# Does it make sense to play the lottery?

A very smart friend of mine told me yesterday that he buys a lottery ticket every week. I’ve encountered other smart people who do the same, and it always confused me. If you’re aware (as these people all are) that the odds are stacked against you, then why do you play?

I raised this question on Facebook recently and got some insightful replies. One economist pointed out that money is different from utility (i.e., happiness, or well-being), and that it’s perfectly legitimate to get disproportionately more utility out of a jackpot win than you lose from buying a ticket.

So for example, let’s say a ticket costs \$1 and gives you a 1 in 10 million chance of winning \$8 million. Then the expected monetary value of the ticket equals \$8 million/ 10 million – \$1 = negative \$0.20. But what if you get 15 million units of utility (utils) from \$8 million, and you only sacrifice one util from losing \$1? In that case, the expected utility value of the ticket equals 15 million utils / 10 million – 1 util = 0.5 utils. Which means buying the ticket is a smart move, because it’ll increase your utility.

I’m sympathetic to that argument in theory — it’s true that we shouldn’t assume that there’s a one-to-one relationship between utility and money, and that someone could hypothetically have a utility curve that makes it a good deal for them to play the lottery. But in practice, the relationship between money and utility tends to be disproportionate in the opposite direction from the example above, in that the more dollars you have, the less utility you get out of each additional dollar. So the \$8 million you would gain if you won the lottery will probably give you less than 8 million times as much utility as the \$1 that you’re considering spending on the ticket. Which would make the ticket a bad purchase even in terms of expected utility, not just in terms of expected money.

Setting that theoretical argument aside, the most common actual response I get from smart people who play the lottery is that they’re buying a fantasy aid. Purchasing the ticket allows them to daydream about what it would be like to win \$8 million, and the experience of daydreaming itself gives them enough utility to make up for the expected monetary loss. “Why can’t you just daydream without paying the \$1?” I always ask. “Because it’s not as satisfying if I know that I have no chance of winning,” they reply.  Essentially, they don’t care how small their chance of winning is, they just need to know that they have some non-zero chance at the jackpot in order to be able to daydream about it.

I used to accept that argument. But in talking with my friend yesterday, it occurred to me that it’s not true that your chances of winning a fortune are zero without a lottery ticket. For example, you could suddenly discover that you have a long-lost wealthy aunt who died and bequeathed her mansion to you. Or you could find a diamond ring in the gutter. Or, for that matter, you could find a winning lottery ticket in the gutter. The probability of any of these events happening isn’t very high, of course, but it is non-zero.

So you simply can’t say that the lottery ticket is worth the money because it increases your chances of becoming rich from zero to non-zero. All it’s really doing is increasing your chances of becoming rich from extremely tiny to very tiny. And if all you need to enable your Scrooge McDuck daydreams is the knowledge that they have a non-zero chance of coming true, then you can keep those daydreams and your ticket money too.

### 21 Responses to Does it make sense to play the lottery?

It might make sense if you’ve discovered a clever exploit that increases your odds by orders of magnitude. Then again, it still might not be worth your time.

2. Barry says:

Very astute. And it reminds me of the story of the exalted rabbi who prayed intensely every night that he would win the lottery, and told God all the ways he could help the poor if he won, but was disappointed every single morning. One night, deep in supplication as usual, he finally heard the voice of God answering him. The answer was as follows: “Rabbi — give me a chance! BUY A TICKET!!”

3. Max says:

Using your example, an \$8 million prize is not worth the \$1 ticket, but a \$12 million dollar prize is worth it, yet in our mind, there’s no qualitative difference between \$8 million and \$12 million. It’s all more than enough to pay off all our debts, remodel the house, travel the world, and do all those things people imagine doing if they win the lottery.

I think buying a lottery ticket greatly increases the chance of winning a fortune within a week compared with doing nothing. You could do other things to try to win a fortune, like trading options or collecting stamps, but it comes with a risk of losing a lot more than \$1.

