# Should You Play The Lottery?

Did you win the Powerball on Saturday? You did? Hey, congratulations, that’s super, can I see your ticket? Can I hold it for a second? Thanks!

The rest of you might wish you had listened to the Wall Street Journal on Friday:

The Powerball jackpot is up to \$600 million, with a day still to go before tomorrow’s drawing. Which means it’s time once again to ask the question: Is this the rare time where it makes economic sense to buy a lottery ticket?

Spoiler alert: No. No it is not.

Every time there’s a big lottery jackpot you get articles like this to do the dreary math for you. The “expected value” of a \$2 ticket is less than \$2: the jackpot amount that you’d actually get, multiplied by the probability of winning, is a little under a dollar. (The Journal thinks it’s around \$0.85; I get more like \$0.98.) A dollar is less than two dollars, so it doesn’t “make economic sense” to buy a ticket.

These articles are always wrong. Or, at least, they make no economic sense. Economics attempts to answer the question: “Is it a good idea for me to do this thing, or would it be better for me not to do this thing?” That is a hard question so a lot of people like to pretend it is equivalent to the much easier question “will I have more money if I do this thing, or will I have less money?” But it isn’t, or at least, it isn’t always. If you buy a sandwich, you will have less money than you do now. Economic reasoning is not about getting more money until you have all the money. Sometimes there are sandwiches.

To decide if you should play the lottery you need to compare the (tiny) chance of winning a giant amount of money, on the one hand, with a dollar, on the other. The simple way to do this is to assume that a one in a million chance of winning a million dollars is “worth” the same as a dollar. But, why?

Among other problems, that assumes that a dollar will do the same amount of good for everyone. It is easy to see that this is wrong. If you give me \$5,000, it will in fact make me very happy. If you give Logan \$5,000, it will make her much happier. If you give Bill Gates \$5,000, it won’t do much for him, probably. He makes that much every 6 seconds. He’s not having a party every 6 seconds.

The more money you have, usually, the less good another dollar will do you: the less “utility” that extra dollar has. One way to express this is to assume that doubling the amount of money someone has always does her the same amount of good, however much money she started with. So \$5,000 is “worth” twice as much to someone who started with \$5,000 as it is to someone who already had \$10,000. Something like this:

Fig. 1

This, as this Slate article from the last big lottery jackpot points out, is a good argument against playing the lottery: because \$600 million (or whatever) is worth less than 600 million times as much as \$1, spending \$1 on a bet to win \$600 million is a bad idea.

But that’s still just a rough assumption. If you double the amount of money Logan has, she’ll be happy and all, but I gather she’ll still be in debt and she probably won’t even quit her second job. If you double the amount of money Bill Gates has, that’ll be great, he can cure more malaria and so forth, but his life won’t change much either. But if you double the amount of money I have, big things happen. I can pay off my mortgage! I can take a year off to travel! My day-to-day life actually changes in a way that neither Logan’s nor Bill’s would.

If you win a \$600 million Powerball jackpot, even if you have to split it and pay taxes and whatever and only end up with, I don’t know, \$70 million, one very important thing happens: you don’t have to work for a living any more. Maybe you already don’t, which is nice for you. But for most people that’s a big deal:

Fig. 2

To be pretentious, although not especially accurate: utility is discontinuous at not having to work for a living.

Other things probably happen too, if you win the Powerball. You can buy a boat or whatever. Maybe a boat is a big deal to you. Something is probably a big deal to you, and if it’s buyable now you can buy it, unless it’s like an aircraft carrier. You can buy love, probably; \$600 million is a lot of money.

On the other hand, here you are with a dollar. You could just keep that dollar. How much is that worth to you? I dunno, like, a dollar. A dollar is like a Snickers bar. But it’s not really. Sometimes it happens that a person says “well, I have a dollar, and if I can spend it on the lottery or I can have a Snickers bar, but not both. If I play the lottery, no Snickers for me.”

Wait, the Powerball actually costs \$2, but you know what I mean.

