Is it rational to play Megamillions, with the current jackpot of a billion dollars or so?
Yes and no. Here are three questions:
- Are you willing to pay $20 for ten MegaMillions tickets?
- Are you willing to pay $20 for seven weeks’ worth of immunity from fatal lightning strikes?
- I propose to flip a coin. Heads, you get the MegaMillions jackpot. Tails, you die. Do you want to play?
There is no irrational answer to any of these questions. Some people like to play the lottery; some don’t. Some people are safety fanatics; some aren’t. Some, but not all, love huge risks with huge potential payouts.
But there is such a thing as an irrational pattern of answers. I know there are millions who answer yes to question 1, because I see them buying tickets. I’m guessing that almost all of those people would answer no to question 2. And I’m guessing that a fair number of those would answer no to question 3 as well. Perhaps you”re one of that fair number. If so, I declare you irrational.
Ten Megamillions tickets give you about a one in thirty-million chance to win the jackpot. One in thirty-million is also pretty close to the chance you’ll be struck by lightning in the next seven weeks. If you’re willing to buy the lottery tickets but not immunity from the lightning, you’re telling me that winning the jackpot means more to you than staying alive. So you should go for the coin flip in question three. If you didn’t, I declare you irrational.
Well, so what? Some economist called you irrational. Why should you care?
You should care because this is exactly the kind of irrationality that will allow me to take all your money. Keep reading to see why.
But first, let’s acknowledge that I’ve ignored a few things, like the possibility that you’ll win one of the lesser prizes in the lottery. I’m assuming that, compared to the jackpot, those are negligible as reasons to buy a ticket. (This is consistent with the observed fact that a whole lot of people buy tickets only when the jackpot is large.) If you don’t like that assumption, we can avoid it by tweaking the numbers in the questions, in ways that I strongly suspect won’t change most people’s answers.
Now let’s get to the fun part where I take all your money. Suppose you’ve answered yes/no/no to the three questions above. Then I’ll rig up the following experiment: I fill a bag with 30 million white balls, one black ball, and one blue ball. I plan to pull a ball from the bag, but before I do, I’ll make you two offers. You can take them or leave them; it’s entirely up to you.
- A. If you give me $20 upfront, and if I draw the blue ball, I’ll give you a billion dollars.
- B. I’ll give you $20 upfront, provided you allow me to electrocute you if I draw the black ball.
You’ve already told me that you’d pay $20 for ten Megamillions tickets, giving you a one in thirty-million chance at roughly a billion dollars. So of course you’ll happily go for A. And you’ve already told you that it’s not worth it to you to give up $20 to avoid a one in thirty-million chance of electrocution. So of course you’ll also happily go for B.
So: You (voluntarily) give me $20 and accept my $20. So far we’re even. And, I might add, you’re very glad to be playing. You got two things and you wanted both of them.
Now the drawing. Most of the time I’ll draw a white ball, and nothing happens. But occasionally I’ll draw a blue or a black. When I do, I’ll tell you (perfectly honestly, because I’m honestly exploiting your irrationality, not trying to trick you) that I’ve drawn a non-white ball. There’s a fifty-fifty chance it’s black, in which case you will die, and a fifty-fifty chance it’s blue, in which case you win the billion. You are effectively facing the very coin flip that you told me in Question 3 that you prefer to avoid. Of course, then, you’ll gladly pay me a few bucks to call everything off. Now I’m ahead.
We might have to play this game several million times before I draw a non-white ball and take your money, but as long as you’re committed to your stated preferences, you should be willing to play several million times — or to let me set my computer to playing a virtual version for us several million times per hour. In a few hours, I’ll win some money from you. In another few hours, I’ll win more. And so on till you’re broke.
When an economist calls you irrational, it almost always means that if you follow through on your stated preferences, a sufficiently clever opponent can take all your money, leaving you smiling along the way. It’s worth being alert to such things.
If you thought this was fun, you should take the ten-question Irrationality Test in Chapter Six of Can You Outsmart an Economist?. Come back to the blog and let me know how you did.
