
The Powerball Jackpot is $425M. Should you play? - jmharvey
https://medium.com/p/28c5a31cd41d
======
modeless
The math for split jackpots assumes that people pick randomly distributed
numbers in their tickets. But that's clearly not true, and that means by
choosing numbers carefully you should be able to significantly reduce the
chance you'll have to split a win _without_ changing your chance of winning.

I'd love to see an analysis of how beneficial this could be and which numbers
you should pick.

~~~
ryandvm
Choose 1, 2, 3, 4, 5, 6. Nobody _ever_ chooses that set.

~~~
jonnathanson
The laws of large numbers make any edge supposedly gained by "picking what
nobody else picks" almost entirely moot. The probability is still incredibly
small that any given sequence will come up. Getting the right sequence is the
_much_ taller hurdle to clear before tackling the problem of how many others
have also picked correctly.

~~~
cmsmith
Of course no one is saying that it's easy to get the winning numbers, the
question is whether it is worth it to play. If the prize is big enough, then
the expected value of a lottery ticket is more than $1, and it makes financial
sense to play. By reducing the chance of sharing the jackpot you can
effectively increase the size of the prize, pushing the odds towards your
favor.

~~~
jonnathanson
_" By reducing the chance of sharing the jackpot you can effectively increase
the size of the prize, pushing the odds towards your favor."_

In theory, sure, but in praxis, whatever edge you gain on expected value by
doing this is still infinitesimal. Furthermore, it's highly unlikely that you
could choose an unlikely-to-be-duplicated number combination any better than
random chance could. I'm not convinced a randomly generated string of numbers
is any more likely to be duplicated by someone else's pick than an
intentionally chosen sequence. Probably less likely, in fact. I would guess
there's someone else out there picking 1,2,3,4,5,6 (for example) more often
than there's someone else out there whose random number generated matches your
random number generated.

~~~
Fuzzwah
But once you accept the small chance of winning, you may as well set it up so
that if this unlikely result came out you'd maximize your winnings. Its a tiny
edge because it'll happen so infrequently, but if it did it would make a huge
difference in the result.

~~~
jonnathanson
_" But once you accept the small chance of winning, you may as well set it up
so that if this unlikely result came out you'd maximize your winnings"_

If there was a way to do so, sure. I'm not convinced there is.

Like I said, I get the logic behind it. I just don't think it's any more
likely to produce an unduplicated sequence than a randomly generated string
would. If anything, an intentionally chosen sequence (e.g., 1,2,3,4,5,6) seems
_more_ likely to be chosen by someone else. Assuming that nobody else (or, at
least, fewer people) will ever choose a specific sequence is a pretty naive
assumption, IMO. I totally understand the thinking behind it, but I don't
think you're likely to do better than a random number generator at
outmaneuvering the X hundred million other tickets in play.

~~~
cmsmith
By definition, if there are numbers which are chosen more often than those
randomly generated, then there must also be numbers which are chosen less
often. I'm not saying that those numbers are 1,2,3,4,5,6 - just that they
exist.

So we are left with an exercise in psychology - should you pick a sequence
that seems common, because the people thinking about it will pick something
else because it seems common? Or should you not pick the sequence because
people will pick the sequence because they think that people thinking about it
will pick something else because it seems common? Some actual data would solve
this, but I imagine the lottery folks would be reluctant to release it.

------
rverghes
In my opinion, using expected value to evaluate lotteries is not correct.
Realistically, there are only two outcomes. Either you win millions, or you
lose a few thousand over several years. The future where your outcome matches
the expected value never actually occurs.

This is unlike games like blackjack, poker, or roulette, where the expected
value does match the eventual outcome.

So if you can live with the second outcome, playing the lottery is fine. But
using expected value does not seem appropriate to me.

~~~
rodly
> In my opinion, using expected value to evaluate lotteries is not correct.
> Realistically,

Expected value is realistic. The relation to casino games is silly seeing as
the timelines for a few hands of blackjack are several orders of magnitude
shorter than buying a ticket for a $300+ million dollar lottery.

------
nullc
Sad to see it not make the Kelly criteria argument for ruin, even where there
is a positive scalar expectation:
[http://r6.ca/blog/20090522T015739Z.html](http://r6.ca/blog/20090522T015739Z.html)

~~~
jmharvey
O.P. here. The Kelly criterion is a product of the logic outlined in the
section about utility being the log of wealth. In short, assuming a jackpot of
under $1 trillion, you'd need to have a bankroll on the order of eight digits
to justify spending $2 on a ticket under the Kelly criterion.

Also, the Kelly criterion doesn't exactly apply here because this is more like
a one-off event than a repeated game. But a lot of the same principles apply.

~~~
RockyMcNuts
not sure I agree with you.

Kelly says, if there is a positive expected value, there is an optimal amount
to bet, which maximizes the growth rate of your stack per bet, if you were to
bet repeatedly. If you overbet, your expected growth rate is negative...
because over the long run, by the time you hit the jackpot you've lost too
much of your stack to get back to even.