4. But the perceived utility of \$8 million, especially for someone who has next to nothing, will be much higher than the actual utility- in that sense, it is still rational to buy a ticket, unless one firmly understands the actual utility of the prize.

5. Val Schuman says:

Don’t forget that a large chunk of that \$8M is taken away in taxes…

• Cory Albrecht says:

Does it change the utility calculation in places like Ontario where lottery winnings are not taxed (you get the whole \$8M placed in you bank account) only the interest you earn from it while deposited in the bank?

6. James says:

I buy a lottery ticket once in a while as randomness insurance. I pay considerably more for life insurance, health insurance, car insurance, and homeowners insurance, none of which have produced one cent of “winnings,” and they will require suffering (or death) on my part when they do.

7. “But in practice, the relationship between money and utility tends to be disproportionate in the opposite direction from the example above, in that the more dollars you have, the less utility you get out of each additional dollar.”

To an extent, but it can also work the other way. If you’re very poor, and you *have* to pay your landlord £400 by the end of the week or you’re going to be evicted, and you only have £20, then an extra pound won’t make any substantial difference at all. Whereas if you’ve got £400, then it makes a difference whether you can eat *as well as* pay rent. In the former situation, it would be rational to buy a lottery ticket, because your chance of winning one of the smaller prizes, say £1000, would be non-zero (though still far from good) while the additional utility of the cost of the ticket would be zero.

Not that this is the actual reasoning from most people who buy lottery tickets, of course. But it’s a valid situation to be in.

8. Eric Robinson says:

You’ve got a slight shift in your argument near the end. You went from “chance to win the jackpot” to “chance to win a fortune”. Those are two very different things. Even if I have a non-zero chance on my own of winning some kind of fortune, the odds and even the outcome are complete unknowns. On the other hand, I know what the jackpot value is, and I know exactly what needs to be done to generally give myself a non-zero chance at it. (And sure, I could potentially find a winning ticket in the gutter, but the odds of that are actually much lower than winning with a ticket I bought.)

9. Barry says:

A number of commenters have argued that there’s a value in knowing that one has taken a concrete step toward winning a vast fortune, because, no matter how remote the odds, that step makes it much easier to believe that one’s material needs could soon be met. Thus, a satisfying fantasy can be entertained much more easily. The fact that the remoteness of the odds doesn’t matter is probably because to most folks “a million to one” is essentially the same as “a hundred million to one.” Here’s an idea that came to me this evening: suppose there were a service to which a person sent \$2 every week (through an automatic deduction, to keep the price of the stamps from cutting too far into the scheme). The service would, at rare random intervals, actually purchase a lottery ticket with the contribution. Most weeks, though, the contribution would simply be salted away in a savings account. If in a given week the ticket was actually purchased, and that ticket won, the lucky contributor would become vastly rich! If not, the account balance would go on growing. At the end of a specified period of time, or if the contributor decided he/she needed an infusion of cash to pay the mortgage or a vital car repair, the funds would be returned (minus a small administrative fee). I really don’t see why the fact that the odds would be an order of magnitude or two less that if a ticket were purchased every week should make any meaningful difference to a fantasizer — he/she could still say “hey, this could be my lucky week!” — and yet most of the money would still be there when it was needed. Seems like a win/win proposition, exploiting the statistical blind spot in the gambler’s favor instead of against it. .

10. Graham says:

In Ontario, the government defends provincial lotteries and gambling on the grounds that they generate revenue that benefits schools, hospitals, etc. But about half of lottery revenues are spent just to keep the lottery going, and while the government highlights such things as schools and hospitals, the fact is that the money generated by the lottery goes into provincial coffers generally and is distributed as part of normal government expenditure. It seems like a form of voluntary taxation: give the lottery folks/provincial government some of your weekly entertainment budget, and they will redistribute it for you. Admittedly, a very, very few people will benefit enormously from this. In Lotto 6/49 (Canada’s major lottery, administered provincially), you have a 1 in 14 million chance of your numbers coming up in any given drawn. At 52 tickets per year that’s, what? An expected win every 270 thousand years or so?

