Winning the Unwinnable

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.

Wikipedia:

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.

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.

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.

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.

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.

Why not use the 40 million and fund your own lottery company?

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.