Why Huge Lottery Jackpots are Not Positive Expected Value

In my previous post, I mentioned that math/trader/arbitrage oriented guys like me see huge nominal lottery jackpots and immediately think “positive expected value situation.”   The problem, of course, is that as more people buy tickets, your risk of getting “pro-rated” – having to split the jackpot – increases.

The easiest way to think about it is that the jackpot is “seeded” with all of the “dead money” from the previous week’s drawing that had no winner.   For example, in the previous drawing from March 27th, the estimated MegaMillions jackpot was $363 MM.   The estimation for tonight’s jackpot is up to $640 MM – up from $500 MM when I wrote my post yesterday.  People are buying tickets at a furious pace.   Someone reading this will know the details better than I do, but I’ve heard that somewhere between 30% and 50% of the ticket purchases go to fund the jackpot.  In other words, for the jackpot to grow from $363 MM to $640 MM, that’s $277 MM in “jackpot dollars,” which equates to between 554MM and 923MM tickets sold.   Le’s just use a nice round number like 700MM.

Anyway – the point is that if you buy a ticket to this lottery drawing, you really want the jackpot to be $363 MM – the size of the previous jackpot plus your lone ticket purchased.   You want to hold the only ticket, so that if you win, you get to keep it all for your own greedy little self.   Each additional ticket purchased by someone else dilutes your “share” of that dead money already in the pot.   Even worse, the entire value of their ticket doesn’t go to the jackpot – only a fraction of it does.   When the jackpot swells to $640 MM, you can win more, but you also have a crapload of “competition” in the form of other tickets that make it more likely you’ll have to split the prize.

Some readers on my initial post tried to quantify this effect.  Commenter Mikecngan extrapolated from a research paper to get an expected 953MM tickets sold.  I noted that his assumption looked a little off, based on the $500 MM jackpot estimate at the time, but now Mikecngan’s estimate is looking pretty darn good, even though his EV math assumed a $500 MM jackpot (corresponding almost perfectly to the upper end of our range based on 30% of proceeds going to the jackpot).   Steve Hamlin‘s comment provided a link to a WSJ post which quoted a study which noted the non-linearity of ticket purchasing as the jackpot grows.  In other words, as the jackpot grows, the rate at which people buy tickets increases, which is bad for the pro-ration risk.

In any case, this is all just a mental exercise.   You’re probably not playing the lottery because you think you’ve got the best of it mathematically.   Despite being a bona-fide EV whore, I’d be happy to split $640 MM several ways, in fact…

related:  A Game Theory Problem



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