I first happened across your blog some months back while looking for my own. You were just getting started then, and I’ve followed it ever since. I love it. I love it. I love it. I have no idea how you maintain this pace. Don’t burn out.

OT, but to respond to your last paragraph, it helps that there are two authors. If one of them starts posting noticeably less often than the other, it would look bad, so they’re constantly competing – and you, dear reader, are the winner!

11. Will Newsome says:

Tangentially, we can show that buying a lottery ticket is most likely a bad idea even in cases where naively considered it seems like the only decision theoretically correct action.

Imagine a man who has ten thousand dollars sitting in the bank. And further imagine that he wishes to have an uncommonly large impact on the world. To make it concrete we’ll say he has a burning desire to help humanity eradicate a dangerous virus. He looks around at various research laboratories to see how he might help with his \$10k. One lab wishes to hire a promising young post-doc who might otherwise go into the biotech industry, but unfortunately they’d need \$50k more to have enough to pay for his trial year salary. Another laboratory wants to run a highly experimental study, but its radical nature makes government grant agencies too nervous, and the lab doesn’t have enough public recognition to drum up enough donations. They’d need \$100k to hire top-notch grant writers or marketers, and many millions of dollars to actually start the study. The man continues to look for places where his \$10k would have a lot of leverage, but doesn’t find anything promising.

I’ve seen people use scenarios like the above as arguments for the plausible rationality of engaging in lotteries, with one ticket or a thousand. Similar arguments are somewhat stronger when it comes to highly speculative stock market endeavors. If your goals can only be attained in possible worlds where you have a lot of money, it’s perfectly sane to accept bets that only very slightly increase your chance at ending up in those worlds instead of the vast majority of possible worlds where you remain poor. But this argument for passive lottery-like strategies, though more sophisticated, still falls to a little bit of creativity.

It is most likely relatively decision theoretically correct for the aforementioned man to do any number of things instead of entering a lottery or blindly day trading on the stock market. He might find or make a list of the English-speaking world’s top 10,000 richest people and then use his \$10k to hire a skilled writer to send a highly persuasive mass email to all of their secretaries. (Or just keep his \$10k for something else and instead persuade one of his most eloquent friends to read a few books on persuasion, human cognitive biases, letter writing, and typography, and then write it.) Or he might leverage his good reputation to catch a visit with a loan shark. Or if he’s deadly serious and willing to wait 2 years, buy a high-yield life insurance policy and then commit suicide via some combination of the most reliable methods. (Yup, that works; I haven’t researched that strategy much though and I haven’t seen it put to the test.)

It will probably take him a long time before the widely applicable meta-strategy ‘save resources while actively exploring the space of possible strategies’ hits diminishing marginal returns substantial enough to justify his engaging in anything very lottery-like. Of course, at the same time, this also applies to his goal of helping humanity eradicate a particular disease — further research into things like optimal philanthropy, existential risks, or even meta-ethics might do him well. Rule one: Always meta-optimize.

This post is right, as far as it goes. But it isn’t about the increase in likelihood from extremely tiny to very tiny. To have a lottery ticket is to have a talisman of possible wealth that, when held and contemplated, brings immediately to mind the fantasy of wealth and makes it feel much more visceral. There is no equivalent talisman for the possibility of a wealthy aunt, and, hence, the fantasy is far weaker.

The “fantasy aid” argument for purchasing a lottery ticket works in light of this very real gain in utiles by comparison to the far more passive fantasizing about one’s wealthy aunt. It isn’t hard to think of other examples of this “talisman effect” in fantasy and memory: physical books are relatively more expensive than ebooks (modulo Amazon’s future-oriented pricing); letters from lovers call to mind more poignant memories than do emails or remembrances; people save chips that are otherwise useless from casinos; etc.