I believe the Jackpot is now well over $1 billion. The odds of winning are 1 in 300 million. After taking the annuity and paying taxes, it strikes me that one would walk away with at last $600 million.
So long term, every dollar I bet returns two dollars. That is, if I live forever and only play when the lottery is at this level, I come out ahead. Isn’t that the right time to gamble, when the odds are in your favor?
Is it an important assumption that the pleasure of buying lottery tickets is left out, i.e., the pleasure of being able to walk around for a few days imagining what it would be like to win a billion dollars? I know I have occasionally felt that pleasure.
We go to movies to imagine what it would be like to be other people and do other things, and we pay money for that experience. That seems rational. We buy a few hours of make-believe for $10. I go to movies too.
So, if the pleasure you get from buying five lottery tickets is about equal to the pleasure you get from going to a movie, is buying the lottery tickets rational?
Most people get minimal pleasure from buying lightning insurance or flipping coins about death, so they aren’t the same.
On the other hand, why would people who buy lottery tickets for the pleasure of it buy more than one ticket? The marginal returns to buying additional tickets diminishes to nearly zero after the first one.
David Pinto: I believe you’re overlooking the fact that you’re not playing for the jackpot; you’re playing for a share of the jackpot. The more tickets sold, the smaller that share is likely to be. (I also ignored this in the post, though it’s easy to correct for it by replacing “a billion dollars” with some fraction thereof throughout.
Zazooba: It seems to me that the difference is that movies help me imagine things I might not have been able to imagine on my own, whereas I don’t see how buying a lottery ticket makes it any easier to imagine what it would be like to have a billion dollars. I can imagine that equally well whether or not I buy the ticket.
(I also don’t understand the bit about marginal returns diminishing to zero after the first one. Two tickets gives me, for all practical purposes, twice the chance of winning the jackpot—unless they happen to be two tickets which exactly duplicate each other.)
Steve #4
The ticket simply makes it more palpable thus the fantasy more real.
One needs neither Playboy nor Penthouse to imagine certain uh things yet they both sold millions in their day?
And, since 2 times (virtually) zero is still (virtually) zero, it might be more productive to buy a glass of wine with the price of the 2nd ticket and let the fantasies rip.
While I heartily concur with the money-pump mathematics, my take on this example is close to David’s: For whatever reason, carrying around a lottery ticket has consumption value; carrying around an insurance card does not. In these cases, my consumption value (positive anticipation or negative worry) is anticipation and is inextricably tied to the status quo.
A little closer to David’s visionary idea, this is how I think of pre-purchasing a ticket to a play that I have not seen or awaiting the denouement of a sports contest. Anyway, I think that the game aspect (vs. lightning and death!) differentiates these examples in a way that the examples in your book avoid.
Steve #4
I speak only from personal experience.
Occasionally, I have bought a lottery ticket and have enjoyed thinking about what would happen if I win. When I don’t buy a lottery ticket, that fantasy is much less palpable. When I buy two lottery tickets, there is very little increase in my enjoyment. The odds are still a zillion to one.
When I check whether I won, it becomes painfully obvious how extreme the odds are, so I feel a little stupid. On the other hand, it is a useful reminder of just how microscopic my place in the universe is (which is kind of a positive thing, intellectually). Net-net buying an occasional lottery ticket has been a modest positive for me.
Where can you get a MegaMillions ticket? Are they available in all states?
Asking for a friend.
To make this a more complete proof of irrationality, maybe you should add a #4: are you willing to play a game where you never win anything, but you have a 1 in 15 million chance of losing a few bucks? Of course everyone is going to say no to that. But then it turns out to be identical to the colored-ball game you invent later on.
Maybe the analysis can be expanded to show irrationality even if we assume the player derives a small amount of utility from the frisson of the game. The irrationality would come from the observation that such people generally buy more than one ticket.
