It seems to me you're misunderstanding how and why we're using lottery examples.
Lotteries are brought up on Less Wrong, first of all, to demonstrate the standard bias of failing to fully internalize how small a probability can be.
Secondly, they make a decent illustration of the horrible LHC inconsistency problem that arises when the probability that "one's model of the probabilities is mistaken" dwarfs the calculated chance of a particularly important outcome. Lottery examples illustrated this well enough to suggest an approach to a solution, in fact.
There are other uses, too, like pointing out that the expected utility of a single ticket can be negative even if the expected value is positive, due to the logarithmic dependence of utility on wealth.
None of these deny that there can exist cases where a particular lottery strategy like yours will win. Your post may be correct, but it is trivial and utterly irrelevant to the conversations we've been having; therefore I don't think it belongs as a top-level post.
In a famous occurrence, a Polish-Irish businessman named Stefan Klincewicz bought up almost all of the 1,947,792 combinations available on the Irish lottery. He and his associates paid less than one million Irish pounds while the jackpot stood at £1.7 million. There were three winning tickets, but with the "Match 4" and "Match 5" prizes, Klincewicz made a small profit overall.
Excellent!
Obviously, most US lotteries have much longer odds (making it more difficult), but good for Klincewicz. It's unsurprising that this has been tried before (and a certainty that it had been thought of before) but I didn't know any specifics of anyone pulling it off.
It's been done before, but remember that you have to pay taxes on the winnings. Also most lotteries in the US force you to choose between a much smaller lump sum or a yearly payment.
Indeed, rollover jackpots can raise the expected value of a ticket (in dollars) above the price of the ticket - as long as nobody else wins and makes you split the prize with them. As for buying up every possible ticket combination, that's deliberately made to be as difficult as possible.
There are also slot machines in Vegas that operate on a similar rollover principle - the more money that is put into the bank of machines, the larger the jackpot gets, and that can eventually give the machine a positive expected value for the player.
Indeed, rollover jackpots can raise the expected value of a ticket (in dollars) above the price of the ticket - as long as nobody else wins and makes you split the prize with them.
Wouldn't the expected value include such occurances?
There are also slot machines in Vegas that operate on a similar rollover principle - the more money that is put into the bank of machines, the larger the jackpot gets, and that can eventually give the machine a positive expected value for the player.
Peter Liston, a prominent figure from MENSA in Melbourne back when I used to be a member there, used this as his primary form of revenue. (Australian slot machines not Vegas ones obviously). He objected somewhat to the term 'Professional Gambler', observing that once you do it professionally the term becomes very nearly oxymoronic. It's the only investment he makes that consistently makes money. Stockmarket investment is far less reliable than a known RNG over a significant number of rolls.
There are also some video poker machines that are beatable if you use the right strategy. (And the right strategy is usually rather counter-intuitive, as it often involves throwing away every card that can't be part of a royal flush from an otherwise promising hand, because about half your total expected payout comes from the royal flush jackpot.)
Here are some figures from http://www.gamemasteronline.com/Archive/VideoPoker/AllAmericanVideoPokerDiary-1.shtml
My understanding is that sometimes you can do better, like when a casino has special offers or promotions. You can also do something like this at online casinos, BTW, although I haven't personally tried it.
The expected value is a function of the jackpot, so it varies relative to the price of the ticket which is constant.
The expected value is a function of the jackpot, so it varies relative to the price of the ticket which is constant.
.... yes.
The expected value is also influenced by the number of other tickets purchased. Neglecting that would give you the wrong expected value (see CronDAS's caveat).
Rereading the comment chain I see that I was misreading your post the first time through. Sorry!
You could do even better than just buying all tickets. Many people use dates and such predictable numbers in lottery, so, if you exclude those that are likely to tie, and fill many of the weird combinations, your expected value goes up.
If the expected value for buying all of the tickets is positive, wouldn't the expected value of any particular ticket be positive? Does the math require you to buy all of the tickets?
A small example:
5 numbers that each cost $1 with payouts of $4 for 1st pick and $2 for 2nd pick. Any ticket has a 1/5 chance of paying $4, a 1/5 chance of paying $2, and a 3/5 chance of paying $0.
.2 $4 + .2 $2 + .6 * 0 = $1.2
Buying all of the tickets will give you $6 for spending $5, which is a profit of $1.2 per dollar invested. So... what am I missing? It seems like if it was good for you to spend $41 million it was good for you to spend $1. Is it a matter of risk management or something like that? This isn't really my area of expertise.