Basically, I think Kelly says you can't just go by expected value, ie your
edge, you have to look at your stack. Overbetting turns a +EV bet into a -EV
bet and eventual loss of your whole stack.

So 1) I don't think Kelly is a product of log utility, it just requires that
you are trying to maximize your growth rate (As an aside, in many settings
people are much more risk averse than log utility implies; and in many
settings people don't have consistent cardinal utility, a time-inconsistent
utility like prospect theory is more predictive) and 2) since you are
presented with interesting bets every day, if not this particular one, Kelly
generally applies, since if you don't heed it you fall victim to gambler's
curse.

~~~
ronaldx
> Kelly says, if there is a positive expected value, there is an optimal
> amount to bet

Yes, and the optimal amount to bet here is for all practical purposes equal to
zero. In short, you don't have a large enough stack to keep on taking this bet
until it pays you back.

I tend to agree with your wider point, that you can treat life as a long-term
sequence of taking risky bets and use Kelly to approximate that.

However, even if you did have an equivalent opportunity to this all day every
day (which is itself hard to assert), Kelly is saying you _definitely_
shouldn't take it here. If you don't have that repeated opportunity, Kelly-
type thinking just leads to an even stronger conclusion - you _definitely_
_definitely_ shouldn't take it.

Good luck to everyone if you're playing the lottery :) Until the numbers are
drawn, you still have a chance ;)

~~~
RockyMcNuts
yes, agree with you and OP! just Kelly makes the OP's advice not to play
stronger, even if you get +EV, unless you're a millionaire you will overbet
and go broke before you win !

------
ChuckMcM
I love this analysis. Growing up in Las Vegas we did probability analysis on
Casino games as a homework exercise. And it explained _exactly_ why Casinos
are a license to print money. It's even weirder when you consider the case
where you gamble $10/week versus save/invest $10/week in a compound interest
account. Guess who is happier 20 years later?

But of course our gambler is more entertained so there is something to be said
for that too.

~~~
seanmcdirmid
Lots of investments are very risky, almost like gambling but the odds aren't
supposed to be stacked (return related to risk). Many people are addicted to
playing the market would be (and often are) the same ones who would be
addicted to games of chance.

I don't see why we have casinos at all. Just replace them with exquisite
trading houses with over blown transaction fees.

------
MrFoof
The odds are beyond terrible, but they are exactly zero if you don't play.

My minimum required jackpot for me to play is a $320,000,000 jackpot. I
increase the minimum every year. Keeps my maximum spending to about $4 or $6 a
year, which is worth it for the entertaining daydreaming it provides.

~~~
gunn
In fact the return is 100% if you don't play.

~~~
jere
$0/$0

The return is undefined actually.

~~~
seanmcdirmid
I thought that was infinity? Ah, but that would cause problems.

~~~
tinco
This concerns real things, not mathematical approximations. So, we divide 0
dollar over 0 people, that means 0 people get 0 dollar. The answer is clearly
0.

~~~
jordanthoms
But _each_ person, of which there aren't any, gets an undefined amount.

------
wcchandler
I've never really thought of playing as an investment. I've always thought of
it as a donation to the education system that also has a potential for a large
payout. I don't play very often -- 2 or 3 times a year. I never feel bad about
losing the money. The way I see it, I would've donated it to some cause. That,
to me, makes it worth playing for.

~~~
DamnYuppie
Until you realize just how little of that money makes it to the schools :-/

~~~
MartinCron
Especially as it becomes an excuse to fund the schools by that much _less_
from the general budget. It's an exquisite accounting lie.

------
stevekemp
Interesting article.

In the past I've used the powerball simulator to get a feel for things. Buying
two tickets a week for hundreds of years, and tracking profit/loss:

[http://justwebware.com/powerball/powerball.html](http://justwebware.com/powerball/powerball.html)

------
feral
In the 'but then it gets even more complicated' spirit:

When the author mentions buying a ticket in the end, for the entertainment of
fantasising about a potential windfall, I was reminded of this interesting
Less Wrong article: "Lotteries: A Waste of Hope" which argues even that is a
bad reason to play lotteries.

[http://lesswrong.com/lw/hl/lotteries_a_waste_of_hope/](http://lesswrong.com/lw/hl/lotteries_a_waste_of_hope/)

~~~
jmharvey
OP here. Interesting article. "Fantasizing" isn't what I was trying to
describe in the "entertainment" section. I 86'ed a longer explanation of what
I meant, but here goes:

When I say, "I also enjoy letting my mind wander and think about what I’d do
with a nine-figure windfall," I actually mean that I think about it, not that
I fantasize about it. Some of my problems would go away. Others wouldn't. I'd
have a whole new set of issues to think about. And I'd still come in to work
[0] tomorrow.

When I think about what it would be like to win the Powerball jackpot, I tend
to reflect on some of the issues raised in PG's "Cities and Ambition" essay
[1], when he talks about the things different cities value: "New York is
pretty impressed by a billion dollars even if you merely inherited it. In
Silicon Valley no one would care except a few real estate agents. What matters
in Silicon Valley is how much effect you have on the world."

The utility I get from buying a lottery ticket isn't about dreams of private
jets and caviar. It's the perspective I gain: if I didn't have to think about
money ever again, what would I want to do with my life? I think I'd keep
working on startups. What would I do differently? I'm not sure, but I'd start
with buying a better set of wheels for my bike. In the bigger picture, I'd
probably spend more time on projects that have the potential to have huge
impact, and think less about whether a particular idea can be a profitable
business.

It's easy to get caught up in the patterns of daily life. Thinking about
winning a jackpot makes me evaluate my life from 30,000 feet. I guess I could
do that for free, but at least for me, that $2 gets me thinking a little
differently. And to me, that's where the value is.

[0] [http://zeromailer.com](http://zeromailer.com) [1]
[http://www.paulgraham.com/cities.html](http://www.paulgraham.com/cities.html)