Zazooba (#10): You won’t get irrationality that way; the second ticket improves your odds by an amount that different people are allowed to value differently, so it can be perfectly rational to plunk down another two bucks for it. To get to irrationality, you’d need to combine this observation with the person’s unwillingness to pay as much for something that improves his odds even more.
A. Yup, Landsburg has demonstrated that people hold irrational views, judged from the perspective of expected returns. Prospect Theory has overthrown Rational Expectations Theory. But I’m not sure that Landsburg has demonstrated an ability to turn this into a cash machine.
Landburg imagines a person who is–
• Willing to pay $20 for ten MegaMillions tickets,
• Unwilling to pay $20 for seven weeks’ worth of immunity from fatal lightning strikes, and
• Unwilling to take a 50% chance to win the MegaMillions jackpot if it also entails a 50% chance of dying.
From this, Landsburg concludes that such a person would be willing to take $20 for a small risk of letting Landsburg electrocute her.
Yet this last statement, even if consistent with Rational Expectations theory, seems inconsistent with Prospect Theory. Prospect Theory tells us that people resist explicit risks of loss when offered the opportunity for comparable (or even slightly higher) gains at comparable risks. Thus, I expect people might well be willing to pay to enter a lottery with a chance for large winnings, yet be unwilling to consciously except a risk of electrocution.
(In addition, people seem averse to suffering harm at the hands of a sentient being; people don’t feel as victimized if comparable harm results from faceless process such as a natural disaster. I don’t understand Landsburg’s hypothetical to be focused on this aspect of irrationality.)
B. Like Zazooba, I understand the joy of buying a lottery ticket as giving people a fantasy of sudden wealth. Landsburg correctly observes that a person can engage in such a fantasy without buying a ticket. (I do this by imagining that my in-laws buy such tickets.) But as others observe, the act of giving yourself a non-zero chance of winning makes the fantasy more present. Perhaps it’s an availability heuristic?
Thus, I also share Zazooba’s puzzlement about people’s propensity to buy multiple tickets. After all, the presumed principle joy of the ticket—the fantasy—seems to be triggered by having a non-zero chance of winning. Each additional ticket doesn’t seem to add more fantasy, just (a trivial bit) more likelihood of winning.
Basically, the fact that people buy multiple tickets seems to challenge my assumption that the principle benefit of buying the ticket is the fantasy. Now, obviously a person can value both the fantasy AND the increasing chance of winning. But I have a harder time projecting myself into that mindset.
“Prospect Theory tells us that people resist explicit risks of loss when offered the opportunity for comparable (or even slightly higher) gains at comparable risks.”
Many people do buy both an insurance policy and a lottery ticket, indicating an inconsistent attitude to risk.
“And you’ve already told you that it’s not worth it to you to give up $20 to avoid a one in thirty-million chance of electrocution. So of course you’ll also happily go for B.”
As nobody really points out, here is the flaw. I may consider it not worth it to give up $20 to avoid a small chance of death, but as I am irrational I may also consider not not worth accepting $20 for a small risk of being killed.
The idea that the value of a ticket is mainly in the fantasy of winning does seem to be dispelled by the multiple ticket thing.
No, it’s not dispelled exactly. But it’s a marginal thing: Once you buy a ticket and get the fantasy, what is the next ticket getting you?
Then again, when my dad wants a letter to be delivered faster, he puts extra stamps on the envelope. So perhaps I needn’t ask….
I buy lottery tickets as insurance against not having to wonder if I would have won had I bought. Having a lottery ticket and losing allays me of that concern.
That provides as much value as a cup of coffee.
I’d probably buy protection against fatal lightning strikes, as well, if that were possible. So would a lot of folks that do outdoor sports that often get canceled when lightning is within 5 miles. Though, that is not possible, while winning the lottery is.
As I understand it, the payoffs accumulate as long as there are no winners. I wonder whether the odds might be more favorable for the less-publicized $300-400 million dollar payouts than for the occasional billion dollars ones that get a lot of publicity. Presumably the worst odds are for the first week when nothing has been accumulated. The people who buy those tickets seem to be the ones willing to subsidize later buyers. Tracking the number of buyers by the size of the pool and attractiveness of the odds might yield an interesting dataset.