It would seem to me that smart, rational people are no less susceptible to the very real utile-increasing effects of talismans. This is a reason to play the lottery: you hold the possibility of riches in your very hand. No hypothetical relatives offer the same pleasure.

• jonamdall says:

Great blog you guys have, I enjoyed browsing the posts.

As for the lottery, I do occasionally buy a ticket, despite the fact that I view winning as being virtually impossible. My life is fairly average, a lottery score doesn’t seem like the type of thing that happens to me.

Great comment from Adam here, I think he nailed it. The driving force that leads to actually buying a ticket for me is simply indulging the fantasy of being extremely wealthy. “Wouldn’t it be cool to be a millionaire, Self?” “Yeah, totally. Why not buy a ticket?”

13. Bao Jingyan says:

If buying a lottery ticket is deemed irrational on the grounds that it has a negative expected value, surely taking out any form of insurance is equally so?

14. Kaleberg says:

I don’t bother with lottery tickets, but I know for a fact that buying one does raise one’s probability of winning. For most people, spending a dollar or two a month will not make any difference in their well being, present or future. But, as James Clavell put it, you can’t get your “f–k you” money out of cash flow. If someone can buy a cup of coffee or a lottery ticket, which has a higher utility value? It depends on how much he or she wants a cup of coffee against how much he or she would like some “f–k you” money. Sadly, for many people a lottery ticket is the price of hope.

For most people, having an unexpected lump sum of cash is much more valuable than having adequate cash flow, and given the unevenness of life, it is very hard to accumulate much of a cash buffer. There is a whole industry based on spending or saving one’s tax refund, and that is expected money. Unexpected money is even more valuable, and can be life changing. (This is especially true in poor countries like the US where winning the lottery can mean things like getting dentures or a tumor removed.)

I used to scoff at the lottery, but a lawyer friend of mine got through law school on the state lottery. She’d buy a few tickets regularly, and won a few hundred here, a few thousand there. It made a big difference.

15. NascentDreaming says:

Time is also relevant in this argument. When you are twenty the chance of you becoming a millionaire in your future are higher than when you are fifty simply because you have less time. The utils are important also but in a slightly different way. Having 100 dollars over 1 dollar is not just 100 times more utils and the more you increase the amount of money you have the more this effect increases. \$100,000 is nice nest egg but if you have a million you u can leverage it more effectively. Lets say that there was a different lottery where you shaved your head and there was a one in ten chance that this particular woman would sleep with you. Lets also say that at the chance of you sleeping with another woman in a different situation (at school or at work ect) was one in a hundred with a shaved and fifty in a hundred without a shaved head at 20, but at fifty the chance with a shaved head is one in a hundred and the chance without hair is still one in a hundred. You might argue that the return on the lottery ticket is negative and so my example is flawed but its important to remember that the utils do not directly translate to dollars. So in my example keeping your hair is equivalent to keeping your dollar. In order for you to want to shave your head you would have to see sex as worth 100 utils and keeping your hair as worth 1 util. And actually you measure those two against each other. Because when you are twenty the hair should be worth 50 utils but at 50 the hair is worth zero (or at least a negligible amount.) That said I apologize for gender bias example. I think the real rationale behind the decision is what are the best available strategies to increase my utils vs their cost.

16. LottoTime says:

Numbers say the larger the lottery ie jackpot the greater mathematical sense to play ie 150M jackpot vs. 15M jackpot.

17. Tweedlover says:

Its a question of how much you can afford to lose versus how much you want to win. Most people don’t even feel € 2 a week. it doesn’t change their lifestyle, and they probably wouldnt do anything else with it. However, a lottery win would completely change their lives. You get the chance to change your life for a negligible marginal cost. Seems very sensible to me! (buying lots of tickets changes things)