~~~
Dylan16807
Your end result is useful, but it's bad that the most effective way to get
there is wasting money to trick your stupid human brain.

~~~
brownbat
I often think about finding a winning lottery ticket for the same reason.

I find the odds indistinguishable from winning after purchasing a ticket.

------
differentView
You should play in a workplace pool situation. $5~$10 is a fair price for
psychological insurance.

If most of the people in your office win the jackpot, no matter how Vulcan you
think you are, it will negatively affect you psychologically for many years.

------
pivotandconquer
This post shows one of the limitations to medium.com, namely, a lack for
math/LaTeX formatting. This post was well written, but occasionally difficult
to parse.

~~~
jmharvey
Agreed. Can you recommend a better alternative?

~~~
dave809
functionspace.org looks nice

------
jrs235
It's darn near impossible to justify buying a ticket as a rational, wise
investment (as the article points out) due to the odds, split jackpots at that
size, and then taxes. But, if you include the intangible of a night or three
of "dreams" of what you would do... then maybe one ticket (and only one
ticket) might be okay... $2 for 60 hours of dreaming or a soda?... But then
again you might be better without that money as most go broke and their
relationships and lives are ruined over money. I'd recommend purchasing a $1
pick 5 game, often times with odds around 1:170,000 if you want to but a dream
and not win enough to completely ruin your life. Or don't buy a ticket at all.

------
erehweb
Betteridge's Law of Headlines continues to apply.
[http://en.wikipedia.org/wiki/Betteridge%27s_law_of_headlines](http://en.wikipedia.org/wiki/Betteridge%27s_law_of_headlines)

------
lifeformed
I don't think a high enough dollar value is enough to make it worth playing.
Over a certain amount, the winnings are simply just "arbitrarily large". If
the prize was a trillion dollars, but each ticket cost $100 to play, does that
mean it's now worth it for everyone to buy a ticket? You're still not going to
win.

I think stuff like this should be weighted towards chance of winning. Given
the choice between a 1 in a million chance of winning $2 million, or a 50%
chance of winning $4, I'd pick the $4 one every time. They have the same
expected value, but one of them gives me a chance of winning anything at all.

------
lcedp
The return is so huge compared to investment that you should play based on...
what ticket cost means to you. Would you play if ticket's cost would be $0.01?
Of course you should - $0.01 won't make any difference to you.

------
arctangent
This doesn't take into account the value of the "hope" that buying a lottery
ticket gets you. Specifically, the owner of a lottery ticket has bought the
right to dream about how their life could suddenly be turned around if only
their numbers came up.

I admit that this "hope" may not be worth much to some people, but to those
who are scraping by it might be worth quite a lot. (I will leave discussion
about whether lotteries exploit the poor to another day.)

------
markdown
> [https://medium.com/p/28c5a31cd41d](https://medium.com/p/28c5a31cd41d)

What kind of URL is that?

~~~
Ackley
"permanent"

------
pallandt
Something interesting for the mathematically inclined.

"Statistical auditing and randomness test of lotto k/N-type games", available
freely at [http://arxiv.org/abs/0806.4595v1](http://arxiv.org/abs/0806.4595v1)

Powerball is not like a typical k/N lotto game though, but you might still
want to skim the paper.

~~~
jmharvey
Interesting, I'll have to give that paper a deeper look. I have noticed
seemingly skewed lottery results in the past, but I've never found a case
where the skewness of the results made a material difference to the optimal
strategy (excluding cases where computer RNGs were configured incorrectly).

------
jackschultz
I've always felt like the value of playing the lottery is not in the numerical
outcome, but rather the enjoyment of thinking about what could happen if you
win. How much enjoyment you get from dreaming about the outcome can, for some
people, be well worth the few dollars it costs for the ticket.

------
eliben
More and more blogs moving to Medium for some reason. This makes me sad,
because it means horrible formatting with anything code or math related. At
least until they fix that...

------
chrisbennet
Does anyone actually consider whether they will be happier if they win the
lottery? (I don't play the lottery on purpose.)

------
sgloutnikov
Interesting article, but hard to put mathematical facts over psychology in
cases like these.

------
coherentpony
If you don't play, you _definitely_ won't win it.