#15. “I’d probably buy protection against fatal lightning strikes, as well, if that were possible.”
It is possible! I pay $50 a month for mine and it is working fine!
Flippancy aside:
“Most experts recommend that outdoor athletic events should be postponed when the thunderstorm approaches from a distance of six miles. The best way to gauge the distance of a thunderstorm is to measure the elapsed time from the flash to bang. Since a count of five seconds equals a distance of one mile, a count of 30 seconds equals a distance of six miles. In most cases, when you can hear thunder, you are no longer safe.”
I guess that must happen pretty often
On a more serious note, if such protection were available, say as part of the signing up for these sports so they could safely go ahead when lightening were near, I guess there would be modest price people would pay.
A solution that would be possible is compensation paid out if the event is cancelled. The relative amounts of price and payout are related to what this post is about, I think.
“I buy lottery tickets as insurance against not having to wonder if I would have won had I bought.”
What do you do for insurance against having to wonder if you would have won had you bought another ticket?
I may well have made a mistake, but here’s a model that I seems consistent with the spirit of the puzzle and where you rationally could answer “yes, no, no.”
Suppose that you have a defined lifespan, maybe seven weeks. In this world, you die from either a lightning strike or when you reach the end of your lifespan. Now you’re buying insurance against lightning for your whole life, but I don’t think that changes the paradox since the odds of death by lightning in this defined period still match the odds of winning the lottery.
You can be reincarnated to exploit the law of large numbers, but not more than once per lifespan; that is, if you die immediately in a coin flip game, you need to wait in limbo seven weeks before you get your next body.
Assume:
* Decreasing marginal utility from money. Say utility(money) = natural log(money)
* You can only enjoy your money if you’re alive, so in any given week, your expected utility = utility(money) * (chance you’re alive)
* Your total utility is the sum of utility from all weeks
* You can choose exactly one of the three options once per lifetime, lottery, insurance or coin flip
* The coin flip and the lottery are decided immediately at the beginning of each life
* The chance of lightning strike is uniform across all time
x = lifespan
s = starting money
c = cost of insurance = cost of lottery
Aside from lottery tickets and lightning insurance, you don’t actually spend your money, you just enjoy Scrooge McDucking or something. So, the simplest case is when you buy insurance. There is no risk of death:
=> Utility (insurance) = ln(s – c) * x
Next, when you do nothing, your expected utility is a power series:
k = chance of living through any given week
Expected utility (nothing) = (ln(s) + ln(s)*k + ln(s)*k^2 … ln(s)*k^x)
Factoring out…
α = (1 – k^x)/(1 – k)
=> Expected utility (nothing) = ln(s) * α
When you play the lottery, prize money enters the picture
p = prize money
Sometimes you win:
Expected utility (win lotto) = ln(s – c + p) * α
Expected utility (lose lotto) = ln(s – c) * α
Given that chance of winning the lottery is equal to chance of death by lighting over the entire lifespan:
ω = chance of win = chance of lighting = (1 – k^x)
=> Expected utility (lotto) = ln(s-c+p)*α*ω + ln(s-c)*α*(1-ω)
Last, when you flip the coin, half your lifetimes you get the prize and the other half you die instantly:
=> Expected utility (flip) = ln(s+p)*α*1/2 + 0
Now, changing numbers a bit, you live to a ripe age of seven weeks, start with some money and insurance is expensive:
x = 7 weeks to live
s = $350 to start
c = $200 to buy lotto or insurance
p = $70,000 lotto prize
k = 0.97 chance of living through a week (~ 20% chance of lottery winning)
Calculate
Utility (lotto) ~ (13.7 + 25.9) ~ 39.6
Utility (nothing) ~ 37.5
Utility (coin) ~ 35.7
Utility (insurance) ~ 35.1
Thus, to maximize utility over time, lottery beats doing nothing, but doing nothing beats both insurance and coin flip.
