How much to spend on a high-variance option?

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37 comments, sorted by Click to highlight new comments since: Today at 9:10 PM

I find this counterintuitive; where are all the economists who should be picking up this free money?

Relevant reading:

- http://www.wired.com/magazine/2011/01/ff_lottery/all/1
- http://en.wikipedia.org/wiki/National_Lottery_%28Ireland%29#Lotto_6.2F36:_1988.E2.80.9392
- http://r6.ca/blog/20090522T015739Z.html

I believe the last link essentially answers your question: the Kelly Criterion, which is an optimal way to invest, advises investing less in a lottery than a single unit (ticket) costs.

I was under the impression that for infinite repeated play, no matter what your actual utility function is (as long as it is increasing and the total number of dollars is bounded), it turns out that the optimal single-turn strategy "looks like" betting with a logarithmic utility function --- hence the Kelly Criterion.

I don't know much about this though so could be mistaken.

I find the results of the Kelly criterion extremely counterintuitive, but it does answer my question. Thanks. I note that, presumably, we are discussing a lifetime strategy rather than a one-off, so the bankroll should not be my current cash reserves but the net present value of my expected lifetime income stream; but the Kelly fraction is so small that the optimal bet still works out to less than a dollar. Fascinating!

Note that if your thinking is something like "I should probably forgo buying a latte this morning and buy a lottery tickets instead", then the Kelly criterion does not apply (it does not affect your lifetime income). Instead you should consider how much of your revenue-neutral funds you can spare and weigh the emotional downside of forgoing one expense (a drink: mmm, feels good) against the actual and potential emotional upside of another (a lottery ticket: what if I win, what if I win! + potentially winning - bummer, I lost!).

I did go a bit further towards alief by putting into my toy MC study, with the simple coin-toss game in your link, a bettor who puts in 50% of his bankroll every time - way, way beyond the Kelly fraction, and then having a think about how he managed to lose all his money. (Not literally, but enough that the remaining bankroll was 0 to my printout accuracy.)In ten thousand iterations my longest win and loss streaks are both of ten games. A loss streak of ten games will reduce this bettor's bankroll by a factor of 1024. But ten winning games will only increase it by about a factor 80. On the other hand, with the Kelly fraction of 4.5%, ten losses reduce your bankroll by about 40%, while ten wins increase it by 62%. The asymmetry in these specific examples is somehow more convincing than the final numbers from the toy MC run.

I mean pretty much exactly that: I plugged in the payoff numbers into the equation, thought hard about my past record of trades & predictions and how calibrated I was in each certainty range to determine my edge, looked at the result of the Kelly Criterion, and felt terror at the idea of committing that much of my Intrade bankroll to one trade. I discuss the KC in http://www.gwern.net/Prediction%20markets#how-much-to-bet

I don't think the straightforward Kelly Criterion quite answers the question. It would tell you how many identical tickets you should buy (in a lottery where the jackpot pays out for each winner rather than being shared). The question at hand is different, because by buying more tickets you increase the chance of winning, rather than the payoff for a win.

I'm sure there's a simple variation of the criterion you can use, but I'm too tired to work it out right now. (I still expect that for most people, the answer is "buy no tickets")

I am trying to understand the implications of the kelly criterion for a real world portfolio. What I get as a result is that if I have free choice on any bet at any odds and chances I should, in total, invest more than I have. (Result by integrating over all probabilities/odds that allow positive expected value) In fact, I should invest infinitely much money. The wikipedia page states that taking out credit to buy a bet would be formalized by the loss formula so the infinity result is not exactly interpretable as taking out a loan, if I can.

One obvious fix is to limit the odds and probabilites to realistic values but that seems quite arbitrary. Intuitively I would expect the Kelly criterion to give a finite sum for all bets with positive expected value, at least it does so for any given odds b in wikipedias terminology.

What I get as a result is that if I have free choice on any bet at any odds and chances I should, in total, invest more than I have.

If you invoke infinities or indefinite sets of bets, it shouldn't surprise you that regular results might not apply: if you decide to invest in a bet of *n* at 99% odds of doubling, wouldn't it be even better to invest *n* at 99% odds of tripling? Or even better than that, invest *n* at 99.9% odds of tripling? Or no, invest *n*+1 at 99.9% odds of tripling! I'm not sure why you'd expect anything useful from a KC or a variant with such arbitrary inputs.

