed at a fairly high rate a Mega Millions ticket is not a good investment, as Ben Casselman of the Wall Street Journal has noted. After taxes the expected value of a Mega Millions ticket is now much less than $1, unless the top marginal tax rate falls substantially over the next twenty years, which is doubtful. However most gambles with long odds, such as a super trifecta bet at the racetrack, have a pre-tax expected value that is lower than the cost of the wager. So the Mega Millions is an unusually good bet for people who enjoy gambling small amounts at long odds, but it would be a bad investment.

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