You cannot treat an iterated decision as identical to a single decision because the human brain takes time into account when computing expected value, which is not irrational and is necessary for learning. It’s basic neuroeconomics. See the “TD algorithm in neuroscience” section here: https://en.m.wikipedia.org/wiki/Temporal_difference_learning
I am pretty sure that most people would recognize that playing a game several million times when the odds are 30M-1 that they get electrocuted is not a good idea and they would not play. Similarly if they were likely to live more than 30 million seven weeks periods they would (if they could afford it) see that paying $20 every 7 weeks to avoid death may be a good deal (this would obviously depend upon a whole host of other factors).
Similarly as long as one has enough money in reserve a 30m-1 punt on a on a billion dollar prize may be a good investment (this would obviously also depend upon a whole host of other factors).
However there seems to be a difference between investing in or insuring against something where you have many millions of iterations compared to when you have only a handful of opportunities. I can’t quite get my head around what that difference is but suspect it has something to do with the essence of gambling as a leisure activity.
Well we have 3 questions here. They essentially ask “Do you prefer A to B”, “Do you prefer A to C” and “Do you prefer B to C”. We’ve got 6 possible orderings on A,B and C and 8 possible answers to the 3 questions. Each ordering gives a different set of answers so 2 of the possible collections of answers are going to be inconsistent.
Given n options instead of 3 we’d have n! orderings and 2^(nC2) sets of answers to questions, which is a nice combinatorial proof that n!<=2^(nC2).
I guess the interesting thing here is that in practice a lot of otherwise rational people seem to give inconsistent answers. I'd imagine that for the most part this is just because we're bad at estimating small probabilities.
Well, not just bad at estimating small probabilities; bad at estimating EXTREMELY CONSEQUENTIAL small probabilities. This was the central insight that led to Cumulative Prospect Theory (CPT). Thus we observe that people pay disproportionately for a 1% chance to win a prize, or to insure against a 1% chance of loss, but are unwilling to pay the same amount to shift their likely outcomes from 55% to 54% or 56%.
From Wikipedia:
“The main modification to Prospect Theory is that, as in rank-dependent expected utility theory, cumulative probabilities are transformed, rather than the probabilities themselves. This leads to the aforementioned overweighting of extreme events which occur with small probability, rather than to an overweighting of all small probability events. The modification helps to avoid a violation of first order stochastic dominance and makes the generalization to arbitrary outcome distributions easier. CPT is therefore on theoretical grounds an improvement over Prospect Theory.”
I think this was the insight sited by the committee that awarded Daniel Kahneman the 2002 Nobel Prize in Econ.
I’m going to say that you cannot prove any irrationality with this example. Let W be initial wealth, W’ be wealth after winning the lottery and p be the probability of winning the lottery or being struck by lightning.
If I biy a lottery ticket I satisfy inequality
p*U(W’-$20} + (1-p)*U(W-$20}>U(W).
If I do not buy lightning insurance I satisfy inequality
(1-p)*U(W)>U(W-$20)
[Without loss in generality I set the utility of dying equal to zero]
If I refuse the bet I satisfy the inequality
.5*U(W’)<U(W)
How does satisfying the first two inequalities violate the third inequality?
Neil:
I made two approximations, both of which I think are reasonable:
A) U(W’) = U(W’-20). In other words, if you win the jackpot, another $20 more or less is not going to significantly affect your utility.
B) p^2=0. In other words, the chance of winning the jackpot (or of being hit by lightning) is very very small.
Given these approximations and your inequalities, I believe the inconsistency should follow.
Thanks, Steve. Since I plan on lighting a cigar with a C-note tonight if I win the Mega, I accept your assumption. ????
???? = :)
‘In a few hours, I’ll win some money from you. In another few hours, I’ll win more. And so on till you’re broke.’
Why would anyone continue to play after the first non-while ball was drawn and they saw your strategy ?
