# Economists Do It With Models

## An Implicit Tax Hike On Irrationality, Lottery Edition…

#### December 13th, 2013 · 12 CommentsBehavioral Econ · Decision Making · Fun With Math

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

### 12 responses so far ↓

• 1 Kyle IsAwesome Almost // Dec 13, 2013 at 6:47 pm

I’ll test your hypothesis by purchasing a last minute ticket and let you know why I did it later. Nice read.

• 2 Nate K // Dec 13, 2013 at 6:49 pm

People play the lottery in order to purchase the feeling/hope that things could be different. This is why lotteries use the slogan, ‘You can’t win if you don’t play.’ It’s not about the odds, it’s essentially buying permission to daydream about a big event that would change your life. (Now that fact that lottery winners are only temporarily happier after winning is a whole other issue).

• 3 Daniel R. Grayson // Dec 13, 2013 at 7:00 pm

I’m glad you mentioned the diminishing marginal utility of wealth, but you applied it only to the winnings. What about applying it also to the \$1 I use to buy the ticket? If that dollar is worth little to me, could it be rational to buy the ticket?

• 4 Evan // Dec 13, 2013 at 7:32 pm

I noticed that the article said that a recent winner paid over \$100 million in taxes on his win.

For this to make sense, then losing lottery tickets should be tax deductible. Is this the case?

• 5 Philip Graves // Dec 13, 2013 at 7:45 pm

Gambling losses can only be deducted from winnings. So the “big winners” get asymmetrically screwed. I also, as with Nate, believe lottery buyers are “buying the ability to dream about what they would do if they won.” Can’t dream if you don’t enter.

• 6 Kyle IsAwesome Almost // Dec 13, 2013 at 8:23 pm

Nate and Philip have pegged my point. “Behavioral economics” is not rational.

• 7 Evan // Dec 13, 2013 at 8:27 pm

I just double checked, and neither country that I have lived in (Australia and Canada) taxes lottery winnings.

Are lottery corporations in the US publicly owned statutory monopolies? If so, the case for taxation of winnings seems rather weak.

• 8 IRS // Dec 13, 2013 at 10:13 pm

Daniel–you apply it to both. However little the dollar is worth to you, the millionth dollar will be worth less. Therefore, if you do place diminishing marginal utility on money, playing a game with negative expected return will never be rational for its monetary benefits alone.

Diminishing marginal utility in isolation implies risk-aversion–a certain quantity of money will always be more valueable than non-point distribution with the same expected value.

• 9 Adam Jacobs // Dec 14, 2013 at 4:46 am

Interesting post. I’m going to come right out and admit that I play the lottery (in the UK, which I guess may have different rules to the one you’re talking about, but no doubt the principle is the same).

I’m also a professional statistician, so I am perfectly well aware it’s a lousy bet. I’d be interested to know if I’m the only statistician in the whole world who does play the lottery!

Anyway, the reason I play it goes something like this. Although there is only a vanishingly small chance I’ll win the jackpot, the fact is that if I did it would be completely life-changing for me. On the other hand, the few pounds a week I spend on tickets
don’t really matter that much (if you want to be all economicsy about it, that money has rather low marginal utility for me).

I like to think of it as analogous to the reason why I have house insurance. It’s really not very likely that my house will be destroyed in a fire or some other catastrophic event. But if it did and I wasn’t insured, then the loss of my house would also be completely life-changing for me, but this time in a bad way. I consider it reasonable to pay my insurance premiums to avoid that risk. And I pay more for my house insurance than I do for playing the lottery.

Let’s face it, insurance is a pretty lousy bet as well. A behavioural economist might argue that it’s a completely different situation, because insurance is about being risk-averse, but as a statistician, I see it as being pretty much the same deal as playing the lottery.

• 10 Erick // Dec 14, 2013 at 11:36 am

I wonder, is it irrational if we change the assumptions about why a person bought the ticket. There could be people who know that the odds are horrible, but paying that dollar (or whatever) a week is worth it for the peace of mind that they at least won’t be left wondering “what if” and they consider a dollar to be worth the price for piece of mind. Do rational choices always have to be measured in monetary terms?

• 11 Evan // Dec 16, 2013 at 2:44 am

@Adam. I think that your behaviour is consistent with, for example, cumulative prospect theory ala Kahneman and Tversky. Two of the key points of their theory are that people overweight small probabilities and are averse to losses.

To clarify, economists define rationality as meaning that an agent has complete and transitive preferences. Playing the lottery does not, in any fashion, imply irrationality.

There are many, many different economic theories of preferences that are consistent with playing the lottery. Prospect theory, as mentioned above, is the most well known.

The basic approach of economists to decision making problems is that a person is entitled to their own preferences – if you like doing something, good on you for doing it. The only time an economist would call you irrational is if your choices are obviously self-contradicting, and Jodi is going to far (according to decision theory) in calling people who play the lottery irrational.

• 12 Clarke // Mar 29, 2015 at 4:05 pm

If we are good enough then this little amount should be nothing in front of that and I believe if we have the skills then we will be able to make endless amount of money from Forex. I use only one company for investment my money in Forex and that’s OctaFX because they are regulated and also have excellent cTrader platform where I am able to get all my trades executed instantly and it also has updated facilities and features.