Lots of people like saying that playing the lottery is a bad bet- and it is, mathematically speaking. If you ever need to convince someone of this, allow me to suggest the following argument:
Let’s say you had the option to put a dollar into a collection along with 9 other people. One of the 10 people is chosen at random to get the $10 collected. Would you go for this? On average, you’d win $10 one out of every ten times, so the expected value of the system is $1, exactly what you are being asked to pay. If a person is neutral to risk, he would probably be indifferent to participating and not participating. In reality, however, most people are risk averse in most situations, which means that they prefer guaranteed outcomes to gambles with the same expected value. For example, a risk-averse person would prefer a guaranteed $50 to a gamble with a 50% chance of getting $100 and a 50% chance of getting nothing, even though both options have the same payout on average (or, in mathematical terms, the same expected value). Therefore, since most people are risk averse in most situations, I wouldn’t expect many people to take up the offer above.
But wait- this isn’t even how most lotteries work. Instead, the analogue of the lotto situation would go something like this: Let’s say you had the option to put a dollar into a collection along with 9 other people. Half of the money collected gets taken by the person collecting it, and the 10 people are asked to pick numbers from 0-9. The person running the lottery also picks a number from 0-9, and the $5 still available for payout gets split among all of the people who matched the lotto czar’s number. If nobody matched the lotto czar’s number, you have to pay again next week to try to match the number in order to get this week’s payout, but at least there’s the upside that you’ll have new people buying in to add to the existing payout.
This proposition sounds terrible to most people, yet there are plenty of people who play the lotto. It’s probably not surprising that economists really like to ponder why this happens- do people just happen to be risk loving in certain contexts? (If this were the case, they would be willing to pay $1 to take a gamble with an expected value of less than $1, which is what the lotto is.) Are they behaving irrationally in some way? (Behavioral economists hypothesize that lotto participation arises due to narrow choice bracketing in which paying $1 for a lotto ticket seems like “peanuts” when one choice to play the lotto is considered in isolation, and the lotto-playing individual doesn’t stop to consider that while $1 might not buy anything meaningful, the same cannot be said for the $52 spent on the weekly lotto tickets over the course of a year, for instance.) Personally, I have a sneaking suspicion that people aren’t even considering the odds when deciding whether to purchase their tickets. Is there a way to test this?
Thanks to the lovely people at Mega Millions, that answer might just be yes:
Your odds of winning the jackpot used to be 1 in 176 million. As of Oct. 22, those odds changed to 1 in 259 million.
That’s because you used to have to pick six numbers from 1 to 56. Now you have to pick them from 1 through 75.
The Mega numbers have decreased to 15 from 46, but your overall chances of winning still are substantially reduced.
John Garnett, a UCLA math professor, explained to me that the changes mean that “for every three winners under the old system, now there will be two.”
Now, this analysis is a bit oversimplified and pessimistic because it doesn’t take into account the fact that, if there are fewer winners, then the jackpots will get larger, and, if there are more numbers to choose from, the chances that a winner would have to split the prize decreases. But here’s something important to consider: while it may be true that, say, cutting one’s probability of winning in half does in fact cut the attractiveness of a gamble by half, the same cannot be said when scaling the size of the prize up and down. Why is this? The simple fact of the matter is that people are generally susceptible to what is known as “diminishing marginal utility of wealth”- basically a fancy way of saying that $300 million is not twice as useful to you as $150 million, since, as you get richer, what on earth are you really going to do with those last dollars on top of the giant pile anyway? Overall, therefore, the increase in expected jackpot size isn’t likely to make up for the decrease in odds.
So why did the Mega Millions people do this? They keep roughly half of the ticket revenue coming in, so they obviously have an incentive to maximize the number of tickets sold. Since the size of the jackpot is far more salient than the odds, the lottery managers are basically betting that people will be distracted by the huge shiny jackpot numbers and not really notice that their lottery deal got crappier overall. On the up side, this change to the lottery system results in a sort of natural experiment (especially since Powerball didn’t change its system, so it can serve as a control group) where we’ll be able to see the change in people’s ticket-purchasing behavior and make inferences about what factors people consider when purchasing lottery tickets.
Or, in shorter form, sucks for people, great for research.
P.S. If you are going to ignore most mathematical reasoning and play the lotto, I humbly request that you at least don’t play the same numbers that my parents insist on playing every week (and apparently convince me to play when they can’t):
I guess the apple sometimes does fall far from the tree in certain ways. =P