Rob Rawlings: The point is that it’s irrational to play in the first place. If you’re irrational enough to play, you’re presumably irrational enough to keep playing.
Or more succinctly: What I’m proving is that if you stick to your stated preferences, I can take all your money. You’re saying “Well, then, I won’t stick to these preferences”. To which my response is: “Good. You’ve learned something.”
re 28: yes, I think your game serves more to demonstrate to people the inconsistency of yes/no/no to the three questions you pose at the beginning rather than as a money making venture :)
I’m not going to say whether or not I won the jackpot yesterday, but this morning I ordered four zeppelins. One in each color.
Seth 15, I once worked with a very rational man who was an excellent poker player because of his intuitive understanding of odds. He went in on the office lottery pool. He knew he was paying a premium that was greater than the expected value but he viewed it as insurance against the unbearable thought that everyone around him would get rich and he wouldn’t.
1 and 2 are insurance, 3 is risk
SL: “There is no irrational answer to any of these questions….
But there is such a thing as an irrational pattern of answers.”
A question of semantics: do you consider irrationality equivalent to inconsistency?
“If you thought this was fun… ”
… look up the Ellsberg paradox, which outsmarted a conference hall of economists in 1960 –
The problem with the original examples is that buying a lottery ticket buys you something more than a chance at a huge jackpot: it buys you a dream. Whenever I buy a lottery ticket (about once every couple of years) I fantasize about what I would do with all that money (even though, of course, I know that I won’t win).
It’s better to think of lottery tickets as “entertainment”.
Buying a lottery ticket offers entertainment whereas buying immunity from lightning strikes does not, therefore SL’s initial example is invalid.
#32 Richard, would it be reasonable to say that consistency was necessary but not sufficient for rationality?
You’re assuming that:
lotterywinlife – currentlife =< currentlife – death.
Neil’s analysis is really thought-provoking. But I can’t respond just now; my office just got a rush job for zeppelins, and we’re scrambling to find enough material. (Honestly, would even want a puce zeppelin?)
Steve, what if I say that I’m only willing to play the game if you agree not to stop in the middle and give me the option to call things off? Or equivalently, what if before the game I precommit not to call things off in the middle, even though at that moment I would prefer to call things off? Then you wouldn’t be able to make money off of me.
Anybody noticed how supermarket trolleys are much smaller than they used to be?
This is typical of the counterintuitive economist who seeks to make a name for himself: Set a false binary scenario and then rip the respondent for his “irrational” answer.
Landsburg’s either-or lottery-ticket/lightning preventions scenario is wrong because it is set on a false premise: A lottery ticket is not a probability ticket. It’s an entertainment ticket. It enables the purchaser to ponder a fantasy that could — however improbable — arise. It permits the imagination to wonder. That’s what the purchaser buys.
The scenario is ridiculously binary and irrelevant: I’ll pay the $20 to daydream. I won’t pay $20 for the impossible — you prevent me from being hit by a fatal lightning strike.
As for the coin flip, it depends on the mental state. I’m happy with life, I’ll pass. If not, I might take it. My wife divorces me to marry another man, I’m fired from my job, I’ve lost my savings in the stock market, and the dog dies. I’m 65 years old. I’m bald, fat, and ugly. This all happens within the past month. I might take the bet
Trick questions? Ok, going out on a limb here, I’m going to guess … yes, someone has noticed.
(Did I win?)
#41. Well, I have noticed, just wondered if anybody else had. It relates to SL observation that until 2000 trolleys were getting bigger but nobody knew why. Perhaps we can settle the reasons now with more data.
https://slate.com/culture/2000/04/attack-of-the-giant-shopping-carts.html
But can you outsmart a medical student?
Assuming your ‘electrocution’ is equivalent to a lightning strike. I’d play.
You only have a 10% chance of dying from a lighting strike.
I have a ~90% chance of becoming a billionaire.
Probably shouldn’t play that game with too many close friends though.