Thursday, December 13, 2007

The lottery: a tax on stupidity?

It is often quipped that the lottery is a tax on stupidity (google "lottery: tax on stupidity" and you will see what I mean). I've always been bothered by this for two reasons: 1) one of the first people who said it to me was an arrogant professor who I always enjoy proving wrong, and 2) someone in my family (who has an advanced degree in mathematics) used to (and may still) play the lottery. So I figured there must be some fallacy in this statement.

I use New York's Lotto as an example, because it is a very simple "jackpot" lottery game. In this game, you choose 6 numbers between 1 and 59 (inclusive). If all 6 match, you win the jackpot. The order of the numbers does not matter. To figure out the chances of a match, you must compute the number of equally likely possible combinations (or just go the Lottos website and skip the next paragraph).

To compute these chance, we first figure the total number of combinations of 6 numbers out of 59. Suppose we choose the numbers in order. Then, we have 59 choices for the first number, 58 for the second number, and so forth until we have 54 for the 6th number. This totals 59*58*57*56*55*54=32,441,381,280 permutations in all. Since we do not care about the order, however, we need to adjust this number (consider, for example that 11,12,13,14,15,16 and 16,15,14,13,12,11 are both the same set of numbers and considered the same in the Lotto drawing). This adjustment is made by dividing by the number of possible orderings of six numbers, which is 6*5*4*3*2*1 = 720. Thus, divide 32,441,381,280 by 720, and you get 45,057,474--the number of possible combinations, of which you choose two for each $1 Lotto ticket. Your chances of winning, then, are about 1 in 22.5 million. The average jackpot is about $9 million (this jackpot amount is only obliquely referred to on the NY Lotto website, in that they say that 40% of revenues go to the jackpot).

Given these odds, how much do you expect to win if you buy a single ticket (good for choosing two six-number combinations)? Well, given the odds of 1 in 22.5 million, you would clearly expect to win absolutely nothing!

But mathematicians don't think this way. Instead, they compute the expectation as the long run average, and by long-run, I mean really really LONG-run (actually infinite-run, but let's not split hairs). To give you some idea of this, you would need to play around 15 million times to have a 50% chance of winning at least one time--this would take 41,000 years or so if you played 1 ticket a day. So, computing the average after a few thousand or even a few million games is likely to get you an average of 0, which is *not* the correct long-run average.

Instead, this expectation is computed by taking the sum of the probabilities of winning multiplied by the amount won. In the case of the Lotto, then you win $0 in (22,499,999/22,500,000) games and $9,000,000 in (1/22,500,000) games. So, the Expected winnings are(22,499,999/22,500,000) *$0 + (1/22,500,000)*$9,000,000 = 40 cents.

So, you pay a dollar, and "expect" to get 40 cents back. This is why some people call the lottery a tax on stupidity. When people say the lottery is a tax on stupidity they are implictly and incorrectly assuming that utility (to throw in an economic term) is based purely on mathematical expectation, and that the utility from $9 million is 9 million times the utility from $1. Yet I doubt that people are playing the lottery based on some mis-guided mathematical expectation calculation. $1 or 40 cents. Who cares? Either way it's barely worth picking up off the ground.

Smart people who play the lottery are valuing 2 things against each other -- $1 versus a miniscule chance of $9 million -- and deciding that the value of $1 to them is less than the value of the chance at the $9 million. Yes, poor people probably value $1 more than average, but they value a chance, even a small one, of forgetting about their financial woes even more.

Let's look at another game that shows the flip-side of this mathematical expectation conundrum. For all you upper-middle class, non-lottery players out there, consider the following: Would you pay your entire net worth for a 1 in 1,000 chance to win $10 billion? If your net worth is less than $10 million, this is a game with positive expectation. For those of us with less than $1 million hanging around the house, the expectation is more than $9 million, but I doubt you'd find any middle-class person willing to play this game.

Why? Because the risk is too great, no matter what the reward. It is widely recognized that people place different values on risk. Risk averse people are willing to lose a small amount of money (or pleasure) to insure they will not lose a large amount of money (or pleasure), even when the mathematical expectation of their transaction is negative. The best example is insurance (Wikipedia's lottery entry points this out). Insurance companies make money not on stupidity but on the fact that people do not want to take large financial risks.

So next time you hear someone say the lottery is a tax on stupidity, tell them about the mathematician who plays, or about the people who turned down a game with an expectation of $9 million.


Anonymous said...

From the mathematician (:-):

1. You can win *something* with fewer matched numbers in most lottery games. (Ex. 4 numbers matching might win $100.)

2. I only buy scratch-off tickets where the odds of winning something are 1:4, but the highest payouts for those are in the thousands, not the millions.

3. On average, I spend about $1 per month on lottery tickets, and, for me, it is purely for entertainment! (:-)

Vincent Granville said...

The sad thing about lottery: I know I'll never win, not even one cent. Because I don't play.

Anonymous said...

The lottery is not a tax on stupidity? Then it's a tax on desperately limited imaginations. Why not just release the dollar bills into the breeze on a downtown street, and watch who grabs for them? Or buy a dimestore bag of rubber balloons. Blow them up and give them to small children. That'll keep you out of the scratcher shop for half an hour, and will deliver entertainment that adults can no longer even dream of.

aileen said...

I recently came accross your blog and have been reading along. I thought I would leave my first comment. I dont know what to say except that I have enjoyed reading. Nice blog. I will keep visiting this blog very often.


Anonymous said...

Concise and accurate analysis. The net winnings are even less, as they are taxable. Take another 1/3 to 1/2 off on any meaningful large win. So the 40% goes to 20%.
The lottery is a decidedly regressive tax that disproportionately appeals to the lower socioeconomic cohorts.