As expected value ≠ expected utility, it's not the case that you should always buy a ticket if expected value is positive. It's a standard result that people actually treat the utility of wealth roughly logarithmically: i.e. that it's better to have a net worth of $1,000,000,000 than $100,000,000, but not that much better compared to how much better $100,000,000 is than $1000 net worth.
To simplify the lottery situation in the case of extreme probabilities and payouts, say that Omega offers a lottery only to you (no worries about split jackpots), in which there are exactly 1,000,000 tickets, each costing $1, and among them there is one winning ticket that pays out $2,000,000.
Now if you can scrounge up a million dollars to buy every ticket, you make a tidy $1 million profit (less interest from your backers) with zero risk, so the expected utility is very positive for this strategy.
If, however, you can only get $100,000 together, you shouldn't buy any tickets (unless you're a millionaire to start), since the utility to you of a 90% chance of losing $100,000 (and having a pretty crappy life being so far in debt) outweighs the utility of a 10% chance of winning $2 million (and a nice standard of living).
It's a standard result that people actually treat the utility of wealth roughly logarithmically
or is it just a standard assumption? I've never heard anything more precise than declining marginal utility.
Logarithmic u-functions have an uncomfortable requirement that you must be indifferent to your current wealth and a 50-50 shot at doubling or halving it (e.g. doubling or halving every paycheck/payment you get for the rest of your life). Most people I know don't like that deal.
That's only a requirement for risk-neutral people. Most people you know are not risk-neutral.
Logarithmic utility functions are already risk-averse by virtue of their concavity. The expected value of a 50% chance of doubling or halving is a 25% gain.
People are often risk-averse in terms of utility. That is, they would sometimes not take a choice with positive expected value in utility because of the possible risk.
For instance, if you have to choose between A and B, where A is a definite gain of 1 utile and B is a 50% chance of staying the same, and a 50% chance of gaining 2 utiles, both choices have the same expected value, but a risk-averse person would prefer choice A because it has smaller risk.
Nitpick: you put the values in utiles, which should include risk-aversion. If you put the values in dollars or something, I would agree.
No, the whole point is that people can be risk averse of utility. This seems to be confusing people (my original post got voted down to -2 for some reason), so I'll try spelling it out more clearly:
Choice X: gain of 1 utile. Choice Y: no gain or loss. Choice Z: gain of 2 utiles.
Choice B was a 50% chance of Y and a 50% chance of Z. To calculate the utility of choice B, we can't just take the expected value of the utility of choice B, because that doesn't include the risk. For a risk-averse person, choice B has a utility of less than 1, although the expected value of choice B is 1.
This would be entirely true if instead of utiles you had said dollars or other resources. As it is, it is false by definition: if two choices have the same expected utility (expected value of the utility function) then the chooser is indifferent between them. You are taking utility as an argument in something like a meta-utility function, which is an interesting discussion to have (which utility function we might want to have) but not the same as standard decision theory.
But the utility is the output of your utility function. If you're not including the risk-aversion cost of choosing B in its expected value in utiles, then you're not listing the expected value in utiles properly.
I would say that such a person doesn't have preferences representable by a utility function.
That's just plain false. Risk-aversion is a valid preference, and can be included as a term in a utility function (at slight risk of circularity, but that's not really a problem).
ETA: well, the stated units were utils, so risk-aversion should be included, so I think you're correct.
The expected value of choice B is 1, but the utility of choice B to a risk-averse person would be less than 1. Risk-averse people just don't equate utility of a choice with the expected value of that choice.
I don't think opportunities to make choices are usually considered to be in the domain of a utility function. (If I'm wrong, educate me. I'd appreciate it.)
Ok, I looked it up and it looks like you and thomblake (ETA: and Technologos. Thanks for correcting me!) are right: the usual way of doing it is to include risk aversion in the utility function. Sorry about that.
Wikipedia on risk-neutral measures does discuss the possibility of adjusting the probabilities, rather than the utility, when calculating the expected value of a choice, but it looks like that's usually done for ease of financial calculation.
So, one explanation for why people don't take the "half or double" gamble is that they do have the log(x) utility function, but don't behave accordingly because of loss aversion (as opposed to risk aversion).
The post is technical, but Stuart_Armstrong analyzed some special cases of not-quite-utility-function agents.
I'm confused about what is uncomfortable about this, or what function of wealth you would measure utility by.
Naively it seems that logarithmic functions would be more risk averse than nth root functions which I have seen Robin Hanson use. How would a u-function be more sensitive to current wealth?
I think the uncomfortable part is that bill's (and my) experience suggests that people are even more risk-averse than logarithmic functions would indicate.
