The other day I was musing about a reasonable approach to playing games like the big lotteries. They don’t cost a lot and losing $40 is not a life changing event for me, but clearly winning a few hundred million dollars is life changing.
My first thought turned to, well if you just play when the expected value is greater than the cost of the ticket that is “rational”. But when I started thinking about it, and even doing some calculations for when that EV condition exists (for things like Mega Millions the jackpot has to be greater then about 550 million) it struck me that the naive EV calculation must be missing something. The odds of actually winning the jackpot are really, really low (as opposed to just really low to rather low for the other prizes). And the payoffs that go into the EV calculation are hugely skewed by the top prices.
I suspect this must be a situation that generalized to other settings and am wondering if anyone knows of better approaches than merely the naive EV calculation. And to be sure I’m using the term as everyone expects, EV just equals the probability weighted payoffs minus the cost of the ticket.
What you actually want is to maximize the growth rate of your bankroll. You can go broke making +EV bets. The Kelly Criterion is the solution you’re looking for for something like a lottery – a bet is “rational” iff the Kelly Criterion says you should make it.
Thanks, I’ve seen that mentioned before but didn’t really pay much attention.
I’m not sure how well it applies to something like a lottery (yes, I get the general point about percentage to put towards something but struggle with just how to assess the performance of the investment strategy part of the equation).
That said, in looking at the approach one thing that seems implied by the criteria is that the performance of the selected strategy seems to dominate a persons individual performance. I might have miscalculated though so wonder if you agree that would be a correct statement.
Yes, I think performance ultimately matters much more than risk preferences. If you really want to take that into account you can just define utility as a function of wealth, and then maximize the growth of utility instead. But I think risk-aversion has been way overemphasized by academics that weren’t thinking about ergodicity, and were thinking along St Petersburg Paradox lines that any +EV bet must be rational, so when people don’t take +EV bets they must be irrationally risk-averse.
It is possible for a lottery to be +EV in dollars and -EV in utility due to the fact of diminishing marginal utility . As you get more of something, the value of gaining another of that thing goes down. The difference between owning 0 homes and owning your first home is substantial, but the difference between owning 99 homes and 100 homes is barely noticeable despite costing just as much money. This is as true of money as it is of everything else since the value of money is in its ability to purchase things (all of which have diminishing marginal utility).
The diminishing value of money is borne out in studies that look for the link between happiness/life satisfaction and income. Additional income almost always improves your life, but the rate of that improvement is approximately at the log scale (i.e. multiplying your income by 10 gives you +1 happiness, regardless of what your income was).
What does this all have to do with a lottery? Well, a lottery gives you a small probability of a massive number of dollars at a fixed cost. Since the hundred millionth dollar is worth much less to you than the first dollar, this can be a bet that has negative expected utility even when you would make money on average.
I agree that both DMV/DMU of money units is true and worth considering. However, I think it might be a bit more complex than that since I think one can make a case for network effects/economies of scale type aspects.
For example, the marginal value of the next dollar I add to my wealth today is pretty small. Clearly if I had 300 million additional dollars the MV of the next dollar absolutely will be smaller. But the MV/MU of having 50 million to put into my “this pays for my day to day life” and this other 100 million goes into some research projects I would like to see done but in no way could pursue that effectively now, and this other 100 million can go to some other useful (in my assessment) efforts that I might want to support and the remaining could be “wasted” on gifts and assistance to those I think deserve more than life has given them. So I think understanding just what the margin is matter a lot in this type of view.
Even with compute power available, the “value” part of EV calculations gets weird outside of the bounds of normal experience. Very small probabilities are susceptible to estimation error, and very large impacts are susceptible to value error.
For the lottery specifically, for most non-impoverished people, the ranges of reasonable estimates are such that it’s rationally justifiable to play or not play, depending on your enjoyment of the act rather than the monetary EV. The actual monetary EV is only a guess anyway—given that more tickets get sold when it’s large, the chance of splitting the win goes up, and that’s ignoring the chance of errors, cheating, or other shenanigans. Add to that the fact that you don’t know how it changes your life and relationships to win—it’s probably quite positive, but it’s impossible to predict how much.
It was popular in the 90s among positive-ev gambling folks (card counters and the like) to search the world for +EV lotteries and pool funds to buy LOTS of non-overlapping tickets. True opportunities were somewhat scarce, and usually the smaller ones, not the giant jackpots.
Just curious, we’re the ones found profitable to your knowledge?
I made some profit in card counting and poker, and made friends with some of the pros—there were definitely profitable teams and syndicates, and part of their profits was in identifying broken games and exploiting them until fixed. I had a decent job at the time, and didn’t really want to commit that much energy to it, so I remained a hobbyist and hanger-on.
I knew about a number of progressive slot jackpots that were +EV for periods of time, a bunch of promo or just miscalculated bonus bets at small casinos, suspiciously-generous Keno payouts, and the like. There was talk of large government lotteries that were exploitable, mostly in too-much-payout to near-misses, but I don’t know of any actual big hits. There were a lot of smaller wins, and perhaps a dozen of my close acquaintances were making a good living at this (and probably 10x that many supplementing a regular job, and I’m sure 100x that many not doing that well but claiming they were).
I seem to remember reading that the overall impact to an individual of winning a large lottery is very frequently overwhelmingly negative; that nearly everybody winning those prizes ends up worse off five or ten years down the road than they were when they started.
… a 5-minute check of the easiest-to-find articles on the subject provides mixed opinions, so grain of salt and all that. But I didn’t see any anybody claiming that winning a lottery is all champagne and rainbows. Rather, most sources seem to be advising a great deal of caution and professional assistance to keep horrible consequences to a minimum.
Most people have non-linear utility of money. Going from 10 thousand to 1 million is less impactful than going from 1 million to 1.99 million, even though it’s the same absolute change.
There’s a tool called “certain equivalent” which can help with answering “how much do I value money?”
It boils down to repeatedly asking and answering question: “If I had a choice between $X and 50% of $Y at which X would I be ambivalent about which side I pick” for different value of $Y, e.g. for Bob, the 25 year old postdoc it may be
Notice how at higher level certain equivalent is no longer just dividing by two. For Bob utility from having $75k is higher than 50% of utility of having $400k. The reason for this nonlinearity is usually downstream effects of having money. E.g. for Bob $75k would be enough to get a downpayment for house he wants and highly increase chance of him a comfortable life down the line. 50% chance of buying the house outright is not worth the risk for Bob.
(I’m going to nix the cost of the ticket as it’s just a constant)
Depends. Do you want to sum the probability weighted payoffs? EV is fine for that. The probability weighting deals with the striking “really, really low” odds (unless you want to further reweight the probabilities themselves by running them through a subjective probability function), and the payoffs are just the payoffs (unless you want to further reweight the payoffs themselves by running them through a subjective utility function). Either or both of these changes may be appropriate to deal with your own subjective views of objective reality, but that’s what they are—personal transformations. However, enough people subscribe to such transformations that EU (expected utility, or see cumulative prospect theory) makes sense more widely than just for you. We indeed perceive probabilities differently from their objective meanings and we indeed value payoffs differently from their mere dollar value.
Now, if you just want a number that best represents the payoff structure, we have candidate central tendencies—mean is a good one (that’s just EV). But since the payoff distribution is highly skewed, maybe you’d prefer the median. Or the mode. It’s a classic problem, but it’s finding what represents the objective distribution rather than what summarizes your possible subjective returns.