The Michigan state lottery

A multi million $ bill laying on the sidewalk

What if one day you are in a convenience store in Michigan. While waiting for the store clerk you notice some state lottery tickets on the counter. They cost $1 a piece and have pay-offs with the odds listed next to them. Because it is roll-down week (nobody has hit the big million $ number for a while) the payout has temporarily been increased and looks as follows:

• 1/54 chance to win $50

• 1/1500 chance to win $1000

• Some tiny tiny tiny chance to win several million $

Notice anything unusual?

Multiplying $50 by 1/54 and adding that to $1000 * 1/1500 will show that the expected value of a $1 lottery ticket is ±$1.6. 

As the store clerk returns you incredulously ask if this is indeed real. He confirms, but not convinced, you check other convenience stores, and find the same thing!

So what do you do? Buy lottery tickets by the truck load? Of course if you make mid to high 6 figures or higher this is probably not going to be worth your time. You will have to buy tens of thousands of lottery tickets, scratch them and check the number to find the winners. So about $3-400/hour given that every ticket should have a $0.5-0.6 average pretax profit. But for most people this will be worth their time and capital. 

Now imagine exploiting this pretty obvious flaw for almost a decade and making millions of dollars while (almost) nobody else seems to notice this. 

If this sounds unrealistic, this actually happened! A convenience store owner named Jerry Selbee in a quiet part of Michigan noticed the exploit and between 2003 and 2012 generated a $7.75 million pretax profit from grinding out the lottery. He even set up a corporation to share the profits with friends and family. And it inspired a movie with Bryan Cranston in the lead.

There are several elements about this story that are quite astonishing. How did the lottery organisers not catch this for nearly a decade? And why does almost every article or video I see about this imply you need to be a maths wizard to notice this exploit? When all that is needed is a simple EV calculation using addition and multiplication that should be taught in high school?

But for a decade, besides MIT (obviously, when aren’t they involved?) and a biomedical researcher from Boston, nobody else seems to have noticed this. When all that was needed was to go into a convenience store in Michigan, glance over to the lottery tickets and notice “hmm 1/54 to win $50 already gets me pretty close to the price of one lottery ticket!”. 

What can we learn from this? Well that simple glaring inefficiencies much larger than a $20 bill on the sidewalk, can lay out in the open for a long time. And they are not immediately exploited and ironed out like efficient market theorists would have you think. 

But they are still pretty rare. You would have had to go over thousands of lotteries and probably only would have found 1-2 obvious exploits like this. 

A similar mechanism exists in the stock market. Except there the odds will not be as clearly advertised. But to find them you need to play the numbers game, and turn over a lot of proverbial rocks. Which of course requires a situation to be simple enough that you can relatively quickly notice the mispricing. But not too simple. If this lottery had advertised the expected value of a lottery ticket during rolldown week it probably would have been noticed by a lot more people. 

And these relatively straight forward exploits did not just exist in lotteries, but also in roulette! This Bloomberg article goes into more detail, a very entertaining read. To get through the paywall press f12, then ctrl + shift + p and type in disable javascript, hit enter and then refresh the page (this works on most paywalls).