But what often happens instead is one of two things. Either you can play the lottery and buy a Snickers bar because you’ve got a bunch of dollars, or money is tight and you weren’t exactly planning to blow your money on Snickers bars anyway before this whole Powerball idea came up.

If you can throw your dollar away on Powerball and still buy all the Snickers you want, how much is that dollar worth to you? If you were going to leave it in the tip jar at the coffee shop, or buy another copy of the Post because you forgot yours at home, how much is it worth to you? Not in dollars—it’s worth a dollar!—but in utility, in things that it will do for your life or that losing it will prevent you from doing. For a lot of people the answer is “a lot.” For a lot of other people the answer is pretty close to “nothing.”

(Oh sure you could skip three \$7 lattes every day for a hundred years and thanks to the magic of compound interest you’ll end up with enough to retire on or whatever but I don’t want to hear it. We’re just talking about a dollar here. For the Powerball.)

Calculating “does it make sense to play the lottery” just means weighing the tiny but perceptible benefit from playing the lottery (a very, very, very tiny probability of a very, very big improvement in your life) against the cost—in utility, not dollars—of buying a ticket.

Satisfyingly, this gives a different answer for different people:

1. If winning the jackpot would not materially change your life because you are stonkingly rich, don’t play the lottery, it’s a waste of time.

2. If winning the jackpot would materially change your life because you are a normal person, and throwing away a few dollars every month or so when you notice there’s a big jackpot doesn’t change your life at all because you are financially secure – in other words, if the cost of those few dollars to you in utility is zero – then: play the lottery, why not.

3. If spending a few dollars on the lottery every once in a while would change your life at all—if spending those dollars worries you at all or causes you to cut back on any other necessity or pleasure—then: don’t play the lottery, it’s a waste of money.

Sadly a lot of people who regularly play the lottery fall into category 3. But a lot of people who buy a ticket every once in a while when Powerball gets really big are solidly in category 2. They should keep buying those tickets. They mostly already knew that. An economics that tells them not to is not a real economics.

All of that oversimplifies by leaving out everything but the dollars involved: the pleasure or pain of buying the ticket, dreaming about winning the lottery, checking the numbers, realizing that you didn’t win the lottery, etc. For most people that utility is probably positive, on balance: it’s really fun to daydream about winning the lottery! But for some people the disappointment of losing outweighs the thrill of anticipation.

For those people I can suggest a strategy I once read about, though I can’t remember where. (If you know, tell me!) Here’s what you do:

2. Write down the numbers.
3. Destroy the ticket. (I recall the person eating it, though that’s optional.)
4. Wait nervously for the drawing.
5. Check the numbers.
6. Feel an immense wave of relief when you confirm that you didn’t destroy a winning ticket.

This strategy, unlike playing the lottery the normal way, has the advantage that you will almost always “win.” Almost.

Matt Levine writes about the financial industry on the internet at Dealbreaker and sometimes at Planet Money. Sometimes he writes about lotteries. He bought \$10 of Powerball tickets on Friday and went back to work again this morning LIKE A CHUMP. Photo: rockinfree

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terrific (#1,532)

yes because I won \$40 on a scratch-off on Saturday and it totally paid for (a) my weekend excesses, and (b) every lottery ticket I’ve ever bought

chic noir (#713)

@terrific – i don’t really play scratch offs. I occasionally buy them to add to Bday cards. Well about a month ago I was on the bus and heard a voice in my head telling me to get off and go get a few scratch offs. So I got off the bus, where I stood between two liquor stores. Decided to go to the liquor store that has winning tickets posted. Brought ten dollars worth of tickets and what do you know… I got a winner. \$500 for me :)

I was a little disappointed that I didn’t win one of the 20k prizes but something is better than nothing.

Two days later, decided to try my luck without the “voice” ,brought four one dollar tickets in 7-11 and won 50.

Have been on scratch off “chill” ever since.

limenotapple (#1,748)

I guess I fall into the #2 category. Mostly I buy a ticket every year in May because my grandmother passed in may a few years back, and buying a ticket just makes me remember her and it makes me feel happy to do this thing she loved to do. I have also pitched in when I worked in offices where people were buying a bunch of tickets that we’d all split, even though I think it’s silly.