Anonymous said...

If you want small entertainment involving money, play bets with your friends and/or go to the casino.. don't waste your money on something as stupid as the lottery.

SP1733 said...

Seriously? You're comparing a game with a 1 in 1,000 chance to a typical lotto? Apples and oranges, my friend. Your blog refutes its own point.
Expected value is a LONG RUN average. A person might reasonably live long enough to experience the long run average of 500 games, not 15 million.
Your first assumption was correct. The lotto is a tax on stupidity.

Alan Salzberg said...

With respect to SP1733's comment and others, I am still surprised by the lack of understanding of the simple point that it is possible that, to some, playing the lottery can be enjoyable. It does not need to have a positive monetary expected value in order for that to be true. Thus, people who understand full well the probabilities can still enjoy it.

The same is true of slot machines. And spectating in baseball, basketball, and any other sport. We pay money, get little or nothing in return, but enjoy the experience. Having said that, I still think we should discourage lotteries because the people who seem to get the most enjoyment tend to be poor, so state sponsored lotterys result in the poor financing (usually) education for the middle class (the kids of the rich are going to private school).

Anonymous said...

it is also the case that rational people would all play the lottery with lowest expected loss and still gain the rewards of being "involved in a social activity" and "a chance at a dream life" if the prize is a similar amount (close substitutes), but this doesn't happen; is there an explanation for this?

Anonymous said...

if lottery players are rational why do we find that lotteries with similar expected payouts (close substitutes) but different expected "losses" are equally played; shouldn't they all flock to the one with lower expected loss?

Alan Salzberg said...

"if lottery players are rational why do we find that lotteries with similar expected payouts (close substitutes) but different expected "losses" are equally played; shouldn't they all flock to the one with lower expected loss?"
Not at all. If they were playing based on expectation, they wouldn't play at all. They could logically play particular ones for convenience, to maximize possible gain (irrespective of expectation), or because they are the most enjoyable (though I don't personally see how pick 5 is more or less fun than pick 6 but I am sure Im missing some subtleties).

Alan Salzberg said...

"it is also the case that rational people would all play the lottery with lowest expected loss and still gain the rewards of being "involved in a social activity" and "a chance at a dream life" if the prize is a similar amount (close substitutes), but this doesn't happen; is there an explanation for this?"
This is an interesting concept. I don't know whether it happens with the lottery. I once played (I think it is the only time I have) when my roommate insisted we should play because the jackpot was really high. Expectation didnt enter into it--he wanted me to play a particular one--they were all teh same to me. I'd imagine the social acceptance would come with playing particular one--similar to the person who doesnt follow sports but checks the scores for the popular team so he can be involved in the conversation at work the next day--it's purely social.

Anonymous said...

Basically you have proved nothing. It is still a tax on stupidity: the only reason a rich person wont bet their net worth is because you would be offering them a one time bet and not the long run expectation.

People pay for a lottery ticket so they can get a licence for some excapist fantasization but intelligent people would dream about somethig they might actually be able to pull off with their own efforts.

Anonymous said...

You're obviously an intelligent man, but it feels as though you are trying to justify your own playing of the lottery since a certain comment damaged your pride. I did enjoy your breakdown of lottery statistics and your way of writing. What was far less satisfying was reading your comments where you threw out everything you said in your article and reduced your argument to "some people play because its fun". Few people play one ticket a month. How many of us are constantly waiting in line behind someone at the gas station who's dropping $20 on tickets? You can't possibly believe that was an intelligent purchase.

Alan Salzberg said...

@anonymous who says: " it feels as though you are trying to justify your own playing of the lottery since a certain comment damaged your pride."
Actually, I've played the lottery exactly once, about 23 years ago when my roommate said we should play because it was at some very high $$ amount.

I don't believe anyone should spend $20 a day on it, but I don't think stupidity is the reason they do. I think there is a much higher correlation with income and lottery playing than there is with intelligence and lottery playing. I have spoken to many people who understand their astronomically small odds of winning but play anyway.

As I said in the original blog, I don't think an Expected value really comes into play with low probability events and thus people play even though they know the expected value is negative.

Jon said...

You go to the store with $1 worth of change in your pocket. You could buy a lottery ticket, and chances are you'd get nothing back in return. Or you could buy an iced tea and gain 120 calories in sugar: a short-term return on enjoyment but negative return to your health. Surely, getting a theoretical $0.40 per dollar is the more logical choice, but the point here is two things:

1. Priorities. Long-term health and economic prospects are not always most highly valued, or we would all lead very boring lives.

2. Logic. It might hurt your ego and senses to accept it, but human beings are not primarily logical, but emotional. Our consciousness is what gives us life, not logic -- we are not walking calculators, and certainly most of us do not live in that mode 24-7.

Furthermore, the last paragraph of the blog is not wholly relevant. We are not spending our life spending for a 1 in X chance of some approaching infinite amount of money. $1 isn't anywhere close to that. So we could spend it hundreds of times over and it wouldn't affect our daily comfort. You could go to the bar for Friday night out and spend up to $100 for dinner and drinks, how's that for a tax on stupidity?

The lottery gets bad rap because of how easy it is to figure the odds. Now figure the odds to your latest expenditures, whether that's vacation, expensive shoes, or furniture. Unless those things have some positive return on investment -- usually not -- you've wasted money, up to -100%. $0.40 to the dollar sounds like a bargain in comparison.

Anonymous said...

I believe many of us should be more rational about our lottery spending. If one is depressed about how much they are spending on lottery they should definitely spend less. If you are o.k. with your spending then there is really no reason to stop, you could win. Probably the best thing to do would be to save your money - but for what? - to spend on something else with no return - no. Save for your retirement or emergencies.
But by all means have some fun in your life.