I'd suggest that any consistent function (prospect theory notwithstanding) for human utility functions is somewhere between log(x) and log(log(x))... If I were given the option of a 50-50 chance of squaring my wealth and taking the square root, I would opt for the gamble.
Pretty sure it's the standard result that people don't consistently assign utilities to levels of wealth.
If I'm reading this analysis correctly, it depends on other players irrationally buying tickets to drive the price up to the point where expected payoff is positive for the next player. Copies of JRMayne could not all play the lottery in this fashion every week for a year and come out ahead.
Right. That's how I've lost $350. Every ticket you buy (so long as you're not buying duplicates) has the same value, assuming others' purchases stay constant. It's just that if you put in $50, your highly likely outcome is a severe loss.
The risk management is a big deal as you get to big numbers. Yes, the expectation for 20 million tickets is equivalently good as far as expected value per ticket bought, but that a 50% chance of a massive loss. The really catastrophic outcome is to screw up in physically inserting the 4.1 million or so paper tickets into the machines if you're forced to buy them that way; you'll likely have some loss to human misfeasance or malfeasance and if you reverse-hit the lottery with a lost ticket, that's officially bad.
Anyway, short version, MrHen's totally right. I'd note to bgrah449's below that you can only buy tickets when the lottery's at the right number and when the rate of purchase is low relative to the jackpot and chance of winning. You do this every week, and you'll get squashed like a bug. But people are now building up that jackpot; if it doesn't hit for long enough, the opportunity's there.
Some friends of mine and I tried a small version of this and bet on several hundred superfecta combinations for the Belmont this last year, when everyone was wondering if Calvin Borel would win riding Mine that Bird again. We did hit the superfecta, but unfortunately the payout for the combination that won was only $800 for a $2 ticket; the final winnings per person were about 1.7x our original "investment," not as much as you would think for hitting the superfecta.
This has occurred to me several times before, although I've never had anything like the sort of contacts one would require to buy 14,000,000 tickets. Also, I was under the impression that the UK national lottery forbade any given syndicate from buying more than some large number (on the order of 10,000) tickets, but I can't seem to find it anywhere in the rules on their website.
There are, of course, also some practical issues with buying 14,000,000 tickets: you'd need to pay an army of people to go out and buy them, and somehow sort out a system for making sure you don't miss any (although there are some lotteries where you can buy tickets online, which might make things easier). The Euro-millions seems to fairly regularly have positive EV draws, although this has never tempted me to buy a ticket.
And store the tickets, and ensure they are organized so you can actually find the one that won.
A Massachusetts lottery also occasionally has positive EV drawings: when the jackpot gets large enough and there is no winner, the jackpot money gets divided among the smaller prizes.
Another thing to factor in: the cost of your time to evaluate the idea. If there's a sufficiently strong probability that it won't work (for example, because the idea obvious enough that it almost certainly will be done by someone else if there's sufficient expected value), then the time required to evaluate the idea can dominate the evaluation, especially since the time could instead be used to evaluate other entrepreneurial ideas in domains that you have a comparative advantage in.
Lotteries are one clear example where many people ignore the expected value of their cough investment -- although the case presented here may be an exception.
Anyway, a very other common form of lottery is called insurance. There, as well, your expected values is negative, still, most people do get (voluntarily, and sometimes involuntarily) insurances. If you had a million houses, it would make little sense to buy fire insurance for them.
I would argue that the expected value is too limited a criterion for rationality; we'd also have to include the subjective feelings of security (insurance) or possible fortunes (lotteries).
I'd be curious to know how you calculated the chance of having to split the jackpot; presumably, as you buy every possible ticket, you drive up the jackpot, incentivizing more people to buy in, which increases the chance you'll have to split the jackpot.
No, you disincentivize people from buying in. They are more likely to match you; fewer buys incentivize purchases.
Edit: Clarifying, they're certain to split the pot; that's a strong disincentive. Optimally, whether buying one ticket or 40 million, you want no other tickets to be purchased for your draw if there's any meaningful carryover.
The main point here seems to me that unless people know that you are making a systematic effort to purchase every ticket they will only see the driven-up jackpot.
Aside from incomplete information, I would posit that lottery players tend to be less rational, and the marginal lottery players would buy tickets given the larger jackpot regardless of expected value.
I'm not sure if this increases or decreases your EV, however (i.e. increases jackpot size "faster enough" than chance of splitting).
