The $656 Mega Millions Jackpot.

The Mega Millions drawing last night was the largest lottery jackpot in history at $656 million, considerably larger than the previous largest jackpot of $390 million. The massive jackpot was in part due to the record sales of tickets nobody won Tuesday night's jackpot of $363 million. The logic behind these tickets sales is simple enough: as the jackpot gets larger, the value of a winning ticket increases, potentially to the point where buying a ticket makes sense from a mathematical point of view.

However, this logic misses an important point: As more tickets are sold, the probability that the pot will have to be split between more than one winner dramatically increases. In general, this effect is more than offset by the increase in the value of the pot. With the incredible hype around yesterdays drawing, though, so many tickets were sold that the splitting effect actually outweighed the pot size effect. I ran the numbers, and the best drawing to play the lottery (in terms of expected value, aka EV, of buying a ticket) was actually last week, as the following table shows:


The first three columns are in millions. The bottom line number is given in the "Actual EV" column.  This number is the amount that you should expect to get back - post taxes - for every dollar you spend if you take the lump sum option. (Nominal EV uses the headline pot size figure rather than the lump sum amount.) As the table shows, playing the lottery is always a bad idea. The "best" drawing to play would have been Tuesday night, and even then the expected loss would be thirty cents on the dollar. Friday night's drawing was the worst in more than two weeks and barely even had 2/3 the return of Tuesday's drawing. The explanation for this lies in the Average Pot Proportion value- with the huge number of tickets sold before that drawing, a winner on average would only keep about 18% of the pot. (As it turned out, there were three winners, so they each got to keep a third of it.)

If you're interested, here's how I got the numbers. The pot sizes and lump amounts were pulled from Lottery Post's news section. I calculated the tickets sold for each drawing by assuming that thirty cents of each ticket goes towards the jackpot, which is consistent with the numbers I've found. Getting the Nominal EV was a bit more difficult: I used the total number of tickets sold to calculate the likelihood of exactly n winners for n=0,1,2... and then calculated the expected share of the jackpot based on the annuity value. (There are two subtleties to this calculation: First, I assumed that everyone selects their numbers at random, which is close enough to being true. Second, an individual who buys a lottery ticket only cares about the total number of winners if he is one of them- ie, conditional on his ticket being a winner. This inflates the expected number of winners in the calculation.) I got the actual EV by considering several other factors: First, I used the lump sum value instead of the annuity value. Second, I included a federal tax of 30% and a state tax of 5% (which is typical.) Finally, I added .14 to the EV to account for non-jackpot prizes (which are independent of the pot size.)

The two obvious sources of error in the calculation are both utility related. First utility is very much non linear by the time you get to hundreds of millions of dollars. Second, people derive utility from imagining the possibility of winning. I suspect that many of the people who bought tickets last night derived 53 cents per ticket of utility from fantasizing about winning.