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.
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.
In that case, you should be playing right after a jackpot is paid out. Your chances of winning are the same, but the chances of splitting a jackpot go way down because there are fewer people playing. Sure, you'll have to suffer with only a $40 million payout, but it's still nice.
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.
Kelly criterion has nothing to do with the marginal utility you gain from more money, it's about the risk of ruin in an iterated game.
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.
The Kelly criterion is designed to optimize the long term growth of your bankroll in an iterated game. And the strategy that optimizes your bankroll in an iterated game is the one that maximizes the expected value of the log of your bankroll after each iteration. Strictly speaking, the Kelly criterion isn't concerned with maximizing expected utility, but the math works out the same.
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.
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... )
> 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 definitelydefinitely shouldn't take it.
Good luck to everyone if you're playing the lottery :) Until the numbers are drawn, you still have a chance ;)
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 !
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.
> 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.
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.
Because choosing that sequence destroys the illusion by highlighting the infinitesimally small chances any given ticket has of winning.
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.
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.
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.
"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.
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.
"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.
Well the one thing which I do believe holds true is that picking numbers which are above 31 would lower the chances of splitting. I'd suspect that even just sticking over 12 would help out.
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.
I remember a California Lottery ad campaign back in the 80's which featured "my numbers" for various famous people and the Steve Jobs version of the ad had these exact numbers along with an explanation of the odds being the same as for any other numbers. So I doubt you're alone.
Many years ago I read an article in a Toronto newspaper that did a study of the least popular numbers. They are essentially prime numbers like 31, 37, 41 etc.
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.
Seems like you want to make sure all your numbers are over 31, so that anyone who encodes dates into his ticket number won't match yours. What other ideas do you have for avoiding collisions?
There was an interesting case in 2005 when 110 people claimed the second prize in Powerball, compared to an expected 4 winners. The lottery officials suspected fraud of some sort since there were so many winners. But it turned out that a whole pile of fortune cookies had been printed with the same numbers and a lot of people use the lucky fortune cookie numbers. Thus 110 people ended up using the same winning numbers.
The point is that the numbers people select can be extremely different from a random distribution.
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.
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.
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.
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.
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.
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.
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.
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...
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.
Entertainment derived from fantasy is fine. I've found that I can cut out the whole "buying a lottery ticket" part and the fantasy is just as rewarding.
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.
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.)
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.
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.
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).
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[ 4.5 ms ] story [ 150 ms ] threadIn 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
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.
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 ;)
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.
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...
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.
What kind of URL is that?
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/
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.
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.)
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.
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.
"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.