But for me, who won’t miss the dollar, it was kind of fun. We didn’t think we’d win, but we still have fun saying, “well, if we win the lottery, we’re totally buying a new couch”. Or, “if we win the lottery, I’m totally going to buy that hedgehog I’ve always wanted”. I think for some people, the daydream is part of the experience. For the record, I’m not even sure I’d want THAT much money, seems like it would bring more pain than it would be worth, though it might be nice to win enough to put in a bitchin’ kitchen.

Whereas under the von Neumann–Morgenstern utility theorem the question isn’t “Should I play the lottery?” but “Which lottery should I play?”

It’s weird to me that there is SO MUCH stigma around the lottery. I mean, I definitely get that there’s an issue around people who are addicted to gambling and are playing irresponsibly, but I remember reading an advice column once where a guy was asking how to get his elderly father (who was not living in poverty) to stop spending \$10/week on lotto tickets, and it was like, if he were spending that much on going to the movies every week, there would be no problem.

chic noir (#713)

@SarcasticFringehead What about other things worth \$520, though? Thinking about it as an annual, rather than weekly, expense puts it into perspective. \$520 could be a date night out with a spouse/SO every 2-3 months, or a season pass at an amusement park for an entire family with candy/refreshment money to spare, or a new TV. Forget the idea of spending the \$10 every week. Would you rather have a bunch of lottery tickets, or would you rather have any of those things I just mentioned? Or many others of the same cost?

RachelW (#2,605)

I am solidly category 2, and I did buy a lottery ticket on Saturday. My fiancé bought one also, and then we had a fun dinner out planning how we would spend our millions. Of course we didn’t win, but we didn’t expect to. Honestly, I’m not sure I actually wanted to win, because I don’t know that I would want my life to change that dramatically. It’s just something that’s fun to think about. And doesn’t a portion of my \$2 go to fixing up the roads in my state, or something like that? I’m ok with that.

OhMarie (#299)

@RachelW Yeah, my husband and I had some awesome conversations! That’s really what you’re getting for your \$2.

chic noir (#713)

I purchase two tickets for Megamillions and Powerball most weeks. I rather spend the money on a chance of winning millions vs another latte, a shirt that falls apart after one wear from H&M, booze , apps for my phone, bottled water etc…

chic noir (#713)

@chic noir I have dreams about the things I would do with the money from a large win like… Go to a Wallmart during Xmas season and pay off the layaway of 100 people. Go to a college or University, invite 200 liberal arts students to a “meeting” and tell them Oprah style that i’m paying off their student loans. Have a couple of scholarships set up in my name. A wing of a hospital with my name on it. Free bus/metro passes for a year for working and missing class workers. Donate to a free clinic in a poor neighborhood. donate to doctor’s without borders. Pay off all of my closest friends student loans and mortgages.

chic noir (#713)

I read the Powrball results. One ticket for the jackpot put 22 people won the second largest prize 1 million dollars. Over one thousand people won the 10k prize.

Kthompson (#1,858)

I never buy lottery tickets, but I admit I had one for this jackpot. I found it in the grocery store parking lot. It wasn’t any sort of winner, though.

nf (#949)

If you’re going to use this reasoning to play the lottery (I do) you should probably try to find a game with a lower payout but better odds. Like Pennsylvania has a game called Millionaire’s Raffle which has a \$1 million payout and 1 in 125,000 odds, I assume other states have their own equivalents. You could easily stop working forever on “just” \$1 million.

j a y (#3,935)

Heh, my very first gamble ever was on a roulette wheel 40 bucks (one black, one number… the number hit) netted like \$800 (split with the person I was with). I have to admit the strongest instinct was to quit while I was ahead.

I never buy lotto tickets but I console myself in that I’m in the lottery pool at work. I did try to persuade them that we should only buy one ticket per draw (‘because 15*0=0!’) but it didn’t go over too well :)

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