I'm not sure if this increases or decreases your EV, however
I'm pretty sure it must decrease it. Clearly best possible for our strategy would be for no-one else at all to buy a ticket, then we win 50% of the $42,000,000 we spend on tickets, plus whatever got rolled over from last week. Every time anyone else buys a ticket, they buy a tiny share of whatever was rolled over (in expectation), thus reducing our EV.
Yes, but lots of people are being incentivized by a larger jackpot, all of whom can count on those other people also increasing the jackpot, but who are only slightly increasing the odds the jackpot is further split.
A few years back, I sent a note to some friends and other smart people. It said, approximately:
"I am trying to raise about $42 million. I have an expected payoff of about 20% in a week. It's not guaranteed; it might be more, or there could be a loss.
"Now, you may be saying to yourself, 'Mayne's got a system to win the lottery or something.'
"Well, it's not, 'something.' I have a system to win the lottery."
At which point I explained the system.
Now, LW is not particularly fond of the lottery. From a social policy/political point of view, I'm not disagreeing.
I specifically represent to you that I do have a system, I use a part of the system, and so far, I've lost about $350 with the system. If you'd loan me your $42 million when I ask for it, though, I'd have not only a high positive expectation, but a much smoother payout matrix, if I could solve the practical problem of buying as many tickets as I want. Which is 41.4 million tickets.
No, I'm not just saying I'll buy all the tickets and therefore guarantee the jackpot and therefore I win. I'm saying my expectation per dollar invested is substantially positive.
Skeptical? Why?
Not only do I have a system, but anyone with math skills and any knowledge of how the lottery works should agree. While it's always easy and obvious to see why other people should think the same as you, the fact that of a large number of math folks, zero here have defended playing the lottery as a cash-plus position, is a sign that people have foregone thinking and simply rejected the lottery outright. Or that I'm an idiot.
Before you get to the post payoff, what's your attitude toward this now? Lottery win is impossible? If you think it's impossible (or close enough to it), why? Unlikely? Probable? Would it change your mind if my self-assessment was that I was likely in the bottom quartile as far as current math skills on LW? Would it alter your probabilities if I told you that I am highly confident that if the practicality of buying 41.4 million tickets is worked out, that using my system to play the lottery is the best investment I know of?
OK, here it is: How to win the lottery.
I'm going to make this relatively brief, but it's really fairly simple. In California, we have a fairly classic lottery setup although the payout rates are a little worse than most states. The immediate payoff of the jackpot is about (carryover + 25% of tickets sold for that draw.) About 25% of the money into the lottery goes to non-jackpot payoffs. Fifty percent goes to valuable or less valuable government programs, including running the lottery.
You have about a 1 in 41 million shot of winning the jackpot with one ticket on any given draw.
A number of factors have seriously decreased purchases of Super Lotto tickets on large jackpots, primarily the cannibalization of the Lotto by the state's participation in the Mega Millions multi-state lottery. This is important to the calculations, and changes the expected payout substantially.
At one point, the Super Lotto jackpot was at $85 million for a prior draw (or $42.5 million in a single immediate payment) and they sold nine million tickets for the next draw.
So, consider: You put in $41.4 million and buy every possible ticket. You get about $10 million back in little prizes. The value of the jackpot is $42.5 million, plus 25% of nine million (the money the others put in) plus 25% of $41 million (the money you put in). You're at about $55 million in jackpot.
If no one ties you for the jackpot, you're at about $65 million in total payout. There's less than a 25% chance that someone ties you. That's a comfortable profit.
If one person ties you, that's about $37 million payout; that's a loss, but presumably a survivable loss. If more tie you that is, well, worse.
I came rather closer to pulling this off than seems likely; the big issue was how to buy the tickets, and the state told me I had to buy them from retailers (and I lack the political pull to find another way.) This opens a wide range of complications, as you can imagine. Then, sadly, some of the investment bankers of my acquaintance were, um, pursuing other interests (not because they were talking to some dude hawking a lottery system, or so I choose to believe), and the project dissolved.
But the math is still basically right. And the lottery has been repeatedly and forcefully brutalized here as something only idiots would play. When the cash expectation is positive, I'm playing again. I'm not saying it's stupid not to play - the marginal utility issues are substantial.
Still, by making the mental equation lottery=stupidity, a lot of people stopped thinking about the potential. I expect someone to pull this off some time, and I expect some political blowback (the lottery's for suckers, not the intelligent rich! What of the poor suckers?) Twenty million should make up for the blowback. Rationality is about winning, right?
Am I wrong on why this has been missed? Does this tell us anything about other opportunities that might be missed? Or, am I wrong that this should work, if the cash is present and the practical problems are solved? Or am I just an idiot?