One obvious fix is to limit the odds and probabilites to realistic values but that seems quite arbitrary.

It does?

I believe I've seen reports that discussed a spike in lottery ticket sales when it's positive EV; I think there are some people that do watch for that and behave accordingly. Putting your "actually risk neutral" investment money into it isn't a terrible idea, but whether or not you have that investment money depends on your situation.

[edit] That is, you should split your financial resources into "expected to be needed in the next 3 months," "expected to be needed in the next year," "expected to be needed in the next 5 years," "expected to be needed in the next 20 years," and "risk neutral." How much of your current income flow to put into each of those buckets depends on your projected income, projected consumption, and so on, and is a fairly complicated calculation if you want to do it carefully. Even so, eyeballing it is better than not running those numbers.

Admittedly this is the weakest part of the argument. I looked at the revenue for 2011, 42 million, and divided by the number of drawings, 3 per week for 52 weeks. Obviously this would miss a recent spike in sales. However, I tried the probability with some theoretical numbers, and to get a probability of someone else winning that significantly affects the expectation value, the number of tickets sold has to go *way, way* up from that baseline quarter million. A full order of magnitude increase in sales, to 2.5 million, only gets you a 17% probability of sharing the jackpot, conditioned on you winning.

I went wandering around ohiolottery.com (For instance, http://www.ohiolottery.com/Games/DrawGames/Classic-Lotto#4) and found this out:

- The cash payoff is half the stated prize.
- The odds to win the jackpot, as noted by the OP, are about 14 million-1.
- The amount of money being spent on individual draws is very low. The jackpot increase was $100K for the last drawing; I don't know exactly what their formula is, but I'd be shocked if they sold more than 400K tickets for the last drawing.
- Ohio is running a lot of lottery games; this is good for players who pick their spots.

There are also payoffs below the jackpot level, so I'm confident there's a positive EV per ticket.

The question as to how many tickets to buy, *assuming you can effectively do so*, is "All of them." Buy each individual ticket, take your 14 million tickets, and probably profit. (Remember, the jackpot kick will include some fraction of your 14 million, also. Plus, you'll have all the side prizes.) In practice, unfortunately, this requires a method to buy them effectively, some armored cars, and a staff of people to do it right. Failure to purchase all tickets results in some drama, for sure.

The execution expenses and risk are troubling; if those could be effectively mitigated, it's a great investment.

Assuming you're a few million short of that, though, it's harder. I buy CA lottery tickets when EV>1.20 per $1 invested. I have no strong justification for that number.

Isn't there a standard reply containing utility functions? Assuming the usual diminishing marginal utility, if p(win) * U(jackpot) > U($1), then 2 p(win) * U(jackpot) > U($2) and so on. You should spend all your money! (Unless you are rich enough to buy so many tickets as to wander out of the domain of approximate linear dependece of p(win) on the number of tickets you own).

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Second attempt (thanks janos):

Isn't there a standard reply containing utility functions? You should buy N tickets where N is the solution to p(win | N tickets) * U(jackpot) + U(-N dollars) is maximal. Assuming the usual diminishing marginal utility of money and that U(jackpot) * p(win | 1 ticket) > -U($-1), a unique solution should exist for N > 0.

I think you're making the wrong comparisons. If you buy $1 worth, you get p(win) * U(jackpot) + (1-p(win)) * U(-$1), which is more-or-less p(win)*U(jackpot)+U(-$1); this is a good idea if p(win) * U(jackpot) > -U(-$1). But under usual assumptions -U(-$2)>-2U(-$1). This adds up to normality; you shouldn't actually spend all your money. :)

So the jackpot in the Ohio lottery is around 25 million, and the chance of winning it is one in roughly 14 million, with tickets at 1 dollar a piece. It appears to me that roughly a quarter million tickets are sold each drawing; so, supposing you win, the probability of someone else also winning is 1 - (1 - 1/14e6)^{250000}=2%, which does not significantly reduce the expectation value of a ticket. So, unless I'm making a silly mistake somewhere, buying lottery tickets has positive expected value. (I find this counterintuitive; where are all the economists who should be picking up this free money? But I digress.)

I pointed this out to my wife, and said that it might be worth putting a dollar into it; and she very cogently asked, "Then why not make it 100 dollars?" Why not, indeed! Is there any sensible way of deciding how much to put into an option that has a positive expected value, but very low chance of payoff?