Having already been critiqued for my economics acumen by Andy earlier today, I figured I might as well wander even deeper into the lion’s den by trying my hand at probability theory. Someone correct me if I’m wrong, but despite the flurry of articles about how buying a lottery ticket is inherently irrational, at this point isn’t a lottery ticket a good investment?^{*}

As I understand it, the expected payoff from a lottery ticket is simply the probability that you win times the value of jackpot. According to CNN, the chances of winning the lottery are 1 in 175 million, and the jackpot is $640 million. Let’s even posit that you lose half of the present value of the jackpot to taxes and the fact that you probably don’t actually get all of the money now, we’re still looking at a 1 in 175 million chance of winning $320 million dollars. By my calculation, that means the expected value of a lottery ticket is $1.83. Since a lottery ticket only costs $1, that would suggest that on this particular day, a Mega Millions lottery ticket is a good investment.

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^{*} Note that I’m assuming that you only care about the financial payout from the lottery ticket. Apparently winning the lottery can also ruin your life. Or not.

[Photo credit: ABC News]

{ 15 comments }

You’re also assuming you don’t have to split the jackpot with anyone else. By your calculations, splitting it with only one other person would instantly turn the ticket into a loser.

jme’s right. Because such a huge volume of tickets are being sold (I saw an estimate of 1.4 billion), the odds of splitting the jackpot are huge. A full analysis is here: http://www.circlemud.org/~jelson/megamillions/.

NPR’s Planet Money blog does the calculations here:

http://www.npr.org/blogs/money/2012/03/30/149685633/is-the-540-million-lottery-a-good-bet?ft=1&f=93559255

Given taxes and the likelihood of splitting the pot, it’s still not a good bet (probably).

The probability of having fun waiting for the numbers and imagining how you’d spend the money is higher if you buy a ticket than if you don’t.

I also wouldn’t neglect the effects of diminishing returns. In absolute terms, the expected value of a given ticket does exceed its price when the jackpot exceeds $174million, provided you elected the phased payout when you win, the market performs generally well in the following 30+ years, and you’re the only person to buy a ticket. But if we assume all of that stuff, the real killer is the diminishing return on large sums.

How much is a dollar worth to you? I might correctly guess the powerball (and no others), and thereby win $2. Discount the one I paid for the ticket, and I’m up a dollar. Which is great! Now think about the average jackpot, (I’m guessing is) circa $40 million? Do you think I’m going to get hung up on winning $40 million plus $1 versus $40 million flat? Not at all. There’s must be some non-linear discounting function. Now ask yourself: what would you do with $500 million that you wouldn’t do with $40 million? The probability of winning the jackpot is small enough (given a non-linear discounting function) that the threshold of rationality is little more than an absurd benchmark with no real meaning.

In terms of actual monetary cost, the above comments are correct – the odds that the winner will have to split is higher than average and a cash payout (as opposed to an annuity) diminishes the jackpot even more.

That said, however, many states have education lotteries, where the state revenues go to pay for K-12 and/or higher ed. Thus, the psychological benefits (minimal though they may be) ought to be considered. When I bought 7 tickets, I thought to myself, that’s some money for K-12 education in GA and some money for the HOPE scholarship for tuition at GA state universities (such as UGA, where I am). While any other return is highly unlikely, it provides the justification I needed to purchase a ticket without cringing.

I think what Tony is getting at is that the rationality of the decision depends on your underlying utility curve, with respect to money.

For most people, $640 million won’t make you 640 million times happier than $1. Lottery winners are happy, but not THAT happy.

Expected utility, not the expected dollar value, is the technically appropriate calculation here.

I love Chris’ explanation.

Funny, I was going to make a crack about endowing the MonkeyCage if I win!

I agree that expected utility is the right calculation, but I think the calculations about are all off. A dollar is worth nothing in marginal value. Sure, I know, it’s worth “a dollar” but what kinds of goods and services can I get for a dollar ? It’s easy me to treat it as virtually worthless.

Since the utility of a dollar is basically zero, then Will, you have it backwards. A megamillions jackpot could transform someone’s life. To place a monetary value on it strikes me as nearly impossible. It’s like asking me what is the value of assuring my own, my family’s, and future generations of the Gronke family monetary security.

This is a behavioral economics problem. This is not a strict dollar to dollar utility calculation. That’s why so many people play.

* Note that I’m assuming that you only care about the financial payout from the lottery ticket. Apparently winning the lottery can also ruin your life. Or not.

The aftermath of a lottery may depend partly on the original financial conditions of the individual pre-lottery. If a person is wealthy and self-sufficient, then the lottery won’t lead to problems down the line. If someone is poor however (and many lottery players are), then the lottery could cause financial difficulties (http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1324845).

Those defending the lottery: Keep in mind that you can spend $1 on things other than lottery tickets. You can buy food for yourself, or give to a charitable organization that you really like, or invest that $1 in a inflation-linked bond, or do anything at all that has a greater chance at benefiting someone other than the people running the lottery.

Okay, an amendment. I should not have said “cause financial difficulties”, but instead “will still lead to continue financial difficulties”, as the paper I linked stated that bankruptcy was postponed (and not caused by the lottery).

Not only did you overlook the strong possibility of multiple winners (with 1.4 billion tickets and 175 million to one odds there will be on average 8 winning tickets or so), you overlooked the fact that, for the ticket to be a good bet, the jackpot would have to exceed the ticket sales. That doesn’t happen. Again $1.4 billion collected, $640 million paid out (even ignoring time values, etc.). Doesn’t that make the return about forty-six cents?

The jackpot includes carryover from prior draws where there was no winner. So in this case yes — the jackpot can be higher than ticket sales. And thus yes, as the level gets high enough it does turn into a “good bet”.

Making sure that you don’t split the jackpot with anyone else is BY FAR the most important consideration in determining the expected value of a lottery ticket. To that end, you need to pick numbers that nobody else will pick, numbers like 23, 24, 25, 26, 27, 28.

Since picking numbers like that (i.e., numbers that everybody knows will never be picked) are the BEST way to maximize your likely winnings, you will now understand why you should NEVER play the lottery.

Paul-

It’s true that a dollar is nearly worthless and probably, for many people, has no significant impact on happiness. And that does make my example suck. And I think you’re right that it’s easier to understand why people buy lottery tickets from a behavioral perspective. (Though I really, really like Chris’ explanation.)

The underlying logic is still correct, though. Winning $640 million doesn’t make a person 6.4 million times happier than winning $100. $100 is worth a lot more to you now than it is after you already have $639,999,900. This is why people are risk-averse!

From what I’ve seen, people who buy lottery tickets get this, and they generally understand that the potential monetary pay-off isn’t worth it. But it’s a fun way to spend a few bucks.

I’m not sure why you write “To place a monetary value on it strikes me as nearly impossible.” It’s absolutely possible– it’s $640 million.

Thanks to everyone who chimed in – this discussion definitely increased the utility I got from the lottery!

So I definitely missed the multiple winners point – thanks to everyone who pointed that out both here, on Facebook, and in discreet text messages. And although I haven’t run those numbers, I’m sure it does change the expected payoff significantly. With that in mind, though, it is interesting that no one brought up the fact that there are lots of other prizes you can win in the lottery, like a 1 in 844 chance of winning $10 and a 1 in 4,000,000 (appr) chance of winning $250,000. Now while none of these are good bets individually, they do, however, add to your expected utility from your $1 bet, which also gets you a shot at all these additional prizes.

Now I doubt this makes up for the multiple winners problem, but it does unambiguously increase the expected value of the ticket (and these prizes do *not* have to be split.)

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