The Powerball Jackpot is $425M. Should you play?
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I'd love to see an analysis of how beneficial this could be and which numbers you should pick.
Rational people who play the lottery know their chances are laughably low, and that a $1 or $2 ticket is just a license to daydream about riches and wealth.
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.
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.
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.
31 is still a potential (birth)date
37 falls heavily into human "randomness" bias, and is often quoted as the most picked psychologically "random" number (a number with distinct odd digits not ending in 5) e.g. http://micro.magnet.fsu.edu/creatures/pages/random.html
>The most random two-digit number is 37, When groups of people are polled to pick a “random number between 1 and 100”, the most commonly chosen number is 37.
http://www.catb.org/~esr/jargon/html/R/random-numbers.html
Unless the lottery pickers aren't trying to pick a "random" number. I have no idea what their logic is anyway.
The point is that the numbers people select can be extremely different from a random distribution.
Sources: http://www.snopes.com/luck/cookie.asp http://www.foxnews.com/story/2005/03/31/fortune-cookie-leads...
According to a mathematician, even numbers over 31 are less common, so you should pick those to avoid splitting the jackpot: http://www.npr.org/2012/03/29/149635815/the-sobering-odds-of...
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.
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.
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.
And indeed, you can say that purchasing one ticket is a one-off event... but if you buy that one, why not every one that happens when the expectation is in your favor? Thus the risk of ruin becomes interesting to consider.
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.
log utility => maximizing the growth rate,
but maximizing the growth rate maximizes utility for many different utility functions, in particular all constant relative risk aversion utility functions, of which log utility is one. so preference for maximizing growth does not imply log utility, could be linear utility too.
(under a CRRA utility function, a risky income stream that varies between $1 and $2 gets the same risk aversion discount as one with a similar distribution that varies between $100 and $200, see e.g. http://ocw.mit.edu/courses/economics/14-123-microeconomic-th... )
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 ;)
But of course our gambler is more entertained so there is something to be said for that too.
I don't see why we have casinos at all. Just replace them with exquisite trading houses with over blown transaction fees.
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.
The return is undefined actually.
No, because there are no people to get 0 dollars; the concept of "each" makes no sense when there are no eachees. The answer is clearly undefined.
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:
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.
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 [1] http://www.paulgraham.com/cities.html
I find the odds indistinguishable from winning after purchasing a ticket.
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.
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.
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.)
What kind of URL is that?
"Statistical auditing and randomness test of lotto k/N-type games", available freely at 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.